Recent zbMATH articles in MSC 11https://zbmath.org/atom/cc/112024-07-25T18:28:20.333415ZUnknown authorWerkzeugBook review of: M. Baake (ed.) et al., The legacy of Kurt Mahler. A mathematical selectahttps://zbmath.org/1537.000082024-07-25T18:28:20.333415Z"Moree, Pieter"https://zbmath.org/authors/?q=ai:moree.pieterReview of [Zbl 1533.01058].MFO-RIMS tandem workshop: Arithmetic homotopy and Galois theory. Abstracts from the MFO-RIMS tandem workshop held September 24--29, 2023https://zbmath.org/1537.000192024-07-25T18:28:20.333415ZSummary: This report presents a general panorama of recent progress in the arithmetic-geometry theory of Galois and homotopy groups and its ramifications. While still relying on Grothendieck's original pillars, the present program has now evolved beyond the classical group-theoretic legacy \textit{to result in an autonomous project that exploits a new geometrization of the original insight and sketches new frontiers between homotopy geometry, homology geometry, and diophantine geometry}.
This panorama ``closes the loop'' by including the last twenty-year progress of the Japanese arithmetic-geometry school via Ihara's program and Nakamura-Tamagawa-Mochizuki's anabelian approach, which brings its expertise in terms of algorithmic, combinatoric, and absolute reconstructions. These methods supplement and interact with those from the classical \textit{arithmetic of covers and Hurwitz spaces and the motivic and geometric Galois representations}.
This workshop has brought together the next generation of arithmetic homotopic Galois geometers, who, with the support of senior experts, are developing new techniques and principles for the exploration of the next research frontiers.Analytical reasoning and problem-solving in Diophantus's \textit{Arithmetica}: two different styles of reasoning in Greek mathematicshttps://zbmath.org/1537.010072024-07-25T18:28:20.333415Z"Christianidis, Jean"https://zbmath.org/authors/?q=ai:christianidis.jean-pSummary: Over the past few decades, the question regarding the proper understanding of Diophantus's method has attracted much scholarly attention. ``Modern (i.e., post-Vietan) algebra'', ``algebraic geometry'', ``arithmetic'', ``analysis and synthesis'', have been suggested by historians as suitable contexts for describing Diophantus's resolutory procedures, while the category of ``premodern algebra'' has recently been proposed by other historians to this end. The aim of this paper is to provide arguments against the idea of contextualizing Diophantus's \textit{modus operandi} within the conceptual framework of the ancient analysis and to examine the few instances, in the preserved books of the \textit{Arithmetica}, which might be regarded as linked to practices belonging to the field of analysis.Soft logic as an extension of Pascal's workhttps://zbmath.org/1537.030212024-07-25T18:28:20.333415Z"Klein, Moshe"https://zbmath.org/authors/?q=ai:klein.moshe"Maimon, Oded"https://zbmath.org/authors/?q=ai:maimon.oded-zSummary: Pascal was a great mathematician and scientist, who contributed to many fields in mathematics and science. When he was 19 years old, he developed the first calculator, and together with Fermat he was the founder of probability theory. He investigated the properties of a triangle of numbers, which is named today ``the Pascal triangle'' and developed the method of proving theorems by mathematical induction. Pascal also investigated the properties of the cycloid, and he conducted the physical experiment that proved the existence of the void. After a spiritual experience at the age of 32, Pascal left mathematics and science altogether and dedicated himself to investigating and writing about religion.
This paper suggests the new language of Soft logic, which is based on the extension of the number 0 to the zero axis. We conclude by an example of the extension of the Pascal triangle with soft numbers. Also, we discuss the possibility to develop a new model of computation.One dimensional groups definable in the \(p\)-adic numbershttps://zbmath.org/1537.030412024-07-25T18:28:20.333415Z"Acosta López, Juan Pablo"https://zbmath.org/authors/?q=ai:acosta-lopez.juan-pabloSummary: A complete list of one dimensional groups definable in the \(p\)-adic numbers is given, up to a finite index subgroup and a quotient by a finite subgroup.Generalized Pell graphshttps://zbmath.org/1537.050232024-07-25T18:28:20.333415Z"Irsic, Vesna"https://zbmath.org/authors/?q=ai:irsic.vesna"Klavzar, Sandi"https://zbmath.org/authors/?q=ai:klavzar.sandi"Tan, Elif"https://zbmath.org/authors/?q=ai:tan.elifSummary: In this paper, generalized Pell graphs \(\Pi_{n, k}\), \(k \geq 2\), are introduced. The special case of \(k = 2\) are the Pell graphs \(\Pi_n\) defined earlier by \textit{E. Munarini} [Discrete Math. 342, No. 8, 2415--2428 (2019; Zbl 1418.05098)]. Several metric, enumerative, and structural properties of these graphs are established. The generating function of the number of edges of \(\Pi_{n, k}\) and the generating function of its cube polynomial are determined. The center of \(\Pi_{n, k}\) is explicitly described; if \(k\) is even, then it induces the Fibonacci cube \(\Gamma_n\). It is also shown that \(\Pi_{n, k}\) is a median graph, and that \(\Pi_{n, k}\) embeds into a Fibonacci cube.Generalized notions of continued fractions. Ergodicity and number theoretic applicationshttps://zbmath.org/1537.110012024-07-25T18:28:20.333415Z"Fernández Sánchez, Juan"https://zbmath.org/authors/?q=ai:fernandez-sanchez.juan"López-Salazar Codes, Jerónimo"https://zbmath.org/authors/?q=ai:lopez-salazar-codes.jeronimo"Seoane Sepúlveda, Juan B."https://zbmath.org/authors/?q=ai:seoane-sepulveda.juan-benigno"Trutschnig, Wolfgang"https://zbmath.org/authors/?q=ai:trutschnig.wolfgangGeneralizations of continued fractions into different directions have been studied widely and the authors give a short ``to-the-point'' introduction in their preface. Moreover they refer in their bibliography to number theory texts, a.o. by \textit{A. Ya. Khinchin} [Continued fractions. Chicago and London: The University of Chicago Press (1964; Zbl 0117.28601)], \textit{C. D. Olds} [Continued fractions. New York: Random House (1963; Zbl 0123.25804)] and \textit{A. M. Rockett} and \textit{P. Szüsz} [Continued fractions. 1st repr. London: World Scientific (1994; Zbl 0869.11058)]. The works by \textit{O. Perron} [Die Lehre von den Kettenbrüchen. Band I. 3. erweiterte und verbesserte Aufl. Stuttgart: Stuttgart: B. G. Teubner Verlagsgesellschaft (1954; Zbl 0056.05901); Die Lehre von den Kettenbrüchen. Band II. Analytisch-funktionentheoretische Kettenbrüche. 3. verbesserte und erweiterte Auflage. Stuttgart: B. G. Teubner Verlagsgesellschaft (1957; Zbl 0077.06602)], \textit{H. S. Wall} [Analytic theory of continued fractions. New York: D. van Nostrand Company, Inc. (1948; Zbl 0035.03601)] and \textit{L. Lorentzen} and \textit{H. Waadeland} [Continued fractions with applications. Amsterdam: North-Holland (1992; Zbl 0782.40001)] are not mentioned.
The book focuses its attention on the Lehner continued fractions, Farey series, Hirzebruch-Jung continued fractions and the connection and interplay with dynamical systems and ergodicity.
The authors have succeeded in writing a book that should be mastered by graduate students and must be on the shelf (nay, on the desk) of senior researchers of the subject.
Information about the contents will be given below.
Preface (\(4\) pages).
Chapter 1: Generalized Lehner continued fractions (\(14\) pages).
Lehner derived the ordinary one (semi-regular continued fractions whose partial denominators
are \(1\) or \(2\)) in [\textit{W. Abikoff} (ed.) et al., The mathematical legacy of Wilhelm Magnus. Groups, geometry and special functions. Conference on the legacy of Wilhelm Magnus, May 1-3, 1992, Polytechnic Univ. Brooklyn, NY, USA. Providence, RI: American Mathematical Society (1994; Zbl 0801.00023)]:
every real number can be expressed by the sum of an integer and a continued fraction
\[
p_0+\frac{q_0|}{|p_0+}\ \frac{q_1|}{|p_2+}\ \frac{q_1|}{|p_2+}\ \frac{q_2|}{|p_3+}\cdots, \eqno{(*)}
\]
with \((p_n,q_n)\in \{(1,1),(2,-1)\}\) for each \(n\).
The authors generalize this by using \((p_n,q_n)\in \{(1,a),(2,-1)\}\) for a fixed positive number \(a\).
The main result is now
Theorem 1.8 Suppose that \((p_n,q_n)\in \{(1,a),(2,-1)\}\) for all \(n\geq 0\). For each \(n\in\mathbb{N}\) assume
\[
\frac{A_n}{B_n}= p_0+\frac{q_0|}{|p_0+}\ \frac{q_1|}{|p_2+}\ \frac{q_1|}{|p_2+}\ \frac{q_2|}{|\dots p_{n-1}+}\ \frac{q_{n-1}|}{|p_n}.
\]
Then the limit \(\lim_{n\rightarrow\infty}\,A_n/B_n\) exists and belongs to \([1,a+1]\). Moreover, if \(x\in [1,1+a]\) and
\(\{p_n\}_{n=0}^{\infty}\) and \(\{q_n\}_{n=0}^{\infty}\) are the sequences associated with \(x\) by \((*)\), then
\[
\lim_{n\rightarrow\infty}\,\frac{A_n}{B_n}=x.
\]
An important role is played by the function \(L_{a}(x)\):
\[
L_{a}(x)=\left\{\begin{matrix} x/(1-ax) & \hbox{if} & 0\leq x\leq 1/(1+a) \cr (1-x)/{ax} & \hbox{if} & 1/(1+a)\leq x\leq 1\end{matrix}\right. .
\]
Chapter 2: \(a\)-modified Farey series (\(9\) pages).
Here the functions \(f_a,g_a: [0,1]\rightarrow[0,1]\) come into play:
\[
f_a(x)=\frac{x}{1+ax},\qquad g_a(x)=\frac{1}{1+ax}.
\]
(\(f_a\) is the inverse of \(L_a\) restricted to \(0,1/(1+a)]\) and \(g_a\) of \(L_a\) restricted to \([1/(1+a),1]\)).
Chapter 3: Ergodic aspects of the generalized Lehner continued fractions (\(18\) pages).
Generalizing existing results on ergodicity to the new continued fractions.
Chapter 4:The \(a\)-simple continued fraction (\(14\) pages).
Here the continued fraction has the form
\[ x=\frac{\qquad 1\qquad |}{|1+m_1a+}\ \frac{\qquad a\qquad|}{|1+m_2a+}\ \frac{\qquad a\qquad|}{|1+m_3a+}\ \frac{\ a\ |}{|\cdots},\]
with \(m_i\in\mathbb{N}\cup \{0\}\) for every \(i\in\mathbb{N}\).
Chapter 5: The generalized Khintchine constant (\(9\) pages).
The ordinary Khintchine constant is extended to the situation of the new continued fractions.
Chapter 6: The entropy of the system \(([0,1],\mathcal{B},\mu_a,T_a)\) (\(7\) pages).
Using finite partitions of the underlying probability space.
Chapter 7: The natural extension of \(([0,1],\mathcal{B},\mu_a,T_a)\) (\(24\) pages).
The situation of two measure-preserving dynamical systems and their probability measures.
Chapter 8: The dynamical system \(([0,1],\mathcal{B},\nu_a,Q_a)\) (\(6\) pages).
Now the following mapping \(Q_a: [0,1]\rightarrow [0,1]\) is used:
\[Q_a(x)=\left\{\begin{matrix} 0 & \hbox{if} & x=0,x=1 \cr \frac{1}{a}\left(\frac{1}{x}-1\right)-\left[\frac{1}{a}\left(\frac{1}{x}-1\right)\right] & \hbox{if} & 0<x<\frac{1}{1+a} \cr \frac{1}{1-x}-\frac{1}{a}-\left[\frac{1}{1-x}-\frac{1}{a}\right] & \hbox{if} & \frac{1}{1+a}<x<1\end{matrix}\right. .\]
Chapter 9: Generalized Hirzebruch-Jung continued fractions (\(14\) pages).
A similar method of generalization as used in Chapter 1.
Chapter 10: The entropy of \(([0,1],\mathcal{B},\vartheta_a,H_a)\) (\(4\) pages).
Chapter 11: The natural extension of \(([0,1],\mathcal{B},\vartheta_a,H_a)\) (\(13\) pages).
Chapter 12: A new generalization of the Farey series (\(7\) pages).
A generalization of Farey series, different from the one treated in Chapter 2.
The starting point is now
\[h_a:[0,1]\rightarrow [0,1],\hbox{ given by }h_a(x)=\frac{1}{1+a-ax}.\]
Bibliography (\(42\) items).
Reviewer: Marcel G. de Bruin (Heemstede)A \(\operatorname{mod} p\) Jacquet-Langlands relation and Serre filtration via the geometry of Hilbert modular varieties: splicing and dicinghttps://zbmath.org/1537.110022024-07-25T18:28:20.333415Z"Diamond, Fred"https://zbmath.org/authors/?q=ai:diamond.fred"Kassaei, Payman"https://zbmath.org/authors/?q=ai:kassaei.payman-l"Sasaki, Shu"https://zbmath.org/authors/?q=ai:sasaki.shuIn the paper under the review, the authors study Hilbert modular varieties, i.e., the Shimura varieties associated to \(\mathrm{Res}_{F / \mathbb{Q}} \mathrm{GL}_2\), where \(F\) stands for the totally real number field. They restric their attention to Hilbert modular varieties in characteristic \(p\) with Iwahori level at \(p\).
As the first main result of the paper, the authors obtain a geometric Jacquet-Langlands relation which provides an isomorphism between the irreducible components of studied varieties and products of projective bundles over quaternionic Shimura varieties of level prime to \(p\). The main new ingredient is the introduction of abelian varieties whose Dieudonné modules are obtained by splicing those of the source and target of the universal isogeny.
Using the obtained Jacquet-Langlands relation, together with the analysis of dualizing sheaves by dividing the schemes into smaller ones of the same dimension, the authors obtain the second main result of the paper. A filtration on the space of mod \(p\) Hilbert modular forms of parallel weight \(2\) and pro-\(p\) Iwahori level at \(p\) is provided, and the graded pieces are identified with spaces of quaternionic modular forms of level prime to \(p\) and weight in \([2, p+1 ]\). This result reflects the representation theory of \(\mathrm{GL}_2\) in characteristic \(p\) and generalizes a result of Serre for classical modular forms.
Reviewer: Ivan Matić (Osijek)A dialectical path to the sum of the cubeshttps://zbmath.org/1537.110032024-07-25T18:28:20.333415Z"Berendonk, Stephan"https://zbmath.org/authors/?q=ai:berendonk.stephanThe author discusses a geometric way of discovering the formula \(1^3 + 2^3 + \ldots + n^3 = (1 + 2 + \ldots + n)^2\).
For the entire collection see [Zbl 1522.01002].
Reviewer: Franz Lemmermeyer (Jagstzell)Note on a binomial coefficient divisorhttps://zbmath.org/1537.110042024-07-25T18:28:20.333415Z"Just, Matthew"https://zbmath.org/authors/?q=ai:just.matthewSummary: Let \(n\) be a positive integer, and \(k\) and integer with \(0 \leq k \leq n\). Let \(g\) be the greatest common divisor of \(k\) and \(n\). We use the cycle construction to give a combinatorial proof that the fraction \(n/g\) divides the binomial coefficient \(\left(\begin{smallmatrix} n \\ k \end{smallmatrix}\right)\).When additive and multiplicative inverses are the samehttps://zbmath.org/1537.110052024-07-25T18:28:20.333415Z"Briggs, Karen S."https://zbmath.org/authors/?q=ai:briggs.karen-s"Spivey, Caylee R."https://zbmath.org/authors/?q=ai:spivey.caylee-rSummary: A professor and her students were working through a routine problem on isomorphisms that required finding the multiplicative inverse of 43 modulo 50. The professor, however, simply asked for ``the inverse'' of 43 modulo 50 and got a surprising response. One student quickly answered that the inverse of 43 modulo 50 was 7. Thinking that the student was responding with the additive inverse, the professor restated her question and asked for the multiplicative inverse. The student checked her work and said that 7 is the multiplicative inverse of 43 mod 50. The observation that the additive and multiplicative inverse of an element could be the same led to a research project into the questions of when, why, and how frequently this phenomenon occurs. Answering these questions led to tools such as the Chinese remainder theorem, the fundamental theorem of cyclic groups, and quadratic residues.A generalization of Bauer's identical congruencehttps://zbmath.org/1537.110062024-07-25T18:28:20.333415Z"Cohen, Boaz"https://zbmath.org/authors/?q=ai:cohen.boazSummary: In this paper we generalize Bauer's Identical Congruence appearing in \textit{G. H. Hardy} and \textit{E. M. Wright}'s book [An introduction to the theory of numbers. 5th ed. Oxford etc.: Oxford at the Clarendon Press (1979; Zbl 0423.10001)], Theorems 126 and 127. Bauer's Identical Congruence asserts that the polynomial \(\prod_t(x-t)\), where the product runs over a reduced residue system modulo a prime power \(p^a\), is congruent (mod \(p^a)\) to the ``simple'' polynomial \((x^{p-1}-1)^{p^{a-1}}\) if \(p>2\) and \((x^2-1)^{2^{a-2}}\) if \(p=2\) and \(a\ge 2\). Our article generalizes these results to a broader context, in which we find a ``simple'' form of the polynomial \(\prod_t(x-t)\), where the product runs over the solutions of the congruence \(t^n\equiv 1\pmod{\text{P}^a}\) in the framework of the ring of algebraic integers of a given number field \(\mathbb{K} \), and where \(\text{P}\) is a prime ideal.An idempotent cryptarithmhttps://zbmath.org/1537.110072024-07-25T18:28:20.333415Z"Seraj, Samer"https://zbmath.org/authors/?q=ai:seraj.samerSummary: Notice that the square of 9376 is 87909376 which has as its rightmost four digits 9376. To generalize this remarkable fact, we show that, for each integer \(n \geq 2\), there exists at least one and at most two positive integers \(x\) with exactly \(n\)-digits in base 10 (meaning the leftmost or \(n\)th digit from the right is nonzero) such that squaring the integer results in an integer whose rightmost \(n\) digits form the integer \(x\). We then generalize the argument to prove that, in an arbitrary number base \(B \geq 2\) with exactly \(m\) distinct prime factors, an upper bound is \(2^m-2\) and a lower bound is \(2^{m-1} -1\) for the number of such \(n\)-digit positive integers. For \(n=1\), there are exactly \(2^m-1\) solutions, including 1 and excluding 0.Partition of quadratic residues and non-residues in \(\mathbb{Z}_p^*\) for an odd prime \(p\)https://zbmath.org/1537.110082024-07-25T18:28:20.333415Z"Yathirajsharma, M. V."https://zbmath.org/authors/?q=ai:yathirajsharma.m-v"Manjunatha, M. R."https://zbmath.org/authors/?q=ai:manjunatha.m-rThe authors determine formulas for the number of quadratic residues that are sums of quadratic residues. These numbers are clearly the same as the well known number of consecutive quadratic residues, traditionally denoted by RR.
Reviewer: Franz Lemmermeyer (Jagstzell)Recursively divisible numbershttps://zbmath.org/1537.110092024-07-25T18:28:20.333415Z"Fink, Thomas M. A."https://zbmath.org/authors/?q=ai:fink.thomas-m-aIn the paper under review, the author studies the recursive divisor function
\[
\kappa_x(n) = n^x + \sum_{d\lfloor n} \kappa_0(d),
\]
where \(d \lfloor n\) means \(d \vert n\) and \(d < n\). First, he introduces a geometric interpretation of the recursive divisor function by drawing the divisor tree for a given value of \(n\). Based on some properties of these tree diagrams, he obtains a relation between \(\kappa_x(n)\) and \(\kappa_0(n)\) as follows
\[
\frac{\kappa_x(n)}{n^x} = \frac{1}{2} + \frac{1}{2} \sum_{d|n} \frac{\kappa_0(d)}{d^x}.
\]
Then, he shows that for \(x > 1\),
\[
\frac{\kappa_x(n)}{n^x} < \frac{1}{2 - \zeta(x)}.
\]
The author obtains some of the properties of \(\kappa_0(n)\), including the exponential generating function of \(\kappa_0(p_1 \cdots p_r)\), from the properties of the function \(K(n)\), providing the number of ordered factorizations into integers greater than one, which satisfies \(K(1)=1\) and \(K(n) = \sum_{d\lfloor n} K(d)\).
Reviewer: Mehdi Hassani (Zanjan)On near-perfect numbers of special formshttps://zbmath.org/1537.110102024-07-25T18:28:20.333415Z"Hasanalizade, Elchin"https://zbmath.org/authors/?q=ai:hasanalizade.elchinFor a positive integer \(n\), let \(\sigma (n)\) denote the sum of all positive divisors of \(n\). A positive integer \(n\) is called \textit{near-perfect} if \(\sigma (n)=2n+d\) for some proper divisor \(d\) of \(n\). In this paper, the author studies the near-perfect numbers in the Fibonacci and Lucas sequences, near-perfect values taken by integer polynomials and near-perfect numbers with identical digits. For example, the author proves that (1) there are no odd near-perfect Fibonacci or Lucas numbers; (2) there are no near-perfect Fibonacci numbers with at most three distinct prime divisors; (3) the only near-perfect Lucas number with two distinct prime factors is \(18\).
Reviewer: Yong-Gao Chen (Nanjing)On prime primitive roots of \(2^kp+1\)https://zbmath.org/1537.110112024-07-25T18:28:20.333415Z"Filipovski, S."https://zbmath.org/authors/?q=ai:filipovski.slobodanLet \(p\) be a prime number and \(k \ge 1\) a natural number. In this note, all pairs \((p,k)\) are determined for which \(p\) is a primitive root modulo \(2^kp+1\): These are \((2,2)\), \((3,3)\), \((3,4)\) and \((5,4)\). Clearly \(2^kp+1 = q^m\) must be a prime power, and the result is proved using the possible factorizations of \(q^m-1 = 2^k p\).
Reviewer: Franz Lemmermeyer (Jagstzell)Bias in the number of steps in the Euclidean algorithm and a conjecture of Ito on Dedekind sumshttps://zbmath.org/1537.110122024-07-25T18:28:20.333415Z"Minelli, Paolo"https://zbmath.org/authors/?q=ai:minelli.paolo"Sourmelidis, Athanasios"https://zbmath.org/authors/?q=ai:sourmelidis.athanasios"Technau, Marc"https://zbmath.org/authors/?q=ai:technau.marcAuthors' abstract: We investigate the number of steps taken by three variants of the Euclidean algorithm on average over Farey fractions. We show asymptotic formulae for these averages restricted to the interval \((0,1 / 2)\), establishing that they behave differently on \((0,1 / 2)\) than they do on \((1 / 2,1)\). These results are tightly linked with the distribution of lengths of certain continued fraction expansions as well as the distribution of the involved partial quotients. As an application, we prove a conjecture of Ito on the distribution of values of Dedekind sums. The main argument is based on earlier work of Zhabitskaya, Ustinov, Bykovskiĭ and others, ultimately dating back to Lochs and Heilbronn, relating the quantities in question to counting solutions to a certain system of Diophantine inequalities. The above restriction to only half of the Farey fractions introduces additional complications.
Reviewer: Alexey Ustinov (Khabarovsk)Construction and categoricity of the real number system using decimalshttps://zbmath.org/1537.110132024-07-25T18:28:20.333415Z"Singh, Arindama"https://zbmath.org/authors/?q=ai:singh.arindama(no abstract)Congruences and integral roots of D'Arcais polynomialshttps://zbmath.org/1537.110142024-07-25T18:28:20.333415Z"Sriram, S."https://zbmath.org/authors/?q=ai:sriram.siddharth|sriram.sridevi|sriram.sundararajan|sriram.sastha"David Christopher, A."https://zbmath.org/authors/?q=ai:christopher.a-davidLet \(z\in \mathbb C\) and consider the product-to-sum representation: \[\displaystyle \prod_{m=1}^{\infty}(1-q^m)^{-z}=\sum_{n=0}^{\infty}P_n(z)q^n,\] where the term \(P_n(z),\) introduced by D'Arcais is a polynomial in \(z\) of degree \(n\) and as Newman found that it satisfies the recurrence relation: \[P_n(z)=\displaystyle\dfrac{z}{n}\sum_{k=1}^{n}\sigma(k)P_{n-k}(z),P_0(z)=1, \sigma(k)=\sum_{d|k}d.\] \textit{J.-P. Serre} [Glasg. Math. J. 27, 203--221 (1985; Zbl 0583.10015)] utilized Newman's result and obtained some facts concerning the lacunarity of even powers \(z\in \{2, 4, 6, 8, 10, 14, 26\}.\) It must be mentioned here that the polynomial representation of \(P_n(z)\) helps in obtaining some congruence properties when \(z\in\mathbb Z.\) Further, the non-vanishing attribute of \(P_n(z)\) at integral values of \(z\) is significant as it is evident from the fact that \(P_n(-24)\) is non-vanishing at all \(n\in \mathbb N\) and this is in fact equivalent to Lehmer conjecture [\textit{D. H. Lehmer}, Duke Math. J. 14, 429--433 (1947; Zbl 0029.34502)]. Also, in this interesting article, Kostant's representation for the coefficients of \(P_n(z)\) is revisited and along with the valuable auxiliary results, corollaries and consequences, the following results regarding roots (and associated bounds) for \(n!P_n(z)\) are elucidated:
Theorem 1. Let \(n\) be a positive integer and define
\[
T_n^{\star}=\displaystyle\max_{0\leq i\leq n-1}\{|a_{n, i}|^{\frac{1}{n-i}}\},\tag{1}
\]
where \(a_{n, r}\) denotes the coefficient of \(z^r\) of the polynomial \(n!P_n(z).\) If \(n!P_n(\beta)=0\) for some integer \(\beta,\) then \(\beta\in \{t\in \mathbb Z~|~t|(n-1)!\sigma(n), |t|<2T_n^{\star}\}.\)
Theorem 2. Let \(n\) be a positive integer. Let \(T_n^{\star}\) be as defined in equation (1). If \(n!P_n(\beta)=0\) for some integer \(\beta,\) then \[\beta\in\{t\in\mathbb Z~|~t^2|\Big\{-t\displaystyle\dfrac{n!}{2}\sum_{a+b=n}\dfrac{\sigma(a)}{a}\dfrac{\sigma(b)}{b}-(n-1)!\sigma(n)\Big\}, |t|<2T_n^{\star}\}.\]
Reviewer: Sanjeev Kumar (Chandigarh)Sidon sets and perturbationshttps://zbmath.org/1537.110152024-07-25T18:28:20.333415Z"Nathanson, Melvyn B."https://zbmath.org/authors/?q=ai:nathanson.melvyn-bernardSummary: A subset \(A\) of an additive abelian group is an \(h\)-Sidon set if every element in the \(h\)-fold sumset \(hA\) has a unique representation as the sum of \(h\) not necessarily distinct elements of \(A\). Let \(\mathbf{F}\) be a field of characteristic 0 with a nontrivial absolute value, and let \(A = \{a_i :i \in \mathbb{N} \}\) and \(B = \{b_i :i \in \mathbb{N} \}\) be subsets of \(\mathbf{F}\). Let \(\varepsilon = \{\varepsilon_i:i \in \mathbb{N} \}\), where \(\varepsilon_i > 0\) for all \(i \in \mathbb{N}\). The set \(B\) is an \(\varepsilon\)-perturbation of \(A\) if \(|b_i-a_i| < \varepsilon_i\) for all \(i \in \mathbb{N}\). It is proved that, for every \(\varepsilon = \{\varepsilon_i:i \in \mathbb{N} \}\) with \(\varepsilon_i > 0\), every set \(A = \{a_i :i \in \mathbb{N} \}\) has an \(\varepsilon\)-perturbation \(B\) that is an \(h\)-Sidon set. This result extends to sets of vectors in \(\mathbf{F}^n\).
For the entire collection see [Zbl 1479.11005].Multiple correlation sequences not approximable by nilsequenceshttps://zbmath.org/1537.110162024-07-25T18:28:20.333415Z"Briët, Jop"https://zbmath.org/authors/?q=ai:briet.jop"Green, Ben"https://zbmath.org/authors/?q=ai:green.ben|green.ben-gSummary: We show that there is a measure-preserving system \((X,\mathscr{B}, \mu, T)\) together with functions \(F_0, F_1, F_2 \in L^{\infty}(\mu)\) such that the correlation sequence \(C_{F_0, F_1, F_2}(n) = \int_X F_0 \cdot T^n F_1 \cdot T^{2n} F_2 \, d\mu\) is not an approximate integral combination of \(2\)-step nilsequences.High-entropy dual functions over finite fields and locally decodable codeshttps://zbmath.org/1537.110172024-07-25T18:28:20.333415Z"Briët, Jop"https://zbmath.org/authors/?q=ai:briet.jop"Labib, Farrokh"https://zbmath.org/authors/?q=ai:labib.farrokhSummary: We show that for infinitely many primes \(p\) there exist dual functions of order \(k\) over \(\mathbb{F}_p^n\) that cannot be approximated in \(L_\infty\)-distance by polynomial phase functions of degree \(k-1\). This answers in the negative a natural finite-field analogue of a problem of Frantzikinakis on \(L_\infty\)-approximations of dual functions over \(\mathbb{N}\) (a.k.a. multiple correlation sequences) by nilsequences.Partition and analytic rank are equivalent over large fieldshttps://zbmath.org/1537.110182024-07-25T18:28:20.333415Z"Cohen, Alex"https://zbmath.org/authors/?q=ai:cohen.alex"Moshkovitz, Guy"https://zbmath.org/authors/?q=ai:moshkovitz.guySummary: We prove that the partition rank and the analytic rank of tensors are equal up to a constant, over finite fields of any characteristic and any large enough cardinality depending on the analytic rank. Moreover, we show that a plausible improvement of our field cardinality requirement would imply that the ranks are equal up to \(1 + o(1)\) in the exponent over every finite field. At the core of the proof is a technique for lifting decompositions of multilinear polynomials in an open subset of an algebraic variety, and a technique for finding a large subvariety that retains all rational points such that at least one of these points satisfies a finite-field analogue of genericity with respect to it. Proving the equivalence between these two ranks, ideally over fixed finite fields, is a central question in additive combinatorics, and was reiterated by multiple authors. As a corollary, we prove -- allowing the field to depend on the value of the norm -- the polynomial Gowers inverse conjecture in the \(d\) versus \(d - 1\) case.A note on the largest sum-free sets of integershttps://zbmath.org/1537.110192024-07-25T18:28:20.333415Z"Jing, Yifan"https://zbmath.org/authors/?q=ai:jing.yifan"Wu, Shukun"https://zbmath.org/authors/?q=ai:wu.shukunLet \(A\) be a set of \(N\) positive integers. Let \(k,\ell\) be two positive integers. A subset \(X\) of \(A\) is called \((k,\ell)\)-sum-free if for every \(k+\ell\) terms \(x_1, x_2, \ldots, x_k, y_1, y_2, \ldots, y_{\ell}\) of \(X\), we always have \(\sum_{i=1}^{k}x_i\neq \sum_{j=1}^{\ell}y_j\). Let \(\mathscr{M}_{(k,\ell)}(A)\) denote the size of maximal \((k,\ell)\)-sum-free subset of \(A\), and let
\[
\mathscr{M}_{(k,\ell)}(N)=\inf_{A\subseteq \mathbb{Z}^{>0}, |A|=N}\mathscr{M}_{(k,\ell)}(A).
\]
\textit{P. Erdős} proved that \(\mathscr{M}_{(2,1)}(N)\geq N/3\) in [Proc. Symp. Pure Math. 8, 181--189 (1965; Zbl 0144.28103)]. Since then, the invariant \(\mathscr{M}_{(k,\ell)}(N)\) was studied by many researchers. In this paper under review, the authors prove that for every \(k\geq 1\), there is a function \(\omega(N)=\log N/\log\log N\) such that for every set \(A\) of \(N\) positive integers, there exists a maximal \((2k,4k)\)-sum-free set \(\Omega(2k,4k)\subset \mathbb{R}/\mathbb{Z}\), and
\[
\max_{x\in \mathbb{R}/\mathbb{Z}}\sum_{n\in A}\left(\1_{\Omega(2k,4k)}-\frac{1}{6k}\right)(nx)\gg \omega(N),
\]
where \(\1_{\Omega(2k,4k)}\) denotes the characteristic function of \(\Omega(2k,4k)\). As a consequence, they prove that there is an absolute constant \(c > 0\), such that \(\mathscr{M}_{(2k,4k)}(N)\geq \frac{N}{6k}+c\omega(N)\). The proof is based on a structural analysis of the given set \(A\).
Reviewer: Jiangtao Peng (Tianjin)Recovering affine linearity of functions from their restrictions to affine lineshttps://zbmath.org/1537.110202024-07-25T18:28:20.333415Z"Khare, Apoorva"https://zbmath.org/authors/?q=ai:khare.apoorva"Tikaradze, Akaki"https://zbmath.org/authors/?q=ai:tikaradze.akakiSummary: Motivated by recent results of \textit{T. Tao} and \textit{T. Ziegler} [Discrete Anal. 2016, Paper No. 13, 60 p. (2016; Zbl 1400.11028)] and \textit{R. Greenfeld} and \textit{T. Tao} [``A counterexample to the periodic tiling conjecture'', Preprint \url{arXiv:2211.15847}] on concatenating affine-linear functions along subgroups of an abelian group, we show three results on recovering affine linearity of functions \(f: V \rightarrow W\) from their restrictions to affine lines, where \(V, W\) are \({\mathbb{F}}\)-vector spaces and \(\dim V \geq 2\).
First, if \(\dim V < |{\mathbb{F}}|\) and \(f: V \rightarrow{\mathbb{F}}\) is affine-linear when restricted to affine lines parallel to a basis and to certain ``generic'' lines through 0, then \(f\) is affine-linear on \(V\). (This extends to all modules \(M\) over unital commutative rings \(R\) with large enough characteristic.)
Second, we explain how a classical result attributed to von Staudt (1850s) extends beyond bijections: If \(f: V \rightarrow W\) preserves affine lines \(\ell\), and if \(f(v) \not \in f(\ell)\) whenever \(v \not \in \ell\), then this also suffices to recover affine linearity on \(V\), but up to a field automorphism. In particular, if \({\mathbb{F}}\) is a prime field \({\mathbb{Z}}/p{\mathbb{Z}}\) \((p>2)\) or \({\mathbb{Q}}\), or a completion \({\mathbb{Q}}_p\) or \({\mathbb{R}}\), then \(f\) is affine-linear on \(V\).
We then quantitatively refine our first result above, via a weak multiplicative variant of the additive \(B_h\)-sets initially explored by \textit{J. Singer} [Trans. Am. Math. Soc. 43, 377--385 (1938; Zbl 0019.00502)], \textit{P. Erdös} and \textit{P. Turán} [J. Lond. Math. Soc. 16, 212--215 (1941; Zbl 0061.07301)], and \textit{R. C. Bose} and \textit{S. Chowla} [Comment. Math. Helv. 37, 141--147 (1962; Zbl 0109.03301)]. Weak multiplicative \(B_h\)-sets occur inside all rings with large enough characteristic, and in all infinite or large enough finite integral domains/fields. We show that if \(R\) is among any of these classes of rings, and \(M = R^n\) for some \(n \geq 3\), then one requires affine linearity on at least \(\binom{n}{\lceil n/2 \rceil}\)-many generic lines to deduce the global affine linearity of \(f\) on \(R^n\). Moreover, this bound is sharp.An energy decomposition theorem for matrices and related questionshttps://zbmath.org/1537.110212024-07-25T18:28:20.333415Z"Mohammadi, Ali"https://zbmath.org/authors/?q=ai:mohammadi.ali"Pham, Thang"https://zbmath.org/authors/?q=ai:pham-van-thang."Wang, Yiting"https://zbmath.org/authors/?q=ai:wang.yitingSummary: Given \(A \subseteq GL_2(\mathbb{F}_q)\), we prove that there exist disjoint subsets \(B, C \subseteq A\) such that \(A = B \sqcup C\) and their additive and multiplicative energies satisfying
\[
\max\{ E_+ (B), E_{\times}(C) \} \ll \frac{|A|^3}{M(|A|)},
\]
where
\[
M(|A|) = \min \Bigg \{ \frac{q^{4/3}}{|A|^{1/3}(\log|A|)^{2/3}}, \frac{|A|^{4/5}}{q^{13/5}(\log|A|)^{27/10}}\Bigg \}.
\]
We also study some related questions on moderate expanders over matrix rings, namely, for \(A, B, C \subseteq GL_2(\mathbb{F}_q)\), we have
\[
|AB + C|, |(A + B)C| \gg q^4,
\]
whenever \(|A\|B\|C| \gg q^{10 + 1/2}\). These improve earlier results due to \textit{Y. D. Karabulut} et al. [Forum Math. 31, No. 4, 951--970 (2019; Zbl 1462.11023)].Solutions of certain meta-Fibonacci recurrenceshttps://zbmath.org/1537.110222024-07-25T18:28:20.333415Z"Sobolewski, Bartosz"https://zbmath.org/authors/?q=ai:sobolewski.bartosz"Ulas, Maciej"https://zbmath.org/authors/?q=ai:ulas.maciejIn most general terms, a \textit{meta-Fibonacci sequence} is a solution of a recurrence relation of the form \(f(n) = f(p(n)) + f (q(n))\) where \(p(n)\), \(q(n)\) are expressions involving \(n\), \(f (n-1),\dots, f (n-k)\) for some \(k\). The class of these sequences is broad and some of these sequence may not be well-defined. For instance, it contains the Hofstadter's sequence defined by \(Q(n)=Q(n-Q(n-1)) + Q(n-Q(n-2))\) and \(Q(1)=Q(2)=1\).
The authors address the following general question: ``Can we expect the existence of a solution of the recurrence \(f(n)= f(n-f(n-u)) + f (n-v)\), where \(u,v\) are positive fixed integers such that at least one of its subsequences along some arithmetic progression is neither a linear recurrence sequence, nor is slow or unpredictable (chaotic)?'' More precisely, they consider the ``smallest interesting case'' which correspond to sequences denoted by \(h_{a,b}\) and satisfying a relation of the form
\[
f (n) = f (n-f (n-1)) + f (n-2)
\]
and with initial conditions given by some integer parameters \(a,b\).
For some cases, they obtain sequences with quite interesting behavior. The sequence \(h_{1,1}\) starts with
\[
1, 1, 2, 2, 4, 3, 6, 4, 10, 5, 13, 6, 19, 7, 23, 8, 33, 9, 38, 10, 51, 11, 57, 12, 76, 13.
\]
We have \(h_{1,1} (2n+1) = n + 1\), but the solution along even indices can be presented as a sum of functions counting certain partitions into powers of \(2\), the sequence \(h_{1,1}\) does not satisfy a linear recurrence equation, is not \(k\)-regular (in the sense of Allouche and Shallit) for any \(k\) and its ordinary generating function is transcendental over the rational function field. They also prove that the sequence reduced modulo 2 is \(2\)-automatic. The cases \(a,b>1\) are then discussed.
In conclusion, this paper highlights interesting behaviour of a family of integer sequences and opens the way to further questions and investigations.
Reviewer: Michel Rigo (Liège)On Horadam finite operator hybrid numbershttps://zbmath.org/1537.110232024-07-25T18:28:20.333415Z"Yağmur, Tülay"https://zbmath.org/authors/?q=ai:yagmur.tulaySummary: In this paper, we introduce and study a new hybrid number sequence with Horadam numbers and a finite operator, called Horadam finite operator hybrid numbers. We derive recurrence relation, Binet-like formula, ordinary generating function, exponential generating function, Poisson generating function, and summation formula for Horadam finite operator hybrid numbers. Moreover, we give matrix representation and Cassini's identities for these numbers.A weighted extension of Fibonacci numbershttps://zbmath.org/1537.110242024-07-25T18:28:20.333415Z"Bhatnagar, Gaurav"https://zbmath.org/authors/?q=ai:bhatnagar.gaurav"Kumari, Archna"https://zbmath.org/authors/?q=ai:kumari.archna"Schlosser, Michael J."https://zbmath.org/authors/?q=ai:schlosser.michael-jThe Fibonacci numbers, one of the best-known integer sequences, denoted by \(F_n\), are defined by \(F_n = 0\) for \(n = 0\), \(F_1 = 1\), and thereafter each \(F_n\) is the sum of the two previous numbers. The first few terms of these numbers are \(0, 1, 1, 2, 3, 5, 8, 13, 21, 34\ldots \)
In literature, many properties (and generalizations) have been discovered and many applications have been studied by many scientists using different methods.
This paper includes a ancient arising history for Fibonacci numbers as follows: According to [\textit{P. Singh}, Hist. Math. 12, 229--244 (1985; Zbl 0574.01005)]), these numbers were obtained in ancient Indian texts (600--800 A.D.) in the context of explaining the rules of poetry composition in Sanskrit and Prakrit (two ancient Indian languages). Knowledge at the time was transferred orally; it was useful to present it in the form of verse, so that it was easier to memorize and replicate. To describe the aforementioned rules, we need the following vocabulary. A mora is a unit of syllables (plural: morae). The basic units of Sanskrit prosody is a letter having one mora (called laghu) and with two morae (called guru). A type of metre (or line) in Sanskrit poetry allows sequences of laghu and guru letters in any order, but the metre should contain exactly \(n\) morae. It is in counting all possible metres of this type that the so-called Fibonacci numbers arose.
Recently, many mathematicians studied generalizations of these numbers and more properties of these numbers.
In the related paper, the authors defined the weighted Fibonacci numbers (with arbitrary weights). Then, they gave the analytic and combinatorial definitions of the weighted Fibonacci numbers, as well as the proposed elliptic weights, which contains weighted analogues of several Fibonacci identities. Lastly, the author derived some Fibonacci identities by using the telescoping methods given in [\textit{G. Bhatnagar}, Electron. J. Comb. 18, No. 2, Research Paper P13, 44 p. (2011; Zbl 1230.33010); Fibonacci Q. 54, No. 2, 166--171 (2016; Zbl 1400.11035)].
Reviewer: Uğur Duran (İskenderun)On matrix sequences represented by negative indices Pell and Pell-Lucas number with the decoding of Lucas blocking error correcting codeshttps://zbmath.org/1537.110252024-07-25T18:28:20.333415Z"Khompurngson, Kannika"https://zbmath.org/authors/?q=ai:khompurngson.kannika"Sompong, Supunnee"https://zbmath.org/authors/?q=ai:sompong.supunneeIn mathematics, the Pell numbers are an infinite sequence of integers, known since ancient times, that comprise the denominators of the closest rational approximations to the square root of 2. This sequence of approximations begins \(\frac{1}{1},\frac{3}{2},\frac{7}{5},\frac{17}{12},\) and \(\frac{41}{29 }\), so the sequence of Pell numbers begins with \(1,2,5,12,\) and \(29\). The numerators of the same sequence of approximations are half the companion Pell numbers or Pell-Lucas (companion Pell) numbers; these numbers form a second infinite sequence that begins with \(2,6,14,34\), and \(82\).
The Pell \(P_{n}\) and Pell-Lucas \(Q_{n}\) sequences are firstly\ defined by \textit{A. F. Horadam} [Fibonacci Q. 9, No. 3, 245--252, 263 (1971; Zbl 0219.10018)] and \textit{A. F. Horadam} and \textit{B. J. M. Mahon} [Fibonacci Q. 23, 7--20 (1985; Zbl 0557.10011)].
In literature, many properties (and generalizations) of Pell \(P_{n}\) and Pell-Lucas \(Q_{n}\) sequences have been discovered and many applications have been studied by many scientists using different methods.
Recently, many mathematicians studied some matrix sequences of these numbers and more matrix properties of these numbers (cf. [Pell and Pell-Lucas numbers with applications. New York: John Wiley \& Sons (2014)] and see the references cited therein).
This paper will present the matrix sequences that represent negative indices Pell and Pell-Lucas number. Important relationships between negative indices Pell and Pell-Lucas number and Pell and Pell-Lucas matrix sequences will be an essential part of this study. Since information security has become a more significant topic in recent years, there have been various studies on coding and decoding which will lead to applied matrices for this topic. Specifically, there will be a focus on the Lucas blocking algorithm which was proposed by \textit{S. Uçar} et al. [``A new application to coding theory via Fibonacci and Lucas number'', Math. Sci. Appl. E-Notes 7, No. 1, 62--70 (2019; \url{doi:10.36753/mathenot.559251})]. The development of this algorithm will allow for the detection of some cases that might mistake between the sender and the receiver by using this algorithm.
Reviewer: Uğur Duran (İskenderun)Some properties of Fibonacci-sigmoid numbers and polynomials matrixhttps://zbmath.org/1537.110262024-07-25T18:28:20.333415Z"Kim, M. S."https://zbmath.org/authors/?q=ai:kim.moon-su|kim.myeung-su|kim.museong|kim.moo-sang|kim.man-sik|kim.min-sub|kim.myung-soo|kim.mun-suk|kim.myun-sik|kim.min-sik|kim.myung-sub|kim.minsoo|kim.myoung-suk|kim.min-soengThe golden ratio is frequently used in many branches of science as well as mathematics. Interestingly, this mysterious number also appears in architecture and art. Miscellaneous properties of Golden calculus (or \(F\)-calculus) have been introduced and investigated in detail by \textit{Ş. Nalcı} and \textit{O. K. Pashaev} [J. Phys. A, Math. Theor. 45, No. 1, Article ID 015303, 23 p. (2012; Zbl 1235.81091)] which are the key references for golden calculus.
Using the \(F\)-analog of exponential function (say, also the golden exponential function), \textit{O. K. Pashaev}, \textit{M. Ozvatan} [``Bernoulli-Fibonacci polynomials''. Preprint, \url{arXiv:2010.15080}] defined the Bernoulli-Fibonacci polynomials and related numbers. Then the Euler-Fibonacci numbers and polynomials and the Apostol Bernoulli-Fibonacci and Apostol Euler-Fibonacci of order \(\alpha \) were introduced in [\textit{E. Gülal} and \textit{N. Tuglu}, Turkish J. Math. Comput. Sci. 15, No. 1, 203--211 (2023; \url{doi:10.47000/tjmcs.1242781}) and \textit{N. Tuglu} et al., Indian J. Pure Appl. Math. \url{doi:10.1007/s13226-023-00413-2}], and also some identities and matrix representations for Bernoulli-Fibonacci polynomials and Euler-Fibonacci polynomials were provided.
Recently, many mathematicians have recently studied various matrices for different types of polynomials and sequences. For instance, Pascal's matrices were studied in depth by \textit{G. S. Call} and \textit{D. J. Velleman} [Am. Math. Mon. 100, No. 4, 372--376 (1993; Zbl 0788.05011)] and \textit{Z. Zhang} [Linear Algebra Appl. 250, 51--60 (1997; Zbl 0873.15014)]. In addition, Fibonacci-Pascal matrices and the inverse of these matrices were studied and organized [\textit{S.-L. Yang} and \textit{Z.-K. Liu}, Int. J. Math. Math. Sci. 2006, No. 1, Article ID 90901, 7 p. (2006; Zbl 1131.15006)]. Furthermore, other matrices such as Bernoulli and Euler matrices were studied extensively as well.
There are two approaches for this topic: The first is to study generalizations of these studies and the second is to derive more properties on the mentioned studies.
In the paper under review, the author focuses mainly on the matrices which contain entries regarding Fibonacci-sigmoid polynomials. The author introduces the Fibonacci-sigmoid polynomials and numbers to show their relation. Then, he defines the Fibonacci-sigmoid polynomials matrix and calculates an example of the matrix. Then, the author checks if the additivity property holds for the matrix and also calculates diverse of the inverse matrices. Finally, the Fibonacci-sigmoid polynomials matrices can be factorized by the Fibonacci matrix by defining a new matrix.
Reviewer: Uğur Duran (İskenderun)Fibonacci convolutionshttps://zbmath.org/1537.110272024-07-25T18:28:20.333415Z"Nacin, David"https://zbmath.org/authors/?q=ai:nacin.davidSummary: We present visual proofs of two results on convolutions of the Fibonacci numbers with related sequences.Arctangents of Fibonacci ratioshttps://zbmath.org/1537.110282024-07-25T18:28:20.333415Z"Nelsen, Roger B."https://zbmath.org/authors/?q=ai:nelsen.roger-bSummary: We illustrate an identity for the sum of two arctangents of ratios of Fibonacci numbers.Identities for Pell numbers: a visual samplerhttps://zbmath.org/1537.110292024-07-25T18:28:20.333415Z"Nelsen, Roger B."https://zbmath.org/authors/?q=ai:nelsen.roger-bSummary: We present some visual arguments for various Pell and Pell-Lucas number identities.On the divisibility of \(F_{n_0k^{p-1}}\) by \(k^p\) and of \(L_{n_0k^{p-1}}\), with \((F)\) the Fibonacci sequence and \((L)\) the Lucas sequencehttps://zbmath.org/1537.110302024-07-25T18:28:20.333415Z"Pichereau, Alain"https://zbmath.org/authors/?q=ai:pichereau.alainThe author studies the divisibility of Fibonacci and Lucas numbers with index \(n_op^{k-1}\) by \(k^p\) using elementary number theory.
Reviewer: Franz Lemmermeyer (Jagstzell)Gaussian binomial coefficients in group theory, field theory, and topologyhttps://zbmath.org/1537.110312024-07-25T18:28:20.333415Z"Chebolu, Sunil K."https://zbmath.org/authors/?q=ai:chebolu.sunil-k"Lockridge, Keir"https://zbmath.org/authors/?q=ai:lockridge.keir-hSummary: In this article we offer group-theoretic, field-theoretic, and topological interpretations of the Gaussian binomial coefficients and their sum. For a finite \(p\)-group \(G\) of rank \(n\), we show that the Gaussian binomial coefficient \(\binom{n}{k}_p\) is the number of subgroups of \(G\) that are minimally expressible as an intersection of \(n - k\) maximal subgroups of \(G\), and their sum is precisely the number of subgroups that are either \(G\) or an intersection of maximal subgroups of \(G\). We provide a field-theoretic interpretation of these quantities through the lens of Galois theory and a topological interpretation involving covering spaces.Asymptotics on a class of Legendre formulashttps://zbmath.org/1537.110322024-07-25T18:28:20.333415Z"Diaz, Maiyu"https://zbmath.org/authors/?q=ai:diaz.maiyuSummary: Let \(f\) be a real-valued function of a single variable such that it is positive over the primes. In this article, we construct a factorial, \(n!_f\), associated to \(f\), called the associated Legendre formula, or \(f\)-factorial, and show, subject to certain criteria, that \(n!_f\) satisfies a weak Stirling approximation. As an application, we will give weak approximations to the Bhargava factorial over the set of primes and to a less well-known Legendre formula.The factorial function and generalizations, extendedhttps://zbmath.org/1537.110332024-07-25T18:28:20.333415Z"Lagarias, Jeffrey C."https://zbmath.org/authors/?q=ai:lagarias.jeffrey-c"Yangjit, Wijit"https://zbmath.org/authors/?q=ai:yangjit.wijitThis paper presents an extension of Bhargava's theory of generalized factorials associated to any nonempty subset \(S\) of the integers \(\mathbb Z\) as presented in [\textit{M. Bhargava}, Am. Math. Mon. 107, No. 9, 783--799 (2000; Zbl 0987.05003)]. Bhargava's generalized factorials \(k!_S\) are produced in factored form as the product over all primes \(p\) of the \(k\)-th invariants, which are defined using a notion of \(p\)-ordering of the elements in \(S\). This paper defines \(b\)-orderings and \(b\)-sequences of \(S\) for all integers \(b\ge 2\), which coincide with Bhargava's definitions when \(b\) is a prime. These notions for the extreme cases \(b=1\) and \(b=0\) are also defined. The main result is the well-definedness of \(b\)-sequences independent of the choice of \(b\)-ordering. It defines generalized factorials \(k!_{S,\mathcal T}\) and generalized binomial coefficients \(\binom{k+\ell}{k}_{S,\mathcal T}\) as nonnegative integers, for all nonempty \(S\) and allowing only \(b\) in \(\mathcal T\subseteq\mathbb N:=\{0,1,2,\dots\}\). Formulas are given for the \(b\)-ordering invariants for \(S=\mathbb Z\) and \(S=\mathbb P\) (the set of all primes).
Reviewer: Takao Komatsu (Hangzhou)Analytic aspects of generalized central trinomial coefficientshttps://zbmath.org/1537.110342024-07-25T18:28:20.333415Z"Liang, Huyile"https://zbmath.org/authors/?q=ai:liang.huyile"Wang, Yaling"https://zbmath.org/authors/?q=ai:wang.yaling"Wang, Yi"https://zbmath.org/authors/?q=ai:wang.yi.10Summary: The divisibility and congruence of usual and generalized central trinomial coefficients have been extensively investigated. The present paper is devoted to analytic properties of these numbers. We show that usual central trinomial polynomials \(\mathcal{T}_n(x)\) have only real roots, and roots of \(\mathcal{T}_n(x)\) interlace those of \(\mathcal{T}_{n+1}(x)\), as well as those of \(\mathcal{T}_{n+2}(x)\), which gives an affirmative answer to an open question of Fisk. We establish necessary and sufficient conditions such that the generalized central trinomial coefficients \(T_n(b,c)\) form a log-convex sequence or a Stieltjes moment sequence.On \(n\)th order Euler polynomials of degree \(n\) that are Eisensteinhttps://zbmath.org/1537.110352024-07-25T18:28:20.333415Z"Filaseta, Michael"https://zbmath.org/authors/?q=ai:filaseta.michael-a"Luckner, Thomas"https://zbmath.org/authors/?q=ai:luckner.thomasThis paper concern Bernoulli numbers, Eisenstein polynomials, Euler polynomials, Genocchi numbers, irreducible polynomials, \(p\)-adic valuation. It has very important applications in probability theory and applied sciences. The authors also give many results and examples with related tables covering the above content.
Reviewer: Yilmaz Simsek (Antalya)Explicit formula for sums related to the generalized Bernoulli numbershttps://zbmath.org/1537.110362024-07-25T18:28:20.333415Z"Mittou, Brahim"https://zbmath.org/authors/?q=ai:mittou.brahim(no abstract)Explicit, determinantal, recursive formulas and relations of the Peters polynomials and numbershttps://zbmath.org/1537.110372024-07-25T18:28:20.333415Z"Dağlı, Muhammet Cihat"https://zbmath.org/authors/?q=ai:dagli.muhammet-cihatThis paper's content covers the Bell polynomial of the second kind, the Boole number, the Boole polynomial, determinantal expression, generating function, the Hessenberg determinant, the Peters number, the Peters polynomial, recursive relation. This content has very important applications in combinatorics, probability theory, and in applied sciences. By applying the Faà di Bruno formula with the Bell polynomials of the second kind, the author gives many formulas for the Peters polynomials and numbers and also derivative formulas. The author also gives recursive relations for the Peters polynomials and numbers. Moreover, he gives some applications for alternative recursive relations and recursive relation for the Hessenberg determinants involving these numbers an polynomials and also the Boole polynomials and numbers.
Reviewer: Yilmaz Simsek (Antalya)Extending Galileo's ratio to hex numbers and beyondhttps://zbmath.org/1537.110382024-07-25T18:28:20.333415Z"Mukherjee, Rajib"https://zbmath.org/authors/?q=ai:mukherjee.rajib"Chakraborty, Manishita"https://zbmath.org/authors/?q=ai:chakraborty.manishitaSummary: In this article we are extending Galileo's ratio to Hex numbers and beyond.Computational aspects of sturdy and flimsy numbershttps://zbmath.org/1537.110392024-07-25T18:28:20.333415Z"Clokie, Trevor"https://zbmath.org/authors/?q=ai:clokie.trevor"Lidbetter, Thomas F."https://zbmath.org/authors/?q=ai:lidbetter.thomas-f"Molina Lovett, Antonio"https://zbmath.org/authors/?q=ai:molina-lovett.antonio"Shallit, Jeffrey"https://zbmath.org/authors/?q=ai:shallit.jeffrey-o"Witzman, Leon"https://zbmath.org/authors/?q=ai:witzman.leonSummary: Following Stolarsky, we say that a natural number \(n\) is \textit{flimsy} in base \(b\) if some positive multiple of \(n\) has smaller digit sum in base \(b\) than \(n\) does; otherwise it is \textit{sturdy}. When \(n\) is proven flimsy by multiplier \(k\), we say \(n\) is \textit{k-flimsy}. We study computational aspects of sturdy and flimsy numbers. We provide some criteria for determining whether a number is sturdy. We study the computational problem of checking whether a given number is sturdy, giving several algorithms for the problem, focusing particularly on the case \(b = 2\). We find two additional, previously unknown sturdy primes. We develop a method for determining which numbers with a fixed number of 0's in binary are flimsy. Finally, we develop a method that allows us to estimate the number of \(k\)-flimsy numbers with \(n\) bits, and we provide explicit results for \(k = 3\) and \(k = 5\). Our results demonstrate the utility (and fun) of creating algorithms for number theory problems, based on methods of automata theory.Curious congruences for cyclotomic polynomials IIhttps://zbmath.org/1537.110402024-07-25T18:28:20.333415Z"Matsusaka, Toshiki"https://zbmath.org/authors/?q=ai:matsusaka.toshiki"Shibukawa, Genki"https://zbmath.org/authors/?q=ai:shibukawa.genkiThis paper is a subsequent research of the paper by \textit{S. Akiyama} and \textit{H. Kaneko} [Res. Number Theory 8, No. 4, Paper No. 102, 10 p. (2022; Zbl 1503.11007)]. The authors of the current paper prove the conjecture proposed by Akiyama and Kaneko and also improve their result.
Let \(\Phi_n (x)\) be the \(n\)-th cyclotomic polynomial. The degree of \(\Phi_n (x)\) is given by Euler's totient function \(\phi (n)\). Define the polynomial \(F_k (x_1, \ldots , x_k) \in \mathbb{Q} [x_1, \ldots , x_k]\) by the generating function:
\[
\sum_{k=0}^{\infty} F_k (x_1, \ldots, x_k) \frac{t^k}{k!} =(1+t)^{x_1} \exp \left( 2 \sum_{\nu=1}^{\infty} \frac{B_{2\nu}}{(2\nu) !} (-\log (1+t))^{2\nu} x_{2\nu }\right)
\]
as per Lemma 3.1 in this paper, where \(B_{2\nu}\) is the \(2\nu\)-th Bernoulli number.
The authors prove that \(F_{2k+1} (x_{1}, \ldots ,x_{2k+1})\) is divisible by \(x_1-k\) in \(\mathbb{Q} [x_1, \ldots, x_{2k+1}]\) for every non-negative integer \(k\) as it was conjectured by Akiyama and Kaneko.
There is a close relation between the polynomial \(F_k\) and the derivative of cyclotomic polynomial found by \textit{D. H. Lehmer} [J. Math. Anal. Appl. 15, 105--117 (1966; Zbl 0168.29304)]:
\[
\frac{ \Phi_n^{(k)}(1)}{\Phi_n (1)} = F_k \left(\frac{\phi(n)}{2}, \frac{J_2 (n)}{4}, \ldots, \frac{J_k (n)}{2k} \right),
\]
where \(J_k (n) = n^k \prod_{p \mid n} (1-p^{-k})\) is Jordan's totient function. By proving a certain integral property of \(F_k\) (Theorem 2.5), the authors succeed to show: (i) \(\Phi_n' (1)/\Phi_n (1) = \phi (n)/2\); (ii) \(\Phi_n^{(3)} (1) /\Phi_n (1) \) is divisible by \(\phi (n)/2-1\); (iii) \(\Phi_n^{(2k+1)} (1) /\Phi_n (1) \) is divisible by \(\phi (n)/2-2k\) for \(k \ge 2\). These are improvements of the result by Akiyama and Kaneko.
Reviewer: Masanari Kida (Tōkyō)Good things come in threes: a ternary tree for triangular tripleshttps://zbmath.org/1537.110412024-07-25T18:28:20.333415Z"Bartz, Jeremiah"https://zbmath.org/authors/?q=ai:bartz.jeremiahSummary: A common saying is that good things come in threes. In [Tidskr. Elem. Mat. Fys. Kemi 17, 129--139 (1934; JFM 60.0121.03)], \textit{B. Berggren} may have agreed when unearthing three matrices which generate a ternary tree of all primitive Pythagorean triples from the initial triple \((3,4,5)\). We show that a similar structure exists when replacing primitive Pythagorean triples with triangular triples. The construction of both ternary trees are intertwined, involving Berggren's three matrices and a third ternary tree. Perhaps good things really do come in threes!An octic Diophantine equation and related families of elliptic curveshttps://zbmath.org/1537.110422024-07-25T18:28:20.333415Z"Choudhry, Ajai"https://zbmath.org/authors/?q=ai:choudhry.ajai"Shamsi Zargar, Arman"https://zbmath.org/authors/?q=ai:shamsi-zargar.armanLet \(\varphi (x_1, x_2, x_3) = x^8_1+x^8_2+x^8_3 -2x_1^4x_2^4-2x_1^4x_3^4-2x_2^4x_3^4\). In connection with equiareal triangles whose sides are perfect squares of integers the author studies the diophantine equation \(\varphi (x_1, x_2, x_3)=\varphi (y_1, y_2, y_3)\). Two parametric solutions are found which lead to elliptic curves of rank 5. One of the families is studied in detail.
Reviewer: István Gaál (Debrecen)On prime powers in linear recurrence sequenceshttps://zbmath.org/1537.110432024-07-25T18:28:20.333415Z"Odjoumani, Japhet"https://zbmath.org/authors/?q=ai:odjoumani.japhet"Ziegler, Volker"https://zbmath.org/authors/?q=ai:ziegler.volkerThe authors consider the Diophantine equation \(U_n=p^s\) where \(U_n\) is a linear recurrence sequence, \(p\) is a prime and \(x\) is a positive integer. Under technical conditions they show that except for finitely many \(p\), the equations has at most one solution \((n,x)\). The set of exceptional primes are determined for the Tribonacci sequence and for the Lucas sequence plus one. Baker's method and Davenport reduction is used.
Reviewer: István Gaál (Debrecen)Integral points of bounded height on a log Fano threefoldhttps://zbmath.org/1537.110442024-07-25T18:28:20.333415Z"Wilsch, Florian"https://zbmath.org/authors/?q=ai:wilsch.florianSummary: We determine an asymptotic formula for the number of integral points of bounded height on a blow-up of \( \mathbb{P}^3\) outside certain planes using universal torsors.On a variant of Pillai's problem with factorials and \(S\)-unitshttps://zbmath.org/1537.110452024-07-25T18:28:20.333415Z"Faye, Bernadette"https://zbmath.org/authors/?q=ai:faye.bernadette"Luca, Florian"https://zbmath.org/authors/?q=ai:luca.florian"Ziegler, Volker"https://zbmath.org/authors/?q=ai:ziegler.volkerThe authors study the representability of an integer \(c\) as the difference of an \(S\)-unit \(s\) and a factorial:
\[
c=s-m!.
\]
For the set of primes 2,3,5,7 generating the group of \(S\)-units all integers \(c\) are determined for which there exist at least two solutions \(s,m\). Among others Baker's method and LLL-reduction is used, both in the complex and \(p\)-adic case.
Reviewer: István Gaál (Debrecen)Correction to: ``The Hasse principle for systems of diagonal cubic forms''https://zbmath.org/1537.110462024-07-25T18:28:20.333415Z"Brüdern, Jörg"https://zbmath.org/authors/?q=ai:brudern.jorg"Wooley, Trevor D."https://zbmath.org/authors/?q=ai:wooley.trevor-dCorrection to the authors' paper [ibid. 364, No. 3--4, 1255--1274 (2016; Zbl 1372.11048)].Inhomogeneous Diophantine approximation for generic homogeneous functionshttps://zbmath.org/1537.110472024-07-25T18:28:20.333415Z"Kleinbock, Dmitry"https://zbmath.org/authors/?q=ai:kleinbock.dmitry-ya"Skenderi, Mishel"https://zbmath.org/authors/?q=ai:skenderi.mishelAuthors' abstract: This paper is a sequel to [Monatsh. Math. 194, No. 3, 523--554 (2021; Zbl 1481.11039)] in which results of that paper are generalized so that they hold in the setting of inhomogeneous Diophantine approximation. Given any integers \(n\geq 2\) and \(\ell\geq 1\), any \(\boldsymbol{\xi}=( \xi_1,\ldots, \xi_\ell)\in \mathbb{R}^\ell \), and any homogeneous function \(f=( f_1,\ldots, f_\ell): \mathbb{R}^n\to \mathbb{R}^\ell\) that satisfies a certain nonsingularity assumption, we obtain a biconditional criterion on the approximating function \(\psi=( \psi_1,\ldots, \psi_\ell): \mathbb{R}_{\geq 0}\to ( \mathbb{R}_{> 0} )^\ell\) for a generic element \(f\circ g\) in the \(\mathrm{SL}_n(\mathbb{R})\)-orbit of \(f\) to be (respectively, not to be) \( \psi \)-approximable at \(\boldsymbol{\xi}=( \xi_1,\ldots, \xi_n)\): that is, for there to exist infinitely many (respectively, only finitely many) \(\mathbf{v} \in \mathbb{Z}^n\) such that \(| \xi_j-( f_j\circ g)(\mathbf{v})|\leq \psi_j(\| \mathbf{v} \|)\) for each \(j\in\{1,\ldots,\ell\} \). In this setting, we also obtain a sufficient condition for uniform approximation. We also consider some examples of \(f\) that do not satisfy our nonsingularity assumptions and prove similar results for these examples. Moreover, one can replace \(\mathrm{SL}_n(\mathbb{R})\) above by any closed subgroup of \(\mathrm{ASL}_n(\mathbb{R})\) that satisfies certain integrability axioms (being of Siegel and Rogers type) introduced by the authors in the aforementioned previous paper.
Reviewer: Ranjeet Sehmi (Chandigarh)On the exceptional set for Diophantine inequality with unlike powers of primeshttps://zbmath.org/1537.110482024-07-25T18:28:20.333415Z"Liu, Huafeng"https://zbmath.org/authors/?q=ai:liu.huafeng.1|liu.huafeng.2|liu.huafeng"Liu, Rui"https://zbmath.org/authors/?q=ai:liu.rui|liu.rui.1The sequence \(\{v_i\}_{i=1}^\infty\) of positive real numbers is said to be \textit{well spaced} if there exist constants \(c,C\) such that \(0<c<v_{i+1}-v_i<C\) for all \(i=1,2,\ldots\). The Diophantine inequality considered in the paper under review is \[ |\lambda_2p_2^2+\lambda_3p_3^3+\lambda_4p_4^4+\lambda_5p_5^5-v|<v^{-\delta}, \] where \(p_i\) are prime variables, \(\lambda_i\) are given nonzero real numbers, \(v\) belongs to a given well-spaced sequence \(B\), and \(\delta>0\) is a fixed real number. More precisely, denoting by \({\mathbb E}(B,N,\delta)\) the set of \(v\in B\), with \(v\le N\), such that the previous inequality has no solutions in primes \(p_i\), the authors have proved that if, in addition, the coefficients \(\lambda_i\) are not all negative with \(\lambda_2/\lambda_3\) algebraic and irrational, then \[ |{\mathbb E}(B,N,\delta)|\ll N^{359/378+2\delta+\varepsilon} \] for any \(\varepsilon>0\). In case \(\lambda_2/\lambda_3\) is just irrational, then they have shown that the there exist a divergent sequence \(N_j\) such that the bound \[ |{\mathbb E}(B,N_j,\delta)|\ll N_j^{359/378+2\delta+\varepsilon} \] holds for any \(\varepsilon>0\) and all \(j\). Moreover, if the convergent denominators \(q_j\) for \(\lambda_2/\lambda_3\) satisfy \(q_{j+1}^{1-\omega}\ll q_j\) for some \(\omega\in[0,1)\), then for all \(N\ge 1\) and \(\varepsilon>0\), one has \[ |{\mathbb E}(B,N,\delta)|\ll N^{2\chi+2\delta+\varepsilon}, \] where \(\chi=\max\{\frac{5-2\omega}{18-12\omega},\frac{359}{756}\}\). This is an improvement on a recent result [\textit{Q. Mu} and \textit{Z. Gao}, Ramanujan J. 60, No. 2, 551--570 (2023; Zbl 1510.11097)].
Reviewer: Maurizio Laporta (Napoli)\(D\)-finite multivariate series with arithmetic restrictions on their coefficientshttps://zbmath.org/1537.110492024-07-25T18:28:20.333415Z"Bell, Jason"https://zbmath.org/authors/?q=ai:bell.jason-p"Smertnig, Daniel"https://zbmath.org/authors/?q=ai:smertnig.danielA well-known result of Bézivin for univariate power series that satisfy a linear recurrence states that their generating series admit a presentation as a finite sum of particular simple generating series. This result was recently generalized to the non-commutative multivariate case by the authors and in this contribution explicit characterizations of such, so called, \(D\)-finite Bézivin or Pólya series over fields of characteristic \(0\) are derived. Indeed the main result exactly characterizes the rational series, so the result might have applications in the area of weighted formal language theory.
More specifically, let us fix a field of characteristic \(0\). Additionally, for every \(r \in \mathbb N\) and multiplicative subgroup \(G\) we let \(rG\) be the set of all sums of at most \(r\) summands of \(G\). A multivariate power series is called Bézivin series if there exists a finitely generated subgroup \(G\) and \(r \in \mathbb N\) such that all series coefficients are in \(rG\). Finally, a power series is \(D\)-finite if all its partial derivatives are contained in a finite-dimensional vector space over the rational function field. The main results obtained in the contribution states that a Bézivin series is \(D\)-finite if and only if it is rational if and only if it is a finite sum of skew geometric series with coefficients in the subgroup \(G\). Two additional characterizations are also provided.
Overall, the paper is very well written, but assumes a very solid background on series. Full proof details are provided and any graduate of mathematics with a solid background on series should be able to fully appreciate this contribution.
Reviewer: Andreas Maletti (Leipzig)Properness of the global-to-local map for algebraic groups with toric connected component and other finiteness propertieshttps://zbmath.org/1537.110502024-07-25T18:28:20.333415Z"Rapinchuk, Andrei S."https://zbmath.org/authors/?q=ai:rapinchuk.andrei-s"Rapinchuk, Igor A."https://zbmath.org/authors/?q=ai:rapinchuk.igor-aSummary: This is a companion paper to [the authors, J. Number Theory 233, 228--260 (2022; Zbl 1489.11059)], where we proved the finiteness of the Tate-Shafarevich group for an arbitrary torus \(T\) over a finitely generated field \(K\) with respect to any divisorial set \(V\) of places of \(K\). Here, we extend this result to any \(K\)-group \(D\) whose connected component is a torus (for the same \(V\)), and as a consequence obtain a finiteness result for the local-to-global conjugacy of maximal tori in reductive groups over finitely generated fields. Moreover, we prove the finiteness of the Tate-Shafarevich group for tori over function fields \(K\) of normal varieties defined over base fields of characteristic zero and satisfying Serre's condition (F), in which case \(V\) consists of the discrete valuations associated with the prime divisors on the variety (geometric places). In this situation, we also establish the finiteness of the number of \(K\)-isomorphism classes of algebraic \(K\)-tori of a given dimension having good reduction at all \(v\in V\), and then discuss ways of extending this result to positive characteristic. Finally, we prove the finiteness of the number of isomorphism classes of forms of an absolutely almost simple group defined over the function field of a complex surface that have good reduction at all geometric places.On \(I^n\)-forms over field extensionshttps://zbmath.org/1537.110512024-07-25T18:28:20.333415Z"Lorenz, Nico"https://zbmath.org/authors/?q=ai:lorenz.nicoLet~\(E\supseteq F\) be a finite field extension with~\(F\) of characteristic not two, and \(\varphi\)
a quadratic form over~\(E\), which is defined over~\(F\), and whose Witt class is in the \(n\)th
power \(\mathrm{I}^{n}(E)\) of the fundamental ideal of the Witt ring~\(\mathrm{W} (E)\) of~\(E\). The paper
studies the question what can be said about forms (if any) in the preimage of~\(\varphi\)
in~\(\mathrm{I}^{n}(F)\) under the natural morphism \(r_{E/F}:\mathrm{W}(F)\to\mathrm{W}(E)\).
Not surprisingly, if the degree~\([E:F]\) is even not much, but in case~\([E:F]\) is odd
the author shows that there is a unique anisotropic form \(\psi\in r_{E/F}^{-1}(\varphi)\cap\mathrm{I}^{n}(F)\),
and this form and~\(\varphi\) have the same dimension and Pfister number.
In case~\(E\) is a quadratic extension and \(n\in\{ 1,2\}\) it is proven that there exists~\(\psi\in\mathrm{I}^{n}(F)\)
with \(r_{E/F}(\psi)=\varphi\) and \(\dim\psi=\dim\varphi\), but for~\(n\geq 3\) the situation is more complicated.
Here the article gives an example of a quadratic extension \(E\supset F\), where~\(F\) is a Laurent extension of~\(\mathbb{Q}\)
in three variables, and of a form~\(\varphi\) in~\(\mathrm{I}^{n}(E)\), such that every~\(\psi\in r_{E/F}^{-1}(\varphi)\cap\mathrm{I}^{n}(F)\) satisfies
\(\dim\psi\geq\dim\varphi +2^{n-2}\).
Reviewer: Stefan Gille (Edmonton)Making the case for pseudodifferential arithmetichttps://zbmath.org/1537.110522024-07-25T18:28:20.333415Z"Unterberger, André"https://zbmath.org/authors/?q=ai:unterberger.andreSummary: Let \(\Gamma =\mathrm{SL}(2,\mathbb Z)\) act in the plane by linear changes of coordinates. The resulting spectral theory of the automorphic Euler operator, which refines the theory of modular forms of non-holomorphic type, has definite advantages. One of these lies in the enrichment gained by interpreting automorphic distributions as symbols in the Weyl pseudodifferential calculus: the formula making the sharp composition of modular distributions explicit is given in terms of \(L\)-function theory. On the other hand, starting from distributions of arithmetic interest for symbols, we obtain operators the structure of which expresses itself nicely in terms of congruence arithmetic, providing a possible new approach to the Riemann hypothesis.
For the entire collection see [Zbl 1446.00025].On the hybrid mean value of generalized Dedekind sums, generalized Hardy sums and Kloosterman sumshttps://zbmath.org/1537.110532024-07-25T18:28:20.333415Z"Tian, Qing"https://zbmath.org/authors/?q=ai:tian.qing"Wang, Yan"https://zbmath.org/authors/?q=ai:wang.yan.128The paper is devoted to the calculation of some sums involving generalized Dedekind sums and Kloosterman sums.
Let \(k\) be a positive integer. For arbitrary integers \(h, m\), and \(n\), the \textit{generalized Dedekind sum} is defined by
\[
S(h, m, n, k)=\sum_{j=1}^k \bar{B}_m\left(\frac{j}{k}\right) \bar{B}_n\left(\frac{h j}{k}\right),
\]
where
\[
\bar{B}_m(x)= \begin{cases}B_m(x-[x]), & \text { if } x \text { is not an integer, } \\
0, & \text { if } x \text { is an integer }\end{cases}
\]
and \(B_m(x)\) is the \(m\)-th Bernoulli polynomial. For \(m=n=1\), \(S(h, 1,1, q)=S(h, q)\) is the classical Dedekind sum. Kloosterman sum is defined by
\[
K(n, q)=\sum_{c=1}^q e\left(\frac{n c+\bar{c}}{q}\right),
\]
where \(\sum_{c=1}^{\prime q}\) denotes the summation over all \(c\) such that \((c, q)=1, e(y)=\) \(\exp (2 \pi i y)\) and \(\bar{c} \cdot c \equiv 1 \bmod q\). The following sums are known as generalized Hardy sums:
\[
\begin{aligned} s_1(h, m, k) & =\sum_{j=1}^k(-1)^{\left[\frac{h j}{k}\right]} \bar{B}_m\left(\frac{j}{k}\right), \\
s_2(h, m, n, k) & =\sum_{j=1}^k(-1)^j \bar{B}_m\left(\frac{j}{k}\right) \bar{B}_n\left(\frac{h j}{k}\right), \\
s_3(h, n, k) & =\sum_{j=1}^k(-1)^j \bar{B}_n\left(\frac{h j}{k}\right). \end{aligned}
\]
They can be expressed in terms of generalized Dedekind sums. The paper gives explicit formulae for the sums
\begin{gather*}
\mathop{{\sum}'}_{a=1}^q \mathop{{\sum}'}_{b=1}^q K(a, q) K(b, q) S(\bar {a} b, m,n, q), \\
\mathop{{\sum}'}_{a=1}^q \mathop{{\sum}'}_{b=1}^q K(a, q) K(b, q) s_j(2 \bar {a} b, m, q)\qquad (j=1,2,3),
\end{gather*}
where \(q\) is a square-full number and \(m \equiv n\equiv 1 \pmod 2\).
Reviewer: Alexey Ustinov (Khabarovsk)Class numbers, cyclic simple groups, and arithmetichttps://zbmath.org/1537.110542024-07-25T18:28:20.333415Z"Cheng, Miranda C. N."https://zbmath.org/authors/?q=ai:cheng.miranda-c-n"Duncan, John F. R."https://zbmath.org/authors/?q=ai:duncan.john-f-r"Mertens, Michael H."https://zbmath.org/authors/?q=ai:mertens.michael-hIn this paper, the authors express their overall desire and interest in demystifying the interrelationships between finite groups and arithmetic-geometric invariants. In this direction, this paper provides the foundations to systematically study and understand these relationships. In general, it is argued that these connections are not just exceptional occurrences, but instead part of a more general phenomena. Manuscripts in preparation are cited to highlight more examples and further work demonstrating the theory that is initiated here, and at the time of this review some of these have already appeared. At the core of the theory introduced in this paper lies the notion of what the authors call a \(c\)-optimal module for a finite group, a structure which by design is related to optimal holomorphic mock Jacobi forms.
Indeed, for integers \(c,k,m\) with \(m\) positive, a notion of a \(c\)-optimal (mock Jacobi) virtual group module of weight \(k\) and index \(m\) is developed. Loosely, for a group \(G\) this is a virtual \(G\)-module that has a grading for which appropriately defined McKay-Thompson series \(\phi_g^W(\tau ,z)\) for any \(g\in G\) are optimal holomorphic mock Jacobi forms of weight \(k\) and index \(m\) satisfying \(\phi_g^W (\tau ,z)=-c+O(q)\) as \(\mathfrak{I} (\tau)\to \infty\), for any fixed \(z\). In terms of explicit results in this paper, the authors restrict to weight \(2\) and index \(1\), and aim to study the full set of \(c\)-optimal \(G\)-modules for some \(c\), denoted \(\mathcal{W}^{\operatorname{opt}}_{2,1}(G)\). As discussed in the introduction, this amounts to understanding a distinguished minimal positive integer \(\operatorname{c}_{2,1}^{\operatorname{opt}}(G)\) and a lattice structure \(\mathcal{L}_{2,1}^{\operatorname{opt}}(G)\) on the subgroup of \(0\)-optimal \(G\)-modules. For a finite group \(G\), the authors dub the computation of \(\operatorname{c}_{2,1}^{\operatorname{opt}}(G)\) and \(\mathcal{L}_{2,1}^{\operatorname{opt}}(G)\) the ``classification problem for optimal (mock Jacobi) \(G\)-modules (of weight \(2\) and index \(1\)).''
The first main result of the paper then provides the computation of \(\operatorname{c}_{2,1}^{\operatorname{opt}}(G)\) when \(G=\mathbb{Z}/N\mathbb{Z}\) for a prime \(N\). Indeed, it states (see Theorem 4.1.1 and the discussion thereafter) that for such \(G\),
\[
\operatorname{c}_{2,1}^{\operatorname{opt}}(G) =\operatorname{num}\left(\frac{\# G +1}{6}\right),
\]
where \(\operatorname{num}\) on the right side denotes taking the numerator of the reduced rational number \((\# G+1)/6\). Meanwhile, for a cyclic group \(G\) of prime order as above, the authors find \(\mathcal{L}_{2,1}^{\operatorname{opt}}(G) =S_2(N)\otimes R(G)_0\), where \(S_2(N)\) is the space of cuspidal modular forms of weight \(2\) on \(\Gamma_0(N)\) and \(R(G)_0\) is the subgroup consisting of virtual \(G\)-modules \(V\) satisfying \(\operatorname{tr}(\operatorname{e}\vert V)=0\) in the Grothendieck group \(R(G)\) of finitely generated \(\mathbb{C}G\)-modules with \(\operatorname{e}\) as the identity.
A second main result connects the aforementioned results and theory to rational points on imaginary quadratic twists of modular varieties. To state this, let \(J_0(N)\) be the Jacobian of the modular curve \(X_0(N)\) and call an abelian variety \(A\) an optimal quotient of \(J_0(N)\) if it admits a surjective map \(J_0(N)\) with connected kernel. Then, for a prime \(N\) and negative fundamental discriminant \(D\) satisfying \((\frac{D}{N})=-1\), the authors find (see Theorem 4.2.1) that for prime \(p\) dividing \(\# J_0(N)(\mathbb{Q})_{\operatorname{tor}}\) there exists an optimal quotient of the \(J_0(N)\) such that \(A\otimes D\) has only finitely many rational points if \(H^{\operatorname{Hur}}(D)\not \equiv 0 \, \operatorname{mod}\, p\). Here, \(J_0(N)(\mathbb{Q})_{\operatorname{tor}}\) is the torsion subgroup of \(J_0(N)(\mathbb{Q})\) of \(\mathbb{Q}\)-rational points on \(J_0(N)\), and \(H^{\operatorname{Hur}}(D)\) is the Hurwitz class number of discriminant \(D\).
The paper also provides applications of the main theorems, a preview of imminent works by the authors, and a discussion of problems for future work.
Reviewer: Matthew Krauel (Sacramento)Exact formulae and Turán inequalities for Vafa-Witten invariants of \(K3\) surfaceshttps://zbmath.org/1537.110552024-07-25T18:28:20.333415Z"Johnston, Daniel R."https://zbmath.org/authors/?q=ai:johnston.daniel-r"Males, Joshua"https://zbmath.org/authors/?q=ai:males.joshuaThis paper is mainly devoted to study three types of Vafa-Witten invariants for smooth projective \(K3\) surfaces: the \(\mathrm{SU}(r)\) Vafa-Witten invariants, the \(\mathrm{SU}(p)/\mathbb{Z}_p\) Vafa-Witten invariants for prime \(p\), and the twisted Vafa-Witten invariants. The main results are the following. (1) Express the coefficients of the partition functions of the three types of Vafa-Witten invariants in exact formula in terms of the Fourier coefficients of \(\eta(q)^{-24}\), where \(\eta(q)\) is the standard Dedekind \(\eta\)-function. (2) Determine the asymptotic behavior of the coefficients of the partition functions of the three types of Vafa-Witten invariants when \(n\to \infty\). In \(\mathrm{SU}(p)/\mathbb{Z}_p\) case, the asymptotic behavior of coefficients, denoted as \(a_{2,p}(n,w)\), depending on whether \(n\) is divisible \(p\), hence is not uniform. (3) Prove that for \(n\to \infty\), the coefficients of the partition functions of \(\mathrm{SU}(r)\) and twisted Vafa-Witten invariants satisfy the Turan inequality (a generalization of log-concavity). In \(\mathrm{SU}(p)/\mathbb{Z}_p\) case, a similar but slightly more subtle result holds, due to the complexity of the coefficients.
Reviewer: Zili Zhang (Shanghai)Diagonal restriction of Eisenstein series and Kudla-Millson theta lifthttps://zbmath.org/1537.110562024-07-25T18:28:20.333415Z"Branchereau, Romain"https://zbmath.org/authors/?q=ai:branchereau.romainLet \(F\) be a totally real number field of degree \(N\), and \(\psi\) a totally odd unitary Hecke character of finite order. To the character \(\psi\) and a Schwartz function \(\varphi\) one may associate an Eisenstein series of parallel weight one.
The paper is concerned with a formula, relating the Fourier coefficients of the diagonal restriction of the aforementioned Eisenstein series and the intersection numbers of a relative \(N\)-cycle \(C\otimes\psi\) and a special cycle of codimension \(N\) associated with \(\varphi\).
The strategy of the proof is to show that both sides of the formula are equal to a regularised integral of the Kudla-Millson form over the \(N\)-cycle \(C\otimes\psi\). This recovers a result of \textit{H. Darmon} et al. [Math. Ann. 379, No. 1--2, 503--548 (2021; Zbl 1482.11087)] concerning real quadratic fields using Kudla-Millson theory, and generalises their result to totally real fields.
Reviewer: Siu Hang Man (Praha)On the Whittaker range of the generalized metaplectic theta lifthttps://zbmath.org/1537.110572024-07-25T18:28:20.333415Z"Friedberg, Solomon"https://zbmath.org/authors/?q=ai:friedberg.solomon"Ginzburg, David"https://zbmath.org/authors/?q=ai:ginzburg.davidIn their recent work [Geom. Funct. Anal. 30, No. 6, 1531--1582 (2020; Zbl 1478.11066)], the authors introduced an extension of the classical theta correspondence to higher degree metaplectic covers of symplectic and special orthogonal groups. In the paper under the review, they study the existence of generic lifts in the introduced theta tower. Let \(r\) denote an odd positive integer, let \(F\) be a global field, and let \(\mathbb{A}\) stand for the ring of adeles of \(F\). We denote by \(\mathrm{Sp}^{(r)}_{2n}(\mathbb{A})\) the \(r\)-fold metaplectic cover of the symplectic group \(\mathrm{Sp}_{2n}(\mathbb{A})\). Let \(\pi^{(r)}\) denote a genuine irreducible cuspidal automorphic representation of \(\mathrm{Sp}^{(r)}_{2n}(\mathbb{A})\) and let us denote by \(\sigma^{(r)}_{n,k}\) the automorphic representation obtained as a lift of \(\pi^{(r)}\) to the \(r\)-fold metaplectic cover of the split special orthogonal group \(\mathrm{SO}_k(\mathbb{A})\). The authors give conditions of \(\pi^{(r)}\) and \(k\) so that the Whittaker coefficients attached to \(\sigma^{(r)}_{n,k}\) are non-zero and relate them to certain periods of \(\pi^{(r)}\). It is shown that the Whittaker range consists of \(r+1\) groups for the lift from \(\mathrm{Sp}^{(r)}_{2n}(\mathbb{A})\).
Reviewer: Ivan Matić (Osijek)Vanishing coefficients in three families of products of theta functions. IIhttps://zbmath.org/1537.110582024-07-25T18:28:20.333415Z"Tang, Dazhao"https://zbmath.org/authors/?q=ai:tang.dazhaoIn this article, the author studies basically arithmetic progressions of vanishing coefficients of the form \(\gamma_{ck,hl-dk,ml,s,t}(mn+bk)=0\) with other moduli in the three families namely, \(\sum_{n=n_{0}}^{\infty}\gamma_{j,k,r,s,t}(n)q^{n}=(-q^{j},-q^{r-j};q^{r} )_{\infty}^{s}(q^{k},q^{2r-k};q^{2r} )_{\infty}^{t}\), \(\sum_{n=n_{0}}^{\infty}\delta_{j,k,r,s,t}(n)q^{n}=(q^{j},q^{r-j};q^{r} )_{\infty}^{s}(-q^{k},-q^{-2r-k};q^{2r} )_{\infty}^{t}\), and \(\sum_{n=n_{0}}^{\infty}\epsilon_{j,k,r,s,t}(n)q^{n}=(q^{j},q^{r-j};q^{r} )_{\infty}^{s}(q^{k},q^{2r-k};q^{2r} )_{\infty}^{t}\); and he also proves several analogous identities, which are listed in Theorem 1.1((1.9)--(1.24)) and Theorem 1.2((1.25)--(1.36)). The author proposes five conjectures (4.1)--(4.5) on the basis of existing results. All the findings discussed in this article are new and useful for further research works.
For Part I see [the author, Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat., RACSAM 117, No. 1, Paper No. 36, 11 p. (2023; Zbl 1523.11077)].
Reviewer: M. P. Chaudhary (New Delhi)Growth of Fourier coefficients of vector-valued automorphic formshttps://zbmath.org/1537.110592024-07-25T18:28:20.333415Z"Bajpai, Jitendra"https://zbmath.org/authors/?q=ai:bajpai.jitendra"Bhakta, Subham"https://zbmath.org/authors/?q=ai:bhakta.subham"Finder, Renan"https://zbmath.org/authors/?q=ai:finder.renanSummary: In this article, we establish polynomial-growth bound for the sequence of Fourier coefficients associated with even integer weight vector-valued automorphic forms of non-cocompact Fuchsian groups of the first kind. In the end, their \(L\)-functions and exponential sums have been discussed.Zagier duality for real weightshttps://zbmath.org/1537.110602024-07-25T18:28:20.333415Z"Lee, Youngmin"https://zbmath.org/authors/?q=ai:lee.youngmin"Lim, Subong"https://zbmath.org/authors/?q=ai:lim.subongThere exists a connection between the \(n\)-th Fourier coefficient of \( F_1(-m;\tau) \) and the \(m\)-th Fourier coefficient of \( F_0(-n;\tau) \), known as the Zagier duality. In the paper under review \( M !+ \) (or \( M !- \)) represents the space of weakly holomorphic modular forms of weight \( \frac{1}{2} \) (or \( \frac{3}{2} \)) on \( \Gamma_0(4) \) satisfying the Kohnen plus condition. Guerzhoy's result proposed the Zagier duality, asserting that Fourier coefficients create a grid. He provided evidence that a unique grid exists for every positive, even, integral weight, and level 1. In this paper under review, it is demonstrated that the grid not only exists but is also unique for every level and real weight \(k\) where \(k>2\). Additionally, through the application of Hecke operators, there is a method provided to calculate the quantity of ratios among the Fourier coefficients of two weakly holomorphic modular forms within the specified grid. The proof is based on calculations on analytic arguments.
Reviewer: Zeynep Demirkol Özkaya (Van)Iterated primitives of meromorphic quasimodular forms for \(\mathrm{SL}_2(\mathbb{Z})\)https://zbmath.org/1537.110612024-07-25T18:28:20.333415Z"Matthes, Nils"https://zbmath.org/authors/?q=ai:matthes.nilsLet \(E_2,E_4,E_6\) be the Eisenstein series of level one and weight 2,4,6 respectively. Let \(K\) be the rational function field \({\mathbb C}(E_2,E_4,E_6,q)\). The paper studies the structure of \(K\)-algebra \({\mathcal I}^{\mathcal QM}\). Here \({\mathcal QM}\) is the set of meromorphic quasimodular forms, and \({\mathcal I}^{\mathcal QM}\) consists of iterated primitives (iterated integrations) of elements in \({\mathcal QM}\).
Let \(K\langle {\mathcal QM}\rangle\) be the shuffle algebra corresponding to \({\mathcal QM}\). There is a canonical homomorphism from \(K\langle {\mathcal QM}\rangle\) to \({\mathcal I}^{\mathcal QM}\). The main result of the paper is that if \({\mathcal C}\) is the \({\mathbb C}\)-linear complement of the derivatives of \({\mathcal QM}\), then the subalgebra \(K\langle {\mathcal C}\rangle\) is isomorphic to \({\mathcal I}^{\mathcal QM}\) under the canonical homomorphism. This implies the primitives of elements in \({\mathcal I}^{\mathcal QM}\) are algebraically independent over \(K\) if and only if the corresponding classes in \({\mathcal QM}/\delta({\mathcal QM})\) are \({\mathbb C}\)-linearly independent. (Here \(\delta\) is the differential operator \(q\frac{d}{dq}\).)
The main results imply a previous result of Pasol-Zudilin regarding the algebraic independence of \(\Delta/E_4^2, E_4\Delta/E_6^2, E_6\Delta/E_4^3\) over \(K\), where \(\Delta=E_4^3-E_6^2\).
Reviewer: Zhengyu Mao (Newark)The cone of minimal weights for \(\bmod p\) Hilbert modular formshttps://zbmath.org/1537.110622024-07-25T18:28:20.333415Z"Diamond, Fred"https://zbmath.org/authors/?q=ai:diamond.fred"Kassaei, Payman L."https://zbmath.org/authors/?q=ai:kassaei.payman-lSummary: We prove that all \(\bmod p\) Hilbert modular forms arise via multiplication by generalized partial Hasse invariants from forms whose weight falls within a certain \textit{minimal cone}. This answers a question posed by Andreatta and Goren and generalizes our previous results that treated the case where \(p\) is unramified in the totally real field. Whereas our previous work made use of deep Jacquet-Langlands type results on the Goren-Oort stratification (not yet available when \(p\) is ramified), here we instead use properties of the stratification at Iwahori level, which are more readily generalizable to other Shimura varieties.Plectic \(p\)-adic invariantshttps://zbmath.org/1537.110632024-07-25T18:28:20.333415Z"Fornea, Michele"https://zbmath.org/authors/?q=ai:fornea.michele"Guitart, Xavier"https://zbmath.org/authors/?q=ai:guitart.xavier"Masdeu, Marc"https://zbmath.org/authors/?q=ai:masdeu.marcThis nice and interesting paper proposes a new approach to the Birch and Swinnerton-Dyer conjecture based on the theory of \textit{plectic} invariants first introduced by \textit{J. Nekovář} and \textit{A. J. Scholl} [Contemp. Math. 664, 321--337 (2016; Zbl 1402.11092)]. The authors combine this plectic approach with \(p\)-adic methods arising from the techniques developed in the last 20 years by many number theorists (including the authors of the paper under review) around Stark-Heegner points, introduced by \textit{H. Darmon} [Ann. Math. (2) 154, No. 3, 589--639 (2001; Zbl 1035.11027)].
The main constructions and conjectures proposed are rather technical, since they are formulated in a rather general setting. However, they are supported (under some simplifying assumptions) by explicit numerical computations which verify in practice the conjectural statements of the paper, and therefore give a strong support to the theoretical part. The paper can therefore be enjoyed from different perspective, both theoretical and computational, and keeping these two perspective together gives a rich and interesting paper.
For precise statements see the paper, here the reviewer only gives some hints on the type of results and conjectures contained in this paper. Fix a number field \(F\) of narrow class number \(1\) and a non-CM quadratic extension \(E/F\), again of narrow class number \(1\). Let \(A/F\) be an elliptic curve of conductor \(\mathfrak{f}_A\), which we assume to be unramified in \(E\). Let \(p\) be a rational prime which is unramified in \(F\) and let \(S=\{\mathfrak{p}_1,\dots,\mathfrak{p}_r\}\) be a set of distinct \(\mathcal{O}_F\)-ideals above \(p\), which are all inert in \(E\). Let \(\widehat{E}_\mathfrak{p}^\times\) denote the maximal torsion-free quotient of the \(p\)-adic completion of \(E_\mathfrak{p}^\times\). Define \(\widehat{E}_{S,\otimes}^\times=\prod_{\mathfrak{p}\in S}\widehat{E}_\mathfrak{p}^\times\). The \emph{plectic \(p\)-adic invariant} associated to the triple \((A/F,E,S)\) is an element \[Q_A\in \widehat{E}_{S,\otimes}^\times.\] This element is defined under suitable arithmetic conditions on the triple \((A/F,E,S)\) (i.e., each \(\mathfrak{p}\in S\) must divide exactly the conductor \(\mathfrak{f}_A\) of \(A\) and we must have a factorization \(\mathfrak{f}_A=(\prod_{\mathfrak{p}\in S}\mathfrak{p})\times \mathfrak{n}^+\times \mathfrak{n^-}\) and \(\mathfrak{n}^+\) is the product of all prime divisors of \(\mathfrak{f}_A\) which are split in \(E\); moreover, if \(t\) is the number of real places of \(F\) and \(n\) is the number of those real places which split in \(E\), then we must have the congruence condition \(\omega(\mathfrak{n}^-)\equiv(t-n)\pmod{2}\), so that the root number of \(A\) is \((-1)^r\)). The element \(Q_A\) is constructed, as hinted before, by exploiting \(p\)-adic integration techniques and crucially using Tate uniformization \[\phi_\mathrm{Tate}:\widehat{E}_{S,\otimes}^\times \longrightarrow \otimes_{\mathfrak{p}\in S}\widehat{A}(E_\mathfrak{p}),\] where \(\widehat{A}(E_\mathfrak{p})\) is the maximal torsion-free quotient of the \(p\)-adic completion of \(A(E_\mathfrak{p})\).
The main conjectures formulated (and verified numerically) in this paper can be stated roughly as follows: If the algebraic rank \(r_\mathrm{alg}(E/F)\) of the Mordell-Weil group \(A(E)\) is \(\geq r\), then there is an element \(w_A\in \wedge^r A(E)\) such that \(\phi_\mathrm{Tate}(Q_A)=\det_S(w_A)\), where the determinant function \(\det_S:\wedge^r A(E)\rightarrow \otimes_{\mathfrak{p}\in S}\widehat A(E_\mathfrak{p})\) is a natural regulator map. In this case, if \(\phi_\mathrm{Tate}(Q_A)\neq 0\) then the algebraic rank \(r_\mathrm{alg}(E/F)\) is exactly \(r\). Other conjectures (especially concerning the algebraic rank of the Mordell-Weil gruup \(A(F)\)) are also formulated.
Reviewer: Matteo Longo (Padova)Jacobi forms, Saito-Kurokawa lifts, their pullbacks and sup-norms on averagehttps://zbmath.org/1537.110642024-07-25T18:28:20.333415Z"Anamby, Pramath"https://zbmath.org/authors/?q=ai:anamby.pramath"Das, Soumya"https://zbmath.org/authors/?q=ai:das.soumyaLet \(S_k^n\) be the space of holomorphic Siegel cusp forms of weight \(k\) on \(\mathrm{Sp}_n({\mathbb Z})\). For \(F\in S_k^n\) normalized so that its Petersson norm is 1, consider the bound of its sup-norm \(\|F\|_\infty\). The conjecture is \(\|F\|_\infty\ll k^{n(n+1)/8+\varepsilon}\). When \(n=2\) the previous known best bound is \(\|F\|_\infty\ll k^{5/4+\varepsilon}\).
The main result of the paper deals with the average of the \(\|F\|_\infty\) over the space of Saito-Kurukawa lifts, which is a subspace of \(S_k^2\). More precisely, consider
\[
B_k(Z)=\sum_{ F\in B^*_k} \det(Y)^k|F(Z)|^2,
\]
where \(B_k^*\) is an orthonormal basis of the space of Saito-Kurukawa lifts of weight \(k\). The main results states that the \(k^{5/2}\ll sup(B_k)\ll k^{5/2+\varepsilon}\).
The paper also studies the sup-norm bounds for Jacobi forms, and translates the results for Jacobi forms to the above result for Saito-Kurukawa lifts. The lower bound in case of Jacobi forms is established for all cases of \(n\).
Reviewer: Zhengyu Mao (Newark)Notes on Atkin-Lehner theory for Drinfeld modular formshttps://zbmath.org/1537.110652024-07-25T18:28:20.333415Z"Dalal, Tarun"https://zbmath.org/authors/?q=ai:dalal.tarun"Kumar, Narasimha"https://zbmath.org/authors/?q=ai:kumar.narasimhaLet \(A=\mathbb{F}_q[t]\) with \(q=p^r\) a power of an odd prime, let \(\mathfrak{m}\) be an ideal of \(A\), and let \(\mathfrak{p}=(P)\) be a prime of \(A\) not dividing \(\mathfrak{m}\). For any \(k\in \mathbb{N}\) and \(l\in\mathbb{Z}/(q-1)\) such that \(k\equiv 2l\pmod{q-1}\), let \(S_{k,l}(\Gamma_0(\mathfrak{m}))\) (respectively, \(S_{k,l}(\Gamma_0(\mathfrak{mp}))\)) be the space of Drinfeld cusp forms of level \(\mathfrak{m}\) (respectively \(\mathfrak{mp}\)), where, for any ideal \(\mathfrak{n}\) in \(A\),
\[
\Gamma_0(\mathfrak{n})=\left \{ \begin{pmatrix} a & b\\
c & d\end{pmatrix} \in \text{GL}_2(A): c\equiv 0\pmod{\mathfrak{n}} \right\}
\]
(in particular, for \(\mathfrak{n}=(1)\), \(\Gamma_0(1)=\text{GL}_2(A)\)).
The paper deals with some conjectures formulated in [the reviewer and \textit{M. Valentino}, Exp. Math. 31, No. 2, 637--651 (2022; Zbl 07566908)] for \(S_{k,l}(\Gamma_0(t))\) and with their generalization to level \(\mathfrak{mp}\). Namely, the conjectures state that the Hecke operator \(T_t\) is injective on \(S_{k,l}(\Gamma_0(1))\), the Atkin-Lehner operator \(U_t\) is diagonalizable on \(S_{k,l}(\Gamma_0(t))\) and, finally,
\[
S_{k,l}(\Gamma_0(t)) = S_{k,l}^{t-new}(\Gamma_0(t))\oplus S_{k,l}^{t-old}(\Gamma_0(t)),
\]
where \(t\)-oldforms are lifts of modular forms of level \(1\) and \(t\)-newforms are defined via the kernels of certain trace maps (everything can be equivalently formulated for a generic prime \(\mathfrak{p}=(P)\)).
The conjectures can be basically reduced to checking that \(T_{\mathfrak{p}}\) is injective and has no eigenform of eigenvalue \(\pm P^{\frac{k}{2}}\), and the authors do this for \(\dim S_{k,l}(\Gamma_0(1))\leqslant 2\) by computing the action of \(T_{\mathfrak{p}}\) on the Fourier expansion of an explicit basis for cusp forms of level 1. They also show that their method applies to some particular cases for \(S_{k,l}(\Gamma_0(\mathfrak{mp}))\) (again with small \(\dim S_{k,l}(\Gamma_0(\mathfrak{m}))\)).
Moreover, the authors show that the conjectures cannot be naively generalized to non-prime level by providing examples of old eigenforms for \(T_{\mathfrak{p}}\) of level \(\mathfrak{qp}\) (with \(\mathfrak{q}\) a prime different from \(\mathfrak{p}\)) with eigenvalue \(P^{\frac{k}{2}}\) using a twist of the false Eisenstein series \(E\) defined in [\textit{E.-U. Gekeler}, Invent. Math. 93, No. 3, 667--700 (1988; Zbl 0653.14012)].
Reviewer: Andrea Bandini (Pisa)On Drinfeld modular forms of higher rank. VI: The simplicial complex associated with a coefficient formhttps://zbmath.org/1537.110662024-07-25T18:28:20.333415Z"Gekeler, Ernst-Ulrich"https://zbmath.org/authors/?q=ai:gekeler.ernst-ulrichThis article is the continuation of the first five papers of the author on the subject, especially of the fifth one [J. Number Theory 222, 75--114 (2021; Zbl 1480.11056)]. It is shown the simplicity of the Drinfeld modular forms \(_a\ell_k\), where \(a\in A:={\mathbb F}_q[T]\) and \(_a\ell_k\) is the coefficient of the generic Drinfeld module \(\phi^{\omega}\) of rank \(r\geq 2\) on the Drinfeld space \(\Omega^r\):
\[
\phi_a^{\omega}(X)=aX+\sum_{1\leq k\leq r\deg a} {_a\ell_k(\omega)X^{q^k}}.
\]
A modular form \(f\) is \textit{simplicial} if the image \({\mathcal {BT}}^r(f)\) of its zero set \(\Omega^r(f)\) under the building map \(\lambda^r: \Omega^r\to {\mathcal {BT}}^r({\mathbb Q})\) to the set of \({\mathbb Q}\)-points of the Bruhat-Tits building \({\mathcal {BT}}^r\) of \(\mathrm{PGL}(r,{\mathbb F}_q((T^{-1})))\) is the set of \({\mathbb Q}\)-points of a full subcomplex of codimension 1. The simplicial complex \({\mathcal {BT}}({_a\ell_k})\) depends only on \(d=\deg a\) of \(a\in A\); it is strongly equidimensional of codimension \(1\) in \({\mathcal {BT}}^r\); it is boundaryless, and satisfies a certain symmetry property with respect to the involution of the Dynkin diagram of the underlying system of type \(A_{r-1}\). The main theorem (Theorem 1.8) collects these results for \(_a\ell_k\) and the similar ones for the related para-Eisenstein series \(\alpha_k\).
In this work, the author deals with the case rank \(r\geq 3\). All the results are still valid for \(r=2\). First, it is established a vanishing criterion (Theorem 2.6). In Section 3, it is shown that \({\mathcal {BT}}({_a\ell_k})\) is a full subcomplex of \({\mathcal {BT}}\). Section 4 introduces the concept of \(d\)-diagram \(\mathrm{diag}^{(d)}({\mathbf n})\) of the \textit{standard apartment} \({\mathcal A}\) of \({\mathcal {BT}}\), which is the full subcomplex with vertex set \({\mathcal A}({\mathbb Z})=\{[L_{\mathbf n}]\mid {\mathbf n}\in {\mathbb Z}^r\}\), where \(L_{\mathbf n}\) is the lattice \(L_{\mathbf n}= \pi^{n_1}{\mathcal O}_{\infty} \oplus\cdots \oplus \pi^{n_r}{\mathcal O}_{\infty}\) with the uniformizer \(\pi:=T^{-1}\) of \(K_{\infty}\), and \({\mathcal O}_{\infty}\) is the ring of integers of \(K_{\infty}\).
In Section 5, the author shows the connectedness of \({\mathcal {BT}}(d,k)\) (Theorem 5.17). The final section is devoted to some examples and to concluding remarks.
Reviewer: Gabriel D. Villa Salvador (Ciudad de México)Spectral decomposition formula and moments of symmetric square \(L\)-functionshttps://zbmath.org/1537.110672024-07-25T18:28:20.333415Z"Balkanova, O. G."https://zbmath.org/authors/?q=ai:balkanova.olga-gThe author obtains a spectral decomposition for
\[
\sum_{\ell\geq1} \omega(\ell)\mathcal{L}_{n^{2}-4\ell^{2}}(s),
\]
where \(\omega\) is a suitable test function. The Zagier \(L\)-function for \(\operatorname{Re}(s)>1\) is
\[
\mathcal{L}_{n}(s)=\frac{\zeta(2s)}{\zeta(s)} \sum_{q\geq1} b_{q}(n)q^{-s},
\]
with
\[
b_{q}(n)=\#\left\{ x(\operatorname{mod}2q)\left\vert x^{2}\equiv n(\operatorname{mod}4q)\right. \right\}.
\]
The sum over \(n\) rather than \(\ell\) had been studied in an earlier paper of the author and and \textit{D. Frolenkov} [Rev. Mat. Iberoam. 35, No. 7, 1973--1995 (2019; Zbl 1472.11149)]. Sums over both \(n\) and \(\ell\) can be applied to study second moments of symmetric square \(L\)-functions.
The explicit results are too complicated to state here. Proofs involve the Kuznetsov trace formula as well as Gauss and Kloosterman sums associated to congruence groups \(\Gamma_{0}(N)\) for \(N=4,16,64.\) The final formulas involve moments of symmetric square \(L\)-functions associated to holomorphic and Maass cusp forms of level \(N\).
Reviewer: Audrey A. Terras (La Jolla)Non-vanishing of symmetric cube \(L\)-functionshttps://zbmath.org/1537.110682024-07-25T18:28:20.333415Z"Hoffstein, Jeff"https://zbmath.org/authors/?q=ai:hoffstein.jeffrey"Jung, Junehyuk"https://zbmath.org/authors/?q=ai:jung.junehyuk"Lee, Min"https://zbmath.org/authors/?q=ai:lee.minLet \({\mathcal O}_3\) be the ring of integers of \(\mathbb Q(\sqrt{-3})\) and \(\Gamma_3=\operatorname{SL}_2({\mathcal O}_3)\). The main result of the paper shows that there are infinitely many Maass-Hecke cuspforms \(\phi_j\) on \(\Gamma_3(3)\backslash H^3\) such that the symmetric cube \(L\)-function \(L(s,\operatorname{sym}^3,\phi_j)\) is nonvanishing at the center of critical strip \(s=\frac12\); here \(\Gamma_3(3)\) is the level 3 principal subgroup of \(\Gamma_3\). It is expected that other than the cases of vanishing due to a negative sign in function equation, most Maass-Hecke cuspforms \(\phi\) satisfy \(L(\frac12, \operatorname{sym}^3,\phi)\not=0\).
The proof of the result is based on Ginzburg-Jiang-Rallis's work
[\textit{D. Ginzburg} et al., Forum Math. 13, No. 1, 109--132 (2001; Zbl 1034.11033)], which shows that \(L(\frac12, \operatorname{sym}^3,\phi)\not=0\) if the pairing \(\langle \phi, |\theta^2|\rangle\) is nonzero, where \(\theta\) is a cubic metaplectic theta series defined by
\textit{S. J. Patterson} [J. Reine Angew. Math. 296, 125--161 (1977; Zbl 0358.10011)]. The authors consider a pairing \(\langle P,|\theta^2|\rangle\) where \(P\) is a Poincare series. They estimate the size of the pairing in two ways: directly and through spectral decomposition of \(P\). By comparing the estimates they show the contribution from the discrete part of the spectral decomposition of the pairing has to be unbounded, thus implying the main result.
The paper also conjectured an explicit relation between the pairing \(\langle \phi, |\theta^2|\rangle\) and the \(L\)-value \(L(\frac12, \operatorname{sym}^3,\phi)\). A heuristic argument with a computation of the pairing in the case \(\phi\) is an Eisenstein series is included to support the conjectured relationship.
Reviewer: Zhengyu Mao (Newark)On the soft \(p\)-converse to a theorem of Gross-Zagier and Kolyvaginhttps://zbmath.org/1537.110692024-07-25T18:28:20.333415Z"Kim, Chan-Ho"https://zbmath.org/authors/?q=ai:kim.chan-hoThe pioneering work of Gross-Zagier and Kolyvagin proves the BSD Conjecture for elliptic curves of analytic rank one:
\[
\mathrm{ord}_{s=1}L(E,s) = 1 \quad \Rightarrow \quad\mathrm{rank}_{\mathbb Z} E(\mathbb Q ) = 1 \text{ and } \#\Sha(E/\mathbb Q) < \infty.
\]
The other implication is known as a \textit{converse theorem} and is the subject of the paper under review.
When the elliptic curve has complex multiplication, such a converse theorem is known by the work of Rubin, Bertrand, and Perrin-Riou. The first such results for non-CM elliptic curves were obtained by \textit{W. Zhang} [Camb. J. Math. 2, No. 2, 191--253 (2014; Zbl 1390.11091)] and \textit{C. Skinner} [Ann. Math. (2) 191, No. 2, 329--354 (2020; Zbl 1447.11071)] under some assumptions on the conductor of the elliptic curve. The main point of the paper under review is to remove all these ramification assumptions, thereby proving the full converse implication:
\[
\mathrm{rank}_{\mathbb Z} E(\mathbb Q ) = 1 \text{ and } \#\Sha(E/\mathbb Q) < \infty \quad \Rightarrow \quad \mathrm{ord}_{s=1}L(E,s) = 1.
\]
The method of proof is \(p\)-adic and the main result of the paper is a \textit{soft} \(p\)-converse theorem (Corollary 1.2). If \(E\) is a non-CM elliptic curve over \(\mathbb Q\) and \(p>3\) is a good ordinary prime for \(E\) such that:
\begin{itemize}
\item[1.] \(E[p]\) is an irreducible mod \(p\) representation,
\item[2.] \(\mathrm{corank}_{\mathbb Z_p}\mathrm{Sel}(\mathbb Q, E[p^\infty]) = 1\),
\item[3.] the restriction map \(\mathrm{res}_p\colon\mathrm{Sel}(\mathbb Q, V_p(E)) \to E(\mathbb Q_p) \otimes \mathbb Q_p\) is an isomorphism,
\end{itemize}
then \(\mathrm{ord}_{s=1}L(E,s) = 1\). In particular, then \(\mathrm{rank}_{\mathbb Z} E(\mathbb Q ) = 1\) and \(\#\Sha(E/\mathbb Q) < \infty\) by Gross-Zagier-Kolyvagin. A \(p\)-converse theorem with Assumption 3 is known as \textit{soft}. It holds under the assumption that \(\#\Sha(E/\mathbb Q)[p^\infty] < \infty\); see Corollary 1.4. Since Assumption 1 holds for sufficiently large \(p\), this proves the desired converse implication above in the non-CM case.
Reviewer: Aleksander Horawa (Oxford)On the standard \(L\)-function for \(\mathrm{GSp}_{2n}\times\mathrm{GL}_1\) and algebraicity of symmetric fourth \(L\)-values for \(\mathrm{GL}_2\)https://zbmath.org/1537.110702024-07-25T18:28:20.333415Z"Pitale, Ameya"https://zbmath.org/authors/?q=ai:pitale.ameya"Saha, Abhishek"https://zbmath.org/authors/?q=ai:saha.abhishek"Schmidt, Ralf"https://zbmath.org/authors/?q=ai:schmidt.ralfThe critical values of \(L\)-functions attached to cohomological cuspidal automorphic representations of algebraic groups are very important arithmetic objects. The simplest example of this kind is the value of the Riemann zeta function at positive even integers. The higher rank cases that have been studied so far are \(\mathrm{GL}(2)\) (By Shimura and Manin) and \(\mathrm{GSp}(2n) \times \mathrm{GL}(1)\) by various authors. However, in the latter case, the automorphic representations considered were always of special kinds.
The purpose of the paper under review is to prove an explicit integral formula for the \(\mathrm{GSp}(2n) \times \mathrm{GL}(1)\) \(L\)-function associated to a holomorphic vector-valued Siegel cusp form of degree \(n\) and arbitrary level, using a novel choice of local vectors at the Archimedean places. This allows the authors to prove the algebracity of \(\mathrm{GSp}(4) \times \mathrm{GL}(1)\) \(L\)-function for vector-valued Siegel cusp forms of degree \(2\) and arbitrary level. As a consequence, algebraicity of the critical values of the symmetric fourth \(L\)-function of a classical new form is also proved.
Reviewer: Yuanqing Cai (Kanazawa)The Kottwitz conjecture for unitary PEL-type Rapoport-Zink spaceshttps://zbmath.org/1537.110712024-07-25T18:28:20.333415Z"Bertoloni Meli, Alexander"https://zbmath.org/authors/?q=ai:bertoloni-meli.alexander"Nguyen, Kieu Hieu"https://zbmath.org/authors/?q=ai:nguyen.kieu-hieuThe Kottwitz conjecture predicts some fine details about the structure of \(\ell\)-adic cohomology of Rapoport-Zink spaces, which are to be viewed as \(p\)-adic analogues of Shimura varieties. As such, it is a local part of the Langlands program. Plenty of special cases are known, but even to state the conjecture requires the existence of a local Langlands correspondence that behaves in more or less the expected way. This paper considers the case of unitary Rapoport-Zink spaces of PEL type over \(\mathbb{Q}_p\) in an odd number of variables.
Thus a major part of the result is the construction of the local Langlands correspondence: once this is achieved, the cases of the Kottwitz conjecture follow by known procedures, though there is significant work to be done at that stage too. In very general terms, the construction of the correspondence uses the existence of such a structure for \(\operatorname{U}(\mathbb{Q}_p)\) as in [\textit{C. P. Mok}, Endoscopic classification of representations of quasi-split unitary groups. Providence, RI: American Mathematical Society (AMS) (2015; Zbl 1316.22018)] and lifts to \(\operatorname{U}\times Z({\operatorname{GU}})\) before descending to \(\operatorname{GU}\). This is where the restriction to an odd number of variables is needed, as otherwise the map fails to be surjective on \({\mathbb{Q}_p}\) points.
Reviewer: G. K. Sankaran (Bath)The global Gan-Gross-Prasad conjecture for unitary groups: the endoscopic casehttps://zbmath.org/1537.110722024-07-25T18:28:20.333415Z"Beuzart-Plessis, Raphaël"https://zbmath.org/authors/?q=ai:beuzart-plessis.raphael"Chaudouard, Pierre-Henri"https://zbmath.org/authors/?q=ai:chaudouard.pierre-henri"Zydor, Michał"https://zbmath.org/authors/?q=ai:zydor.michalThis is a landmark paper on the global Gan-Gross-Prasad conjecture for unitary groups. Let \(E/F\) be a quadratic extension of number fields. A (generic) Hermitian Arthur parameter for \(\mathrm{GL}_{n, E}\) in this article is defined as an irreducible automorphic representation \(\Pi = \Pi_1 \times \cdots \times \Pi_r\) of \(\mathrm{GL}_n(\mathbb{A}_E)\), where each \(\Pi_i\) is a conjugate-selfdual irreducible cuspidal automorphic representation such that \(L(s, \Pi_i, \mathrm{As}^{(-1)^{n+1}})\) has a pole at \(s=1\). The same notion also exists for \(G = \mathrm{GL}_{n, E} \times \mathrm{GL}_{n+1, E}\). The \(F\)-counterpart \(G'\) embeds into \(G\).
The main achievement is the global GGP conjecture (Theorem 1.1.5.1). It states that for a Hermitian Arthur parameter \(\Pi\) for \(G\), the following are equivalent: (i) The complete Rankin-Selberg \(L\)-function \(L(\frac{1}{2}, \Pi) \neq 0\). (ii) There exist a non-degenerate Hermitian form \(h\) of rank \(n\), with \(U'_h = U(h)\) embeds diagonally into \(U_h = U(h) \times U(h \oplus h_0)\) (where \(h_0\) is the norm form of rank \(1\)), and a cuspidal irreducible automorphic representation \(\sigma\) of \(U_h\) which lifts (weakly) to \(\Pi\), such that the period integral \(\mathcal{P}_h\) along \([U'_h]\) is nontrivial on \(\sigma\). The previous results on global GGP are limited to simple parameters \(\Pi\).
The refined conjecture due to Ichino-Ikeda is also obtained (Theorem 1.1.6.1). This is a precise factorization of \(|\mathcal{P}_h(\varphi)|^2\), where \(\varphi \in \sigma\), assuming that \(\sigma\) is an everywhere tempered cuspidal irreducible automorphic representation of \(U_h\). We refer to the article itself for the detailed statement.
As in the previous works, the proof is based on the comparison of full relative trace formulas (Chaudouard-Zydor). Here are two differences, at least. First, one uses Schwartz test functions instead of \(C^\infty_c\) ones. Secondly, one needs an in-depth study of the contribution of \(\Pi\) to the spectral side of relative trace formula for \(G\), which is highly intricate for non-cuspidal \(\Pi\).
Let \(\chi\) be the cuspidal datum for \(G\) corresponding to \(\Pi\). The main technical result, Theorem 1.3.1.1, is a relation between the contribution \(I_\chi\) in the relative trace formula and the relative character \(I_\Pi\). Note that the contribution of \(\chi\) is purely discrete, as opposed to Arthur's trace formula. Two proofs are offered for this theorem. The first one (see \S 1.3.2) is based on truncations, whilst the second (\S 1.3.3) is based on zeta integrals for Asai and Rankin-Selberg \(L\)-functions.
Finally, the appendix contains results of analytic nature, which are crucial in the proofs.
Reviewer: Wen-Wei Li (Beijing)The generalized doubling method: \((k,c)\) modelshttps://zbmath.org/1537.110732024-07-25T18:28:20.333415Z"Cai, Yuanqing"https://zbmath.org/authors/?q=ai:cai.yuanqing"Friedberg, Solomon"https://zbmath.org/authors/?q=ai:friedberg.solomon"Gourevitch, Dmitry"https://zbmath.org/authors/?q=ai:gourevitch.dmitry"Kaplan, Eyal"https://zbmath.org/authors/?q=ai:kaplan.eyalAuthors' abstract: One of the key ingredients in the recent construction of the generalized doubling method is a new class of models, called \((k,c)\) models, for local components of generalized Speh representations. We construct a family of \((k,c)\) representations, in a purely local setting, and discuss their realizations using inductive formulas. Our main result is a uniqueness theorem which is essential for the proof that the generalized doubling integral is Eulerian.
Reviewer: Goran Muić (Zagreb)Weakly unramified representations, finite morphisms, and Knapp-Stein \(R\)-groupshttps://zbmath.org/1537.110742024-07-25T18:28:20.333415Z"Choiy, Kwangho"https://zbmath.org/authors/?q=ai:choiy.kwanghoLet \(\mathbf{G}\) be a connected reductive group over a \(p\)-adic field \(F\). Let \(G_1\) be the intersection of \(G = G(F)\) with the kernel of Kottwitz's homomorphism \(\kappa: \mathbf{G}(\breve{F}) \to X^*(\mathbf{Z}(\hat{\mathbf{G}}))_I^\Phi\). The group \(X^w(G)\) of weakly unramfied characters of \(G\) is \(\mathrm{Hom}(G/G_1, \mathbb{C}^{\times})\). It agrees with the group of unramified characters when \(\mathbf{G}\) is simply-connected or unramified over \(F\).
The paper extends \textit{C. D. Keys}' description [Ann. Sci. Éc. Norm. Supér. (4) 20, No. 1, 31--64 (1987; Zbl 0634.22014)] of Knapp-Stein \(R\)-groups of unitary unramified principal series for simply-connected, almost simple \(\mathbf{G}\) to the following setting: \(\mathbf{G}_{\mathrm{der}}\) is simply-connected and almost simple, and the inducing characters of the minimal Levi subgroup are only unitary and weakly unramified; the proof makes use of the transfer of \(R\)-groups for such induced representations across inner twists. Such induced representations are of arithmetic interest as they figure in the Satake isomorphism for representations with parahoric fixed vectors.
Reviewer: Wen-Wei Li (Beijing)Asymptotics for Hecke eigenvalues of automorphic forms on compact arithmetic quotientshttps://zbmath.org/1537.110752024-07-25T18:28:20.333415Z"Ramacher, Pablo"https://zbmath.org/authors/?q=ai:ramacher.pablo"Wakatsuki, Satoshi"https://zbmath.org/authors/?q=ai:wakatsuki.satoshiSummary: In this paper, we describe the asymptotic distribution of Hecke eigenvalues in the Laplace eigenvalue aspect for certain families of Hecke-Maass forms on compact arithmetic quotients. Instead of relying on the trace formula, which was the primary tool in preceding studies on the subject, we use Fourier integral operator methods. This allows us to treat not only spherical, but also non-spherical Hecke-Maass forms with corresponding remainder estimates. Our asymptotic formulas are available for arbitrary simple and connected algebraic groups over number fields with cocompact arithmetic subgroups.Galois representations over pseudorigid spaceshttps://zbmath.org/1537.110762024-07-25T18:28:20.333415Z"Bellovin, Rebecca"https://zbmath.org/authors/?q=ai:bellovin.rebeccaSummary: We study \(p\)-adic Hodge theory for families of Galois representations over pseudorigid spaces. Such spaces are non-archimedean analytic spaces which may be of mixed characteristic, and which arise naturally in the study of eigenvarieties at the boundary of weight space. We introduce perfect and imperfect overconvergent period rings, and we use the Tate-Sen method to construct overconvergent \((\varphi, \Gamma)\)-modules for Galois representations over pseudorigid spaces.Gelfand-Kirillov dimension and mod \(p\) cohomology for \(\mathrm{GL}_2 \)https://zbmath.org/1537.110772024-07-25T18:28:20.333415Z"Breuil, Christophe"https://zbmath.org/authors/?q=ai:breuil.christophe"Herzig, Florian"https://zbmath.org/authors/?q=ai:herzig.florian"Hu, Yongquan"https://zbmath.org/authors/?q=ai:hu.yongquan"Morra, Stefano"https://zbmath.org/authors/?q=ai:morra.stefano"Schraen, Benjamin"https://zbmath.org/authors/?q=ai:schraen.benjamin``Let \(p\) be a prime number, \(F\) a totally real number field unramified at places above \(p\) and \(D\) a quaternion algebra of center \(F\) split at places above \(p\) and at no more than one infinite place. Let \(v\) be a fixed place of \(F\) above \(p\) and \(\overline{v}: \text{Gal}(\overline{F}/F) \to \text{GL}_2(\overline{\mathbb F}_p)\) an irreducible modular continuous representation which, at the place \(v\), is semisimple and sufficiently generic (and satisfies some weak genericity conditions at a few other finite places). We prove that many of the admissible smooth representations of \(\text{GL}_2(F_v)\) over \(\overline{\mathbb F}_p\) associated to \(\overline{v}\) in the corresponding Hecke-eigenspaces of the mod \(p\) cohomology have Gelfand-Kirillov dimension \([F_v:\mathbb Q_p]\), as well as several related results.''
``Besides applications to the flatness of completed homology over a big Hecke algebra (Theorem 1.2 below) and on the candidate of \textit{A. Caraiani} et al. [Camb. J. Math. 4, No. 2, 197--287 (2016; Zbl 1403.11073)] for the \(p\)-adic Langlands correspondence (Theorem 1.3 below), our methods also lead us to an abelian subcategory of the category of smooth representations of \(\text{GL}_2(F_v)\) that has desirable finiteness property, with further applications to a functor towards Galois representations; cf. our subsequent work [\textit{C. Breuil} et al., ``Conjectures and results on modular representations of $\mathrm{GL}_n(K)$ for a $p$-adic field $K$'', Preprint, \url{arXiv:2102.06188}; ``Multivariable ($\varphi,\mathcal{O}_K^\times$)-modules and local-global compatibility'', Preprint, \url{arXiv:2211.00438}]).''
Reviewer: Andrzej Dąbrowski (Szczecin)Ordinary deformations are unobstructed in the cyclotomic limithttps://zbmath.org/1537.110782024-07-25T18:28:20.333415Z"Burungale, Ashay"https://zbmath.org/authors/?q=ai:burungale.ashay-a"Clozel, Laurent"https://zbmath.org/authors/?q=ai:clozel.laurent``The deformation theory of ordinary representations of the absolute Galois groups of totally real number fields (over a finite field \(k\)) has been studied for a long time, starting with the work of Hida, Mazur and Tilouine, and continued by Wiles and others. Hida has studied the behaviour of these deformations when one considers the p-cyclotomic tower of extensions of the field. In the limit, one obtains a deformation ring \(R_{\infty}\) classifying the ordinary deformations of the (Galois group of) the \(p\)-cyclotomic extension.''
The authors show, that if \(R_{\infty}\) is Noetherian and certain adjoint \(\mu\)-invariants vanish (as is often expected), then \(R_{\infty}\) is free over the ring of Witt vectors of \(k\) (Theorem 1.6). It is expected that the \(\mu\)-invariant typically vanishes if the underlying Galois representation is residually irreducible (see [\textit{R. Sujatha}, ``WIN-Women in numbers'', Fields Inst. Commun. 60, 265--276 (2011)]).
``One may ask if the main result can be proved for \({}^{L}G\)-valued deformations of a \({}^{L}G\)-valued mod \(p\) Galois representation with \(G\) a reductive group. To follow the current approach, it seems essential to impose adequacy for the image of the mod \(p\) Galois representation and suppose the vanishing of certain adjoint \(\mu\)-invariants.''
Reviewer: Andrzej Dąbrowski (Szczecin)Breuil-Kisin modules and integral \(p\)-adic Hodge theory (with Appendix A by Yoshiyasu Ozeki, and Appendix B by Hui Gao and Tong Liu)https://zbmath.org/1537.110792024-07-25T18:28:20.333415Z"Gao, Hui"https://zbmath.org/authors/?q=ai:gao.huiThis paper is concerned with certain algebraic structures in integral \(p\)-adic Hodge theory. The main innovation is the construction of a new category of so called \textit{Breuil-Kisin \(G_K\)-modules}. This category is then used to classify \textit{integral semi-stable Galois representations}.
Comparing cohomology theories is an essential part of the (using the author's words) \textit{geometric direction} of classical \(p\)-adic Hodge theory as is recalled in Theorem 1.1.2 of the introduction. Comparing cohomology theories is paralleled in the \textit{algebraic direction} by understanding \(p\)-adic Galois representations, i.e., \(p\)-adic continuous representations of the absolute Galois group \(G_K\) of a \(p\)-adic field \(K\), via categories of semi-linear objects (Theorem 1.1.3).
Intergral \(p\)-adic Hodge theory (in the geometric direction) has recently seen tremendous progress through the definition of integral \(p\)-adic cohomology theories of Bhatt-Morrow-Scholze and Bhatt-Scholze and respective comparison theorems (recalled in Theorem 1.1.7). In the context of Galois representations (i.e., in the algebraic direction) one is led to the following question:
Can one construct a category of semi-linear objects that is equivalent to the category of \(G_K\)-stable \(\mathbb{Z}_p\)-lattices in semi-stable \(G_K\)-representations?
The main theorem of the paper at hand (Theorem 1.1.11) answers this affirmatively with the category of Breuil-Kisin \(G_K\)-modules (see Definition 1.1.8). It generalizes various prior results on the above question by different authors (see Remark 1.1.13).
As the author explains in the paper, the category Breuil-Kisin \(G_K\)-modules should be understood as an \textit{algebraic avatar} of the integral \(p\)-adic cohomology theories of Bhatt-Morrow-Scholze and Bhatt-Scholze. The category and the results of the paper at hand are also of interest in the context of the Emerton-Gee stack of \((\varphi, \Gamma)\)-modules. This relationship is explained in Section 7.3 of the paper.
Finally, the paper contains an appendix (Appendix B) where the authors (Gao, Liu) use the category of Breuil-Kisin \(G_K\)-modules to fix a gap in their previous work. The paper also contains a second appendix (Appendix A, by Ozeki), which gives a counter-example to Proposition 3.7 in [\textit{X. Caruso}, Duke Math. J. 162, No. 13, 2525--2607 (2013; Zbl 1294.11207)]. In Section 7.4 it is explained that the paper at hand is unaffected by that mistake.
Reviewer: Judith Ludwig (Heidelberg)The local-global principle for divisibility in CM elliptic curveshttps://zbmath.org/1537.110802024-07-25T18:28:20.333415Z"Creutz, Brendan"https://zbmath.org/authors/?q=ai:creutz.brendan"Lu, Sheng (Victor)"https://zbmath.org/authors/?q=ai:lu.shengLet \(E/k\) be an elliptic curve over a number field \(k\). Given a positive integer \(N\), the authors say that the local-global principle holds for \((E/k, N)\) if
\[
\mathrm{Ker} (H^{1} (k, E[N])\rightarrow\prod_{v\not\in S}H^{1} (k_{v}, E[N])) =0
\]
for every finite set of places \(S\) of \(k\), where \(H^{1} (k, E[N])\) denotes the Galois cohomology of the \(N\)-torsion subgroup \(E[N]\) of \(E\). Let \(E\) be an elliptic curve defined over a number field \(L\) with complex multiplication by an order \(O\) of a quadratic number field \(K, f\) be the conductor of \(O, j\) be the \(j\)-invariant of \(E\), and \(p^{n}\) be an odd prime power.
The authors prove then that the local-global principle holds for \((E/L, p^{n})\) in any of the following cases: (a) \(p\) does not divide \(f\) and \(p\) splits in \(K\); (b) \(p\) does not divide \(f\), \(p\) is inert in \(K\), and \([L:K]<(p^{2}-1)/u\), where \(u=2\) if \(j\neq 0\) and \(u=3\) if \(j=0\); (c) \(p\) divides \(f\) or \(p\) ramifies in \(K\), and \([L:K]<(p-1)/2\), and that their bounds are sharp. Let \(d\) be a positive integer, \(p\) be a prime, \(p\geq 17\), and \(p>2d+1\); the authors prove then that there are at most finitely many elliptic curves \(E\) defined over a number field \(L\) of degree \(d=[L:\mathbb{Q}]\) such that the local-global principle for \((E/L, p^{n})\) fails for some positive integer \(n\). They prove further that the the local-global principle holds for \((E/L, 7^{n})\) for every elliptic curve \(E\) defined over a quadratic number field \(L\) and every positive integer n. In the last section of their work, the authors analyse the local-global principle for a few explicitly defined CM elliptic curves.
Reviewer: B. Z. Moroz (Bonn)On 2-Selmer groups of twists after quadratic extensionhttps://zbmath.org/1537.110812024-07-25T18:28:20.333415Z"Morgan, Adam"https://zbmath.org/authors/?q=ai:morgan.adam|morgan.adam.1"Paterson, Ross"https://zbmath.org/authors/?q=ai:paterson.ross-aLet \(E\) be an elliptic curve over \(\mathbb{Q}\) with full rational \(2\)-torsion. In this paper, the authors study the behaviour of the quadratic twists \(E_d\) over a fixed quadratic extension \(K\), as \(d\) varies over squarefree integers. They prove that for 100\% of twists the dimension of the \(2\)-Selmer group over \(K\) is given by an explicit local formula, and use this to show that this dimension follows an Erdős-Kac type distribution. They also prove that, for 100\% of twists \(d\), the action of \(\mathrm{Gal}(K/\mathbb{Q})\) on the \(2\)-Selmer group of \(E_d\) over \(K\) is trivial, and the Mordell-Weil group \(E_d(K)\) splits integrally as a direct sum of its invariants and anti-invariants. On the other hand, they give examples of thin families of quadratic twists in which a positive proportion of the \(2\)-Selmer groups over \(K\) have non-trivial \(\mathrm{Gal}(K/\mathbb{Q})\)-action.
Reviewer: Andrej Dujella (Zagreb)On the modularity of elliptic curves over the cyclotomic \(\mathbb{Z}_p\)-extension of some real quadratic fieldshttps://zbmath.org/1537.110822024-07-25T18:28:20.333415Z"Zhang, Xinyao"https://zbmath.org/authors/?q=ai:zhang.xinyaoLet \(E_i(i=1,2)\) be two elliptic curves given by the equations
\[
E_1: y^2+xy+y=x^3+x^2-10x-10,
\]
\[
E_2:y^2+xy+y=x^3+x^2-5x+2.
\]
Let \(K\) be a real quadratic field, and \(p\) an odd prime satisfying the following assumptions: (1) \(K\neq \mathbb{Q}(\sqrt{5})\), (2) \(\mathrm{ Sel}_{p^\infty}(E_i/K)=0\), and (3) \(p\) splits in \(K\).
The main result of the paper is the following theorem: for \(K\) and \(p\) satisfy the above assumptions, let \(F\) be a finite field extension contained in the cyclotomic \(\mathbb{Z}_p\)-extension of \(K\). Let \(E\) be an elliptic curve over \(F\), then \(E\) is modular.
The method is to reduce the problem to the previous methods of \textit{J. A. Thorne} [J. Eur. Math. Soc. (JEMS) 21, No. 7, 1943--1948 (2019; Zbl 1443.11103)] and Freitas-Le Hung-Siksek
[\textit{N. Freitas} et al., Invent. Math. 201, No. 1, 159--206 (2015; Zbl 1397.11086)], one key ingredient in the arguments is to use Iwasawa theory for elliptic curves, mainly the algebraic descent formulas of Schneider, Greenberg and Iovita-Pollack, to control the Mordell-Weil group of \(E_i\,(i=1,2)\) in a cyclotomic \(\mathbb{Z}_p\)-extension over \(K\).
Reviewer: Yong-Xiong Li (Beijing)Twisted Edwards curve over the ring \(\mathbb{F}_q [e]\), \(e^2 = 0\)https://zbmath.org/1537.110832024-07-25T18:28:20.333415Z"Taleb El Hamam, Moha Ben"https://zbmath.org/authors/?q=ai:taleb-el-hamam.moha-ben"Chillali, Abdelhakim"https://zbmath.org/authors/?q=ai:chillali.abdelhakim"El Fadil, Lhoussain"https://zbmath.org/authors/?q=ai:el-fadil.lhoussainSummary: Let \(\mathbb{F}_q\) be a finite field of \(q\) elements, where \(q\) is a power of an odd prime number. In this paper, we study the twisted Edwards curves denoted \(E_{E_{a,d}}\) over the local ring \(\mathbb{F}_q [e]\), where \(e^2 = 0\). In the first time, we study the arithmetic of the ring \(\mathbb{F}_q [e], e^2 = 0\). After that we define the twisted Edwards curves \(E_{E_{a,d}}\) over this ring and we give essential properties and we define the group \(E_{E_{a,d}}\), these properties. Precisely, we give a bijection between the groups \(E_{E_{a,d}}\) and \(E_{E_{a_0,d_0}} \times \mathbb{F}_q\), where \(E_{E_{a_0,d_0}}\) is the twisted Edwards curves over the finite field \(\mathbb{F}_q\).Weil polynomials of abelian varieties over finite fields with many rational pointshttps://zbmath.org/1537.110842024-07-25T18:28:20.333415Z"Berardini, Elena"https://zbmath.org/authors/?q=ai:berardini.elena"Giangreco-Maidana, Alejandro J."https://zbmath.org/authors/?q=ai:giangreco-maidana.alejandro-jThe article studies the finite set of isogeny classes of \(g\)-dimensional abelian varieties defined over a finite field of cardinality \(q\) with endomorphism algebra being a field. The main result is Theorem 1.1, where the authors prove that within this set, the class whose varieties have the maximal number of rational points is unique, for any prime even power \(q\) bigger than a positive quantity \(c_g\) depending only on the dimension \(g\), and \(q\) satisfying the following extra condition: if \(\mathfrak{f}_g\) denotes the minimal polynomial of a totally positive algebraic integer of degree \(g\) with minimal trace, and which is maximal in a sense that is made precise in Lemma 3.1, then \(q\) needs to be coprime with \(\mathfrak{f}_g(0)\).
They moreover describe the associated Weil polynomial and prove that the class is ordinary and cyclic outside the primes dividing the integer \(N_g=\mathfrak{f}_g(4)\mathfrak{f}_g(0)\mathrm{discriminant}(\mathfrak{f}_g)\). The article also contains tables with explicit computations illustrating the main result.
Reviewer: Fabien Pazuki (København)Hyperelliptic \(\mathcal{S}_7\)-curves of prime conductorhttps://zbmath.org/1537.110852024-07-25T18:28:20.333415Z"Brumer, Armand"https://zbmath.org/authors/?q=ai:brumer.armand"Kramer, Kenneth"https://zbmath.org/authors/?q=ai:kramer.kennethSummary: An abelian threefold \(A_{/\mathbb{Q}}\) of prime conductor \(N\) is favorable if its 2-division field \(F\) is an \(\mathcal{S}_7\)-extension over \(\mathbb{Q}\) with ramification index 7 over \(\mathbb{Q}_2\). Let \(A\) be favorable and let \(B\) be a semistable abelian variety of conductor \(N^d\) with \(B[2]\) filtered by \(d\) copies of \(A[2]\). We obtain a class field theoretic criterion on \(F\) to guarantee that \(B\) is isogenous to \(A^d\) and a fortiori, \(A\) is unique up to isogeny.Finiteness criteria and uniformity of integral sections in some families of abelian varietieshttps://zbmath.org/1537.110862024-07-25T18:28:20.333415Z"Phung, Xuan Kien"https://zbmath.org/authors/?q=ai:phung.xuan-kienSummary: Let \(A\) be an abelian variety over the function field \(K\) of a compact Riemann surface \(B\). Fix a model \(f :{\mathcal{A}} \rightarrow B\) of \(A/K\) and an effective horizontal divisor \({\mathcal{D}}\subset{\mathcal{A}} \). We give a sufficient condition on the divisor \({\mathcal{D}}\) for the finiteness of the set of \((S, {\mathcal{D}})\)-integral sections of \({\mathcal{A}}\) for every finite subset \(S \subset B\). These integral sections \(\sigma\) are algebraic and satisfy the geometric condition \(f ( \sigma (B) \cap{\mathcal{D}})\subset S\). The notion of \((S, {\mathcal{D}})\)-integral sections is the geometric variant of integral solutions of a system of Diophantine equations. When \({\mathcal{A}}= A_0 \times B\) for some complex abelian variety \(A_0\), we also give a certain uniform bound on the number of \((S, {\mathcal{D}})\)-integral sections. For trivial families of abelian surfaces, a numerical criterion on \({\mathcal{D}}\) for the finiteness of \((S, {\mathcal{D}})\)-integral sections is obtained.The Manin-Mumford conjecture and the Tate-Voloch conjecture for a product of Siegel moduli spaceshttps://zbmath.org/1537.110872024-07-25T18:28:20.333415Z"Qiu, Congling"https://zbmath.org/authors/?q=ai:qiu.conglingSummary: We use perfectoid spaces associated to abelian varieties and Siegel moduli spaces to study torsion points and ordinary CM points. We reprove the Manin-Mumford conjecture, i.e., Raynaud's theorem. We also prove the Tate-Voloch conjecture for a product of Siegel moduli spaces, namely ordinary CM points outside a closed subvariety can not be \(p\)-adically too close to it.The least degree of a CM point on a modular curvehttps://zbmath.org/1537.110882024-07-25T18:28:20.333415Z"Clark, Pete L."https://zbmath.org/authors/?q=ai:clark.pete-l"Genao, Tyler"https://zbmath.org/authors/?q=ai:genao.tyler"Pollack, Paul"https://zbmath.org/authors/?q=ai:pollack.paul"Saia, Frederick"https://zbmath.org/authors/?q=ai:saia.frederickSummary: For a modular curve \(X = X_0(N)\), \(X_1(N)\) or \(X_1(M,N)\) defined over \(\mathbb{Q}\), we denote by \(d_{\mathrm{CM}}(X)\) the least degree of a CM point on \(X\). For each discriminant \(\Delta < 0\), we determine the least degree of a point on \(X_0(N)\) with CM by the order of discriminant \(\Delta\). This places us in a position to study \(d_{\mathrm{CM}}(X)\) as an `arithmetic function' and we do so, obtaining various upper bounds, lower bounds and typical bounds. We deduce that all but finitely many curves in each of the families have sporadic CM points. Finally, we supplement these results with a computational study, for example, computing \(d_{\mathrm{CM}}(X_0(N))\) and \(d_{\mathrm{CM}}(X_1(N))\) exactly for \(N \leqslant 10^6\) and determining whether \(X_0(N)\) (respectively, \(X_1(N)\), \(X_1(M,N))\) has sporadic CM points for all but 106 values of \(N\) (respectively, 227 values of \(N, 146\) pairs \((M,N)\) with \(M \geqslant 2)\).Algebraicity of higher Green functions at a CM pointhttps://zbmath.org/1537.110892024-07-25T18:28:20.333415Z"Li, Yingkun"https://zbmath.org/authors/?q=ai:li.yingkunThe study of the values of automorphic functions at CM points of Shimura varieties is an important problem with deep arithmetical significance. Motivated by earlier results and conjectures by Gross, Zagier, and many other authors (see for example [\textit{B. H. Gross} and \textit{D. B. Zagier}, J. Reine Angew. Math. 355, 191--220 (1985; Zbl 0545.10015)]), the author proves that for a class of functions called higher Green functions, the values at CM points are logarithms of algebraic numbers. This result was conjectured by Zagier in Zagier's 1986 ICM proceeding. The main results of the paper are stated in the introduction where in particular the main Theorem of the paper is stated, that is Theorem 1.3. This says that the difference in the values of higher Green functions on certain orthogonal Shimura varieties at two CM points can be written as sums of logarithms of algebraic numbers. It is then conjectured (Conjecture 1.6) that under the condition that the singular locus of the higher Green function does not contain the CM points of the Shimura varieties, the same formula should hold for a single value of the higher Green function at a CM point. Then, Theorem 1.7 is a specific case of this conjecture, and from it, some of the conjectures of Zagier on the algebraicity of the values of higher Green functions can be deduced.
The paper is organized as follows. Section 2 contains preliminary material from the theory of modular forms and Shimura varieties. Section 3, recalls material from the theory of automorphic functions and higher Green functions. Section 4 is where the main technical results are proved. In particular, Theorem 4.3 constructs a specific family of real-analytic Hilbert modular form, and Theorem 4.10 proves an algebraicity result about linear combinations of their Fourier coefficients. In Section 5, the main theorems of the paper are proved with the crucial input of Theorem 4.10.
Reviewer: Federico Bambozzi (Padova)A survey on recursive towers and Ihara's constanthttps://zbmath.org/1537.110902024-07-25T18:28:20.333415Z"Beelen, Peter"https://zbmath.org/authors/?q=ai:beelen.peterSummary: Since Serre gave his famous Harvard lectures in 1985 on various aspects of the theory of algebraic curves defined over a finite field, there have been many developments. In this survey article, an overview will be given on the developments concerning the quantity \(A(q)\), known as Ihara's constant. The main focus will be on explicit techniques and in particular recursively defined towers of function fields over a finite field, which have given good lower bounds for Ihara's constant in the past.
For the entire collection see [Zbl 1530.11002].A user's guide to the local arithmetic of hyperelliptic curveshttps://zbmath.org/1537.110912024-07-25T18:28:20.333415Z"Best, Alex J."https://zbmath.org/authors/?q=ai:best.alex-j"Betts, L. Alexander"https://zbmath.org/authors/?q=ai:betts.luke-alexander"Bisatt, Matthew"https://zbmath.org/authors/?q=ai:bisatt.matthew"van Bommel, Raymond"https://zbmath.org/authors/?q=ai:van-bommel.raymond"Dokchitser, Vladimir"https://zbmath.org/authors/?q=ai:dokchitser.vladimir"Faraggi, Omri"https://zbmath.org/authors/?q=ai:faraggi.omri"Kunzweiler, Sabrina"https://zbmath.org/authors/?q=ai:kunzweiler.sabrina"Maistret, Céline"https://zbmath.org/authors/?q=ai:maistret.celine"Morgan, Adam"https://zbmath.org/authors/?q=ai:morgan.adam"Muselli, Simone"https://zbmath.org/authors/?q=ai:muselli.simone"Nowell, Sarah"https://zbmath.org/authors/?q=ai:nowell.sarahSummary: A new approach has been recently developed to study the arithmetic of hyperelliptic curves \(y^2=f(x)\) over local fields of odd residue characteristic via combinatorial data associated to the roots of \(f\). Since its introduction, numerous papers have used this machinery of `cluster pictures' to compute a plethora of arithmetic invariants associated to these curves. The purpose of this user's guide is to summarise and centralise all of these results in a self-contained fashion, complemented by an abundance of examples.Ternary and quaternary curves of small fixed genus and gonality with many rational pointshttps://zbmath.org/1537.110922024-07-25T18:28:20.333415Z"Faber, Xander"https://zbmath.org/authors/?q=ai:faber.xander"Grantham, Jon"https://zbmath.org/authors/?q=ai:grantham.jonSummary: We extend the computations from our previous work to determine the maximum number of rational points on a curve over \(\mathbb{F}_3\) and \(\mathbb{F}_4\) with fixed gonality and small genus. We find, for example, that there is no curve of genus 5 and gonality 6 over a finite field. We propose two conjectures based on our data. First, an optimal curve of genus \(g\) has gonality at most \(\big\lfloor \frac{g+3}{2} \big\rfloor\). Second, an optimal curve of gonality \(\gamma\) and large genus over \(\mathbb{F}_q\) has \(\gamma (q+1)\) rational points.Efficient computation of \((2^n, 2^n)\)-isogenieshttps://zbmath.org/1537.110932024-07-25T18:28:20.333415Z"Kunzweiler, S."https://zbmath.org/authors/?q=ai:kunzweiler.sabrinaSummary: Elliptic curves are abelian varieties of dimension one; the two-dimensional analogues are abelian surfaces. In this work we present an algorithm to compute \((2^n, 2^n)\)-isogenies between abelian surfaces defined over finite fields. These isogenies are the natural generalization of \(2^n\)-isogenies of elliptic curves. The efficient computation of such isogeny chains gained a lot of attention as the runtime of the attacks on SIDH (Castryck-Decru, Maino-Martindale, Robert) depends on this computation. Different results deduced in the development of our algorithm are also interesting beyond these applications. For instance, we derive a formula for the evaluation of (2, 2)-isogenies. Given an element in Mumford coordinates, this formula outputs the (unreduced) Mumford coordinates of its image under the (2, 2)-isogeny. Furthermore, we study 4-torsion points on Jacobians of hyperelliptic curves and explain how to extract square roots of coefficients of 2-torsion points from these points.On loop Deligne-Lusztig varieties of Coxeter-type for inner forms of \(\mathrm{GL}_n\)https://zbmath.org/1537.110942024-07-25T18:28:20.333415Z"Chan, Charlotte"https://zbmath.org/authors/?q=ai:chan.charlotte"Ivanov, Alexander B."https://zbmath.org/authors/?q=ai:ivanov.alexander-bSummary: For a reductive group \(G\) over a local non-archimedean field \(K\) one can mimic the construction from classical Deligne-Lusztig theory by using the loop space functor. We study this construction in the special case that \(G\) is an inner form of \(\text{GL}_n\) and the loop Deligne-Lusztig variety is of Coxeter type. After simplifying the proof of its representability, our main result is that its \(\ell \)-adic cohomology realizes many irreducible supercuspidal representations of \(G\), notably almost all among those whose \(L\)-parameter factors through an unramified elliptic maximal torus of \(G\). This gives a purely local, purely geometric and -- in a sense -- quite explicit way to realize special cases of the local Langlands and Jacquet-Langlands correspondences.Corrigendum to: ``On a problem of Lang for matrix polynomials''https://zbmath.org/1537.110952024-07-25T18:28:20.333415Z"Ostafe, Alina"https://zbmath.org/authors/?q=ai:ostafe.alinaSummary: This note is a corrigendum to the author's paper [ibid. 54, No. 5, 1552--1567 (2022; Zbl 1532.11082)].
{{\copyright} 2023 The Authors. The publishing rights in this article are licensed to the London Mathematical Society under an exclusive licence.}Arithmetic deformation theory of Lie algebrashttps://zbmath.org/1537.110962024-07-25T18:28:20.333415Z"Rastegar, Arash"https://zbmath.org/authors/?q=ai:rastegar.arashSummary: This paper is devoted to deformation theory of graded Lie algebras over \(\mathbb{Z}\) or \(\mathbb{Z}_l\) with finite dimensional graded pieces. Such deformation problems naturally appear in number theory. In the first part of the paper, we use Schlessinger criteria for functors on Artinian local rings in order to obtain universal deformation rings for deformations of graded Lie algebras and their graded representations. In the second part, we use a version of Schlessinger criteria for functors on the Artinian category of nilpotent Lie algebras which is formulated by Pridham, and explore arithmetic deformations using this technique.Computing Euclidean Belyi mapshttps://zbmath.org/1537.110972024-07-25T18:28:20.333415Z"Radosevich, Matthew"https://zbmath.org/authors/?q=ai:radosevich.matthew"Voight, John"https://zbmath.org/authors/?q=ai:voight.johnThis article studies the Belyi maps \(\varphi : X \to \mathbb{P}^1_{\mathbb{C}}\), that is, those finite coverings of \(\mathbb{P}^1_{\mathbb{C}}\) whose ramification locus is contained in \(\{ 0, 1, \infty \}\), which arise from finite index subgroups \(\Gamma\) of Euclidean triangle groups. The latter groups are those triangle groups \(\Delta = \Delta (a, b, c) \subset \text{PSL}_2 (\mathbb{R}))\) for which \(1 / a + 1 / b + 1 / c = 1\). Algorithmically, such Belyi maps \(\varphi\), as well as the corresponding groups \(\Gamma\), are most efficiently encoded by transitive permutation triples \(\sigma = (\sigma_0, \sigma_1, \sigma_{\infty}) \in S_n^3\); this article shows how \(\Gamma\) and \(\varphi\) can be calculated once such a triple \(\sigma\) is specified.
After showing how it can be checked whether a given permutation triple \(\sigma\) is Euclidean, the authors give an explicit description of both the group \(\Delta\) and the subgroup \(\Gamma\) corresponding to \(\sigma\). This is accomplished by means of a semidirect product of a group of translations (respectively called \(T (\Delta)\) and \(T (\Gamma)\)) and the finite cyclic stabilizer of a special point, called a vertex of maximal rotation. Using such vertices, one obtains fundamental domains for \(\Gamma\) and \(\Delta\) that are both aesthetically pleasing and algorithmically easy to describe.
On the level of algebraic curves, the inclusion of translation subgroups \(T (\Gamma) \subset T (\Delta)\) corresponds to an isogeny of elliptic curves \(E (\Gamma) \to E (\Delta)\); the authors show how to compute a corresponding field of definition, and explain how an application of Vélu's formulas can then be used to obtain explicit equations for this isogeny. Finally, the original Belyi map \(\varphi : X (\Gamma) \to X (\Delta)\) is obtained by an elegant Galois-theoretic argument; this describes \(X (\Gamma)\) as a quotient of \(E (\Gamma)\) (or one of its translates) by an explicit cyclic group of automorphisms, and similarly for \(E (\Delta)\) and \(X (\Delta)\). In this way, one obtains an expression for \(\varphi : X (\Gamma) \to X (\Delta) = \mathbb{P}^1_{\mathbb{C}}\) by writing the composition \(E (\Gamma) \to E (\Delta) \to X (\Delta)\) of the aforementioned isogeny with the quotient map for \(\Delta\) as a rational function in the generator of the fixed field \(X (\Gamma)\) of \(E (\Gamma)\).
The authors give clear and unambiguous descriptions of their algorithms and include many illustrative examples.
Reviewer: Jeroen Sijsling (Ulm)Arithmetic duality for two-dimensional local rings with perfect residue fieldhttps://zbmath.org/1537.110982024-07-25T18:28:20.333415Z"Suzuki, Takashi"https://zbmath.org/authors/?q=ai:suzuki.takashi.1|suzuki.takashi|suzuki.takashi.2Summary: We give a refinement of Saito's arithmetic duality for two-dimensional local rings by giving algebraic group structures for arithmetic cohomology groups.Pencils of norm form equations and a conjecture of Thomashttps://zbmath.org/1537.110992024-07-25T18:28:20.333415Z"Amoroso, F."https://zbmath.org/authors/?q=ai:amoroso.francesco-antonio"Masser, D."https://zbmath.org/authors/?q=ai:masser.david-william"Zannier, U."https://zbmath.org/authors/?q=ai:zannier.umberto-mSummary: We introduce a new method to deal with families of norm form equations. These generalize the Thue equations studied first by Thomas using Baker's Method (which, however, we do not use here). We show that for all large integer values of the parameter \(t\), every solution over \(\mathbb{Z}\) arises from specializing a solution over \(\mathbb{Z} [ T]\) by \(T = t\). The results are completely effective.
{{\copyright} 2021 The Authors. The publishing rights for this article are licensed to University College London under an exclusive licence.}Totally \(p\)-adic algebraic numbers of degree 4https://zbmath.org/1537.111002024-07-25T18:28:20.333415Z"Ault, Melissa"https://zbmath.org/authors/?q=ai:ault.melissa"Doud, Darrin"https://zbmath.org/authors/?q=ai:doud.darrinLet~\(p\) be a prime number~\(p\). An algebraic number~\(\alpha\) is said to be totally \(p\)-adic if all the roots of its minimal polynomial lie in~\(\mathbb Q_p\). Denote by~\(\tau_{n,p}\) the minimal logarithmic Weil height of a totally \(p\)-adic number of degree~\(n\) which is not a root of unity. \textit{E. Stacy} proved in [Open Book Ser. 4, 387--401 (2020; Zbl 1472.11325)] that if \(p>3\), one has that \(\tau_{n,p}\leq 0.703762\). This article establishes that
\[
\tau_{4,p}\leq \frac{\log(5)}{4}=0.40236.
\]
Reviewer: Ratko Darda (Basel)Kneser's method of neighborshttps://zbmath.org/1537.111012024-07-25T18:28:20.333415Z"Voight, John"https://zbmath.org/authors/?q=ai:voight.johnThe article under review gives a nice overview over many facets of Kneser's neighborhood method. The author starts with the historic origins of this tool which has been invented to get insight into finding all lattices belonging to the genus of a given positive lattice in a euclidean space. He explains many details and gives examples which nicely illustrate the theory. He goes on to explain more recent topics related to the neighborhood method in the theory of (algebraic) modular forms, involving some new algorithms for calculating Hecke operators.
The list of references is exhaustive with but one omission (according to the reviewers taste): \textit{M. Kneser}'s book on quadratic forms [Quadratische Formen. Neu bearbeitet und herausgegeben in Zusammenarbeit mit Rudolf Scharlau. Berlin: Springer (2002; Zbl 1001.11014)] has not been included.
Reviewer: Stefan Kühnlein (Karlsruhe)Algebraic and arithmetical properties of Mahler infinite products generated by the second degree polynomialshttps://zbmath.org/1537.111022024-07-25T18:28:20.333415Z"Duverney, Daniel"https://zbmath.org/authors/?q=ai:duverney.daniel"Kurosawa, Takeshi"https://zbmath.org/authors/?q=ai:kurosawa.takeshiLet \(r\geq 2\), \(\mathbb K\) be a field, and \((a_k), (b_k)\in\mathbb K^{\mathbb N}\); consider the infinite product \[ \Psi_0(X) = \prod_{k=0}^\infty \Big( 1+a_k X^{r^k} + b_k X^{2r^k}\Big)\] as a formal power series in \(X\). The main result of the paper under review is an explicit necessary and sufficient condition for \(\Psi_0\) to be rational (i.e., to belong to \(\mathbb K(X)\)). When \(r\geq 3\) this was known already, as a special case of a result of \textit{Y. Tachiya} [J. Number Theory 125, No. 1, 182--200 (2007; Zbl 1116.11053)]. But the case \(r=2\) is new, and rather surprising: complicated examples exist for which \(\Psi_0\) is rational, and the authors provide counter-examples to a general result of \textit{M. Amou} and \textit{K. Väänänen} [J. Number Theory 153, 283--303 (2015; Zbl 1317.11077)].
Applying the equivalence between the algebraicity of the values of a Mahler function at algebraic points and the rationality of the Mahler function itself, established by Tachiya in this setting [loc. cit.], the authors deduce transcendence results.
The proofs rely on explicit Padé-type approximants.
Reviewer: Stéphane Fischler (Paris)On the number of algebraic points on the graph of the Weierstrass sigma functionshttps://zbmath.org/1537.111032024-07-25T18:28:20.333415Z"Sena, Gorekh Prasad"https://zbmath.org/authors/?q=ai:sena.gorekh-prasad"Kumar, K. Senthil"https://zbmath.org/authors/?q=ai:kumar.k-senthilLet \(\Omega=\mathbb{Z}\omega_1 + \mathbb{Z}\omega_2\) be a lattice in \(\mathbb{C}\), and let \(\sigma_{\Omega}\) be its associated Weierstrass sigma function. Let \(\eta_1\) and \(\eta_2\) be the quasi-periods of \(\Omega\), and assume that \(\rho:=\eta_1/\eta_2\) is a non-zero real number. Let \(P\) be the fundamental domain enclosed by the parallelogram with vertices \(\frac12(\pm\omega_1\pm \omega_2)\).
Define a set \(A_\rho\) as follows: let \(\Omega^+ = \{m\omega_1 + n\omega_2 : mn\geq 0\}\), \(\Omega^-= \{m\omega_1 + n\omega_2 : mn\leq 0\}\), \(A^+ = \{z \in \mathbb{C} : \text{ exists } z_0 \in P \text{ such that }z- z_0 \in \Omega^+\}\), \(A^- = \{z \in \mathbb{C} : \text{ exists } z_0 \in P\text{ such that }z- z_0 \in \Omega^-\}\), and finally set \(A_\rho=A^+\) if \(\rho>0\) and \(A_\rho=A^-\) if \(\rho<0\).
The first result of the paper is that when \(\omega_1\) and \(\omega_2\) are both algebraic, there exists a constant \(C = C(\Omega) > 0\) such that for all \(d \ge e\) and \(H \ge e^e\), there are at most
\[
Cd^6(\log d)(\log H)^2 \log \log(H)
\]
algebraic numbers \(z\) such that \(z \in A_\rho\), \([\mathbb{Q}(z,\sigma_{\Omega}(z)) : \mathbb{Q}]\leq d\) and \(H(z,\sigma_{\Omega}(z))\leq H\).
A second similar result is obtained when it is assumed that the invariants \(g_2\) and \(g_3\) are both algebraic (instead of \(\omega_1\) and \(\omega_2\) both algebraic): the upper bound is
\[
C'd^{20}(\log d)^5(\log H)^2 \log \log(H)
\]
for some constant \(C' = C'(\Omega) > 0\). A similar upper bound is also obtained when no algebraicity assumption on \(\omega_1, \omega_2, g_2\) or \(g_3\) is made, but only when \(\text{dist}(z, \Omega)\ge\delta\) for a fixed \(\delta>0\).
These results extend, under the assumption \(\rho\in \mathbb R^*\), those obtained by \textit{G. Boxall} et al. [Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat., IX. Ser., Rend. Lincei, Mat. Appl. 32, No. 4, 819--833 (2021; Zbl 1483.11152)]. The proof uses similar arguments as well as a non-algebraicity measure for the elements of \(\Omega\) due to \textit{S. David} and \textit{N. Hirata-Kohno} [J. Reine Angew. Math. 628, 37--89 (2009; Zbl 1169.11034)].
Reviewer: Tanguy Rivoal (Grenoble)The metric theory of the pair correlation function for small non-integer powershttps://zbmath.org/1537.111042024-07-25T18:28:20.333415Z"Rudnick, Zeév"https://zbmath.org/authors/?q=ai:rudnick.zeev"Technau, Niclas"https://zbmath.org/authors/?q=ai:technau.niclasSummary: For \(0<\theta <1\), we show that for almost all \(\alpha \), the pair correlation function of the sequence of fractional parts of \(\lbrace \alpha n^\theta :n\geqslant 1 \rbrace\) is Poissonian.Intermediate-scale statistics for real-valued lacunary sequenceshttps://zbmath.org/1537.111052024-07-25T18:28:20.333415Z"Yesha, Nadav"https://zbmath.org/authors/?q=ai:yesha.nadavSummary: We study intermediate-scale statistics for the fractional parts of the sequence \((\alpha a_n)_{n=1}^{\infty}\), where \((a_n)_{n=1}^{\infty}\) is a positive, real-valued lacunary sequence, and \(\alpha\in\mathbb{R}\). In particular, we consider the number of elements \(S_N\!(L,\alpha)\) in a random interval of length \(L/N\), where \(L=O\!\left(N^{1-\epsilon}\right)\), and show that its variance (the number variance) is asymptotic to \(L\) with high probability w.r.t. \(\alpha\), which is in agreement with the statistics of uniform i.i.d. random points in the unit interval. In addition, we show that the same asymptotic holds almost surely in \(\alpha\in\mathbb{R}\) when \(L=O\!\left(N^{1/2-\epsilon}\right)\). For slowly growing \(L\), we further prove a central limit theorem for \(S_N\!(L,\alpha)\) which holds for almost all \(\alpha\in\mathbb{R}\).\(\times 2\) and \(\times 3\) invariant setshttps://zbmath.org/1537.111062024-07-25T18:28:20.333415Z"De Saxcé, Nicolas"https://zbmath.org/authors/?q=ai:de-saxce.nicolasThe author of the present survey considers sets which are invariant by multiplication by \(2\) and \(3\) or any other pair of multiplicative independent integers. He considers recent progress on a weak form of a conjecture of Furstenberg.
Let \(\mathbb{T}=\mathbb{R}/\mathbb{Z}\) be the circle. Furthermore let \(p\) and \(q\) be positive integers and let \(T_p\) and \(T_q\) denote the multiplication by \(p\) and \(q\) on \(\mathbb{T}\), respectively. We call \(p\) and \(q\) multiplicatively independent if the only solution to the equation \(p^x=q^y\) in integers \(x\) and \(y\) is the trivial one \(x=y=0\). Finally we denote by \(\langle T,x\rangle\) the orbit of \(x\) under the map \(T\colon \mathbb{T}\to\mathbb{T}\) as well as by \(\dim_H\) the Hausdorff dimension.
\textit{H. Furstenberg} [Math. Syst. Theory 1, 1--49 (1967; Zbl 0146.28502)] obtained that if \(p\) and \(q\) are multiplicatively independent, then the only set that is invariant with respect to \(T_p\) and \(T_q\) (at the same time) is the whole circle \(\mathbb{T}\). In the very same paper he conjectured that the Lebesgue measure is the only diffuse invariant probability measure (invariant with respect to \(T_p\) and \(T_q\)). A weaker form of this conjecture is to show that for \(x\in\mathbb{T}\) the inequality \(\dim_H\overline{\langle T_p,x\rangle}+\dim_H\overline{\langle T_q,x\rangle}<1\) implies that \(x\) is rational. This latter conjecture has been proven independently by \textit{P. Shmerkin} [Ann. Math. (2), 189, No. 2, 319--391 (2019; Zbl 1426.11079)] and \textit{M. Wu} [Ann. Math. (2), 189, No. 3, 707--751 (2019; Zbl 1430.11106)] and is the central topic of the present survey.
The survey starts with a motivation of Furstenberg's theorem. In the first section the author presents the major steps of its proof. He does not show Furstenberg's proof but the refined version by \textit{M. D. Boshernitzan} [Proc. Am. Math. Soc. 122, No. 1, 67--70 (1994; Zbl 0815.11036)]. The second section deals with the connection of Hausdorff dimension of invariant sets, the projection of invariant measures and the projection on invariant sets. In the third section he considers Cantor type sets and their role in the proof of \textit{P. Shmerkin} [Ann. Math. (2), 189, No. 2, 319--391 (2019; Zbl 1426.11079)]. By a Cantor set we mean for example sets of reals in whose expansion in base 4 only the digits 0 and 3 occur. The survey is rounded up by showing the leap between the weak version and Furstenberg's original conjecture.
Reviewer: Manfred G. Madritsch (Vandœuvre-lès-Nancy)Generalization of a density theorem of Khinchin and Diophantine approximationhttps://zbmath.org/1537.111072024-07-25T18:28:20.333415Z"Beck, József"https://zbmath.org/authors/?q=ai:beck.jozsef"Chen, William W. L."https://zbmath.org/authors/?q=ai:chen.william-w-lA classical result of Khinchin implies that any half-infinite geodesic is superdense on the unit torus \([0,1]^2\) if and only if the slope of the geodesic is a badly approximable number. The authors extend this result to so-called finite polysquare translation surfaces. A polysquare region is a connected polygon on the plane which is built from unit squares such that two unit squares are either disjoint, intersect at a common point or have a common edge. Furthermore, any two squares are joined by a chain of squares in which neighboring squares share an edge. A finite polysquare translation surface is a polysquare region in which horizontal and vertical boundary edges are pairwise identified. The main result of the paper (Theorem 1.1) shows that a half-infinite geodesic that does not hit a vertex of a given arbitrary, finite polysquare translation surface is superdense on this surface if and only if the slope of the geodesic is a badly approximable number. Importantly, this result can be shown using only traditional methods from number theory such as continued fractions and the Three-Distance Theorem.
Reviewer: Florian Pausinger (Belfast)Finite \(A_2\)-continued fractions in the problems of rational approximations of real numbershttps://zbmath.org/1537.111082024-07-25T18:28:20.333415Z"Pratsiovytyi, M."https://zbmath.org/authors/?q=ai:pratsiovytyi.mykola"Goncharenko, Ya."https://zbmath.org/authors/?q=ai:goncharenko.ya-v"Lysenko, I."https://zbmath.org/authors/?q=ai:lysenko.iryna|lysenko.i-m"Ratushnyak, S."https://zbmath.org/authors/?q=ai:ratushnyak.s-pAn \(A_2\)-continued fraction (\(A_2\)-fraction) is a continued fraction \(1/a_1+1/a_2+\dots+1/a_n=[0;a_1,a_2,\dots,a_n]\) all elements \(a_i\) of which belong to a two-element set \(\{e_0, e_1\}\), with \(0<e_0<e_1\). Consider a finite \(A_2\)-continued fraction where \(e_0=\frac{1}{2}\) and \(e_1=1\), and study the structure of the set \(F\) of values of finite \(A_2\)-continued fractions and the problem of the number of representations of numbers from the segment \(\left[\frac{1}{2},1\right]\) by fractions of this kind. It is proved that the set \(F\subset\left[\frac{1}{3},2\right]\) has a scale-invariant structure and is dense in the segment \(\left[\frac{1}{2},1\right]\) the set of its elements that are greater than \(1\) is the set of terms of two decreasing sequences approaching \(1\), while the set of its elements that are smaller than \(\frac{1}{2}\) is the set of terms of two increasing sequences approaching \(\frac{1}{2}\). The fundamental difference between the representations of numbers with the help of finite and infinite \(A_2\)-fractions is emphasized. The following hypothesis is formulated: every rational number from the segment \(\left[\frac{1}{2},1\right]\) can be represented in the form of a finite \(A_2\)-continued fraction.
Reviewer: Takao Komatsu (Hangzhou)On a Turán conjecture and random multiplicative functionshttps://zbmath.org/1537.111092024-07-25T18:28:20.333415Z"Angelo, Rodrigo"https://zbmath.org/authors/?q=ai:angelo.rodrigo"Xu, Max Wenqiang"https://zbmath.org/authors/?q=ai:xu.max-wenqiangLet \(f\) be a random completely multiplicative function, i.e., a completely multiplicative function such that \(f(p)=1\) and \(f(p)=-1\) with probabilities \(1/2\) independently at each prime. The authors of this paper are interested in the probability that for every \(x\) the partial sum \[\sum_{n\leqslant x}\frac{f(n)}{n}\] is positive. One of the two main results is the following assertion:
Let \(f\) be a random completely multiplicative function. The probability that \[ \sum_{n\leqslant x}\frac{f(n)}{n}\] is positive for every \(x\) is at least \(1-10^{-45}\).
Reviewer: Jonas Šiaulys (Vilnius)A new two-term exponential sums and its fourth power meanhttps://zbmath.org/1537.111102024-07-25T18:28:20.333415Z"Xuexia, Wang"https://zbmath.org/authors/?q=ai:xuexia.wang"Li, Wang"https://zbmath.org/authors/?q=ai:li.wangIn the paper under review, the authors prove that for any odd prime \(p\),
\[
C_4(p):=\sum_{m=0}^{p-1}\left|\sum_{n=0}^{p-1}\mathrm{e}\left(\frac{n^2(m+n)}{p}\right)\right|^4=2p^3+O(p^{5/2}),
\]
where \(\mathrm{e}(x)=e^{2\pi ix}\). They consider two cases when \(p-1\) is divisible by \(3\) or not. When \(3|(p-1)\), they provide the following explicit and better approximation for \(C_4(p)\),
\[
C_4(p)=2p^3-p^2\cdot\left(4-\left(\frac{-3}{p}\right)\sum_{a=1}^{p-1}\left(\frac{a+a'-1}{p}\right)\right),
\]
where \((\frac{\cdot}{p})\) denotes the Legendre's symbol modulo \(p\), and \(a'\) is the multiplicative inverse of \(a\) modulo \(p\).
Reviewer: Mehdi Hassani (Zanjan)Discorrelation of multiplicative functions with nilsequences and its application on coefficients of automorphic \(L\)-functionshttps://zbmath.org/1537.111112024-07-25T18:28:20.333415Z"He, Xiaoguang"https://zbmath.org/authors/?q=ai:he.xiaoguang"Wang, Mengdi"https://zbmath.org/authors/?q=ai:wang.mengdi.1Let \(G\) be a connected, simply-connected nilpotent Lie group, and let \(\Gamma\leq G\) be a lattice. A filtration \(G_{\bullet}=(G_i)_{i=0}^{\infty}\) on \(G\) is a descending sequence of groups \(G=G_1\supseteq G_2\supseteq\cdots\supseteq G_d\supseteq G_{d+1}=\{\mathrm{id}_G\}\) such that \([G,G_{i-1}]\subseteq G_i\) for all \(i\geq 2\). The number \(d\) is the degree of the filtration \(G_\bullet\). The step \(s\) of \(G\) is the degree of the lower central filtration defined by \(G_{i+1}=[G,G_i]\). A lattice \(\Gamma\) must be cocompact, and the compact quotient \(G/\Gamma\) is called a nilmanifold. We say that \(g\) is a polynomial sequence with coefficients in \(G_{\bullet}\), and write \(g\in\mathrm{poly}(\mathbb{Z},G_{\bullet})\), if \(g:\mathbb{Z}\to G\) satisfies the derivative condition \(\partial_{h_1}\cdots \partial_{h_i}g(n) \in G_i\) for all \(i\geq 0\), \(n\in\mathbb{Z}\) and all \(h_1,\dots, h_i\in \mathbb{Z}\), where \(\partial_h g(n):=g(n+h)g(n)^{-1}\) is the discrete derivative with shift \(h\). The Mal'cev basis \(\mathcal{X}\) induces a right invariant metric \(d_G\) on \(G\), which is the largest metric such that \(d(x,y)\leq|\psi_\mathcal{X}(xy^{-1})|\) always holds, where \(|\cdot|\) denotes the \(l^\infty\)-norm on \(\mathbb{R}^m\), and \(\psi_\mathcal{X}:G\to\mathbb{R}\) is the Mal'cev coordinate map. For a function \(F:G/\Gamma\to\mathbb{C}\), its Lipschitz norm is defined by
\[
\|F\|_{\mathrm{Lip}}=\|F\|_{\infty}+\sup_{\substack{x,y\in G/\Gamma\\
x\neq y}} \frac{|F(x)-F(y)|}{d_{G/\Gamma}(x,y)}
\]
with respect to \(d_{G/\Gamma}\). If \(F:G/\Gamma\to \mathbb{C}\) is a Lipschitz function (that is \(\|F\|_{\mathrm{Lip}}<\infty\)), we call a sequence of the form \(n \mapsto F(g(n)\Gamma)\) a nilsequence.
In the the paper under review, the authors prove that if \(G/\Gamma\) is a nilmanifold of dimension \(m_G\geq1\), \(\mathcal X\) is a \(M_0\)-rational Mal'cev basis adapted to \(G/\Gamma\) for some \(2\leq M_0\leq \log N\), and \(G_\bullet\) is a filtration of \(G\) of degree \(d\geq 1\), then assuming that \(g\in\mathrm{poly}(\mathbb{Z},G_\bullet)\) is a polynomial sequence and \(F:G/\Gamma\to\mathbb{C}\) is a 1-bounded Lipschitz function, for every function \(f\in\mathcal M'\), one has
\[
\frac{\phi(W)}{WN}\sum_{n\in[N]}({f(Wn+b)-\mathbb{E}_f(N;W,b)}F(g(n)\Gamma)\ll_{m_G,d}(1+\|{F}\|_{\mathrm{Lip}})\frac{1}{\log N},
\]
where \(\mathbb{E}_f(N;W,b)=\frac{\phi(W)}{WN}\sum_{n\in[N]}f(Wn+b)\), and
\[
\frac{1}{N}\sum_{n\in[N]}\lambda_\pi(n)F(g(n)\Gamma)\ll_{m_G,d,\pi}{1+\|F\|_{\mathrm{Lip}}}\frac{1}{\log N},
\]
where \(\lambda_\pi (n)\) the Dirichlet coefficients of automorphic \(L\)-function \(L(s,\pi)\) attached to \(\pi\). Also, they show that if \(\pi\) is self-dual and \(\pi \ncong \pi\otimes\chi\) for any quadratic primitive character \(\chi\), then
\[
\sum_{n \leqslant N} \mu(n) \lambda_{\pi}(n) F(g(n)\Gamma) \ll_{m_G,d,\pi} \frac{N}{\log N}.
\]
Reviewer: Mehdi Hassani (Zanjan)An additive problem over Piatetski-Shapiro primes and almost-primeshttps://zbmath.org/1537.111122024-07-25T18:28:20.333415Z"Li, Jinjiang"https://zbmath.org/authors/?q=ai:li.jinjiang"Zhang, Min"https://zbmath.org/authors/?q=ai:zhang.min.1"Xue, Fei"https://zbmath.org/authors/?q=ai:xue.fei.2|xue.fei.1|xue.feiSummary: Let \(\mathcal{P}_r\) denote an almost-prime with at most \(r\) prime factors, counted according to multiplicity. In this paper, we establish a theorem of Bombieri-Vinogradov type for the Piatetski-Shapiro primes \(p=[n^{1/\gamma}]\) with \(\frac{85}{86}<\gamma <1\). Moreover, we use this result to prove that, for \(0.9989445<\gamma <1\), there exist infinitely many Piatetski-Shapiro primes such that \(p+2=\mathcal{P}_3\), which improves the previous results of \textit{Y. M. Lu} [Acta Math. Sin., Engl. Ser. 34, No. 2, 255--264 (2018; Zbl 1446.11185)], \textit{X. Wang} and \textit{Y. Cai} [Int. J. Number Theory 7, No. 5, 1359--1378 (2011; Zbl 1231.11122)], and \textit{T. P. Peneva} [Monatsh. Math. 140, No. 2, 119--133 (2003; Zbl 1117.11045)].A dichotomy for extreme values of zeta and Dirichlet \(L\)-functionshttps://zbmath.org/1537.111132024-07-25T18:28:20.333415Z"Bondarenko, Andriy"https://zbmath.org/authors/?q=ai:bondarenko.andriy-v"Darbar, Pranendu"https://zbmath.org/authors/?q=ai:darbar.pranendu"Hagen, Markus V."https://zbmath.org/authors/?q=ai:hagen.markus-valas"Heap, Winston"https://zbmath.org/authors/?q=ai:heap.winston-p"Seip, Kristian"https://zbmath.org/authors/?q=ai:seip.kristianAuthors' abstract: We exhibit large values of the Dedekind zeta function of a cyclotomic field on the critical line. This implies a dichotomy whereby one either has improved lower bounds for the maximum of the Riemann zeta function, or large values of Dirichlet \(L\)-functions on the level of the Bondarenko-Seip bound.
{\copyright} 2023 The Authors. \textit{Bulletin of the London Mathematical Society} is copyright {\copyright} London Mathematical Society.
Reviewer: Giovanni Coppola (Napoli)A weighted one-level density of families of \(L\)-functionshttps://zbmath.org/1537.111142024-07-25T18:28:20.333415Z"Fazzari, Alessandro"https://zbmath.org/authors/?q=ai:fazzari.alessandro\textit{N. M. Katz} and \textit{P. Sarnak} [Bull. Am. Math. Soc., New Ser. 36, No. 1, 1--26 (1999; Zbl 0921.11047); Random matrices, Frobenius eigenvalues, and monodromy. Providence, RI: American Mathematical Society (1999; Zbl 0958.11004)] describe the \textit{Density Conjecture}, that the distributions of zeros in many natural families of \(L\)-functions seem to coincide with eigenvalue distributions of certain groups of random matrices. One aspect concerns ``one-level density'' for a family \(\mathcal{F}\) of \(L\)-functions, where the Density Conjecture would predict that
\[
\lim_{X \to \infty} \frac{1}{\# \mathcal{F}_X} \sum_{L \in \mathcal{F}_X} \sum_{\gamma_L} f(c_L \gamma_L) \stackrel{?}{=} \int_{\mathbb{R}} f(x) W_{\mathcal{F}}(x) dx.
\]
Here, the inner sum is over imaginary parts of nontrivial zeros of \(L\) (denoted \(\gamma_L\)); the outer sum is over \(L\)-functions \(L\) in the family \(\mathcal{F}\) with log-conductor at \(1/2\) bounded by \(X\) (denoted by \(L \in \mathcal{F}_X\)); the constants \(c_L\) act to normalize the spacings of zeros; the function \(f\) is an even Schwartz function of rapid decay; and \(W_{\mathcal{F}}\) on the right is a one-level density function on an appropriate matrix group.
This can be stated more succinctly as stating that the distribution of low-lying zeros (conjecturally) agrees with the distribution of low eigenvalues of random matrices in appropriate families. In this paper, the author studies a weighted version of this one-level density. The author weights (or \textit{tilts}, as stated in the paper) by certain functions \(V\) of the central value \(L(1/2)\), specifically
\[
\mathcal{D}_k^{\mathcal{F}}(f, X) := \frac{1}{ \sum_{L \in \mathcal{F}_X} V\big( L(\tfrac{1}{2}) \big)^k } \sum_{L \in \mathcal{F}_X} \sum_{\gamma_L} f(c_L \gamma_L) V\big(L(\tfrac{1}{2})\big)^k.
\]
The weighting function \(V\) depends on the symmetry type of the family, but in all cases it has the effect of emphasizing \(L\)-functions where \(L(1/2)\) is large, where low-lying zeros should generically be less common.
Broadly, this paper extends the Density Conjecture to a \textit{Weighted Density Conjecture}, asserting that \(\mathcal{D}_k^{\mathcal{F}}(f)\) should also agree with a \textit{weighted} average over random matrices. In support of this new conjecture, the author investigates three families of \(L\)-functions: the unitary family \(\{ \zeta(\tfrac{1}{2} + it) : t \in \mathbb{R} \}\), the symplectic family of quadratic Dirichlet \(L\)-functions, and the orthogonal family of quadratic twists of \(L(s, \Delta)\), the \(L\)-function associated to the discriminant modular form. Assuming the associated Riemann hypotheses and the Ratios Conjectures (of \textit{B. Conrey} et al. [Commun. Number Theory Phys. 2, No. 3, 593--636 (2008; Zbl 1178.11056)]), this paper proves a weighted form of the Density Conjecture for \(\mathcal{D}_k^{\mathcal{F}}\) for weight exponents \(k \leq 4\). The author also includes unconditional proofs of random matrix analogs.
Finally, in sections 2A and 2B, the author gives conjectured shapes of the analogous results for all \(k \in \mathbb{N}\) and describes the matrix weights in terms of hypergeometric functions near \(0\).
The structure of the proofs build on ideas introduced in the work of Conrey, Farmer, and Zirnbauer, as well as on strategies using the Ratios Conjecture described by \textit{J. B. Conrey} and \textit{N. C. Snaith} [Proc. Lond. Math. Soc. (3) 94, No. 3, 594--646 (2007; Zbl 1183.11050)].
Reviewer: David Lowry-Duda (Providence)Hypergeometric type extended bivariate zeta functionhttps://zbmath.org/1537.111152024-07-25T18:28:20.333415Z"Pathan, M. A."https://zbmath.org/authors/?q=ai:pathan.mohammed-ahmed|pathan.mahmood-ahmad"Shahwan, Mohannad J. S."https://zbmath.org/authors/?q=ai:shahwan.mohannad-jamal-said"Bin-Saad, Maged G."https://zbmath.org/authors/?q=ai:bin-saad.maged-gumaanSummary: Based on the generalized extended beta function, we shall introduce and study a new hypergeometric-type extended zeta function together with related integral representations, differential relations, finite sums, and series expansions. Also, we present a relationship between the extended zeta function and the Laguerre polynomials. Our hypergeometric type extended zeta function involves several known zeta functions including the Riemann, Hurwitz, Hurwitz-Lerch, and Barnes zeta functions as particular cases.On two conjectures of Sun concerning Apéry-like serieshttps://zbmath.org/1537.111162024-07-25T18:28:20.333415Z"Charlton, Steven"https://zbmath.org/authors/?q=ai:charlton.steven"Gangl, Herbert"https://zbmath.org/authors/?q=ai:gangl.herbert"Lai, Li"https://zbmath.org/authors/?q=ai:lai.li"Xu, Ce"https://zbmath.org/authors/?q=ai:xu.ce"Zhao, Jianqiang"https://zbmath.org/authors/?q=ai:zhao.jianqiang|zhao.jianqiang.1Let \(\zeta(s) := \sum_{n=1}^{\infty} n^{-s}\) be the Riemann zeta function for \(\Re (s) >1\). In the 1979's proof of the irrationality of \(\zeta(3)\),
\textit{R. Apéry} [Astérisque 61, 11--13 (1979; Zbl 0401.10049)] made use of the following infinite series involving central binomial coefficients:
\[
\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n^{3}\binom{2n}{n}}=\frac{2}{5}\zeta(3).
\]
Since then, the \textit{Apéry-like} series have attracted much attention and many tools and theories have been developed to evaluate these series in closed forms. In particular, since late 1980s Z.-W. Sun has discovered many beautiful and highly nontrivial identities of infinite series, often involving interesting objects such as Harmonic numbers, Fibonacci numbers, Bernoulli numbers, Euler numbers, etc.
The paper under review is to prove two conjectures of Z.-W. Sun concerning Apéry-like series:
\begin{align*}
&\sum_{n=0}^\infty\frac{\binom{2n}{n}}{(2n+1)^3 16^n}\left(9H_{2n+1}+\frac{32}{2n+1}\right)=40\beta(4)+\frac{5}{12}\pi \zeta(3),\\
&\sum_{n=0}^\infty\frac{\binom{2n}{n}}{(2n+1)^2(-16)^n}\left(5H_{2n+1}+\frac{12}{2n+1}\right)=14\zeta(3),
\end{align*}
where \(H_n\) is classical harmonic number defined by
\[
H_n:=\sum_{k=1}^{n}\frac{1}{k},\quad\text{for~}n=1,2,3,\ldots,
\]
and \(\beta(s)\) is Dirichlet beta function defined by
\[
\beta(s):=\sum_{n=0}^{\infty}\frac{(-1)^n}{(2n+1)^s},\quad\text{for~}\Re (s)>0.
\]
Their main strategy is to convert the series (resp. the alternating series) to log-sine-cosine (resp. log-sinh-cosh) integrals. Then they express all these integrals using single-valued Bloch-Wigner-Ramakrishnan-Wojtkowiak-Zagier polylogarithms. The conjectures then follow from a few rather non-trivial functional equations of those polylogarithms in weight \(3\) and \(4\).
Reviewer: Jiangtao Li (Beijing)Global series for height 1 multiple zeta functionshttps://zbmath.org/1537.111172024-07-25T18:28:20.333415Z"Young, Paul Thomas"https://zbmath.org/authors/?q=ai:young.paul-thomasFor positive integers \(s_2,\ldots,s_j\), let
\[
\zeta(s,s_2,\ldots,s_j)=\sum\limits_{n_1>n_2>\cdots>n_j>0}\frac{1}{n_1^{s}n_2^{s_2}\cdots n_j^{s_j}}
\]
be the single-variable function on the variable \(s\in\mathbb{C}\) defined for \(\operatorname{Re}(s)>1\), which is a multiple zeta value in the case when \(s=s_1\in\mathbb{Z}_{>1}\). As the quantity \(\sharp\{i\mid s_i>1\}\) is the height of the multiple zeta value \(\zeta(s_1,s_2,\ldots,s_j)\), the multiple zeta function \(\zeta(s,\{1\}^{j-1})\) is called the height \(1\) zeta function. The paper under review studies the analytic properties of the height \(1\) zeta function. By applying the higher order derivatives with respect to \(r\) to the multiple Hurwitz zeta function \(\zeta_r(s,a)\), the author obtains a meromorphic continuation of the height \(1\) zeta function to the complex plane and finds all the poles. Then the author determines the singular parts of the Laurent series at each of the poles, and gives expressions for the linear Laurent coefficients, which are series involving the Bernoulli numbers of the second kind. The results generalize the classical Mascheroni series for Euler's constant. Finally, the author gives the linear Laurent coefficients at \(s=1\) and at \(s=0\) in terms of the Ramanujan summation of multiple harmonic star sums \(\zeta^{\star}_n(1,\ldots,1)\) defined as
\[
\zeta_{n}^{\star}(\{1\}^j)=\sum\limits_{n\geq n_1\geq n_2\geq \cdots\geq n_j\geq 1}\frac{1}{n_1n_2\cdots n_j}.
\]
Reviewer: Zhonghua Li (Shanghai)Uniqueness of meromorphic functions sharing two sets of least cardinalities with finite weighthttps://zbmath.org/1537.111182024-07-25T18:28:20.333415Z"Banerjee, Abhijit"https://zbmath.org/authors/?q=ai:banerjee.abhijit"Kundu, Arpita"https://zbmath.org/authors/?q=ai:kundu.arpitaSummary: Using the notion of weighted sharing of sets we investigate the uniqueness problem of a special class of meromorphic function sharing two or three sets containing least number of elements. Our results will provide the best possible answer of a question raised in \textit{J.-F. Chen} [Open Math. 15, 1244--1250 (2017; Zbl 1376.30023)] as well as in \textit{J.-F. Chen} [Georgian Math. J. 28, No. 3, 349--354 (2021; Zbl 1469.30062)]. Our results have also improved those in \textit{P. Sahoo} and \textit{H. Karmakar} [Acta Univ. Sapientiae, Math. 10, No. 2, 329--339 (2018; Zbl 1412.30116)] and \textit{P. Sahoo} and \textit{A. Sarkar} [Bol. Soc. Mat. Mex., III. Ser. 26, No. 2, 417--423 (2020; Zbl 1435.30108)] to a largo extent. We have exhibited a number of examples to show that some conditions used in the results are essential.Ideal growth in amalgamated powers of nilpotent rings of class two and zeta functions of quiver representationshttps://zbmath.org/1537.111192024-07-25T18:28:20.333415Z"Bauer, Tomer"https://zbmath.org/authors/?q=ai:bauer.tomer"Schein, Michael M."https://zbmath.org/authors/?q=ai:schein.michael-mLet \(A\) be a discrete valuation ring that is compact or, equivalently, complete with finite residue field, and an \(A\)-algebra means a free \(A\)-module \(L\) of finite rank equipped with an \(A\)-bilinear multiplication \([ , ] : L \times L \to L\). Let \(A\)-algebras be the nilpotent \(A\)-algebras of class at most two (i.e., when the inclusion \([L,L]\leq Z(L)=\{x \in L: \text{ for all } y \in L, [x,y]=[y,x]= 0\}\) holds, where \([L,L]\) is the \(A\)-submodule generated by the image of multiplication map). Recall that an \(A\)-ideal of \(L\) is an \(A\)-submodule \(I \leq L\) such that \([x,y]\in I\) and \([y,x] \in I\) for all \(x \in I\) and \(y \in L\). In a frame of such definitions and notations, define the ideal zeta function by the series \[ \zeta_L^{\triangleleft A}(s)=\sum_{n=1}^{\infty}a_n^{\triangleleft A}(L)n^{-s}, \] with a complex variable \(s\), and polynomially growing in \(n\) sequence \(a_n^{\triangleleft A}(L)\) if the ring \(A\) is either finitely generated (as a \(\mathbb{Z}\)-algebra) or semi-local. In both cases, the function \(\zeta_L^{\triangleleft A}(s)\) converges on some right half-plane of complex plane.
In the paper, it is shown that there is a rational function in \(q\), \(q^m\), \(q^{-s}\), and \(q^{-ms}\) giving the ideal zeta function of the amalgamation of \(m\) copies of \(L\) over the derived subring, for every \(m \geq 1\), up to an explicit factor. More precisely, this is proved for the zeta functions of nilpotent quiver representations of class two defined by Lee and Voll.
Reviewer: Roma Kačinskaitė (Vilnius)Conjectures for moments associated with cubic twists of elliptic curveshttps://zbmath.org/1537.111202024-07-25T18:28:20.333415Z"David, Chantal"https://zbmath.org/authors/?q=ai:david.chantal"Lalín, Matilde"https://zbmath.org/authors/?q=ai:lalin.matilde-n"Nam, Jungbae"https://zbmath.org/authors/?q=ai:nam.jungbaeSummary: We extend the heuristic introduced by Conrey, Farmer, Keating, Rubinstein, and Snaith [\textit{J. B. Conrey} et al., Commun. Math. Phys. 237, No. 3, 365--395 (2003; Zbl 1090.11055)] in order to formulate conjectures for the \((k, \ell)\)-moments of \(L\)-functions of elliptic curves twisted by cubic characters. We also apply the work of Keating and Snaith on the \((k, \ell)\)-moments of characteristic polynomials of unitary matrices to extend our conjecture to \(k\), \(\ell \in \mathbb{C}\) such that \(\mathrm{Re}(k)\), \(\mathrm{Re}(\ell)\) and \(\mathrm{Re}(k + \ell) > -1\). Our conjectures are then numerically tested for two families. A novelty of our conjectures is that cubic twists naturally lead us to consider the possibility \(k \neq \ell\).Spectral triples and \(\zeta\)-cycleshttps://zbmath.org/1537.111212024-07-25T18:28:20.333415Z"Connes, Alain"https://zbmath.org/authors/?q=ai:connes.alain"Consani, Caterina"https://zbmath.org/authors/?q=ai:consani.caterinaThis paper is a new and significant addition to the series of papers by the two authors aimed at approaching the famous Riemann Hypothesis through the language and machinery of operators and noncommutative geometry. The basic idea is to approximate the set of low-lying zeros of the Riemann zeta function by the spectrum of some operator of the form \(\frac{1}{2} +\mathrm{i} D\), where \(D\) is (unbounded) self-adjoint. Moreover, to connect this to a geometric perspective, \(D\) should be the so-called `Dirac operator' associated to a spectral triple, meaning there is some underlying \(\ast\)-algebra \(\mathcal{A} \subseteq \mathcal{B}(\mathcal{H})\) (where \(\mathcal{H}\) is the Hilbert space on which \(D\) acts) such that \([D,a]\) extends to a bounded operator for every \(a \in \mathcal{A}\). For \(\lambda>1\) and positive integer \(k < 2 \lambda^2\), a family of spectral triples \((\mathcal{A}(\lambda), \mathcal{H}(\lambda),D(\lambda, k))\), indexed by \((\lambda, k)\) has been constructed so that the spectrum of \(\frac{1}{2}+\mathrm{i} D(\lambda, k)\) turns out to be very similar to the low-lying zeros of the Riemann zeta function.
To give some details of the main construction, let \(\mathcal{S}^0_\mathrm{ev}\) be the space of real-valued, even Schwartz functions \(f\) satisfying \(f(0)=0=\int_{-\infty}^{\infty} f(x) dx\). For such an\(f\), the authors define a function \( \Sigma_\mu \mathcal{E} (f)\) on the circle \(C_\mu \equiv \mathbb{R}^*_+/ \mu \mathbb{Z}\) given by,
\[
\Sigma_\mu \mathcal{E}(f)(u)=u^{\frac{1}{2}} \sum_{k \in \mathbb{Z}} \sum_{n>0} \mu^{\frac{k}{2}} f(n \mu^k u),
\]
proving that the series does converge and we get a bounded measurable function. If the range of the linear map \(\Sigma_\mu \mathcal{E}\) is not dense in \(L^2(C\mu)\) the authors call the circle a \(\zeta\)-cycle and take the Hilbert space \(\mathcal{H}(\lambda)\) as the orthogonal complement of the above range in \(L^2(C_\lambda)\). There is a natural action of the multiplicative group \(\mathbb{R}^*_+\) on this Hilbert space and they prove the following result.
Theorem. The spectrum of the above action is formed by the imaginary parts of the zeros of the Riemann zeta function \(\zeta(z)\) on the critical line. Conversely, if \(s>0\) is such that \(\zeta(\frac{1}{2}+\mathrm{i}s)=0\), then any real circle of length an integral multiple of \(\frac{2 \pi}{s}\) is a \(\zeta\)-cycle and the spectrum of the corresponding \(\mathbb{R}^*_+\) action contains \(\mathrm{i}s\).
The Dirac operators \(D(\lambda, k)\) are suitable perturbations of the canonical Dirac operator \(D_0=-\mathrm{i} u \partial_u\) on the circle by certain finite rank operators. A key ingredient of this construction is classical prolate spheroidal wave functions. The Riemann-Weil explicit formulas give a concrete and finite expression of the semi-local Weil quadratic form has also been used crucially.
The paper is a deep and beautiful addition to our understanding of operator theoretic and geometric perspectives of the Riemann Hypothesis.
Reviewer: Debashish Goswami (Kolkata)On the distribution of the greatest common divisor of the elements in integral part setshttps://zbmath.org/1537.111222024-07-25T18:28:20.333415Z"Srichan, Teerapat"https://zbmath.org/authors/?q=ai:srichan.teerapatSummary: It is a classical result that the probability that two positive integers \(n, m \leq x\) are relatively prime tends to \(1/ \zeta (2) = 6/\pi^2\) as \(x \to \infty\). In this paper, the same result is still true when \(n\) and \(m\) are restricted to sub-sequences, i.e. Piatetski-Shapiro sequence, Beatty sequence and the floor function set.Two upper bounds for the Erdős-Hooley delta-functionhttps://zbmath.org/1537.111232024-07-25T18:28:20.333415Z"de la Bretèche, Régis"https://zbmath.org/authors/?q=ai:de-la-breteche.regis"Tenenbaum, Gérald"https://zbmath.org/authors/?q=ai:tenenbaum.geraldFor integer \(n\geqslant 1\) and real \(u\), let \(\Delta(n,u):=|\{d:d\mid n, e^u<d\leqslant e^{u+1}\}|\). The Erdős-Hooley Delta-function is then defined by \(\Delta(n):=\max_{u\in\mathbb{R}}\Delta(n,u)\). In the paper under review, the authors study the normal order of the \(\Delta\)-function, and approximate the weighted sum \(S(x;g):=\sum_{n\leqslant x}g(n) \Delta(n)\) for any arithmetic function \(g\) belonging to the class \(\mathcal{M}_A(y,c,\eta)\), where for \(A>0\), \(y\geqslant 1\), \(c>0\), \(\eta\in]0,1[\), consists of the arithmetic functions \(g\) that are multiplicative, non-negative, and satisfy the conditions (i) \(g(p^\nu)\leqslant A^\nu\) for \(\nu\geqslant 1\), (ii) \(g(n)\ll_\varepsilon n^\varepsilon\) for any \(\varepsilon>0\) and \(n\geqslant 1\), (iii) \(\sum_{p\leqslant x}{g(p)}=y\,\mathrm{li}(x)+O\big(x e^{-c(\log x)^\eta}\big)\) for any \(x\geqslant 2\). The authors prove that for any \(g\) in \(\mathcal{M}_A(y,c,\eta)\) with the above mentioned constants \(A, y, c, \eta\), and for any \(a>\sqrt{2}\log 2\) and \(x\geqslant 3\),
\[
S(x;g)\ll x (\log x)^{2y-2}e^{a\sqrt{\log\log x}}.
\]
Also, regarding to the normal order of the \(\Delta\)-function, they prove that letting \(b>(\log 2)/(\log 2+1/\log 2-1)\),
\[
\Delta(n)\leqslant (\log\log n)^{b}.
\]
Reviewer: Mehdi Hassani (Zanjan)On the primes in floor function setshttps://zbmath.org/1537.111242024-07-25T18:28:20.333415Z"Ma, Rong"https://zbmath.org/authors/?q=ai:ma.rong"Wu, Jie"https://zbmath.org/authors/?q=ai:wu.jie.3|wu.jie.5|wu.jie.7|wu.jie.1|wu.jie.14|wu.jie.4|wu.jie.9|wu.jie.6Consider two a set parametrized by \(x>1\):
\[
{\mathcal S}(x)=\{[x/n]:1\leq n\leq x\},
\]
where \([\cdot]\) denotes the floor function, and \(n\) a positive integer. The authors study the prime-counting function
\[
\pi_{\mathcal S}(x) = \lvert\{p\in {\mathcal S}(x):p\text{ is a prime}\}\rvert,
\]
and its weighted version
\[
S_{\mathbb{1_P}}(x) = \lvert\{n: 1\leq n\leq x \text{ and }[x/n]\text{ is a prime}\}\rvert.
\]
They show, improving upon the results of \textit{R. Heyman} [Integers 22, Paper A59, 10 p. (2022; Zbl 1504.11095)], that
\[
\pi_{\mathcal S}(x) = \int_2^{\sqrt{x}} \frac{d t}{\log t} + \int_2^{\sqrt{x}} \frac{d t}{\log(x/t)} + O\Big(\sqrt{x}\,\mathrm{e}^{-c(\log x)^{3/5}(\log\log x)^{-1/5}}\Big)
\]
for some \(c>0\), as \(x\to\infty\), and
\[
S_{\mathbb{1_P}}(x) = \left(\sum_{p} \frac{1}{p(p+1)}\right) x + O_{\varepsilon}(x^{9/19+\varepsilon})
\]
as \(x\to\infty\). Notably in the latter estimate (as well as its prime-power version, which is also given) the exponent in the remainder term is smaller than \(1/2\).
Reviewer: Maciej Radziejewski (Poznań)Remark on the Farey fraction spin chainhttps://zbmath.org/1537.111252024-07-25T18:28:20.333415Z"Technau, Marc"https://zbmath.org/authors/?q=ai:technau.marc\textit{P. Kleban} and \textit{A. E. Özlük} [Commun. Math. Phys. 203, No. 3, 635--647 (1999; Zbl 1019.82004)] introduced the notion of ``Farey fraction spin chain'', and made a conjecture regarding its asymptotic number of states with given energy. Essentially, the number \(\Phi(N)\) of $2\times 2$ matrices arising as products of matrices having the lines \((1,0) \) and \((1,1)\) as well as the one with lines \((1,1)\) and \((0,1)\) respectively, whose trace equals \(N\), should be studied. Their conjecture was disproved by
\textit{M. Peter} [J. Number Theory 90, No. 2, 265--280 (2001; Zbl 1002.11072)], but on average a precise result was proved by \textit{J. Kallies} et al. [Commun. Math. Phys. 222, No. 1, 9--43 (2001; Zbl 1004.11051)], \textit{F. P. Boca} [J. Reine Angew. Math. 606, 149--165 (2007; Zbl 1135.11050)], as well as \textit{A. V. Ustinov} [Sb. Math. 204, No. 5, 762--779 (2013; Zbl 1288.11092); translation from Mat. Sb. 204, No. 5, 143--160 (2013)]. The author of the paper under review shows that, the problem of estimating \(\Phi(N)\) can be reduced to a problem on divisors of quadratic polynomials, which was solved by \textit{C. Hooley} [Math. Z. 69, 211--227 (1958; Zbl 0081.03904)] in a special case, and by \textit{V. A. Bykovskiĭ} and \textit{A. V. Ustinov} [Dokl. Math. 99, No. 2, 195--200 (2019; Zbl 1429.11184); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 485, No. 5, 539--544 (2019)]) in full generality-thus obtaining an unconditional estimate (which is too complicated to be stated here) for \(\Phi(N)\).
Reviewer: József Sándor (Cluj-Napoca)On a variant of the prime number theoremhttps://zbmath.org/1537.111262024-07-25T18:28:20.333415Z"Zhang, Wei"https://zbmath.org/authors/?q=ai:zhang.wei.12Let \(\Lambda(n)\) be the von Mangdolt function, \(\tau_{k}(n)\) be the generalized divisor function denoting the number of representations of \(n\) as product of \(k\) natural numbers, and
\[
\tau_{(2)}(n)= \sum_{\substack{d|n\\
d\in\mathcal{Q}_{2}}}1
\]
be the square-free divisor function, where \(\mathcal{Q}_{2}\) is the set of positive square-free integers. Improving some previously known results, in the paper under review, the author proves that for \(f=\Lambda\) and \(f=\tau_k\),
\[
S_f(x)=\sum_{1\leq n\leq x}f \left(\left[\frac{x}{n}\right]\right)=c_fx +O\left(x^{7/15+1/195+\varepsilon}\right),
\]
where \(c_f=\sum_{n=1}^{\infty} f(n)/(n(n+1))\) and \(\varepsilon\) is an arbitrary small positive constant. For \(f=\tau_{(2)}\), the author proves similar result with \(O\left(x^{107/229+\varepsilon}\right)\). The proof of the above approximations based on some results on the exponential sums. Accordingly, the author gives possibility of some improvements on the exponent in the error terms, choosing some certain exponential pairs.
Reviewer: Mehdi Hassani (Zanjan)Notes on the distribution of roots modulo primes of a polynomial. IVhttps://zbmath.org/1537.111272024-07-25T18:28:20.333415Z"Kitaoka, Yoshiyuki"https://zbmath.org/authors/?q=ai:kitaoka.yoshiyukiSummary: For polynomials \(f (x),g_1(x),g_2(x)\) over \(\mathbb{Z}\), we report several observations about the density of primes \(p\) for which \(f (x)\) is fully splitting at \(p\) and \(\left\{\frac{g_1(r)}{p}\right\} < \left\{\frac{g_2(r)}{p}\right\}\) for some root \(r\) of \(f (x) \equiv 0 \bmod p\).
For Part I, see [the author, Unif. Distrib. Theory 12, No. 2, 91--117 (2017; Zbl 1448.11177)].
For Part II, see [the author, ibid. 14, No. 1, 87--104 (2019; Zbl 1469.11390)].
For Part III, see [the author, ibid. 15, No. 1, 93--104 (2020; Zbl 1475.11178)].Slim exceptional sets of Waring-Goldbach problems involving squares and cubes of primeshttps://zbmath.org/1537.111282024-07-25T18:28:20.333415Z"Han, X."https://zbmath.org/authors/?q=ai:han.xue.1"Liu, H."https://zbmath.org/authors/?q=ai:liu.huafengAuthors' abstract: Let \(p_1,p_2,\dots,p_6\) be prime numbers. First we show that, with at most \(O(N^{1/12+\varepsilon})\) exceptions, all even positive integers not exceeding \(N\) can be represented in the form \(p_1^2+p_2^2+p_3^3+p_4^3+p_5^3+p_6^3\), which improves the previous result \(O(N^{1/4+\varepsilon})\) obtained by \textit{J. Liu} [Proc. Steklov Inst. Math. 276, 176--192 (2012; Zbl 1297.11130)]. Moreover, we also prove that, with at most \(O(N^{5/12+\varepsilon})\) exceptions, all even positive integers not exceeding \(N\) can be represented in the form \(p_1^2+p_2^3+p_3^3+p_4^3+p_5^3+p_6^3\).
Reviewer: Giovanni Coppola (Napoli)Metric decomposability theorems on sets of integershttps://zbmath.org/1537.111292024-07-25T18:28:20.333415Z"Bienvenu, Pierre-Yves"https://zbmath.org/authors/?q=ai:bienvenu.pierre-yvesThis contribution fits inside the area of additive combinatorics. A set \( \mathcal{A} \subseteq \mathbb{N} \) is said to be \emph{additively decomposable} (resp., \emph{asymptotically additively decomposable}) if there exist sets \( \mathcal{B}, \mathcal{C} \subseteq \mathbb{N} \), each with a cardinality of at least two, such that \( \mathcal{A} = \mathcal{B} + \mathcal{C} \) (resp., \( \mathcal{A} \triangle (\mathcal{B} + \mathcal{C}) \) is finite).If none of these properties hold, the set \( \mathcal{A} \) is referred to as \emph{totally primitive}. In this context, \textit{E. Wirsing} [Arch. Math. 4, 392--398 (1953; Zbl 0052.04804)] showed that all subsets of \( \mathbb{N} \) are totally primitive.
The main contribution of the paper starts extending these definitions to \( \mathbb{Z} \)-decomposability with subsets \( \mathcal{A}, \mathcal{B}, \mathcal{C} \subseteq \mathbb{Z}\). Later, following the spirit of Wirsing, the author explores decomposability from a probabilistic perspective. Initially, he establishes that almost all symmetric subsets of \( \mathbb{Z} \) are \( \mathbb{Z} \)-decomposable. Subsequently, he demonstrates that almost all small perturbations of the set of primes yield a totally primitive set. Moreover, this result remains valid when replacing the set of primes with the set of sums of two squares, which is, by definition, decomposable.
{\copyright} 2023 The Authors. \textit{Bulletin of the London Mathematical Society} is copyright {\copyright} London Mathematical Society.
Reviewer: Juanjo Rué Perna (Barcelona)Computational study of non-unitary partitionshttps://zbmath.org/1537.111302024-07-25T18:28:20.333415Z"Akande, A. P."https://zbmath.org/authors/?q=ai:akande.agbolade-patrick"Genao, Tyler"https://zbmath.org/authors/?q=ai:genao.tyler"Haag, Summer"https://zbmath.org/authors/?q=ai:haag.summer"Hendon, Maurice D."https://zbmath.org/authors/?q=ai:hendon.maurice-d"Pulagam, Neelima"https://zbmath.org/authors/?q=ai:pulagam.neelima"Schneider, Robert"https://zbmath.org/authors/?q=ai:schneider.robert-j|schneider.robert-f|schneider.robert-b|schneider.robert-p"Sills, Andrew V."https://zbmath.org/authors/?q=ai:sills.andrew-vSummary: Following Cayley, MacMahon, and Sylvester, define a non-unitary partition to be an integer partition with no part equal to one, and let \(\nu (n)\) denote the number of non-unitary partitions of size \(n\). In [\textit{R. Schneider}, J. Ramanujan Math. Soc. 36, No. 1, 33--37 (2021; Zbl 1480.11127)], the sixth author proved a formula to compute \(p(n)\) by enumerating only non-unitary partitions of size \(n\), and recorded a number of conjectures regarding the growth of \(\nu (n)\) as \(n \rightarrow \infty\). Here we refine and prove some of these conjectures. For example, we prove \(p(n) \sim \nu (n) \sqrt{n/\zeta (2)}\) as \(n \rightarrow \infty\), and give Ramanujan-like congruences between \(p(n)\) and \(\nu (n)\) such as \(p(5n) \equiv \nu (5n) \pmod{5}\).Corrigendum to: ``The discriminant of compositum of algebraic number fields''https://zbmath.org/1537.111312024-07-25T18:28:20.333415Z"Khanduja, Sudesh Kaur"https://zbmath.org/authors/?q=ai:khanduja.sudesh-kSummary: We point out that there is an error in the proof of Theorem 1.1 in our paper [ibid. 15, No. 2, 353--360 (2019; Zbl 1441.11269)]. We also prove that the result of this theorem holds with an additional hypothesis. However, it is an open problem whether the result of the theorem is true in general or not.Structure of relative genus fields of cubic Kummer extensionshttps://zbmath.org/1537.111322024-07-25T18:28:20.333415Z"Aouissi, Siham"https://zbmath.org/authors/?q=ai:aouissi.siham"Azizi, Abdelmalek"https://zbmath.org/authors/?q=ai:azizi.abdelmalek"Ismaili, Moulay Chrif"https://zbmath.org/authors/?q=ai:ismaili.moulay-chrif"Mayer, Daniel C."https://zbmath.org/authors/?q=ai:mayer.daniel-c"Talbi, Mohamed"https://zbmath.org/authors/?q=ai:talbi.mohamedConsider the number field \(N={\mathbb Q}(\sqrt[3]{D}, \zeta_3)\), a cubic Kummer extension of the cyclotomic field \(K={\mathbb Q}(\zeta_3)\), where \(D\) is a cube free positive integer and \(\zeta_3\) a primitive cubic root of unity.
The main result of the article establishes the exhaustive list of the 13 forms of the integer \(D\) --in terms of its prime decomposition-- for which the Galois group over \(N\) of the relative 3-genus field of the extension \(N/K\) is isomorphic to \({\mathbb Z}/3{\mathbb Z} \times {\mathbb Z}/3{\mathbb Z}\).
The relative 3-genus field of the extension \(N/K\) is the maximal abelian extension of \(K\) contained in the maximal abelian unramified 3-extension of \(N\), and studying its Galois group over \(N\) is a first step into the determination of the structure of \(Cl_3(N)\), the 3-class group of the number field \(N\).
The authors extend in the present article the results of their previous paper [\textit{S. Aouissi} et al., Period. Math. Hung. 81, No. 2, 250--274 (2020; Zbl 1474.11186)] based on Gerth's methods, see [\textit{F. Gerth III}, J. Reine Angew. Math. 278/279, 52--62 (1975; Zbl 0334.12011); J. Number Theory 8, 84--98 (1976; Zbl 0329.12006)]. Moreover, they also study and classify the different extensions \(N/K\) in terms of their conductor.
Reviewer: Isabelle Dubois (Metz)Distribution of genus numbers of abelian number fieldshttps://zbmath.org/1537.111332024-07-25T18:28:20.333415Z"Frei, Christopher"https://zbmath.org/authors/?q=ai:frei.christopher"Loughran, Daniel"https://zbmath.org/authors/?q=ai:loughran.daniel"Newton, Rachel"https://zbmath.org/authors/?q=ai:newton.rachelLet \(K\) be an algebraic extension of the number field \(k\). The genus field of \(K\) with respect to \(k\) is the largest extension \(F_{K/k}\) of \(K\) that is unramified at all places of \(K\) such that \(F_{K/k}\) is an abelian extension of \(k\). The genus group of \(K/k\) is the Galois group Gal\((F_{K/k}/K)\). The genus number \(g_{K/k}\) is the cardinality of the genus group. Then \(k \subseteq K\subseteq F_{K/k} \subseteq H_K\) where \(H_K\) is the Hilbert class field of \(K\). The Galois group of \(H_K\) over \(K\) is the class group \(\mathcal{C}\ell(K)\) of \(K\). The principal genus of \(K/k\) is the subgroup Gal\((H_K/F_{K/k})\) of \(\mathcal{C}\ell(K)\). The genus group is the quotient group \(\mathcal{C}\ell(K)/\mathrm{Gal}(H_K/F_{K/k})\).
The main theorem of the authors concerns the average size of the genus numbers as one sums over all abelian extension of bounded conductor with given Galois group. Define \(\phi(K/k)\) to be the absolute norm of the conductor of \(K\). Let \(G\) be a non-trivial abelian group and let \(B\) be a fixed constant. Then the authors give an asymptomatic value of the sum of the genus numbers \(g_{K/k}\), when the sum is taken over all groups \(G\) isomorphic to the Galois group Gal\((K/k)\) with \(\phi(K/k)\leq B\). They obtain an asymptotic value involving the constant \(B\) and also log\((B)\) to a power given by a sum over all \(g\in G\setminus \{I\!d_G\}\). They consider some Dirichlet series which they rewrite in terms of idelic series and also in terms of Dirichlet zeta functions of \(k\). The whole proof involves harmonic analysis, Poisson summation, local and global Fourier transforms, Frobenius functions. They use also some purely adelic interpretations of genus numbers. In short, the paper is rather technical and there is no doubt about the abilities of the authors.
Reviewer: Claude Levesque (Québec)On Ihara's conjectures for Euler-Kronecker constantshttps://zbmath.org/1537.111342024-07-25T18:28:20.333415Z"Dixit, Anup B."https://zbmath.org/authors/?q=ai:dixit.anup-b"Murty, M. Ram"https://zbmath.org/authors/?q=ai:murty.maruti-ramThe Euler-Kronecker constant \(\gamma_K\) of an algebraic number field \(K\) has been defined by \textit{Y. Ihara} [Prog. Math. 253, 407--451 (2006; Zbl 1185.11069)] as
\[
\gamma_K = \frac{c_0}{c_{-1}},
\]
with \(c_{-1},c_0\) given by the Laurent expansion at \(s=1\) of the Dedekind zeta-function \(\zeta_K(s)\)
\[
\zeta_K(s)= \frac{c_{-1}}{s-1} + c_0 +\sum_{m=1}^\infty c_m(s-1)^m.
\]
In Theorem 1.1 one finds connections of \(\gamma_K\) with the error term in the prime ideal theorem. In particular it is shown that
\[
\gamma_K=1-\int_1^\infty\frac{\Delta_K(x)}x^2,
\]
where
\[
\Delta_K(x)=\sum_{n\le x}\Lambda_K(n)-x
\]
and
\[
-\frac{\zeta_K'(s)}{\zeta_K(s)}=\sum_{n=1}^\infty\frac{\Lambda_K(n)}{n^s}.
\]
Ihara [loc. cit.] showed that \(GRH\) implies
\[
\gamma_K\le 2\log\log\left(|d_K|^{1/2}\right),
\]
where \(d_K\) denotes the discriminant of \(K\).
Theorem 1.2 shows that if \(\zeta_K(s)\) has no Siegel zeros, then
\[
\gamma_K=O\left(\log(d_K)|\right),
\]
and when \(\beta\) is a Siegel zero of \(\zeta_K(s)\), then
\[
\gamma_K=\frac1{2\beta(1-\beta)}+O\left(\log(d_K)|\right).
\]
Denote by \(\gamma_m\) the constant \(\gamma_K\) when \(K\) is the \(m\)-th cyclotomic field. \textit{Y. Ihara} [Sémin. Congr. 21, 79--102 (2009; Zbl 1242.11091)] conjectured that there are constants \(0<a,b\le 2\) such that for any \(\varepsilon>0\) one has for sufficiently large \(m\)
\[
(a-\varepsilon)\log m <\gamma_m < (b+\varepsilon)\log m.
\]
This conjecture is still open, and one knows only that it holds for almost all \(m\) [\textit{K. Ford} et al., Math. Comput. 83, No. 287, 1447--1476 (2014; Zbl 1294.11164)], so the question arises whether it is true on average. It has been shown by \textit{É. Fouvry} [J. Number Theory 133, No. 4, 1346--1361 (2013; Zbl 1282.11143)] that for \(Q\ge3\) one has
\[
\sum_{m=Q}^{2Q}\gamma_m = Q\log Q +O(Q\log\log Q),
\]
and the authors obtain (Theorem 1.3) a similar assertion for \(\gamma_q\) with prime \(q\) under the assumption of the Elliott-Halberstam conjecture:
\[
\frac 1Q\sum_{Q\le q<2Q}\gamma_q = (1+o(1)Q).
\]
Reviewer: Władysław Narkiewicz (Wrocław)Trinomial equations of degree 6 over \(\mathbb{Q}_p \)https://zbmath.org/1537.111352024-07-25T18:28:20.333415Z"Alp, M."https://zbmath.org/authors/?q=ai:alp.murat"Ismail, M."https://zbmath.org/authors/?q=ai:ismail.m-kh|ismail.mohamed-f|ismail.mustafa|ismail.muhammad-faizal|ismail.mohamed-a|ismail.m-a-h|ismail.mohd-tahir|ismail.mardhiyah|ismail.mohammad-s|ismail.m-n|ismail.mahmoud-h|ismail.mohamed-mahmoud|ismail.mohammed|ismail.m-i|ismail.mohammad-vaseem|ismail.munira|ismail.mehmet-s|ismail.mohd-azmi|ismail.mahamod|ismail.mat-rofa-bin|ismail.mohammad|ismail.mohd-vaseem|ismail.moshira-a|ismail.m-ghazie"Saburov, M."https://zbmath.org/authors/?q=ai:saburov.mansur-khBased on authors' abstract: This article has as a main objective finding roots of a single variable polynomial, which is among the oldest problems of mathematics. In the field of real numbers, the solution for this problem has been found, but in the field of \(p\)-adic numbers, it is still not completely elucidated. The solvability criteria and local descriptions of roots of lower degree polynomial equations over \(\mathbb{Q}_p\) with applications were explored in the literature. However, no information is available on the \(p\)-adic absolute value or the first digit of the roots of the quadratic equation over \(\mathbb{Q}_p\). In this paper, the authors introduce the cube root function over \(\mathbb{Q}_p\) to calculate the \(p\)-adic absolute value and the first digit of roots of the trinomial equation of degree \(6\) over \(\mathbb{Q}_p\) afterwards.
Reviewer: Mouad Moutaoukil (Fès)Combinatorics of Serre weights in the potentially Barsotti-Tate settinghttps://zbmath.org/1537.111362024-07-25T18:28:20.333415Z"Caruso, Xavier"https://zbmath.org/authors/?q=ai:caruso.xavier"David, Agnès"https://zbmath.org/authors/?q=ai:david.agnes"Mézard, Ariane"https://zbmath.org/authors/?q=ai:mezard.arianeThis paper studies certain deformation spaces of a fixed absolutely irreducible mod \(p\), \(2\)-dimensional Galois representation of \(F\), the finite unramified extension of \(\mathbb{Q}_p\) of degree \(f\). The authors conjecture these deformation spaces are determined by their associated Kisin variety and their main result gives evidence towards this conjecture.
More precisely, let \(E\) be a sufficiently large finite extension of \(\mathbb{Q}_p\) with ring of integers \(O_E\) and residue field \(k_E\). Let \(\overline{\rho}\) be a \(2\)-dimensional representation of the absolute Galois group \(G_F\) with coefficients in \(k_E\) and \(\psi: G_F \rightarrow O^{\times}_E\) a character. Then \(R^{\psi}(\overline{\rho})\) is the universal deformation ring parametrizing \(O_E\)-liftings of \(\overline{\rho}\) with determinant \(\psi\).
Let \(t\) be a tame inertial type of level \(f\), which means that \(t = \omega^{\gamma}_f \oplus \omega^{\gamma'}_f\), where \(\gamma, \gamma'\) are integers and \(\omega_f\) is the fundamental character of level \(f\) of the inertia group \(I_F \subset G_F\). In [Ann. Math. (2) 170, No. 3, 1085--1180 (2009; Zbl 1201.14034)] \textit{M. Kisin} showed that \(R^{\psi}(\overline{\rho})\) has a unique reduced quotient \(R^{\psi}(t, \overline{\rho})\) whose \(E\)-points parametrize potentially crystalline lifts \(\rho\) of type \(t\) with Hodge-Tate weights \(0, 1\) for each embedding \(F \hookrightarrow E\). The scheme \(R^{\psi}(t, \overline{\rho})\) is constructed to be the reduced image of a scheme \(\mathcal{GR}^{\psi}(t, \overline{\rho})\) (a moduli space of Breuil-Kisin modules) under its natural map to \(R^{\psi}(\overline{\rho})\).
The current paper is organized around Conjecture 1 which asserts that \(R^{\psi}(t, \overline{\rho})\) only depends on the special fiber of \(\mathcal{GR}(t, \overline{\rho})\) (called the \textit{Kisin variety} attached to \((t, \overline{\rho})\)) and satisfies a certain compatibility with products.
The main result of the paper is then Theorem 2 which proves a numerical version of Conjecture 1 on the special fiber. Working on the special fiber, the Breuil-Mézard conjecture predicts that \(\mathrm{spec}(R^{\psi}(t, \overline{\rho}))_{k_E}\) is given as a sum of certain cycles in \(\mathrm{spec}(R^{\psi}(\overline{\rho}))\) indexed by Serre weights (i.e., irreducible representations of \(\mathrm{GL}_2(k_F)\)). Then Theorem \(2\) says the number of Serre weights appearing in the Breuil-Mezard conjecture is determined by the associated Kisin variety. Moreover, the set of Serre weights is produced in a combinatorial way from the Kisin variety of \((t, \overline{\rho})\). Actually, the authors prove more, including a certain non-decreasing property whose precise statement is given in Corollary 3B.8.
The proof uses the combinatorial theory of genes developed in [\textit{X. Caruso} et al., J. Inst. Math. Jussieu 17, No. 5, 1019--1064 (2018; Zbl 1450.11050)]. In [Zbl 1450.11050] a gene is attached to the datum \((t, \overline{\rho})\) and in this paper, a set of Serre weights is attached to a gene. Both steps are highly combinatorial in nature. Then Theorem 2 is deduced from Theorem 3, which proves that the set of Serre weights attached to the gene of \((t, \overline{\rho})\) is precisely the set of Serre weights appearing in the Breuil-Mézard conjecture. The proof is given in Section 4 and involves intricate weight combinatorics arguments
Reviewer: Alexander Bertoloni Meli (Ann Arbor)Torsion of algebraic groups and iterate extensions associated with Lubin-Tate formal groupshttps://zbmath.org/1537.111372024-07-25T18:28:20.333415Z"Ozeki, Yoshiyasu"https://zbmath.org/authors/?q=ai:ozeki.yoshiyasuSummary: We show finiteness results on torsion points of commutative algebraic groups over a \(p\)-adic field \(K\) with values in various algebraic extensions \(L/K\) of infinite degree. We mainly study the following cases: (1) \(L\) is an abelian extension which is a splitting field of a crystalline character (such as a Lubin-Tate extension). (2) \(L\) is a certain iterate extension of \(K\) associated with points on Lubin-Tate formal groups.Lubin-Tate formal modules over higher local fieldshttps://zbmath.org/1537.111382024-07-25T18:28:20.333415Z"Vostokov, S. V."https://zbmath.org/authors/?q=ai:vostokov.sergei-vladimirovich"Leonova, E. O."https://zbmath.org/authors/?q=ai:leonova.ekaterina-olegovnaSummary: An analog of Lubin-Tate formal groups for higher local fields of characteristic 0 is considered. The modules formed by the roots of the automorphisms of these formal groups are studied. The corresponding field extensions are constructed and their Galois groups are calculated.Determination of a class of permutation quadrinomialshttps://zbmath.org/1537.111392024-07-25T18:28:20.333415Z"Ding, Zhiguo"https://zbmath.org/authors/?q=ai:ding.zhiguo"Zieve, Michael E."https://zbmath.org/authors/?q=ai:zieve.michael-eLet \(\mathbb{F}_q\) be a finite field with \(q\) elements. A polynomial \(f(X)\in\mathbb{F}_q[X]\) is called a permutation polynomial if the function \(\alpha\mapsto f(\alpha)\) defines a bijection of \(\mathbb{F}_q\). For a given positive integer \(r\) and \(A(X)\in\mathbb{F}_{q^2}[x]\), many researchers have studied permutation polynomials over \(\mathbb{F}_{q^2}\) of the form \(f(X):=X^rA(X^{q-1})\) and arose out several conjectures and open problems concerning the choices of \(r\) and the degrees of \(A(X)\). In this paper, the authors deal with these conjectures and open problems and present several classifications of permutation polynomials \(f(X)\) over \(\mathbb{F}_{q^2}\) by composing \(f(X)\) with permutation monomials and reducing mod \(X^{q^2}-X\) under some conditions. They resolve eight conjectures and open problems from the literature by combining geometric techniques and a series of elementary computations in many situations.
Reviewer: Manjit Singh (Murthal)Condition \((\mathbf{S}+)\) in ranks \(4, 8\), and \(9\)https://zbmath.org/1537.111402024-07-25T18:28:20.333415Z"Katz, Nicholas M."https://zbmath.org/authors/?q=ai:katz.nicholas-m"Tiep, Pham Huu"https://zbmath.org/authors/?q=ai:tiep.pham-huuThe authors work over an algebraically closed field \(\mathbb{C}\) of characteristic zero, which they take to be \(\bar{\mathbb{Q}}_{\ell}\) for a suitable prime \(\ell\). For a nonzero finite-dimensional \(\mathbb{C}\)-vector space \(V\), a group \(\Gamma\) and a representation \(\Phi :\Gamma \rightarrow\)GL\((V)\), the pair \((\Gamma ,V)\) satisfies condition \((\mathbf{S}+\)) if the \(\Gamma\)-module \(V\) is irreducible, primitive, indecomposable, not tensor induced and the determinant \(\det (\Gamma |V)\) is finite. Condition\((\mathbf{S}+\)) plays a key role in the study of Kloosterman and hypergeometric \(\ell\)-adic local systems in positive characteristic \(p\); the latter have been studied in previous papers of the authors. Prior results establish \((\mathbf{S}+\)) for primitive Kloosterman and hypergeometric sheaves, except possibly in ranks 4, 8, and 9. In this paper the authors study \((\mathbf{S}+\)) in these remaining ranks, and completely determine when \((\mathbf{S}+\)) does or does not hold.
Reviewer: Vladimir P. Kostov (Nice)Certain diagonal equations and conflict-avoiding codes of prime lengthshttps://zbmath.org/1537.111412024-07-25T18:28:20.333415Z"Hsia, Liang-Chung"https://zbmath.org/authors/?q=ai:hsia.liang-chung"Li, Hua-Chieh"https://zbmath.org/authors/?q=ai:li.hua-chieh"Sun, Wei-Liang"https://zbmath.org/authors/?q=ai:sun.weiliangSummary: We study the construction of optimal conflict-avoiding codes (CAC) from a number theoretical point of view. The determination of the size of optimal CAC of prime length \(p\) and weight 3 is formulated in terms of the solvability of certain twisted Fermat equations of the form \(g^2 X^{\ell} + g Y^{\ell} + 1 = 0\) over the finite field \(\mathbb{F}_p\) for some primitive root \(g\) modulo \(p\). We treat the problem of solving the twisted Fermat equations in a more general situation by allowing the base field to be any finite extension field \(\mathbb{F}_q\) of \(\mathbb{F}_p\). We show that for \(q\) greater than a lower bound of the order of magnitude \(O(\ell^2)\) there exists a generator \(g\) of \(\mathbb{F}_q^{\times}\) such that the equation in question is solvable over \(\mathbb{F}_q\). Using our results we are able to contribute new results to the construction of optimal CAC of prime lengths and weight 3.Transformations for Appell series over finite fields and traces of Frobenius for elliptic curveshttps://zbmath.org/1537.111422024-07-25T18:28:20.333415Z"Kalita, Gautam"https://zbmath.org/authors/?q=ai:kalita.gautam"Azharuddin, Shaik"https://zbmath.org/authors/?q=ai:azharuddin.shaikSummary: Using properties of certain character sums over finite fields, we here obtain some transformations for finite field Appell series \(F_2^{\ast}\) and \(F_4^{\ast}\) in terms of \({}_4 F_3\)-Gaussian hypergeometric series. These motivate us to find a finite field analog for certain transformations satisfied by the classical Appell series \(F_2\) and \(F_4\). We also deduce an expression for the finite field Appell series \(F_4^{\ast}\) in terms of \({}_3 F_2\)-Gaussian hypergeometric series. Consequently, we investigate connections of trace of Frobenius endomorphism for certain families of elliptic curves and the finite field Appell series \(F_4^{\ast}\).Existential definability and diophantine stabilityhttps://zbmath.org/1537.111432024-07-25T18:28:20.333415Z"Mazur, Barry"https://zbmath.org/authors/?q=ai:mazur.barry"Rubin, Karl"https://zbmath.org/authors/?q=ai:rubin.karl"Shlapentokh, Alexandra"https://zbmath.org/authors/?q=ai:shlapentokh.alexandraSummary: Let \(K\) be a number field, let \(L\) be an algebraic (possibly infinite degree) extension of \(K\), and let \(\mathcal{O}_K \subset \mathcal{O}_L\) be their rings of integers. Suppose \(A\) is an abelian variety defined over \(K\) such that \(A(K)\) is infinite and \(A(L) / A(K)\) is a torsion group. If at least one of the following conditions is satisfied:
\begin{itemize}
\item[1.] \(L\) is a number field,
\item[2.] \(L\) is totally real,
\item[3.] \(L\) is a quadratic extension of a totally real field,
\end{itemize}
then \(\mathcal{O}_K\) has a diophantine definition over \(\mathcal{O}_L\).On the Diophantine equation \(U_n - b^m = c\)https://zbmath.org/1537.111442024-07-25T18:28:20.333415Z"Heintze, Sebastian"https://zbmath.org/authors/?q=ai:heintze.sebastian"Tichy, Robert F."https://zbmath.org/authors/?q=ai:tichy.robert-franz"Vukusic, Ingrid"https://zbmath.org/authors/?q=ai:vukusic.ingrid"Ziegler, Volker"https://zbmath.org/authors/?q=ai:ziegler.volkerThe authors show that for any linear recurrence sequence \((U_n)_{n\in\mathbb N}\) there are effectively computable \(B,N_0\) such that for any \(c\) and \(b>B\) the equation \(U_n-b^m=c\) has at most two distinct solutions \((n,m)\) with \(n\ge N_0\) and \(m\ge 1\). Using Baker's method and reduction algorithms for the Tribonacci sequence the authors deduce \(N_0=2\) and \(B=e^{438}\).
Reviewer: István Gaál (Debrecen)Solving fifth-degree algebraic equations using modular elliptic functionshttps://zbmath.org/1537.120012024-07-25T18:28:20.333415Z"Carletti, Ettore"https://zbmath.org/authors/?q=ai:carletti.ettoreAfter the results of Galois and Abel on the solvability of polynomial equations by radicals, mathematicians turned their attention to solving The quintic by other means. A lot of beautiful and surprising results were discovered when Hermite and Kronecker applied the theory of elliptic functions for solving quintics; see Klein's famous book on the icosahedron [\textit{F. Klein}, Vorlesungen über das Ikosaeder und die Auflösung der Gleichungen vom fünften Grade. Leipzig: Teubner (1884; JFM 16.0061.01)].
In this beautifully written article, the author presents a modern version of the solution of the quintic in Bring's form \(x^5 + 5x - a = 0\) given by \textit{H. Weber} in his algebra [Lehrbuch der Algebra. Zweite Auflage. Dritter Band: Elliptische Funktionen und algebraische Zahlen. (Zugleich 2. Auflage des Werkes: Elliptische Funktionen und algebraische Zahlen.). Braunschweig: F. Vieweg \& Sohn (1898; JFM 39.0508.06)].
Reviewer: Franz Lemmermeyer (Jagstzell)Surreal birthdays and their arithmetichttps://zbmath.org/1537.120082024-07-25T18:28:20.333415Z"Roughan, Matthew"https://zbmath.org/authors/?q=ai:roughan.matthewSummary: This paper is about the structure underlying surreal numbers, namely, the birthdays of surreal number forms. The results are intriguing because (i) addition of surreal forms results in simple addition of their birthdays, but (ii) multiplication results in a much more complicated structure of birthdays, given by a kind of two-dimensional Fibonacci sequence. The paper also provides many examples and illustrations designed to help a student become familiar with this interesting set of numbers.Necessary and sufficient conditions for unique factorization in \(\mathbb{Z} [(-1 + \sqrt{d})/2]\)https://zbmath.org/1537.130032024-07-25T18:28:20.333415Z"Ramírez Viñas, Víctor Julio"https://zbmath.org/authors/?q=ai:ramirez-vinas.victor-julioSummary: Let \(d\) be an integer, \(\alpha = (-1+ \sqrt{d})/2\) if \(d\equiv 1 \pmod{4}\), and \(\alpha = \sqrt{d}\) otherwise. In this note we present elementary necessary and sufficient conditions for \(\mathbb{Z}[\alpha]\) to be a unique factorization domain. We then apply this result to produce sufficient conditions for \(\mathbb{Z}[\alpha]\) to be a unique factorization domain, in terms of prime-producing quadratic polynomials. We also apply this criterion to give an improvement of Rabinowitsch's result that provides necessary and sufficient conditions for the imaginary quadratic field \(\mathcal{K} = \mathbb{Q}(\sqrt{1-4m}), m \in \mathbb{N}\), to have class number one. We also give two non-trivial applications to real quadratic number fields.Absorbing prime multiplication modules over a pullback ringhttps://zbmath.org/1537.130152024-07-25T18:28:20.333415Z"Farzalipour, Farkhondeh"https://zbmath.org/authors/?q=ai:farzalipour.farkhonde"Ghiasvand, Peyman"https://zbmath.org/authors/?q=ai:ghiasvand.peymanRings \(R\) are commutative domains, with unity and all modules are unitary. A proper submodule \(N\) of an \(R\)-module \(M\) is said to be 1-absorbing prime, if for all non-unit elements \(a, b\in R\) and every \(m\in M\), \(abm\in N\) implies that either \(m\in N\) or \(ab\in (N:M)\). An \(R\)-module \(M\) is defined to be an absorbing prime multiplication module (apmm) if it does not have any 1-absorbing prime submodules.
This paper concentrates on this subclass of pure injective \(R\) modules, where \(R\) is either a discrete valuation domain or a pullback of two discrete valuation domains. A classification of possible indecomposable apmm in case when \(R\) is a discrete valuation domain (with maximal ideal \(P\)) is as follows: Either \(R\), or the indecomposable torsion modules, the injective hull of \(R/P\) or the field of fractions of \(R\). If \(R\) is a pullback of two discrete valuation domains over a common field, then a description of an indecomposable apmm with finite-dimensional top over \(R/\mathrm{Rad}(R)\) is given as any of the three types of modules; this result generalizes the corresponding result in [\textit{S. E. Atani} and \textit{F. Farzalipour}, Colloq. Math. 114, No. 1, 99--112 (2009; Zbl 1157.13007)]. As a consequence, the authors also show that if \(R\) is the pullback of two discrete valuation domains with a common factor field, then every indecomposable apmm with finite-dimensional top is pure-injective.
Almost half of the cited references is by S. Ebrahimi Atani and his coauthors.
Reviewer: Radoslav M. Dimitrić (New York)A freeness criterion without patching for modules over local ringshttps://zbmath.org/1537.130182024-07-25T18:28:20.333415Z"Brochard, Sylvain"https://zbmath.org/authors/?q=ai:brochard.sylvain"Iyengar, Srikanth B."https://zbmath.org/authors/?q=ai:iyengar.srikanth-b"Khare, Chandrashekhar B."https://zbmath.org/authors/?q=ai:khare.chandrashekharSummary: It is proved that if \(\varphi \colon A\to B\) is a local homomorphism of commutative noetherian local rings, a nonzero finitely generated \(B\)-module \(N\) whose flat dimension over \(A\) is at most \(\operatorname{edim} A - \operatorname{edim} B\) is free over \(B\) and \(\varphi\) is a special type of complete intersection. This result is motivated by a `patching method' developed by Taylor and Wiles and a conjecture of de Smit, proved by the first author, dealing with the special case when \(N\) is flat over \(A\).Galois closures and elementary components of Hilbert schemes of pointshttps://zbmath.org/1537.140072024-07-25T18:28:20.333415Z"Satriano, Matthew"https://zbmath.org/authors/?q=ai:satriano.matthew"Staal, Andrew P."https://zbmath.org/authors/?q=ai:staal.andrew-pLet \(A\) be a commutative \(B\)-algebra which is free as a \(B\)-module. For \(a \in A\), let \(a^{(i)} = 1 \otimes 1 \dots \otimes a \otimes \dots \otimes 1 \in A^{\otimes n}\) with \(a\) in the \(i\)th position and the tensor product taken over \(B\). Let \(p_a (t) \in B[t]\) be the characteristic polynomial of the multiplication map \(m_a: A \to A\) and write \(p_a (t) = \sum_{i=0}^n s_i (a) t^{n-i}\). Letting \(e_j (a)\) be the \(j\)th elementary symmetric function in \(a^{(1)}, \dots a^{(n)}\) and \(I = \langle s_j (a) - e_j (a) | a \in A \rangle\),
\[
G^{(n)} (A/B) = A^{\otimes n}/I
\]
is the \(n\)th higher Galois closure of the extension \(B \subset A\). When \(n = \mathrm{rank}_B A\), this is the Galois closure studied by \textit{M. Bhargava} and \textit{M. Satriano} [J. Eur. Math. Soc. (JEMS) 16, 1881--1913 (2014; Zbl 1396.13007)]. The permutation action of \(S_n\) on \(A^{\otimes n}\) descends to \(G^{(n)} (A/B)\). Further, if \(A\) is a quotient of \(B[x_1, \dots, x_r]\), then the images \(x_{i,j}\) of \(x_j^{(i)}\) generate \(G^{(n)} (A/B)\) as a \(B\)-algebra and the images of \(\sum_j x_{i,j}\) vanish, so one can eliminate the variables \(x_{i,n}\). Thus if \(B=k\) is a field, \([A] \in \mathrm{Hilb}^n (\mathbb A^r_k)\) yields \([G(A/k)] \in \mathrm{Hilb}^d (\mathbb A_k^{r(n-1)})\) where \(d\) is the length of \(G^{(n)}(A/k)\). This raises questions. What is the length \(d\)? Under what conditions does \([G(A/k)]\) have negative tangents in the sense of \textit{J. Jelisiejew} [J. Lond. Math. Soc. 100, 249--272 (2019; Zbl 1441.14023)]? More generally, when does \([G(A/k)]\) lie on an elementary component of \(\mathrm{Hilb}^d (\mathbb A_k^{r(n-1)})\)?
The authors answer these questions for the \(k\)-algebra of rank \(m+1\) given by
\[
A=A_{m}= k[x_1, \dots, x_{m}]/(x_1, \dots, x_{m})^2,
\]
the most degenerate \(k\)-algebra in the moduli space of rank \(m+1\) rings of \textit{B. Poonen} [Eur. Math. Soc. (JEMS) 10, 817--836 (2008; Zbl 1151.14011)]. For \(n \leq 3\), both \(A\) and \(G^{(n)} (A/k)\) lie on the main component, so the question is only interesting for \(n \geq 4\). Here the authors prove that \(G^{(n)} (A_m/k) \in \mathrm{Hilb}^d (\mathbb A_k^{m(n-1)})\) has trivial negative tangents if and only if (a) \(n=1\), (b) \(m=1\) and \(n \leq 2\), (c) \(n=3\) and \(m \geq 3\) or (d) \(n \geq 4\) and \(m \geq 2\) and further give a combinatorial formula for \(d = d(n,m)\), the length of \(G^{(n)} (A_m/k)\). In cases (c) and (d), they show that the obstruction space \(T^2 (G^{(n)} (A_m/k)/k)\) to smoothness vanishes if and only if \(n=3\) and \(m \geq 3\) or \(n=4\) and \(2 \leq m \leq 3\), giving families in which \(G^{(n)} (A_m/k)\) corresponds to a smooth point in \(\mathrm{Hilb}^d (\mathbb A_k^{m(n-1)})\). Continuing an idea from their earlier paper [Forum Math. Sigma 11, Paper No. e45, 37 p. (2023; Zbl 1522.14009)], the authors give conditions under which the quotient of \(G^{(n)} (A_m/k)\) by socle elements of certain degree also has trivial negative tangents. They give tables showing the corresponding Hilbert functions for their new examples.
Reviewer: Scott Nollet (Fort Worth)Arithmetic inflection formulae for linear series on hyperelliptic curveshttps://zbmath.org/1537.140342024-07-25T18:28:20.333415Z"Cotterill, Ethan"https://zbmath.org/authors/?q=ai:cotterill.ethan"Darago, Ignacio"https://zbmath.org/authors/?q=ai:darago.ignacio"Han, Changho"https://zbmath.org/authors/?q=ai:han.changhoSummary: Over the complex numbers, \textit{Plücker's formula} computes the number of inflection points of a linear series of fixed degree and projective dimension on an algebraic curve of fixed genus. Here, we explore the geometric meaning of a natural analog of Plücker's formula and its constituent local indices in \(\mathbb{A}^1\)-homotopy theory for certain linear series on hyperelliptic curves defined over an arbitrary field.
{{\copyright} 2023 Wiley-VCH GmbH.}Representation of non-special curves of genus 5 as plane sextic curves and its application to finding curves with many rational pointshttps://zbmath.org/1537.140362024-07-25T18:28:20.333415Z"Kudo, Momonari"https://zbmath.org/authors/?q=ai:kudo.momonari"Harashita, Shushi"https://zbmath.org/authors/?q=ai:harashita.shushiIn this paper, the authors study the problem of parametrising genus 5 curves. The ultimate goal is to find a parametrisation where the number of parameters equals the dimension of the moduli space. Such a parametrisation allows one to effectively enumerate such curves over a finite field. For hyperelliptic curves over an algebraically closed field, this is not so hard to establish. For trigonal curves, the problem was studied before in [\textit{M. Kudo} and \textit{S. Harashita}, Exp. Math. 31, No. 3, 908--919 (2022; Zbl 1497.14039)]. The current paper studies the case of non-hyperelliptic and non-trigonal curves of genus 5.
The idea is to use a geometric construction that gives a birational map to a sextic curve in the projective plane. Generically, this sextic curve has at five double points as its singularities, in which case the original genus 5 curve is called non-special by the authors. Now one can instead enumerate sextics with five prescribed double points in order to construct genus 5 curves. The authors implemented an algorithm in Magma to do this and used this to find new genus 5 curves over \(\mathbb{F}_3\) that have 32 points over \(\mathbb{F}_9\).
Reviewer: Raymond van Bommel (Cambridge, MA)Weak approximation versus the Hasse principle for subvarieties of abelian varietieshttps://zbmath.org/1537.140382024-07-25T18:28:20.333415Z"Creutz, Brendan"https://zbmath.org/authors/?q=ai:creutz.brendanSummary: For varieties over global fields, weak approximation in the space of adelic points can fail. For a subvariety of an abelian variety one expects this failure is always explained by a finite descent obstruction, in the sense that the rational points should lie dense in the set of unobstructed (modified) adelic points. We show that this follows from a priori weaker assumptions concerning descent obstructions to the Hasse principle, i.e., to the existence of rational points. We also prove a similar statement for the obstruction coming from the Mordell-Weil sieve.Weak approximation for del Pezzo surfaces of low degreehttps://zbmath.org/1537.140392024-07-25T18:28:20.333415Z"Demeio, Julian"https://zbmath.org/authors/?q=ai:demeio.julian-lawrence"Streeter, Sam"https://zbmath.org/authors/?q=ai:streeter.samSummary: We prove, via an ``arithmetic surjectivity'' approach inspired by work of \textit{J. Denef} [Algebra Number Theory 13, No. 9, 1983--1996 (2019; Zbl 1432.14020)], that weak weak approximation holds for surfaces with two conic fibrations satisfying a general assumption. In particular, weak weak approximation holds for general del Pezzo surfaces of degrees \(1\) or \(2\) with a conic fibration.Graded quotients of ramification groups of local fields with imperfect residue fieldshttps://zbmath.org/1537.140402024-07-25T18:28:20.333415Z"Saito, Takeshi"https://zbmath.org/authors/?q=ai:saito.takeshiSummary: We prove that the graded quotients of the filtration by ramification groups of any henselian discrete valuation field of residue characteristic \(p > 0\) are \(\mathbb{F}_p\)-vector spaces. We define an injection of the character group of each graded quotient to a twisted cotangent space defined using a cotangent complex.On the étale cohomology of Hilbert modular varieties with torsion coefficientshttps://zbmath.org/1537.140412024-07-25T18:28:20.333415Z"Caraiani, Ana"https://zbmath.org/authors/?q=ai:caraiani.ana"Tamiozzo, Matteo"https://zbmath.org/authors/?q=ai:tamiozzo.matteoThe main theorem of this paper states that for Hilbert modular varieties, the étale cohomology with torsion coefficents vanishes outside the middle degree after localizing at a generic maximal ideal. This builds on earlier work on unitary Shimura varieties by \textit{A. Caraiani} and \textit{P. Scholze} [Ann. Math. (2) 186, No. 3, 649--766 (2017; Zbl 1401.11108); ``On the generic part of the cohomology of non-compact unitary Shimura varieties'', Preprint, \url{arXiv:1909.01898}]. The nongeneric case is also investigated, and certain bounds found on the degrees where the torsion cohomology can be nonzero.
A key ingredient of the proof is the geometric Jacquet-Langlands functoriality inspired by work of \textit{Y. Tian} and \textit{L. Xiao} [Compos. Math. 152, No. 10, 2134--2220 (2016; Zbl 1405.14061)] on Goren-Oort stratifications. In the final section of the paper, the completed homology of Hilbert modular varieties is related to the \(p\)-adic Langlands correspondence, under certain conditions.
Reviewer: Salman Abdulali (Greenville)Tautological rings of Shimura varieties and cycle classes of Ekedahl-Oort stratahttps://zbmath.org/1537.140422024-07-25T18:28:20.333415Z"Wedhorn, Torsten"https://zbmath.org/authors/?q=ai:wedhorn.torsten"Ziegler, Paul"https://zbmath.org/authors/?q=ai:ziegler.paulSummary: We define the tautological ring as the subring of the Chow ring of a Shimura variety generated by all Chern classes of all automorphic bundles. We explain its structure for the special fiber of a good reduction of a Shimura variety of Hodge type and show that it is generated by the cycle classes of the Ekedahl-Oort strata as a vector space. We compute these cycle classes. As applications we get the triviality of \(\ell\)-adic Chern classes of flat automorphic bundles in characteristic \(0\), an isomorphism of the tautological ring of smooth toroidal compactifications in positive characteristic with the rational cohomology ring of the compact dual of the hermitian domain given by the Shimura datum, and a new proof of Hirzebruch-Mumford proportionality for Shimura varieties of Hodge type.Strongly semistable reduction of syzygy bundles on plane curveshttps://zbmath.org/1537.140522024-07-25T18:28:20.333415Z"Hahn, Marvin Anas"https://zbmath.org/authors/?q=ai:anas-hahn.marvin"Werner, Annette"https://zbmath.org/authors/?q=ai:werner.annetteSummary: We investigate degenerations of syzygy bundles on plane curves over \(p\)-adic fields. We use Mustafin varieties which are degenerations of projective spaces to find a large family of models of plane curves over the ring of integers such that the special fibre consists of multiple projective lines meeting in one point. On such models we investigate vector bundles whose generic fibre is a syzygy bundle and which become trivial when restricted to each projective line in the special fibre. Hence these syzygy bundles have strongly semistable reduction. This investigation is motivated by the fundamental open problem in \(p\)-adic Simpson theory to determine the category of Higgs bundles corresponding to continuous representations of the étale fundamental group of a curve. Faltings' \(p\)-adic Simpson correspondence and work of Deninger and the second author shows that bundles with Higgs field zero and potentially strongly semistable reduction fall into this category. Hence the results in the present paper determine a class of syzygy bundles on plane curves giving rise to a \(p\)-adic local system. We apply our methods to a concrete example on the Fermat curve suggested by Brenner and prove that this bundle has potentially strongly semistable reduction.K3 surfaces with two involutions and low Picard numberhttps://zbmath.org/1537.140552024-07-25T18:28:20.333415Z"Festi, Dino"https://zbmath.org/authors/?q=ai:festi.dino"Nijgh, Wim"https://zbmath.org/authors/?q=ai:nijgh.wim"Platt, Daniel"https://zbmath.org/authors/?q=ai:platt.daniel-eMotivated by the construction of \(G_2\)-manifolds, the problem considered in the paper under review is to find the minimum Picard number of a \(K3\) surface with holomorphic and non-holomorphic involutions that commute, and the answer is given in Theorem 1.1 as the main theorem. In particular, they study \(K3\) surfaces with such involutions and of Picard number \(2\).
Hinted by an inplication of the statements in Proposition 3.3 that a double cover of \(\mathbb{P}^2\) branching at a smooth sextic curve admits a holomorphic involution, they give examples of \(K3\) surfaces with Picard number \(2\), of degrees \(2\), \(8\), \(6\), and a smooth quartic of degree \(2\). They also produce Magma programmes that give many \(K3\) surfaces that have these structures.
Lastly, they also investigate \(K3\) surfaces defined over \(\mathbb{R}\); motivated by a key observation that a complex \(K3\) surface admits commuting holomorphic and non-holomorphic involutions if and only if the underlying algebraic \(K3\) surface is defined over \(\mathbb{R}\) and admits an automorphism of order \(2\). In fact, they prove that for any primitive even lattice \(N\), in the \(K3\) lattice, of rank \(r\) with \(1\leq r\leq 10\), there exists a \(K3\) surface \(X\) defined over \(\mathbb{R}\) with non-empty \(\mathbb{R}\)-points, such that the Picard lattice of \(X\) and that of the associated complex \(K3\) surface are isometric to \(N\). This is an analogue of Morrison's result over \(\mathbb{R}\).
Reviewer: Makiko Mase (Tōkyō)Rigid analytic \(p\)-divisible groups. IIhttps://zbmath.org/1537.140672024-07-25T18:28:20.333415Z"Fargues, Laurent"https://zbmath.org/authors/?q=ai:fargues.laurentSummary: Let \(K\) be a \(p\)-adic field. We continue to develop the theory of rigid analytic \(p\)-divisible groups over \(K\). For example, we explain how to recover the category of Banach-Colmez spaces from rigid analytic \(p\)-divisible groups ``at finite level'' without perfectoid spaces. We then establish some results about families of rigid analytic \(p\)-divisible groups. This allows us to prove a ``minimality'' result in the sense of birational geometry for integral models of unramified Rapoport-Zink spaces.
For Part I see [the author, ibid. 374, No. 1--2, 723--791 (2019; Zbl 1441.14146)]The distribution of defective multivariate polynomial systems over a finite fieldhttps://zbmath.org/1537.140692024-07-25T18:28:20.333415Z"Giménez, Nardo"https://zbmath.org/authors/?q=ai:gimenez.nardo"Matera, Guillermo"https://zbmath.org/authors/?q=ai:matera.guillermo"Pérez, Mariana"https://zbmath.org/authors/?q=ai:perez.mariana"Privitelli, Melina"https://zbmath.org/authors/?q=ai:privitelli.melinaLet \(K\) be a field, \(x_1,\ldots ,x_n\) a sequence of variables and \(R=K[x_1,\ldots ,x_n]\). For any \(d \ge 1\), denote by \(R_{\le d}\) the set of all polynomials in \(R\) of degree at most \(d\). Furthermore, given a sequence \(D:= (d_1, \ldots , d_s)\) of non-negative integers, we denote by \(R^D\) the set of \(R_{\le d_1}\times \cdots \times R_{\le d_s}\).
The paper investigates upper bounds on the dimensions of sets \(A\) and \(B\), where \(A\) is defined as the collection of all \(F\in R^D\) that might fail to define a set-theoretic or an ideal-theoretic complete intersection, while \(B\) contains those failing to define an irreducible variety. In particular, the authors discuss these problems in the case that \(K\) is the finite field of \(q\) elements where \(q\) is a prime power.
Reviewer: Amir Hashemi (Isfahan)Codes on subgroups of weighted projective torihttps://zbmath.org/1537.140722024-07-25T18:28:20.333415Z"Şahin, Mesut"https://zbmath.org/authors/?q=ai:sahin.mesut.1"Yayla, Oğuz"https://zbmath.org/authors/?q=ai:yayla.oguzThe paper under review provides a comprehensive study of algebraic invariants essential for analyzing codes on subgroups of weighted projective tori within an \(n\)-dimensional weighted projective space. The authors specifically focus on computing the primary parameters of generalized toric codes located in a weighted projective plane \(P(1, 1, a)\). The introduction offers a thorough overview of the theoretical framework, including the definition of weighted projective spaces and their representation through the Geometric Invariant Theory quotient. The paper elucidates how these spaces are smooth only when they coincide with the classical projective space \(P^n\), establishing a foundational understanding for the subsequent analysis.
The paper advances by detailing the evaluation map for subsets of \( F_q \)-rational points and the formation of linear codes through this map. A significant contribution of this work is the calculation of the basic parameters (length, dimension, and minimum distance) for the codes on these subgroups of tori. For instance, the evaluation map is defined as:
\[
\text{ev}_Y : S_{\alpha} \rightarrow F_q^N, \quad F \mapsto (F(P_1), \dots, F(P_N))
\]
where the image \( C_{\alpha, Y} = \text{ev}_Y(S_{\alpha}) \) forms a linear code with specific parameters determined by the structure of the weighted projective space and the subset of rational points chosen. The authors also explore the properties of degenerate tori and provide explicit formulas for the Hilbert function and the a-invariant, which play crucial roles in the dimension and minimum distance calculations. This meticulous approach yields practical insights into constructing efficient coding schemes within the weighted projective space framework.
Reviewer: Behrooz Mosallaei (Las Cruces)Hausdorff approximations and volume of tubes of singular algebraic setshttps://zbmath.org/1537.140782024-07-25T18:28:20.333415Z"Basu, Saugata"https://zbmath.org/authors/?q=ai:basu.saugata"Lerario, Antonio"https://zbmath.org/authors/?q=ai:lerario.antonioIn this work, the volume of a tubular neighborhood of a real algebraic set \(Z\) is estimated in a probabilistic form. In [\textit{M. Lotz}, Proc. Am. Math. Soc. 143, No. 5, 1875--1889 (2015; Zbl 1349.53108)], this problem is studied in the case \(Z\) is a smooth complete intersection in \(\mathbb{R}^n\), and in [\textit{P. Bürgisser} et al., Math. Comput. 77, No. 263, 1559--1583 (2008; Zbl 1195.65018)], in the case \(Z\) is a hypersurface (possibly singular) in the sphere \(S^n\). Here, the authors prove bounds with no smoothness assumption and no restriction on the dimension of \(Z\):
Theorems 1.1 and 3.2. Let \(F\) in \(R[X_1,...,X_n]\) be a finite set of polynomials with degrees bounded by \(\delta\) and \(Z \subset \mathbb{R}^n\) be their common zero set. Assume \(\dim_\mathbb{R}(Z) \leq m\). Given \(p\in \mathbb{R}^n\) and \(\sigma > 0\), let \(B(p,\sigma)\) be the euclidean ball of radius \(\sigma\) centered at \(p\). Let \(x\in B(p,\sigma)\) be a uniformly distributed point and denote by \(\mathrm{dist}(x, Z)\) the euclidean distance between \(x\) and \(Z\). Then, for every \(\varepsilon > 0\)
\[
\mathbb{P}(\mathrm{dist}(x, Z) \leq \varepsilon)\leq 4 \left(\dfrac{4n\delta\varepsilon}{\sigma}\right)^{n-m}\left(1+\dfrac{(4\delta+1)\varepsilon}{\sigma} \right)^m
\]
In particular, if \(\varepsilon\leq \sigma/((4\delta+1)m)\),
\[
\mathbb{P}(\mathrm{dist}(x, Z) \leq \varepsilon)\leq 4e\left(\dfrac{4n\delta\varepsilon}{\sigma}\right)^{n-m}
\]
In the case the ambient space is the sphere, the authors prove a similar theorem, which generalizes [\textit{P. Bürgisser} and \textit{F. Cucker}, Condition. The geometry of numerical algorithms. Berlin: Springer (2013; Zbl 1280.65041), Theorem 21.1]:
Theorems 1.3. Let \(\mathcal P\) in \(\mathbb{R}[X_1,\ldots,X_n]\) be a finite set of homogeneous polynomials of degree bounded by \(\delta\) and \(Z \subset S^n\) be their common zero set. Assume \(\dim_\mathbb{R}(Z) \leq m\). Given \(p\in S^n\) and \(\sigma > 0\), let \(B(p,\sigma)\) be the euclidean ball of radius \(\sigma\) centered at \(p\). Let \(x\in B(p,\sigma)\) be a uniformly distributed point and denote by \(dist(x, Z)\) the euclidean distance between \(x\) and \(Z\). Then, for every \(\varepsilon > 0\)
\[
\mathbb{P}(\mathrm{dist}(x, Z) \leq \varepsilon)\leq 2e \left(1+\frac{8\pi^3}{15} \right) \left(\frac{8n\delta \sin\varepsilon}{\sin \sigma} \right)^{n-m}\left(1+(8n\delta+8\delta+1 )\frac{\sin\varepsilon}{\sin\delta} \right)^m
\]
In particular, if \(\sin\varepsilon\leq \sin \sigma/((8n\delta+8\delta+1)m)\),
\[
\mathbb{P}(\mathrm{dist}(x, Z) \leq \varepsilon) \leq 2e \left(1+\frac{8\pi^3}{15} \right) \left(\frac{8n\delta \sin\varepsilon}{\sin \sigma} \right)^{n-m}
\]
In their proofs, they consider the Hausdorff metric and apply Weyl's Tube Formula [\textit{H. Weyl}, Am. J. Math. 61, 461--472 (1939; JFM 65.0796.01)].
Furthermore, in Section 1.1, it is showed how to interpret the previous results to give a solution to [\textit{P. Bürgisser} and \textit{F. Cucker}, Condition. The geometry of numerical algorithms. Berlin: Springer (2013; Zbl 1280.65041), Problem 17], ``Conic Condition Numbers of Real Problems with High Codimension of Ill-posedness''.
Reviewer: Gema Maria Diaz Toca (Murcia)Counting real roots in polynomial-time via Diophantine approximationhttps://zbmath.org/1537.140812024-07-25T18:28:20.333415Z"Rojas, J. Maurice"https://zbmath.org/authors/?q=ai:rojas.j-mauriceFor a set \(A:=\{ a_1,\ldots ,a_{n+2}\}\subset \mathbb{Z}^n\) of cardinality \(n+2\), with all coordinates of \(a_j\) having absolute value at most \(d\) and the \(a_j\) not all lying in the same affine hyperplane, one sets \(x^{a_j}:=x_1^{a_{1,j}}\cdots x_n^{a_{n,j}}\) and one considers the \(n\) functions \(f_i:=\sum _jc_{i,j}x^{a_j}\in \mathbb{Z}[x_1^{\pm 1},\ldots ,x_n^{\pm 1}]\).
Suppose that \(F:=(f_1,\ldots ,f_n)\) is an \(n\times n\) polynomial system with generic integer coefficients at most \(H\) in absolute value, and \(A\) the union of the sets of exponent vectors of the \(f_i\). The author gives the first algorithm that, for any fixed \(n\), counts exactly the number of real roots of \(F\) in time polynomial in log(d\(H\)). He also discusses a number-theoretic hypothesis that would imply a further speed-up to time polynomial in \(n\) as well.
Reviewer: Vladimir P. Kostov (Nice)Coordinate-free exponentials of general multivector in \(Cl_{p, q}\) algebras for \(p + q = 3\)https://zbmath.org/1537.150162024-07-25T18:28:20.333415Z"Acus, Arturas"https://zbmath.org/authors/?q=ai:acus.arturas"Dargys, Adolfas"https://zbmath.org/authors/?q=ai:dargys.adolfasSummary: Closed form expressions in real Clifford geometric algebras \(Cl_{0,3}\), \(Cl_{3, 0}\), \(Cl_{1, 2}\), and \(Cl_{2, 1}\) are presented in a coordinate-free form for exponential function when the exponent is a general multivector. The main difficulty in solving the problem is connected with an entanglement (or mixing) of vector and bivector components \(a_i\) and \(a_{jk}\) in a form \((a_i - a_{jk})^2\), \(i \neq j \neq k\). After disentanglement, the obtained formulas simplify to the well-known de Moivre-type trigonometric/hyperbolic function for vector or bivector exponentials. The presented formulas may find wide application in solving GA differential equations, in signal processing, automatic control and robotics.
{\copyright} 2023 John Wiley \& Sons, Ltd.On generalization of Lipschitz groups and spin groupshttps://zbmath.org/1537.150172024-07-25T18:28:20.333415Z"Filimoshina, Ekaterina"https://zbmath.org/authors/?q=ai:filimoshina.ekaterina"Shirokov, Dmitry"https://zbmath.org/authors/?q=ai:shirokov.dmitry-sSummary: This paper presents some new Lie groups preserving fixed subspaces of geometric algebras (or Clifford algebras) under the twisted adjoint representation. We consider the cases of subspaces of fixed grades and subspaces determined by the grade involution and the reversion. Some of the considered Lie groups can be interpreted as generalizations of Lipschitz groups and spin groups. The Lipschitz groups and the spin groups are subgroups of these Lie groups and coincide with them in the cases of small dimensions. We study the corresponding Lie algebras.
{\copyright} 2022 John Wiley \& Sons, Ltd.Extending Lasenby's embedding of octonions in space-time algebra \(Cl(1, 3)\), to all three- and four dimensional Clifford geometric algebras \(Cl(p, q)\), \(n = p + q = 3, 4\)https://zbmath.org/1537.150182024-07-25T18:28:20.333415Z"Hitzer, Eckhard"https://zbmath.org/authors/?q=ai:hitzer.eckhardSummary: We study the embedding of octonions in the Clifford geometric algebra for space-time STA \(Cl(1, 3)\), as suggested by \textit{A. Lasenby} at [``Some recent GA results in mathematical physics and the GA approach to the fundamental forces of nature'', Presentation, \url{https://www.youtube.com/watch?v=fFj4E7q4hbY}]. As far as possible, we extend the approach to similar octonion embeddings for all three- and four-dimensional Clifford geometric algebras \(Cl(p, q)\), \(n = p+q = 3,4\). Noticeably, the lack of a quaternionic subalgebra in \(Cl(2, 1)\) seems to prevent the construction of an octonion embedding in this case and necessitates a special approach in \(Cl(2, 2)\). As examples, we present for \(Cl(3, 0)\) the nonassociativity of the octonionic product in terms of multivector grade parts with cyclic symmetry and show how octonion products and involutions can be combined to make the opposite transition from octonions to the Clifford geometric algebra \(Cl(3, 0)\) and how octonionic multiplication can be represented with (complex) biquaternions or Pauli matrix algebra.
{\copyright} 2022 John Wiley \& Sons, Ltd.Some recent results for SU(3) and octonions within the geometric algebra approach to the fundamental forces of naturehttps://zbmath.org/1537.150242024-07-25T18:28:20.333415Z"Lasenby, Anthony"https://zbmath.org/authors/?q=ai:lasenby.anthony-nSummary: Different ways of representing the group \(SU(3)\) within a Geometric Algebra approach are explored. As part of this, we consider characteristic multivectors for \(SU(3)\) and how these are linked with decomposition of generators into commuting bivectors. The setting for this work is within a 6d Euclidean Clifford Algebra. We then go on to consider whether the fundamental forces of particle physics might arise from symmetry considerations in just the 4d geometric algebra of spacetime -- the STA. As part of this, a representation of \(SU(3)\) is found wholly within the STA, involving preservation of a bivector norm. We also show how Octonions can be fully represented within the Spacetime Algebra, which we believe will be useful in making them understandable and accessible to a new community in Physics and Engineering. The two strands of the paper are drawn together in showing how preserving the octonion norm is the same as preserving the timelike part of the Dirac current of a particle. This suggests a new model for the symmetries preserved in particle physics. Following on from work by \textit{M. Guenaydin} and \textit{F. Guersey} [J. Math. Phys. 14, 1651--1667 (1973; Zbl 0338.17004)] on the link between quarks, and octonions, and by Furey on chains of octonionic multiplications, we show how both of these fit well within our scheme and give some wholly STA versions of the operations involved, which in the cases considered have easily understandable equivalents in terms of 4d geometry. Links with larger groups containing \(SU(3)\), such as \(G_2\) and \(SU(8)\), are also considered.
{\copyright} 2023 The Authors. Mathematical Methods in the Applied Sciences published by John Wiley \& Sons, Ltd.Matrices in \(\mathbb{M}_2 [\mathbb{F}_q [T]]\) with quadratic minimal polynomialhttps://zbmath.org/1537.150322024-07-25T18:28:20.333415Z"van Zyl, Jacobus Visser"https://zbmath.org/authors/?q=ai:van-zyl.jacobus-visserThe field of polynomial equations over a ring with matrix solutions or matrices that satisfy certain polynomials over a ring is a fascinating area of study within linear algebra, dating back to the time of Cayley and Hamilton. In 1933, \textit{C. G. Latimer} and \textit{C. C. MacDuffee} [Ann. Math. (2) 34, 313--316 (1933; Zbl 0006.29002)] established a one-to-one correspondence between matrix solutions and ideals. In this paper the authors developed an algorithm for constructing representatives of matrix solutions to \(p(X) = 0\) when \(R = F_{q}[T]\).
Let \(p(X) \in R[X]\) be a monic, irreducible polynomial of degree \(n\) over a principal ideal domain \(R\). If \(C \in M_{n}[R]\) is a matrix solution to the equation \(p(X) = 0\), then the solution set of this equation is exactly
\[\{S^{-1}CS : S \in \mathrm{GL}_{n}[R]\} \cap M_{n}[R],\]
where \(R^{\prime}\) is the field of fractions of \(R\). We call two solutions \(A\) and \(B\) \emph{equivalent} if \(B = S^{-1}AS\) for some \(S \in \mathrm{GL}_{n}[R]\). Two ideals \(\mathfrak{a}\) and \(\mathfrak{b}\) of \(R[\beta]\) are \emph{equivalent} if there exist \(a, b \in R\) such that \(a\mathfrak{a} = b\mathfrak{b}\), where \(\beta\) is a root of \(p(X)\) in the algebraic closure of \(R^{\prime}\). Let \(p(X) = X^{2} -\Gamma X -\Delta \in F_{q}[T][X]\) be irreducible. Let \(\mathrm{deg}(\Gamma) = g\) and \(\mathrm{deg}(\Delta) = d\) and consider the \(2\times 2\) matrices over \(F_{q}[T]\) which satisfy the equation
\[ X^{2} - \Gamma X - \Delta =0. \tag{1}\]
A matrix solution \[ A= \left( \begin{array}{cc} b & -c \\ a & \Gamma -b \end{array} \right) \]
to (1) is said to be \emph{reduced} if \(\mathrm{deg}(b) < \mathrm{deg}(a) < \max \{ \frac{1}{2}d, g\}\), and is said to be \emph{almost reduced} if \(\mathrm{deg}(b) < \mathrm{deg}(a) = \max \{ \frac{1}{2}d, g \}\).
Define the mapping \(\phi\) to map the reduced matrix \[ A= \left( \begin{array}{cc} b & -c \\ a & \Gamma -b \end{array} \right) \] to the matrix \(S^{-1}AS\), where \[S =\left( \begin{array}{cc} x & -1 \\ 1 & 0 \end{array} \right)\]
and \(x\) is the unique nonzero polynomial \(ax^{2} + (\Gamma - 2b)x + c\) has degree less than \(g\) for exactly two distinct values of \(x\).
The main results of the article are as follows:
(a) If \(\mathrm{deg}(\Delta)\) is odd and \(\mathrm{deg}(\Delta) > 2 \ \mathrm{deg}(\Gamma)\), then every matrix solution to (1) is equivalent to a unique reduced matrix;
(b) Two reduced matrix solutions to (1) are equivalent if and only if \(B = \phi^{k}(A)\) for some integer \(k\).
The above results can be extended to higher order matrices and over other types of rings.
Reviewer: Telveenus Antony (Kingston upon Thames)Representatives of similarity classes of matrices over PIDs corresponding to ideal classeshttps://zbmath.org/1537.150332024-07-25T18:28:20.333415Z"Knight, Lucy"https://zbmath.org/authors/?q=ai:knight.lucy"Stasinski, Alexander"https://zbmath.org/authors/?q=ai:stasinski.alexanderThe bijection established in 1933 by \textit{C.G. Latimer} and \textit{C.C. MacDuffee} [Ann. Math. (2) 34, 313--316 (1933; JFM 59.0110.02)] involving integer numbers, can be extended between the similarity classes of matrices in \(M_n(A)\), with irreducible characteristic polynomial \(f(x)\), and the ideal classes of the order \(A[x]/(f(x))\), where \(A\) is a principal ideal domain. In this paper, the authors prove that if \(A[x]/(f(x))\) is maximal, then every similarity class contains a representative close to being a companion matrix.
Reviewer: Carlos M. da Fonseca (Safat)Representations of the Laurent series Rota-Baxter algebras and regular-singular decompositionshttps://zbmath.org/1537.160222024-07-25T18:28:20.333415Z"Lin, Zongzhu"https://zbmath.org/authors/?q=ai:lin.zongzhu"Qiao, Li"https://zbmath.org/authors/?q=ai:qiao.liSummary: There is a Rota-Baxter algebra structure on the field \(A = \mathbf{k}((t))\) with \(P\) being the projection map from \(A = \mathbf{k} [[t]] \oplus t^{- 1} \mathbf{k} [ t^{- 1}]\) onto \(\mathbf{k} [[t]]\). We study the representation theory and regular-singular decompositions of any finite dimensional \(A\)-vector space. The main result shows that the category of finite dimensional representations is semisimple and consists of exactly three isomorphism classes of irreducible representations which are all one-dimensional. As a consequence, the number of \(\mathrm{G L}_A(V)\)-orbits in the set of all regular-singular decompositions of an \(n\)-dimensional \(A\)-vector space \(V\) is \((n + 1)(n + 2) / 2\). We also use the result to compute the generalized class number, i.e., the number of the \(\mathrm{G L}_n(A)\)-isomorphism classes of finitely generated \(\mathbf{k} [[t]]\)-submodules of \(A^n\).McKay matrices for pointed rank one Hopf algebras of nilpotent typehttps://zbmath.org/1537.160372024-07-25T18:28:20.333415Z"Cao, Liufeng"https://zbmath.org/authors/?q=ai:cao.liufeng"Xia, Xuejun"https://zbmath.org/authors/?q=ai:xia.xuejun"Li, Libin"https://zbmath.org/authors/?q=ai:li.libinFor a finite-dimensional Hopf algebra \(H\), the McKay matrix of an \(H\)-module \(V\) encodes the relations for tensoring the simple \(H\)-modules with \(V\).
In the paper under review, the authors investigate the McKay matrix of a finite-dimensional pointed rank one Hopf algebra \(H\) of nilpotent type over a finite group \(G\) for tensoring with the 2-dimensional indecomposable \(H\)-module \(V:= M(2,0)\). It turns out that the characteristic polynomial, eigenvalues and eigenvectors of the McKay matrix are related to the character table of the finite group and a kind of generalized Fibonacci polynomial. Moreover, as an example, the authors explicitly compute the characteristic polynomial and eigenvalues of the McKay matrix and give all eigenvectors of each eigenvalue for the McKay matrix when \(G\) is a dihedral group of order \(4N + 2\).
Reviewer: Vida Milani (North Logan)New fundamental relations in hyperrings and the corresponding quotient structureshttps://zbmath.org/1537.160462024-07-25T18:28:20.333415Z"Ghiasvand, Peyman"https://zbmath.org/authors/?q=ai:ghiasvand.peyman"Mirvakili, Saeed"https://zbmath.org/authors/?q=ai:mirvakili.saeed"Davvaz, Bijan"https://zbmath.org/authors/?q=ai:davvaz.bijanThe fundamental relations are one of the most important and interesting concepts in algebraic hyperstructures establishing that ordinary algebraic structures are derived from algebraic hyperstructures using them. In this paper, the authors introduce and analyse the smallest equivalence binary relation in a hyperring such that the set of all equivalence classes is a commutative ring with identity and of characteristic \(m\). Also, they introduce and analyse a new smallest strongly regular relation on a hyperring \(R\), denoted by \(\sigma^*_p\) such that \(R/\sigma^*_p\) is a \(p\)-ring.
Reviewer: Dariush Heidari (Mahallat)Milnor \(K\)-theory of \(p\)-adic ringshttps://zbmath.org/1537.190032024-07-25T18:28:20.333415Z"Lüders, Morten"https://zbmath.org/authors/?q=ai:luders.morten"Morrow, Matthew"https://zbmath.org/authors/?q=ai:morrow.matthew-t|morrow.matthewAuthors' abstract: We study the mod \(p^r\) Milnor \(K\)-groups of \(p\)-adically complete and \(p\)-henselian rings, establishing in particular a Nesterenko-Suslin-style description in terms of the Milnor range of syntomic cohomology. In the case of smooth schemes over complete discrete valuation rings we prove the mod \(p^r\) Gersten conjecture for Milnor \(K\)-theory locally in the Nisnevich topology. In characteristic \(p\) we show that the Bloch-Kato-Gabber theorem remains true for valuation rings, and for regular formal schemes in a pro sense.
From the introduction:
Our fundamental results from which the others follow are the following ``Gersten injectivity'' and Bloch-Kato isomorphism:
Theorem 0.1. Let \(V\) be a complete discrete valuation ring of mixed characteristic, and \(R\) a local, \(p\)-henselian, ind-smooth \(V\)-algebra with infinite residue field; let \(j\), \(r \geq 1\). Then
(i) the canonical map \(K^M_j (R)/p^r \to K^M_j (R[ 1/p ])/p^r\) is injective,
(ii) and the Galois symbol \(K^M_j (R[ 1/p ])/p^r \to H^j_{\mbox{\'et}}(R[ 1/p ], \mu^{\otimes j}_{p^r} )\) is an isomorphism.
When the residue field is finite but sufficiently large, one replaces Milnor \(K\)-theory with the improved Milnor \(K\)-theory of Gabber and Kerz.
The proofs use ``a convoluted reduction to a special case which we treat by hand.''
Reviewer: Wilberd van der Kallen (Utrecht)Addendum to: ``Real topological Hochschild homology of schemes''.https://zbmath.org/1537.190042024-07-25T18:28:20.333415Z"Hornbostel, Jens"https://zbmath.org/authors/?q=ai:hornbostel.jens"Park, Doosung"https://zbmath.org/authors/?q=ai:park.doosungFrom the text: For commutative rings \(A\) in which 2 is invertible, as well as for \(A=\mathbb Z\), [\textit{J. Hornbostel} and \textit{D. Park}, J. Inst. Math. Jussieu 23, No. 3, 1461--1518 (2024; Zbl 07851602), Proposition 2.3.5] is true. However, for some rings \(A\) in which 2 is not invertible, the result is not correct as stated. The reason is the description of the ideal \(T\) in the proof of loc. cit. is not correct in this case, and this might imply that the map \(\alpha\) in loc. cit. is not an isomorphism.
The only statement in [loc. cit.] where this proposition is used is the following one in the proof of [loc. cit., Proposition 3.2.2], for which we now provide an alternative proof.Geometric construction of Heisenberg-Weil representations for finite unitary groups and Howe correspondenceshttps://zbmath.org/1537.200302024-07-25T18:28:20.333415Z"Imai, Naoki"https://zbmath.org/authors/?q=ai:imai.naoki"Tsushima, Takahiro"https://zbmath.org/authors/?q=ai:tsushima.takahiroSummary: We give a geometric construction of the Heisenberg-Weil representation of a finite unitary group by the middle étale cohomology of an algebraic variety over a finite field, whose rational points give a unitary Heisenberg group. Using also a Frobenius action, we give a geometric realization of the Howe correspondence for \((\mathrm{Sp}_{2n},\mathrm{O}_2^-)\) over any finite field including characteristic two. As an application, we show that unipotency is preserved under the Howe correspondence.A generalization of Szep's conjecture for almost simple groupshttps://zbmath.org/1537.200332024-07-25T18:28:20.333415Z"Gill, Nick"https://zbmath.org/authors/?q=ai:gill.nick"Giudici, Michael"https://zbmath.org/authors/?q=ai:giudici.michael"Spiga, Pablo"https://zbmath.org/authors/?q=ai:spiga.pabloLet \(G\) be a finite group. For every \(x \in G\) let \(C_{G}(x)\) be the centralizer of \(x\) in \(G\) and \(N_{G}(x)\) be the normalizer of the cyclic group \(\langle x \rangle\) in \(G\). It was conjectured by \textit{J. Szep} [Rend. Sem. Mat. Fis. Milano 38, 228--230 (1968; Zbl 0187.29202)] that if \(G= C_{G}(x)C_{G}(y)\) with \(x,y\in G \setminus \{1\}\), then \(G\) is not a non-abelian simple group. Using the classification of the finite simple groups, \textit{E. Fisman} and \textit{Z. Arad} [J. Algebra 108, 340--354 (1987; Zbl 0614.20013)] (this item is not reported in the references) gave a positive answer to Szep's question.
In the paper under review, the authors prove a natural generalization of Szep's conjecture. Let \(G\) be an almost simple group with socle not isomorphic to an orthogonal group having Witt defect zero, we classify all possible group elements \(x,y \in G \setminus \{1 \}\) with \(G=N_{G}(x)N_{G}(y)\). As a consequence of this result, they classify all possible group elements \(x,y \in G \setminus \{1\}\) with \(G=C_{G}(x)C_{G}(y)\).
Reviewer: Enrico Jabara (Venezia)Primitive prime divisors of orders of Suzuki-Ree groupshttps://zbmath.org/1537.200352024-07-25T18:28:20.333415Z"Grechkoseeva, M. A."https://zbmath.org/authors/?q=ai:grechkoseeva.maria-aleksandrovnaA primitive prime divisor of \(q^{m}-1\), where \(q\) and \(m\) are integers larger than 1, is a prime that divides \(q^{m}-1\) and does not divide \(q^{i}-1\) for all \(1 \leq i < m\). From a result by \textit{K. Zsigmondy} [Monatsh. f. Math. 3, 265--284 (1892; JFM 24.0176.02)] primitive prime divisors exist except in the following two cases (a) \(m=2\), \(q=2^{s}-1\), where \(s \geq 2\), (b) \(m=6\), \(q=2\).
Primitive prime divisors play an important role in the theory of finite groups of Lie type. Indeed, for a finite group of Lie type over a field of size \(q\), properties of its elements of prime order \(r\) largely depend not on \(r\) itself but on the smallest integer \(m\) such that \(r\) divides \(q^{m} -1\). In the paper under review, the author focuses on prime divisors of orders of groups \(^{2}B_{2}(2^{m})\) (Suzuki groups), \(^{2}G_{2}(3^{m})\) (Ree groups) and \(^{2}F_{4}(2^{m})\) (Tits group), where \(m\) is an odd prime (and which the author calls groups of Suzuki-Ree type).
The prime graph (or Gruenberg-Kegel graph) of a finite group \(G\) is the graph \(\mathrm{GK}(G)\) whose vertex set is the set \(\pi(G)\) of prime divisors of \(|G|\) and in which two primes \(r\) and \(s\) are adjacent iff \(r\not = s\) and \(G\) contains an element of order \(rs\). Denote by \(t(G)\) the largest size of an independent set of vertices in the graph \(\mathrm{GK}(G)\) and by \(t(2,G)\) the largest size of an independent set containing 2.
The numbers \(t(L)\) and \(t(2,L)\) for all finite simple groups \(L\) were found in [\textit{A. V. Vasil'ev} and \textit{E. P. Vdovin}, Algebra Logika 44, No. 6, 682--725 (2005; Zbl 1104.20018)]. The next natural step is to find these numbers for all almost simple groups, that is, for all \(G\) such that \(L \leq G \leq \Aut(G)\), where \(L\) is a non-abelian simple group. The main result is (Theorem 3): Let \(L\in \{^{2}B_{2}(2^{m}),^{2}G_{2}(3^{m}),^{2}F_{4}(2^{m})\}\) and suppose \(L<G \leq \Aut(G)\). Then one of the following holds:
\begin{itemize}
\item[(a)] \(t(G)=t(L)\) and \(t(2,G)=t(2,L)\);
\item[(b)] \(L=\, ^{2}B_{2}(32)\), \(t(G)=t(L)-1\) and \(t(2,G)=t(2,L)-1\);
\item[(c)] \(L= \,^{2}F_{4}(2^{m})\), \(\pi(G) \not \subseteq \pi(L)\), \(t(G)=t(L)+1\) and \(t(2,G)=t(2,L)\).
\end{itemize}
Reviewer: Egle Bettio (Venezia)Affine Bruhat order and Demazure productshttps://zbmath.org/1537.200922024-07-25T18:28:20.333415Z"Schremmer, Felix"https://zbmath.org/authors/?q=ai:schremmer.felixAuthor's abstract: We give new descriptions of the Bruhat order and Demazure products of affine Weyl groups in terms of the weight function of the quantum Bruhat graph. These results can be understood to describe certain closure relations concerning the Iwahori-Bruhat decomposition of an algebraic group. As an application towards affine Deligne-Lusztig varieties, we present a new formula for generic Newton points.
Reviewer: Egle Bettio (Venezia)Existence and rotatability of the two-colored Jones-Wenzl projectorhttps://zbmath.org/1537.201222024-07-25T18:28:20.333415Z"Hazi, Amit"https://zbmath.org/authors/?q=ai:hazi.amitSummary: ``The two-colored Temperley-Lieb algebra \(2\,\mathrm{TL}_R({}_s n)\) is a generalization of the Temperley-Lieb algebra. The analogous two-colored Jones-Wenzl projector \(\mathrm{JW}_R({}_s n) \in 2\,\mathrm{TL}_R({}_s n)\) plays an important role in the Elias-Williamson construction of the diagrammatic Hecke category. We give conditions for the existence and rotatability of \(\mathrm{JW}_R({}_s n)\) in terms of the invertibility and vanishing of certain two-colored quantum binomial coefficients. As a consequence, we prove that Abe's category of Soergel bimodules is equivalent to the diagrammatic Hecke category in complete generality.''
Ordinary quantum numbers \([n]_q\) depend on a parameter \(q\). We now need two parameters: the `red' parameter \(s\) and the `blue' parameter \(t\).
Theorem A: The two-colored Jones-Wenzl projector \(\mathrm{JW}_R({}_s n)\) exists if and only if \({\binom{n}{k}}_s\) is invertible in \(R\) for each integer \(0\leq k\leq n\).
The two-colored Temperley-Lieb projector \(\mathrm{JW}_R({}_s n)\) is described by a two-colored Temperley-Lieb diagram. Such a diagram may be left or right rotated. (The author uses the terms clockwise/counterclockwise). One calls \(\mathrm{JW}_R({}_s n)\) rotatable if there is a scalar \(\lambda\) so that both the left and right rotation by one string of \(\mathrm{JW}_R({}_s n)\) equals \(\lambda \mathrm{JW}_R({}_t n)\).
Theorem B: The two-colored Temperley-Lieb projectors \(\mathrm{JW}_R({}_s n)\) and \(\mathrm{JW}_R({}_t n)\) exist and are rotatable if and only if \({\binom{n+1}{k}}_s={\binom{n+1}{k}}_t=0\) for each integer \(1\leq k\leq n\).
Reviewer: Wilberd van der Kallen (Utrecht)Generalized Bockstein maps and Massey productshttps://zbmath.org/1537.201282024-07-25T18:28:20.333415Z"Lam, Yeuk Hay Joshua"https://zbmath.org/authors/?q=ai:lam.yeuk-hay-joshua"Liu, Yuan"https://zbmath.org/authors/?q=ai:liu.yuan|liu.yuan.1"Sharifi, Romyar"https://zbmath.org/authors/?q=ai:sharifi.romyar-t"Wake, Preston"https://zbmath.org/authors/?q=ai:wake.preston"Wang, Jiuya"https://zbmath.org/authors/?q=ai:wang.jiuyaLet \(p\) be a prime and let \(G\) be a profinite group of \(p\)-cohomological dimension \(d\). Let \(N\) be a closed normal subgroup of \(G\) such that \(H = G/N\) is finitely generated and pro-\(p\). Let \(I\) be the augmentation ideal of the completed group algebra \(\Omega := \mathbb{Z}_p[[H]]\). Then the Iwasawa cohomology of \(N\) has a filtration by powers of \(I\). The authors relate the graded pieces to the continuous group cohomology of \(G\) via `generalized Bockstein maps' and the values of the latter to Massey products introduced by \textit{W. S. Massey} [in: Sympos. Int. Topologia Algebraica 145--154 (1958; Zbl 0123.16103)]. \par More precisely, it is shown that for a finitely generated \(\mathbb{Z}_p\)-module \(T\) with a continuous action of \(G\), there are canonical isomorphisms
\[
\frac{I^n H^d_{\mathrm{Iw}}(N,T)}{I^{n+1} H^d_{\mathrm{Iw}}(N,T)} \simeq \frac{H^d(G,T) \otimes_{\mathbb{Z}_p} I^n / I^{n+1}}{\mathrm{Im}(\Psi^{(n)})}
\]
for all \(n \geq 1\). Here,
\[
\Psi^{(n)}: H^{d-1}(G,T \otimes_{\mathbb{Z}_p} \Omega/I^n) \rightarrow H^d(G,T) \otimes_{\mathbb{Z}_p} I^n/I^{n+1}
\]
is what the authors call a generalized Bockstein map and is just a connecting homomorphism associated to the short exact sequence
\[
0 \rightarrow T \otimes_{\mathbb{Z}_p} I^n/I^{n+1} \rightarrow T \otimes_{\mathbb{Z}_p} \Omega/ I^{n+1} \rightarrow T \otimes_{\mathbb{Z}_p} \Omega/I^n \rightarrow 0.
\]
(It is assumed that \(\Omega/ I^n\) is \(\mathbb{Z}_p\)-flat.) The proof is based on a spectral sequence argument and generalizes a result of \textit{R. Sharifi} [J. Reine Angew. Math. 603, 1--33 (2007; Zbl 1163.11077)] (where \(d=2\) and \(H \simeq \mathbb{Z}_p\)). \par Then the authors relate the image of \(\Psi^{(n)}\) to Massey products under some hypotheses. This works, for instance, whenever \(H\) is procyclic, pro-bicyclic or elementary abelian, or if \(H\) is a Heisenberg group and \(n<4\). \par If \(H \simeq \mathbb{Z}_p\), the image is generated by Massey products with respect to certain `proper' defining systems. We refrain from giving a precise definition of this notion here. We only recall that a Massey product of \(\chi_1, \dots, \chi_n \in H^1(G, \mathbb{Z}_p)\) is an element of \(H^2(G, \mathbb{Z}_p)\) that measures the obstruction of lifting a homomorphism \(\rho: G \rightarrow U_{n+1}'(\mathbb{Z}_p)\) (the quotient of \(U_{n+1}(\mathbb{Z}_p)\) by its centre) with \(i\)th off-diagonal entry equal to \(\chi_i\) to a homomorphism \(G \rightarrow U_{n+1}(\mathbb{Z}_p)\). Then \(\rho\) is called a defining system for the Massey product. Note that a Massey product does not always exist and, even if it does, it may depend upon the choice of \(\rho\). If \(n=2\), then a Massey product always exists and equals the cup product \(\chi_1 \cup \chi_2\). For more general \(T\), a defining system consists of a pair of homomorphisms
\[
\phi: H \rightarrow U_{a+1}(\mathbb{Z}_p), \, \theta: H \rightarrow U_{b+1}(\mathbb{Z}_p)
\]
with \(a+b=n\) and a \(1\)-cocylce \(f: G \rightarrow T \otimes_{\mathbb{Z}_p}\Omega/ I^n\). Note that each pair \((\phi, \theta)\) gives rise to an \(n\)-tuple of elements of \(H^1(G, \mathbb{Z}_p)\) by precomposinng with \(G \rightarrow H\). \par Finally, the authors give two applications of their results. They prove lower bounds on the \(p\)-ranks of class groups of certain \(p\)-ramified Galois extensions of \(\mathbb{Q}(\mu_p)\) and provide a new proof of the triple Massey vanishing theorem of \textit{I. Efrat} and \textit{E. Matzri} [J. Eur. Math. Soc. 19, No. 12, 3629--3640 (2017; Zbl 1425.12004)] and of \textit{J. Mináč} and \textit{N. D. Tân} [J. Lond. Math. Soc., II. Ser. 94, No. 3, 909--932 (2016; Zbl 1378.12002)]. For further applications in Iwasawa theory, see [\textit{R. Sharifi}, Trans. Am. Math. Soc. 375, No. 8, 5361--5392 (2022; Zbl 1523.11205)].
Reviewer: Andreas Nickel (München)Some further remarks on characterized subgroups generated by modular simple densityhttps://zbmath.org/1537.220072024-07-25T18:28:20.333415Z"Ghosh, Ayan"https://zbmath.org/authors/?q=ai:ghosh.ayanLet \(t_{(2^n)}(\mathbb{T})=\mathbb{Z}(2^\infty)\) be characterized by \((2^n)\) subgroup of \(\mathbb{T}\) (the Prüfer group) and \(t_{(2^n)}^s(\mathbb{T})\) the statistically characterized subgroup of \(\mathbb{T}\). In the notion of \(f^g\)-statistically characterized subgroups \(t_{(2^n)}^{f,g}(\mathbb{T})\) of \(\mathbb{T}\) (see [\textit{K. Bose} et al., Indag. Math., New Ser. 29, No. 5, 1196--1209 (2018; Zbl 1439.28002)]), \(t_{(2^n)}^s(\mathbb{T})\) coincides with the case of \(f(x)=f_1(x)=x\) and \(g(n)=g_1(n)=n\). For \(\alpha\in(0,1]\), by replacing \(g_1\) by \(g_\alpha(n)=n^\alpha\), we define the \(\alpha\)-statistically characterized subgroup \(B_\alpha=t_{(2^n)}^{f_1,g_\alpha}(\mathbb{T})\) of \(\mathbb{T}\). In [Period. Math. Hung. 84, No. 1, 47--55 (2022; Zbl 1499.22001)], \textit{P. Das} and \textit{K. Bose} showed that \(B_\alpha\) are Borel subgroups of cardinality continuum between \(t_{(2^n)}(\mathbb{T})\) and \(t_{(2^n)}^s(\mathbb{T})\) satisfying \(B_\alpha\subsetneq B_\beta\) whenever \(\alpha<\beta\) and \(t_{(2^n)}(\mathbb{T})\subsetneq\cap_{\alpha\in(0,1)}B_\alpha, \cup_{\alpha\in(0,1)}B_\alpha\subsetneq t^s_{(2^n)}(\mathbb{T})\).
In this paper, \((2^n)\) is generalized to an arbitrary arithmetic sequence \((a_n)\) and \(f_1\) to a strictly increasing unbounded modular function \(f:[0,\infty)\to[0,\infty)\). The author shows that for \(\alpha\in(0,1)\), there exists a weight function \(g_\alpha:\mathbb{N}\to[0,\infty)\) such that \(B_\alpha=t^{f,g_\alpha}_{(a_n)}(\mathbb{T})\) are Borel subgroups of cardinality continuum between \(t_{(a_n)}(\mathbb{T})\) and \(t_{(a_n)}^f(\mathbb{T})=t_{(a_n)}^{f,g_1}(\mathbb{T})\) satisfying \(B_\alpha\subsetneq B_\beta\) whenever \(\alpha<\beta\) and \(t_{(a_n)}(\mathbb{T})\subsetneq\bigcap_{\alpha\in(0,1)}B_\alpha\), \(\bigcup_{\alpha\in(0,1)}B_\alpha\subsetneq t^{f}_{(a_n)}(\mathbb{T})\).
Reviewer: Takeshi Kawazoe (Yokohama)On the geometry of a Picard modular grouphttps://zbmath.org/1537.220252024-07-25T18:28:20.333415Z"Deraux, Martin"https://zbmath.org/authors/?q=ai:deraux.martinIn the group \(\mathrm{PU}(2,1)\) of holomorphic isometries of the complex hyperbolic plane \(H_{\mathbb{C}}^2\) (which is a symmetric space with 1/4-pinched sectional curvature) the author considers one discrete subgroup -- Picard modular group \(\Gamma=\mathrm{PU}(2,1,\mathcal O_7)\), where \(\mathcal O_7\) denotes the ring of algebraic integers in \(\mathbb{Q}(i\sqrt 7)\). Computer calculations are used. At the same time, the author notes that some other subgroups of a similar form (based on \(\mathbb{Q}(i\sqrt d)\)) can be studied in a similar way (for other integer values of \(d\) other than d=7 discussed in this article), but this involves a lot of computation. For the indicated subgroup \(\Gamma\) (which is the non-uniform lattice in \(\mathrm{PU}(2,1)\)) its fundamental domain, conjugacy classes of torsion elements, maximal finite subgroups, Fuchsian subgroups that occur as stabilizers of mirrors of complex reflections in \(\Gamma \), are described. A representation of the group \(\Gamma\) with four generators and ten relations is indicated. The author also finds an explicit torsion-free subgroup of index 336 in \(\Gamma\). This subgroup is a principal congruence subgroup (the kernel of the reduction modulo the ideal \((i\sqrt 7\))). It is noted that \(\Gamma\) has exactly two normal subgroups of index 336, but only one of them is torsion-free. There is a computer code to find this subgroup, applicable also for other values of a parameter \(d\).
Reviewer: V. V. Gorbatsevich (Moskva)Diameter of homogeneous spaces: an effective accounthttps://zbmath.org/1537.220282024-07-25T18:28:20.333415Z"Mohammadi, A."https://zbmath.org/authors/?q=ai:mohammadi.amir"Salehi Golsefidy, A."https://zbmath.org/authors/?q=ai:salehi-golsefidy.alireza"Thilmany, F."https://zbmath.org/authors/?q=ai:thilmany.francoisThis paper contains another powerful application of \textit{G. Prasad}'s volume formula [Publ. Math., Inst. Hautes Étud. Sci. 69, 91--117 (1989; Zbl 0695.22005)] and allied results due to \textit{A. Borel} and \textit{G. Prasad} [Publ. Math., Inst. Hautes Étud. Sci. 69, 119--171 (1989; Zbl 0707.11032)]. The authors consider arithmetic quotients \(G/\Gamma\) for semisimple groups \(G\) and arithmetic subgroups \(\Gamma\). The main aim is to give an estimate for the size of a so-called ``small'' lift in \(G\) of a point \(x \in G/\Gamma\); the estimate is polynomially dependent on the injectivity radius at \(x\). The size also depends on the arithmetic datum that defines the arithmetic group. In contrast to relying on effectivizing reduction theory for their purpose, the authors use a uniform spectral gap for arithmetic quotients. Further, they obtain and use effective versions of Levi decompositions to extend their results to also groups that are not semisimple. They formulate and prove their results first in the adelic setting and then deduce the \(S\)-arithmetic case. Apart from the results of Prasad [loc. cit.] and Borel-Prasad [loc. cit.], another deep result used by the authors is a result due to \textit{M. Einsiedler} et al. [J. Am. Math. Soc. 33, No. 1, 223--289 (2020; Zbl 1466.11049)] on (qualitative as well as quantitative) equidistribution theorems for adelic homogeneous subsets and property \(\tau\) for arithmetic groups.
Reviewer: Balasubramanian Sury (Bangalore)Non-admissible irreducible representations of \(p\)-adic \(\mathrm{GL}_n\) in characteristic \(p\)https://zbmath.org/1537.220472024-07-25T18:28:20.333415Z"Ghate, Eknath"https://zbmath.org/authors/?q=ai:ghate.eknath-p"Le, Daniel"https://zbmath.org/authors/?q=ai:le.daniel"Sheth, Mihir"https://zbmath.org/authors/?q=ai:sheth.mihirLet \(p>3\) be prime and let \(F\) be a non-Archimedean local field whose residue field is isomorphic to \(\mathbb{F}_{p^f}\) for some \(f\ge2\). This paper constructs some limiting examples of representations of \(\mathrm{GL}_n(F)\) on vector spaces over \(\mathbb{F}_{p^f}\). Specifically, for every \(n\ge2\) the authors construct absolutely irreducible non-admissible smooth \(\mathbb{F}_{p^f}\)-representations of \(\mathrm{GL}_n(F)\). They also construct irreducible smooth \(\mathbb{F}_{p^f}\)-representations of \(\mathrm{GL}_2(F)\) whose endomorphism algebras contain an algebraically closed field. These constructions extend earlier work by the first and the third author [C. R., Math., Acad. Sci. Paris 358, No. 5, 627--632 (2020; Zbl 1465.11228)] and the second author [Math. Res. Lett. 26, No. 6, 1747--1758 (2019; Zbl 1469.20050)] in which \(F\) is assumed to be a finite unramified extension of \(\mathbb{Q}_p\) of degree \(f\ge2\).
Reviewer: Kevin Keating (Gainesville)Depth-preserving property of the local Langlands correspondence for quasi-split classical groups in large residual characteristichttps://zbmath.org/1537.220482024-07-25T18:28:20.333415Z"Oi, Masao"https://zbmath.org/authors/?q=ai:oi.masaoLet \(F\) be a \(p\)-adic field and \(G\) a quasi-split classical group over \(F\). The local Langlands correspondence for \(G\) gives a disjoint partition of the set of isomorphism classes of irreducible smooth representations of \(G(F)\) into finite subsets, i.e., \(L\)-packets, parametrized by the Langlands parameters of \(G\).
\[
\Pi(G(F)) = \bigsqcup_{\phi \in \Phi(G)} \Pi_{\phi}.
\]
This is due to \textit{J. Arthur} [The endoscopic classification of representations. Orthogonal and symplectic groups. Providence, RI: American Mathematical Society (AMS) (2013; Zbl 1310.22014)] when \(G\) is special orthogonal or symplectic and due to \textit{C. P. Mok} [Endoscopic classification of representations of quasi-split unitary groups. Providence, RI: American Mathematical Society (AMS) (2015; Zbl 1316.22018)] when \(G\) is unitary. The paper under review concerns the relation of depths between a Langlands parameter \(\phi\) and the representations in the corresponding \(L\)-packet \(\Pi_{\phi}\). The depth of \(\phi \in \Phi(G)\) is defined to be
\[
\mathrm{depth}(\phi) = \mathrm{inf}\{ r \in \mathbb{R}_{\geq 0} \mid \phi(w, 1) = 1 \rtimes w \text{ for every } w \in I^{r+}_{F}\},
\]
where \(I^{\bullet}_{F}\) is the upper ramification filtration of the inertial subgroup \(I_{F}\). The depth of \(\pi \in \Pi(G(F))\) is defined to be
\[
\mathrm{depth}(\pi) = \mathrm{inf}\{ r \in \mathbb{R}_{\geq 0} \mid \pi^{G_{x, r+}} \neq 0 \text{ for some } x \in \mathcal{B}(G, F)\},
\]
where \(\mathcal{B}(G, F)\) is the Bruhat-Tits building of \(G\) and \(G_{x, \bullet}\) is the Moy-Prasad filtration of \(G\). The author proves that
\[
\mathrm{max}\{\mathrm{depth}(\pi) \mid \pi \in \Pi_{\phi} \} = \mathrm{depth}(\phi)
\]
when the residual characteristic of \(F\) is large enough (specified in the paper). When \(G\) is unitary, the author further proves the constancy of depth on \(\Pi_{\phi}\). The proof follows the same strategy of [\textit{R. Ganapathy} and \textit{S. Varma}, J. Inst. Math. Jussieu 16, No. 5, 987--1074 (2017; Zbl 1398.11079)]. It uses the twisted character relation by viewing \(G\) as a twisted endoscopic group of certain \(\mathrm{GL}(n)\). It also explores the regions of validity for the local (twisted) character expansions for both \(G\) and \(\mathrm{GL}(n)\), and their relations with depths. Along the way, the author also proves a generalization of the fundamental lemma to positive depth in the case of base change for unitary groups, which has independent interest.
Reviewer: Bin Xu (Beijing)Faber polynomial coefficient inequalities for bi-Bazilevič functions associated with the Fibonacci-number series and the square-root functionshttps://zbmath.org/1537.300142024-07-25T18:28:20.333415Z"Srivastava, H. M."https://zbmath.org/authors/?q=ai:srivastava.hari-mohan"Khan, Shahid"https://zbmath.org/authors/?q=ai:khan.shahid-ali"Malik, Sarfraz Nawaz"https://zbmath.org/authors/?q=ai:malik.sarfraz-nawaz"Tchier, Fairouz"https://zbmath.org/authors/?q=ai:tchier.fairouz"Saliu, Afis"https://zbmath.org/authors/?q=ai:saliu.afis"Xin, Qin"https://zbmath.org/authors/?q=ai:xin.qin(no abstract)On the algebraic differential independence of \(\Gamma\) and \(\zeta\)https://zbmath.org/1537.330022024-07-25T18:28:20.333415Z"Lü, Feng"https://zbmath.org/authors/?q=ai:lu.feng"Jiang, Shuxin"https://zbmath.org/authors/?q=ai:jiang.shuxinThe authors extend earlier results concerning the problem of algebraic-differential independence between the Riemann zeta function and the Euler gamma function. They introduce a general class of functions, which contains in particular the gamma function and some exponential-type functions. In the proofs, a theorem of \textit{S. M. Voronin} [Proc. Steklov Inst. Math. 128, 153--175 (1972; Zbl 0294.10026); translation from Trudy Mat. Inst. Steklov 128, 131--150 (1972); Sov. Math., Dokl. 16, 410 (1975; Zbl 0326.10035); translation from Dokl. Akad. Nauk SSSR 221, 771 (1975)]
related to the distribution of nonzero values of the zeta function is used, combined with some results of complex function theory, as the Cartan-Boutroux lemma (see \textit{H. Cartan} [Ann. Sci. Éc. Norm. Supér. (3) 45, 255--346 (1928; JFM 54.0357.06)]), or the classical Poisson-Jensen integral formula. The statements of exact results are too complicated to be stated here.
Reviewer: József Sándor (Cluj-Napoca)Two general series identities involving modified Bessel functions and a class of arithmetical functionshttps://zbmath.org/1537.330042024-07-25T18:28:20.333415Z"Berndt, Bruce C."https://zbmath.org/authors/?q=ai:berndt.bruce-c.1"Dixit, Atul"https://zbmath.org/authors/?q=ai:dixit.atul"Gupta, Rajat"https://zbmath.org/authors/?q=ai:gupta.rajat"Zaharescu, Alexandru"https://zbmath.org/authors/?q=ai:zaharescu.alexandruThe paper examines two series of the form \[\varphi(s)= \sum_{n=1}^{\infty} \frac{a(n)}{\lambda_n^s }, \qquad \psi(s)= \sum_{n=1}^{\infty} \frac{b(n)}{\mu_n^s}. \]
These series are assumed to satisfy a functional equation of the form
\[\chi(s) = (2\pi)^{-s} \Gamma(s) \varphi(s) = (2\pi)^{s-\delta}\Gamma(\delta -s) \psi(\delta-s).\]
Prior work on functions satisfying this relation dates back to the paper [\textit{K. Chandrasekharan} and \textit{R. Narasimhan}, Ann. Math. (2) 74, 1--23 (1961; Zbl 0107.03702)].
Define the Bessel function for a complex number by \[I_{\nu}(z) = \sum_{n=0} \frac{\left(\frac{z}{2}\right)^{{\nu}+2n}}{n! \Gamma({\nu}+n+1)}.\]
Define \(K_{\nu}(z)\) as follows.
When \(\nu\) is a complex number which is not an integer, \[K_{\nu}(z)= \frac{\pi}{2} \frac{I_{-\nu}(z) - I_{\nu}(z)}{\sin \nu \pi }.\]
When \(n\) is an integer, we have
\[K_{n}(z) = \lim_{\nu \rightarrow n} K_{\nu}(z).\]
The main result of this paper is a theorem which relates \(a(n)\) and \(b(n)\) with a series involving certain integrals along with various arguments of \(K_{\nu}(z).\)
They then specialize to specific choices of \(a(n)\) and \(b(n)\), deriving related theorems for a wide variety of functions. These include:
(1) The number of representations of \(n\) as a sum of \(k\) squares, \(r_k(n)\);
(2) The sum of the \(k\)-th powers of positive divisors of \(n\), \(\sigma_k(n)\);
(3) The Ramanujan tau function, \(\tau(n)\);
(4) A primitive character mod \(q\) for fixed \(q\), \(\chi(n)\);
(5) The number of integral ideals of norm \(n\) in an imaginary quadratic field. This case is closely connected to the behavior of the Dedekind zeta function of the corresponding field.
Some of the results have special cases which are classical.
Reviewer: Joshua Zelinsky (New Haven)Two-variable \(q\)-Laguerre polynomials from the context of quasi-monomialityhttps://zbmath.org/1537.330222024-07-25T18:28:20.333415Z"Cao, Jian"https://zbmath.org/authors/?q=ai:cao.jian"Raza, Nusrat"https://zbmath.org/authors/?q=ai:raza.nusrat"Fadel, Mohammed"https://zbmath.org/authors/?q=ai:fadel.mohammedThe Laguerre polynomials are a class of important orthogonal polynomials with many applications in a wide variety of fields, including quantum group theory, harmonic oscillator theory, and coding theory. Among such applications, is the expression of covariant oscillator algebra.
The interest in two-variable Laguerre polynomials is due to their substantial mathematical significance. They appear as natural solutions of a particular set of partial differential equations, such as a natural solution of the heat diffusion equation, and also appear in the treatment of radiation physics problems such as electromagnetic wave propagation and quantum beam lifetime in a storage ring.
In this work, the authors introduce and study the theory of two-variable \(q\)-Laguerre polynomials by means of a generating function involving \(0\)-th order \(q\)-Bessel Tricomi functions, and they establish two-variable \(q\)-Laguerre polynomials from the context of quasi-monomial. Moreover, they introduce the \(m\)-th order two-variable \(q\)-Laguerre polynomials, deduce the \(q\)-partial differential equations and \(q\)-integro-differential equations for the \(m\)-th order two-variable \(q\)-Laguerre polynomials and examine the quasi-monomiality characteristics of these new \(q\)-polynomials.
The authors conclude this work by providing graphical representations of the \(q\)-Laguerre polynomials.
Reviewer: Roberto S. Costas-Santos (Sevilla)Effective equidistribution for multiplicative Diophantine approximation on lineshttps://zbmath.org/1537.370122024-07-25T18:28:20.333415Z"Chow, Sam"https://zbmath.org/authors/?q=ai:chow.sam"Yang, Lei"https://zbmath.org/authors/?q=ai:yang.lei.3The Littlewood conjecture in Diophantine approximation asks if
\[\liminf_{n\to\infty}n\langle n\alpha\rangle\langle n\beta\rangle=0\]
for all \(\alpha,\beta\in\mathbb{R}\) where \(\langle\cdot\rangle\) denotes the distance to the nearest integer. Partial progress has been made but the full conjecture remains open. A different direction is to ask for improvements in the rate, and here \textit{P. Gallagher} [J. Lond. Math. Soc. 37, 387--390 (1962; Zbl 0124.02902)] showed that \(\liminf_{n\to\infty}n(\log n)^2\langle n\alpha\rangle\langle n\beta\rangle=0\) for Lebesgue almost every \((\alpha,\beta)\in\mathbb{R}^2\). This is sharp in the sense that for any \(\kappa>2\) the set of \((\alpha,\beta)\in\mathbb{R}^2\) for which \(\liminf_{n\to\infty}n(\log n)^{\kappa}\langle n\alpha\rangle\langle n\beta\rangle=0\) is a Lebesgue null set. Here related problems for points on lines are studied. \textit{V. Beresnevich} et al. [Sums of reciprocals of fractional parts and multiplicative Diophantine approximation. Providence, RI: American Mathematical Society (AMS) (2020; Zbl 1445.11002)] considered vertical lines and showed that for any \(\alpha\) almost every \(\beta\) satisfies Gallagher's result. Subsequent results relied on the lines being vertical, and the framework developed here show Gallagher's result for almost every point on arbitrary lines. Once again the exponent \(2\) is sharp. The approach builds on the ergodic approach of \textit{M. Einsiedler} et al. [Ann. Math. (2) 164, No. 2, 513--560 (2006; Zbl 1109.22004)] and involves proving an effective asymptotic equidistribution result for one-parameter unipotent orbits in \(\mathrm{SL}_3(\mathbb{R})/\mathrm{SL}_3(\mathbb{Z})\). Further results develop the structural theory of dual Bohr sets to provide a complementary convergence result, and refine the theory of logarithm laws in homogeneous spaces.
Reviewer: Thomas B. Ward (Durham)An arithmetic Kontsevich-Zorich monodromy of a symmetric origami in genus 4https://zbmath.org/1537.370422024-07-25T18:28:20.333415Z"Gong, Xun"https://zbmath.org/authors/?q=ai:gong.xun"Sanchez, Anthony"https://zbmath.org/authors/?q=ai:sanchez.anthonySummary: We demonstrate the existence of a certain genus four origami whose Kontsevich-Zorich monodromy is arithmetic in the sense of Sarnak. The surface is interesting because its Veech group is as large as possible and given by \(\mathrm{SL}(2, \mathbb{Z})\). When compared to other surfaces with Veech group \(\mathrm{SL}(2, \mathbb{Z})\) such as the Eierlegende Wollmichsau and the Ornithorynque, an arithmetic Kontsevich-Zorich monodromy is surprising and indicates that there is little relationship between the Veech group and monodromy group of origamis. Additionally, we record the index and congruence level in the ambient symplectic group which gives data on what can appear in genus 4.Equidistribution for matings of quadratic maps with the modular grouphttps://zbmath.org/1537.370512024-07-25T18:28:20.333415Z"Matus De La Parra, V."https://zbmath.org/authors/?q=ai:matus-de-la-parra.vSummary: We study the asymptotic behavior of the family of holomorphic correspondences \(\lbrace \mathcal{F}_a\rbrace_{a\in \mathcal{K}}\), given by
\[
\bigg(\frac{az+1}{z+1}\bigg)^2+\bigg(\frac{az+1}{z+1}\bigg)\bigg(\frac{aw-1}{w-1}\bigg)+\bigg(\frac{aw-1}{w-1}\bigg)^2=3.
\]
It was proven by \textit{S. Bullett} and \textit{L. Lomonaco} [Invent. Math. 220, No. 1, 185--210 (2020; Zbl 1437.37054)] that \(\mathcal{F}_a\) is a mating between the modular group \(\operatorname{PSL}_2(\mathbb{Z})\) and a quadratic rational map. We show for every \(a\in \mathcal{K}\), the iterated images and preimages under \(\mathcal{F}_a\) of non-exceptional points equidistribute, in spite of the fact that \(\mathcal{F}_a\) is weakly modular in the sense of \textit{T.-C. Dinh} et al. [Int. J. Math. 31, No. 5, Article ID 2050036, 21 p. (2020; Zbl 1450.37041)], but it is not modular. Furthermore, we prove that periodic points equidistribute as well.Quantum \(K\)-theory and \(q\)-difference equationshttps://zbmath.org/1537.390082024-07-25T18:28:20.333415Z"Ruan, Yong Bin"https://zbmath.org/authors/?q=ai:ruan.yongbin"Wen, Yao Xiong"https://zbmath.org/authors/?q=ai:wen.yaoxiongSummary: This is a set of lecture notes for the first author's lectures on the difference equations in 2019 at the Institute of Advanced Study for Mathematics at Zhejiang University. We focus on explicit computations and examples. The convergence of local solutions is discussed.Perturbed interpolation formulae and applicationshttps://zbmath.org/1537.410012024-07-25T18:28:20.333415Z"Ramos, João P. G."https://zbmath.org/authors/?q=ai:ramos.joao-p-g"Sousa, Mateus"https://zbmath.org/authors/?q=ai:sousa.mateusFrom the Abstract:
``We employ functional analysis techniques in order to deduce some versions of classical and recent interpolation results in Fourier analysis with perturbed nodes''.
For the `classical' they refer to Kadec's \(\frac{1}{4}\)-theorem for interpolation formulae in the Paley-Wiener space both in the real and complex case
([\textit{M. I. Kadets}, Sov. Math., Dokl. 5, 559--561 (1964; Zbl 0196.42602); translation from Dokl. Akad. Nauk SSSR 155, 1253--1254 (1964)])
and for the `recent' results to the papers
[\textit{D. Radchenko} and \textit{M. Viazovska}, Publ. Math., Inst. Hautes Étud. Sci. 129, 51--81 (2019; Zbl 1455.11075)] and
[\textit{H. Cohn} et al., Ann. Math. (2) 185, No. 3, 1017--1033 (2017; Zbl 1370.52037)].
The layout of the paper is as follows:
\begin{itemize}
\item[{\S1. Introduction}] (\(9\frac{1}{2}\) pages)
Contains the results in the form of 6 Theorems.
\item[{\S2. Preliminaries}] (\(3\frac{1}{2}\) pages)
Subsections: Band-limited functions, Modular forms and Functional analysis.
\item[{\S3. Perturbed interpolation formulae for band-limited functions}] (\(9\frac{1}{2}\) pages)
Subsections: Perturbed interpolation formulae for band-limited functions, From Shannon to Vaaler (the proof of Theorem 1.2), Perturbed interplation formulae with derivatives
\item[{\S4. Perturbed Fourier interpolation on the real line}] (\(14\) pages)
Contains the proofs of results (specifically Theorem 1.6).
\item[{\S5. Applications of the main results and techniques}] (\(15\) pages)
This long section contains the subsections Interpolation formulae perturbing the origin (with the proof of theorem 5.3), Uniqueness for small powers of integers, Annihilating pairs (proof of Theorem1.7), The Cohn-Kummer- Miller- Radchenko-Viazovska result and perturbed interpolation formulae with derivatives (proof of Theorem 5.11), Perturbed interpolation for odd functions.
\item[{\S6. Comments and remarks}] (\(6\) pages)
Addresses: Asymmetric perturbations, Maximal perturbed interpolation formulae for band-limited functions, Theorem 1.6 and optimal decay rates for interpolating functions and maximal perturbations.
\item[{References}] (\(41\) items)
\end{itemize}
This paper brings the reader to the edge of research on Fourier analysis with perturbed nodes.
Reviewer: Marcel G. de Bruin (Heemstede)A new type of Szász-Mirakjan operators based on \(q\)-integershttps://zbmath.org/1537.410142024-07-25T18:28:20.333415Z"Sabancigil, Pembe"https://zbmath.org/authors/?q=ai:sabancigil.pembe"Mahmudov, Nazim"https://zbmath.org/authors/?q=ai:mahmudov.nazim-idrisoglu"Dagbasi, Gizem"https://zbmath.org/authors/?q=ai:dagbasi.gizem(no abstract)On the best approximation of non-integer constants by polynomials with integer coefficientshttps://zbmath.org/1537.410202024-07-25T18:28:20.333415Z"Trigub, Roald M."https://zbmath.org/authors/?q=ai:trigub.roald-mIt's well known that function different from a polynomial admits a uniform approximation by polynomials qn with integer coeffcients \((\mathbb{Z}+i\mathbb{Z})\) on a compactum \(K \subset \mathbb{C}\). Let \(\lambda \in \mathbb{C}\) be a non-integer, that is, \(\lambda \not\in \mathbb{Z}+i \mathbb{Z}\). The best approximations of constants in the cases where the compactum \(K\) is a disk in \(\mathbb{C}\) and a ball and a cube in \(\mathbb{R}^d\) were studied in the paper. The following theorems are proved in the article.
Theorem 1. For any \(n \in \mathbb{N}\) we have
\[
E^e_n (\lambda; K_r)=\min_{q_n} \max_{K_r} |\lambda -q_n(z)| \leq (n+1) \rho^n,
\]
where \(K_r =\{z \in \mathbb{C} : |z-z_0|\leq r\}, 0< r < 1/\sqrt{2}, \rho=\max\{r/|z_0|, |z_0|+r \}. \) In general, \(\rho\) cannot be lessened. Moreover, if a function \(f\) is analytic in the closed disk \(K_R(z_0)\) of radius \(R>r\) and with the same center \(z_0\), then there holds the equality
\[
E^e_n (f; K_r)=\min_{q_n} \max_{K_r} |f(z) -q_n(z)|=O\big( \big(r/R)^n +(n+1)\rho^n\big).
\]
Theorem 2. For any \(n \in \mathbb{N}\) we have
\[
E^e_n(\lambda; \Pi_{a,b}) \leq c(d) n^d \rho^n, \qquad \rho= \max\{(\sqrt{b}-\sqrt{a})/(\sqrt{b}+\sqrt{a}), b\}, \quad \Pi_{a,b}=\{x \in \mathbb{R}^d: x\in [a,b], 1\leq j \leq d\},
\]
and \(\rho\) cannot be taken smaller in general.
Theorem 3. Let \(\lambda \in \mathbb{R}\backslash \mathbb{Z}\), \(r\in (0,2)\), \(p\in [1,+\infty)\), and let the degrees of polynomials in all variables \(n\) tend to infinity. We have
\[
E^e_n(\lambda; K_r)_p=\min_{q_n} \Bigg(\int\limits_{K_r} |\lambda-q_n(x)|^p dx \Bigg)^{\frac{1}{p}} \asymp n^{-\frac{d}{p}}, \qquad K_r= \{x=(x_1,\dots,x_d) : |x|< r \},
\]
and
\[
E^e_n(\lambda; \Pi_{-r,r})_p=\min_{q_n} \Bigg(\int\limits_{\Pi_{-r,r}} |\lambda-q_n(x)|^p dx \Bigg)^{\frac{1}{p}} \asymp n^{-\frac{d}{p}}, \qquad \Pi_{-r,r}=\{x=(x_1,\dots,x_d): |x_j|\leq r\},
\]
and for \(r \in (0,1]\) we have
\[
E^e_n(\lambda; \Pi_{0,r})_p \asymp n^{-\frac{2d}{p}}.
\]
For \(r\geq 2\) we have
\[
E^e_n(\lambda; \Pi_{-r,r})_p \geq E^e_n(\lambda; K_{r})_p \geq c(\alpha,r,p)>0.
\]
Reviewer: Mykhaylo Pahirya (Uzhhorod)Some uncertainty principles for the right-sided multivariate continuous quaternion wavelet transformhttps://zbmath.org/1537.420062024-07-25T18:28:20.333415Z"Hleili, Manel"https://zbmath.org/authors/?q=ai:hleili.manelSummary: For the right-sided multivariate continuous quaternion wavelet transform (CQWT), we analyse the concentration of this transform on sets of finite measure. We also establish an analogue of Heisenberg's inequality for the quaternion wavelet transform. Finally, we extend local uncertainty principle for a set of finite measure to CQWT.Inequalities pertaining to quaternion ambiguity functionhttps://zbmath.org/1537.420112024-07-25T18:28:20.333415Z"Sembe, Imanuel Agung"https://zbmath.org/authors/?q=ai:sembe.imanuel-agung"Bahri, Mawardi"https://zbmath.org/authors/?q=ai:bahri.mawardi"Bachtiar, Nasrullah"https://zbmath.org/authors/?q=ai:bachtiar.nasrullah"Zakir, Muhammad"https://zbmath.org/authors/?q=ai:zakir.muhammad-saqlainSummary: The quaternion ambiguity function is an expansion of the standard ambiguity function using quaternion algebra. Various properties such as linearity, translation, modulation, Moyal's formula and inversion identity are studied in detail. In addition, an interesting interaction between the quaternion ambiguity function and the quaternion Fourier transform is demonstrated. Based on these facts, we seek for several versions of the uncertainty inequalities associated with the proposed quaternion ambiguity function.A class of quaternionic Fourier orthonormal baseshttps://zbmath.org/1537.420542024-07-25T18:28:20.333415Z"Li, Yun-Zhang"https://zbmath.org/authors/?q=ai:li.yunzhang|li.yunzhang.1"Zhang, Xiao-Li"https://zbmath.org/authors/?q=ai:zhang.xiaoliSummary: Due to its applications in signal analysis and image processing, the quaternionic Fourier analysis has received increasing attention. In particular, quaternionic Gabor frames (QGFs) attracted some mathematicians' interest. From the literatures, some results on QGFs are based on quaternionic Fourier orthonormal bases. But those used so-called quaternionic Fourier orthonormal bases have a gap that they are all incomplete. In this paper, we present a class of quaternionic Fourier orthonormal bases, and using them derive the corresponding Gabor orthonormal bases.On Salem's integral equation and related criteriahttps://zbmath.org/1537.450012024-07-25T18:28:20.333415Z"Patkowski, Alexander E."https://zbmath.org/authors/?q=ai:patkowski.alexander-ericSummary: We extend Salem's integral equation to the non-homogenous form, and offer the associated criteria for the Riemann Hypothesis. Explicit solutions for the non-homogenous case are given, which in turn give further insight into Salem's criteria for the RH. As a conclusion, we show these results follow from a corollary relating the uniqueness of solutions of the non-homogenous form with Wiener's theorem.Advances in functional analysis and operator theory. AMS-EMS-SMF special session, Université de Grenoble-Alpes, Grenoble, France, July 18--22, 2022https://zbmath.org/1537.460022024-07-25T18:28:20.333415ZPublisher's description: This volume contains the proceedings of the AMS-EMS-SMF Special Session on Advances in Functional Analysis and Operator Theory, held July 18--22, 2022, at the Université de Grenoble-Alpes, Grenoble, France.
The papers reflect the modern interplay between differential equations, functional analysis, operator algebras, and their applications from the dynamics to quantum groups to number theory. Among the topics discussed are the Sturm-Liouville and boundary value problems, axioms of quantum mechanics, \(C^*\)-algebras and symbolic dynamics, von Neumann algebras and low-dimensional topology, quantum permutation groups, the Jordan algebras, and the Kadison-Singer transforms.
The articles of this volume will be reviewed individually.
Indexed articles:
\textit{Kerimov, Nazim; Aliyev, Yagub}, Minimality conditions for Sturm-Liouville problems with a boundary condition depending affinely or quadratically on an eigenparameter, 1-12 [Zbl 07840369]
\textit{Assanova, Anar; Trunk, Carsten; Uteshova, Roza}, On the solvability of boundary value problems for linear differential-algebraic equations with constant coefficients, 13-19 [Zbl 07840370]
\textit{Brix, Kevin Aguyar}, Invertible and noninvertible symbolic dynamics and their \(C^*\)-algebras, 21-52 [Zbl 07840371]
\textit{Derkach, Volodymyr; Trunk, Carsten}, \(\mathcal{PT}\)-symmetric couplings of dual pairs, 53-76 [Zbl 07840372]
\textit{Dragović, Vladimir; Shramchenko, Vasilisa}, Chebyshev dynamics on two and three intervals and isomonodromic deformations, 77-123 [Zbl 07840373]
\textit{Etesi, Gábor}, The universal von Neumann algebra of smooth four-manifolds revisited, 125-151 [Zbl 07840374]
\textit{Freslon, Amaury}, Advances in quantum permutation groups, 153-197 [Zbl 07840375]
\textit{Friedman, Oleg; Katz, Alexander A.}, A note on the quotient of a locally \(JC\)-algebra by a closed Jordan ideal, 199-208 [Zbl 07840376]
\textit{Ivanescu, Cristian; Kucerovsky, Dan}, Villadsen idempotents, 209-219 [Zbl 07840377]
\textit{Kucerovsky, Dan}, A remark on the Kadison-Singer transform, 221-227 [Zbl 07840378]
\textit{Nikolaev, Igor V.}, Shafarevich-Tate groups of abelian varieties, 229-239 [Zbl 07840379]Plane curves giving rise to blocking sets over finite fieldshttps://zbmath.org/1537.510052024-07-25T18:28:20.333415Z"Asgarli, Shamil"https://zbmath.org/authors/?q=ai:asgarli.shamil"Ghioca, Dragos"https://zbmath.org/authors/?q=ai:ghioca.dragos"Yip, Chi Hoi"https://zbmath.org/authors/?q=ai:yip.chi-hoiLet \(\mathbb{P} = \mathrm{PG}(2, q)\) be the Desarguesian projective plane over the finite field \(\mathbb{F}_q\), \(q\) a prime pover. A \textit{blocking set} \(B\) of \(\mathbb{P}\) is a set of points \(B\) of \(\mathbb{P}\) intersecting any \(\mathbb{F}_q\)-line. Namely, a set \(B\) of points of \(\mathbb{P}\) consisting of \(q+1\) points of an \(\mathbb{F}_q\)-line is a (trivial) blocking set. A blocking set \(B\) is \textit{nontrivial} if it does not contain all the \(\mathbb{F}_q\)-points of any \(\mathbb{F}_q\)-line. In the study of blocking sets, Rédei-type polynomials are a very useful tool, and so the (highly reducible) algebraic curves associated with them have applications in this area of finite geometry.
In the nice paper under review, the authors study when blocking sets can arise from rational points on plane curves over finite fields.
They start by considering the following question: When is the set \({\mathcal C}(\mathbb{F}_q)\) of the \(\mathbb{F}_q\)-points on a curve \(\mathcal C\) of \(\mathbb{P}\) not a blocking set (\textit{not blocking curve})? And they show that an irreducible curve of \(\mathbb{P}\) of low degree is not blocking, and for curves of degree \(3\) and \(4\) they prove that if \(q >5\) an irreducible cubic curve is not blocking and if \(q >19\) an irreducible curve of degree \(4\) is not blocking.
Then, they consider the question to give examples of smooth or irreducible blocking plane curves and provide a general construction of irreducible blocking plane curves using Frobenius nonclassical curves.
Also, they consider the question of determining the minimum degree of an irreducible curve passing through a specific blocking set in the case of the projective triangle.
Reviewer: Vito Napolitano (Caserta)Parallelograms and the VC-dimension of the distance setshttps://zbmath.org/1537.520262024-07-25T18:28:20.333415Z"Pham, Thang"https://zbmath.org/authors/?q=ai:pham.thang-van|pham-van-thang.The paper is about the affine plane over the finite field \(\mathbb{F}_q\). The distance of the points \(a=(x_1,y_1)\) and \(b=(x_2,y_2)\) of this plane is defined as \(||a-b||=(x_1-x_2)^2+(y_1-y_2)^2\). Given four sets of points, \(E_1,E_2,E_3,E_4\), of this plane, the paper is concerned with paralelograms and rhombi, whose four vertices fall into the given four sets. Under suitable conditions, it is shown that the number of rhombi that have distance \(t\) between its \(E_1\) and \(E_2\) vertices is close to \(1/q\) times the number of parallelograms that have distance \(t\) between its \(E_1\) and \(E_2\) vertices. Under suitable conditions, it is also shown that the number of parallelograms that have distance \(t\not= 0\) between its \(E_1\) and \(E_2\) vertices, is close to \(1/q\) times the number of parallelograms. An application of these results is the following. Fix a large set \(E\) of points in the plane, which does not contain the origin. Consider the family of finite sets \(\mathcal{F}(E)= \{ \{y\in E: ||x-y||=t\}:x\in E\}\). If \(|E|>>q^{13/7}\), then the VC dimension of \(\mathcal{F}(E)\) is three. This is an improvement on a previous result that required \(|E|>>q^{15/8}\) for the same conclusion. The proof uses Fourier analysis.
Reviewer: László A. Székely (Columbia)Improved Elekes-Szabó type estimates using proximityhttps://zbmath.org/1537.520272024-07-25T18:28:20.333415Z"Solymosi, Jozsef"https://zbmath.org/authors/?q=ai:solymosi.jozsef"Zahl, Joshua"https://zbmath.org/authors/?q=ai:zahl.joshuaSummary: We prove a new Elekes-Szabó type estimate on the size of the intersection of a Cartesian product \(A \times B \times C\) with an algebraic surface \(\{f = 0\}\) over the reals. In particular, if \(A, B, C\) are sets of \(N\) real numbers and \(f\) is a trivariate polynomial, then either \(f\) has a special form that encodes additive group structure (for example, \(f(x, y, x) = x + y - z\)), or \(A \times B \times C \cap \{f = 0\}\) has cardinality \(O(N^{12/7})\). This is an improvement over the previous bound \(O(N^{11/6})\). We also prove an asymmetric version of our main result, which yields an Elekes-Ronyai type expanding polynomial estimate with exponent \(3/2\). This has applications to questions in combinatorial geometry related to the Erdős distinct distances problem.
Like previous approaches to the problem, we rephrase the question as an \(L^2\) estimate, which can be analyzed by counting additive quadruples. The latter problem can be recast as an incidence problem involving points and curves in the plane. The new idea in our proof is that we use the order structure of the reals to restrict attention to a smaller collection of proximate additive quadruples.Increasing sequence of \(T_0\)-topologies on the set of positive integershttps://zbmath.org/1537.540162024-07-25T18:28:20.333415Z"Krasiński, Dawid"https://zbmath.org/authors/?q=ai:krasinski.dawid"Szyszkowska, Paulina"https://zbmath.org/authors/?q=ai:szyszkowska.paulinaLet \(\mathbb{N}\) be the set of positive integers and, for \(a\in\mathbb{N}\), let \(\Theta(a)\) be the set of prime factors of \(a\) and \(l_p(a)\) be the largest number \(k\) such that \(p^k\) divides \(a\). In this paper, the authors define for each \(m\in\mathbb{N}\) a topology \(\mathcal{T}_m\) whose basis is the set of all arithmetic progressions \(\{an+b\mid n\in\mathbb{N}\}\) such that \(\Theta(a)\subseteq\Theta(b)\) and such that \(l_p(a)\leq m\) for all \(p\in\Theta(a)\). These topologies generalize the division topology introduced by \textit{G. B. Rizza} [Riv. Mat. Univ. Parma, V. Ser. 2, 179--185 (1993; Zbl 0834.11006)] and are analogous to a similar construction introduced by the second author of the present paper to generalize the Golomb and the Kirch topology [\textit{P. Szyszkowska}, Topology Appl. 314, Article ID 108096, 8 p. (2022; Zbl 1495.54015)].
The authors study the main properties of the topology \(\mathcal{T}_m\): they show that the topological space \((\mathbb{N},\mathcal{T}_m)\) is always compact, connected and locally connected, and that is has a unique closed point (namely, the number 1). They also show that, for all \(m\), the set of all prime numbers is disconnected but locally connected in the topology \(\mathcal{T}_m\).
Reviewer: Dario Spirito (Padova)Magic squares, the symmetric group and Möbius randomnesshttps://zbmath.org/1537.600122024-07-25T18:28:20.333415Z"Gorodetsky, Ofir"https://zbmath.org/authors/?q=ai:gorodetsky.ofirSummary: Diaconis and Gamburd computed moments of secular coefficients in the CUE (circular unitary ensemble) ensemble. We use the characteristic map to give a new combinatorial proof of their result. We also extend their computation to moments of traces of symmetric powers, where the same result holds but in a wider range. Our combinatorial proof is inspired by gcd matrices, as used by Vaughan and Wooley and by Granville and Soundararajan. We use these CUE computations to suggest a conjecture about moments of characters sums twisted by the Liouville (or by the Möbius) function, and establish a version of it in function fields. The moral of our conjecture (and its verification in function fields) is that the Steinhaus random multiplicative function is a good model for the Liouville (or for the Möbius) function twisted by a random Dirichlet character. We also evaluate moments of secular coefficients and traces of symmetric powers, without any condition on the size of the matrix. As an application we give a new formula for a matrix integral that was considered by Keating, Rodgers, Roditty-Gershon and Rudnick in their study of the \(k\)-fold divisor function.An efficient algorithm for integer lattice reductionhttps://zbmath.org/1537.650392024-07-25T18:28:20.333415Z"Charton, François"https://zbmath.org/authors/?q=ai:charton.francois"Lauter, Kristin"https://zbmath.org/authors/?q=ai:lauter.kristin-e"Li, Cathy"https://zbmath.org/authors/?q=ai:li.cathy"Tygert, Mark"https://zbmath.org/authors/?q=ai:tygert.markSummary: A lattice of integers is the collection of all linear combinations of a set of vectors for which all entries of the vectors are integers and all coefficients in the linear combinations are also integers. Lattice reduction refers to the problem of finding a set of vectors in a given lattice such that the collection of all integer linear combinations of this subset is still the entire original lattice and so that the Euclidean norms of the subset are reduced. The present paper proposes simple, efficient iterations for lattice reduction which are guaranteed to reduce the Euclidean norms of the basis vectors (the vectors in the subset) monotonically during every iteration. Each iteration selects the basis vector for which projecting off (with integer coefficients) the components of the other basis vectors along the selected vector minimizes the Euclidean norms of the reduced basis vectors. Each iteration projects off the components along the selected basis vector and efficiently updates all information required for the next iteration to select its best basis vector and perform the associated projections.Monochromatic arithmetic progressions in binary Thue-Morse-like wordshttps://zbmath.org/1537.681562024-07-25T18:28:20.333415Z"Aedo, Ibai"https://zbmath.org/authors/?q=ai:aedo.ibai"Grimm, Uwe"https://zbmath.org/authors/?q=ai:grimm.uwe"Nagai, Yasushi"https://zbmath.org/authors/?q=ai:nagai.yasushi"Staynova, Petra"https://zbmath.org/authors/?q=ai:staynova.petraSummary: We study the length of monochromatic arithmetic progressions in the Thue-Morse word and in a class of generalised Thue-Morse words. In particular, we give exact values or upper bounds for the lengths of monochromatic arithmetic progressions of given fixed differences inside these words. Some arguments for these are inspired by van der Waerden's proof for the existence of arbitrary long monochromatic arithmetic progressions in any finite colouring of the (positive) integers. We also establish upper bounds for the length of monochromatic arithmetic progressions of certain differences in any fixed point of a primitive binary bijective substitution.Lie complexity of wordshttps://zbmath.org/1537.681582024-07-25T18:28:20.333415Z"Bell, Jason P."https://zbmath.org/authors/?q=ai:bell.jason-p"Shallit, Jeffrey"https://zbmath.org/authors/?q=ai:shallit.jeffrey-oSummary: Given a finite alphabet \(\Sigma\) and a right-infinite word \textbf{w} over \(\Sigma\), we define the Lie complexity function \(L_{\mathbf{w}} : \mathbb{N} \to \mathbb{N}\), whose value at \(n\) is the number of conjugacy classes (under cyclic shift) of length-\(n\) factors \(x\) of \textbf{w} with the property that every element of the conjugacy class appears in \textbf{w}. We show that the Lie complexity function is uniformly bounded for words with linear factor complexity. As a result, we show that words of linear factor complexity have at most finitely many primitive factors \(y\) with the property that \(y^n\) is again a factor for every \(n\). We then look at automatic sequences and show that the Lie complexity function of a \(k\)-automatic sequence is also \(k\)-automatic.Self-similar structure of \(k\)- and biperiodic Fibonacci wordshttps://zbmath.org/1537.681602024-07-25T18:28:20.333415Z"Bortz, Darby"https://zbmath.org/authors/?q=ai:bortz.darby"Cummings, Nicholas"https://zbmath.org/authors/?q=ai:cummings.nicholas"Gao, Suyi"https://zbmath.org/authors/?q=ai:gao.suyi"Jaffe, Elias"https://zbmath.org/authors/?q=ai:jaffe.elias"Mai, Lan"https://zbmath.org/authors/?q=ai:mai.lan"Steinhurst, Benjamin"https://zbmath.org/authors/?q=ai:steinhurst.benjamin"Tillotson, Pauline"https://zbmath.org/authors/?q=ai:tillotson.paulineSummary: Defining the biperiodic Fibonacci words as a class of words over the alphabet \(\{0,1\}\), and two specializations the \(k\)-Fibonacci and classical Fibonacci words, we provide a self-similar decomposition of these words into overlapping words of the same type. These self-similar decompositions complement the previous literature where self-similarity was indicated but the specific structure of how the pieces interact was left undiscussed.Transionospheric autofocus for synthetic aperture radarhttps://zbmath.org/1537.780152024-07-25T18:28:20.333415Z"Gilman, Mikhail"https://zbmath.org/authors/?q=ai:gilman.mikhail"Tsynkov, Semyon V."https://zbmath.org/authors/?q=ai:tsynkov.semyon-vThis paper presents an algorithm to mitigate the distortions caused by ionospheric turbulence on SAR space images. To quote the authors: ``In the current work, we propose a new optimization-based autofocus algorithm that helps correct the turbulence-induced distortions of spaceborne SAR images''. The new algorithm extends the potential of these corrections beyond traditional methods. Mathematically, this SAR processing consists of finding solutions to inverse problems of reconstructing target reflectivity from the radar return signal and adding functions to correct for clutter as the radar passes through the Earth's ionosphere. The authors of this reconstruction proposal call it ``the transionosphere autofocus''. To reduce the complexity of the problem, certain plausible assumptions are made. The analyzed scenario is one where reflectivity depends only on azimuth. The performance of the proposed method is demonstrated through various simulated numerical experiments. Several target scattering point scenarios are analyzed. The case of targets without scattering points will be analyzed in the future. A detailed presentation of interesting future work on this topic concludes the main part of the paper. To emphasize the practical importance of the situations analyzed, three appendices are included. The reference list contains 63 articles. Most are directly related to SAR imaging. More than 55 are from after 2000.
Reviewer: Oscar Bustos (Córdoba)Equations and character sums with matrix powers, Kloosterman sums over small subgroups, and quantum ergodicityhttps://zbmath.org/1537.810282024-07-25T18:28:20.333415Z"Ostafe, Alina"https://zbmath.org/authors/?q=ai:ostafe.alina"Shparlinski, Igor E."https://zbmath.org/authors/?q=ai:shparlinski.igor-e"Voloch, José Felipe"https://zbmath.org/authors/?q=ai:voloch.jose-felipeSummary: We obtain a nontrivial bound on the number of solutions to the equation
\[
\sum\limits_{i=1}^{\nu}A^{x_i}=\sum\limits_{i=\nu+1}^{2\nu}A^{x_i},\qquad 1\leqslant x_i\leqslant\tau,
\]
with a fixed \(n\times n\) matrix \(A\) over a finite field \(\mathbb{F}_q\) of \(q\) elements of multiplicative order \(\tau\). We apply our result to obtain a new bound for additive character sums with a matrix exponential function, nontrivial beyond the square-root threshold. For \(n=2\), this equation has been considered by Kurlberg and Rudnick (for \(\nu=2\)) and Bourgain (for large \(\nu\)) in their study of quantum ergodicity for linear maps over residue rings. We use a new approach to improve their results and also obtain a bound on Kloosterman sums over small subgroups, of size below the square-root threshold.The special case of cyclotomic fields in quantum algorithms for unit groupshttps://zbmath.org/1537.810582024-07-25T18:28:20.333415Z"Barbulescu, Razvan"https://zbmath.org/authors/?q=ai:barbulescu.razvan"Poulalion, Adrien"https://zbmath.org/authors/?q=ai:poulalion.adrienSummary: Unit group computations are a cryptographic primitive for which one has a fast quantum algorithm, but the required number of qubits is \(\tilde{O}(m^5)\). In this work we propose a modification of the algorithm for which the number of qubits is \(\tilde{O}(m^2)\) in the case of cyclotomic fields. Moreover, under a recent conjecture on the size of the class group of \(\mathbb{Q}(\zeta_m+\zeta_m^{-1})\), the quantum algorithm is much simpler because it is a hidden subgroup problem (HSP) algorithm rather than its error estimation counterpart: continuous hidden subgroup problem (CHSP). We also discuss the (minor) speed-up obtained when exploiting Galois automorphisms thanks to the Buchmann-Pohst algorithm over \(\mathcal{O}_K\)-lattices.
For the entire collection see [Zbl 1529.94003].Routing strategy for distributed quantum circuit based on optimized gate transmission directionhttps://zbmath.org/1537.810632024-07-25T18:28:20.333415Z"Chen, Zilu"https://zbmath.org/authors/?q=ai:chen.zilu"Chen, Xinyu"https://zbmath.org/authors/?q=ai:chen.xinyu"Jiang, Yibo"https://zbmath.org/authors/?q=ai:jiang.yibo"Cheng, Xueyun"https://zbmath.org/authors/?q=ai:cheng.xueyun"Guan, Zhijin"https://zbmath.org/authors/?q=ai:guan.zhijinSummary: Due to the limitations of quantum device manufacturing technology, the number of qubits in current quantum computing devices is still relatively limited. Distributed quantum computing, as an emerging quantum computing paradigm, aims to achieve larger-scale quantum computation. To reduce the cost of state transfer and routing in distributed quantum circuits, this paper focuses on the transmission direction of global gates generated after quantum circuit partitioning. We provide a threshold for the number of state transfers required for converting global gates to local gates in the entire circuit. We propose a method to evaluate the quality of transmission directions and utilize a genetic algorithm to find the optimal transmission directions for minimizing the transmission cost of distributed quantum circuit. Based on the optimized transmission directions, this paper maps the qubits in the logical circuit to a distributed quantum architecture model and presents a method for converting global gates to local gates in the quantum circuit. The routing process of distributed quantum circuits is simulated. Experimental results show that compared to existing methods for reducing the number of state transfers, the proposed algorithm can achieve a lower overall cost for executing distributed quantum circuits.Algebraic identities between families of (elliptic) modular graphshttps://zbmath.org/1537.811012024-07-25T18:28:20.333415Z"Basu, Anirban"https://zbmath.org/authors/?q=ai:basu.anirbanSummary: Consider an algebraic identity between elliptic modular graphs where several vertices are at fixed locations (and hence unintegrated) while the others are integrated over the toroidal worldsheet. At any unintegrated vertex, we can glue an arbitrary expression involving elliptic modular graphs which has the same unintegrated vertex. Integrating over that vertex, we obtain new algebraic identities between elliptic modular graphs. Hence this elementary process of convoluting the original ``seed'' identity with other graphs yields infinite number of new identities. We consider various seed identities in which two of the vertices are unintegrated. Convoluting them with families of elliptic modular graphs, we obtain new identities. Each identity is parametrized by an arbitrary number of links in the graphs as well as the positions of unintegrated vertices. On identifying the unintegrated vertices, this leads to an algebraic identity involving modular graphs where all the vertices are integrated over the worldsheet.Improved energy formula for highly excited vibrational states of Kratzer-Fues oscillatorhttps://zbmath.org/1537.811072024-07-25T18:28:20.333415Z"Molski, Marcin"https://zbmath.org/authors/?q=ai:molski.marcinSummary: A mixed supersymmetric-algebraic approach is employed to derive an improved Kratzer-Fues energy formula, which describes the highly excited states of vibrating diatomic systems up to the dissociation limit. The approach proposed has been used to reproduce the coherent anti-Stokes Raman spectra generated by the vibrational transitions of the nitrogen molecule \(^{14}\mathrm{N}_2\) in the ground electronic state \(X^1\Sigma_g^+\) and the energy levels of dioxygen \(^{16}\mathrm{O}_2\) in the ground electronic state \(X^3\Sigma_g^-\). The model includes \(v\)-dependence of the potential depth \(D_0\rightarrow D_v\) as well as interatomic equilibrium separation \(r_0\rightarrow r_v\) and can be used to describe vibrations of diatomic molecules in which nonadiabatic vibrational effects play a significant role. Exact analytical formulae relating the vibrational spectroscopic constants \(\omega_e\), \(\omega_e x_e\) and \(\omega_ey_e\) to the parameters defining the model proposed are derived. They enable calculation of the spectral parameters or determination of the model parameters from the experimental data using the inverse spectroscopic procedure. It has been proven that the improved Kratzer-Fues oscillator has a finite number of vibrational quantum states, which distinguishes it from the original model, endowed with infinite number of states.Quantum statistical mechanics and the boundary of modular curveshttps://zbmath.org/1537.811372024-07-25T18:28:20.333415Z"Marcolli, Matilde"https://zbmath.org/authors/?q=ai:marcolli.matilde"Panangaden, Jane"https://zbmath.org/authors/?q=ai:panangaden.janeSummary: The theory of limiting modular symbols provides a noncommutative geometric model of the boundary of modular curves that includes irrational points in addition to cusps. A noncommutative space associated to this boundary is constructed, as part of a family of noncommutative spaces associated to different continued fractions algorithms, endowed with the structure of a quantum statistical mechanical system. Two special cases of this family of quantum systems can be interpreted as a boundary of the system associated to the Shimura variety of \(\mathrm{GL}_2\) and an analog for \(\mathrm{SL}_2\). The structure of equilibrium states for this family of systems is discussed. In the geometric cases, the ground states evaluated on boundary arithmetic elements are given by pairings of cusp forms and limiting modular symbols.
{\copyright 2024 American Institute of Physics}Mirror symmetry and new approach to constructing orbifolds of Gepner modelshttps://zbmath.org/1537.811642024-07-25T18:28:20.333415Z"Belavin, Alexander"https://zbmath.org/authors/?q=ai:belavin.aleksandr-abramovich"Parkhomenko, Sergey"https://zbmath.org/authors/?q=ai:parkhomenko.sergei-evgenevichSummary: Motivated by the principles of the conformal bootstrap, primarily the principle of Locality, simultaneously with the requirement of space-time supersymmetry, we reconsider constructions of compactified superstring models. Starting from requirements of space-time supersymmetry and mutual locality, we construct a complete set of physical fields of orbifolds of Gepner models. To technically implement this, we use spectral flow generators to construct all physical fields from the chiral primary fields. The set of these spectral flow operators forms a so-called admissible group \(G_{adm}\), which defines a given orbifold. The action of these operators produces a collection of physical fields consistent with the action of supersymmetry generators. The selection of mutually local fields from this collection is carried out using the mirror group \(G_{adm}^\ast\). The permutation of \(G_{adm}\) and \(G_{adm}^\ast\) replaces the original orbifold with a mirror one that satisfies the same conditions as the original one. This also implies that the resulting model is modular invariant.A game-theoretic implication of the Riemann hypothesishttps://zbmath.org/1537.910102024-07-25T18:28:20.333415Z"Ewerhart, Christian"https://zbmath.org/authors/?q=ai:ewerhart.christianSummary: The Riemann hypothesis (RH) is one of the major unsolved problems in pure mathematics. In the present paper, a parameterized family of non-cooperative games is constructed with the property that, if RH is true, then any game in the family admits a unique Nash equilibrium. We argue that this result is not degenerate. Indeed, neither is the conclusion a tautology, nor is RH used to define the family of games.Arithmetic crosscorrelation of binary \(\mathfrak{m}\)-sequences with coprime periodshttps://zbmath.org/1537.940392024-07-25T18:28:20.333415Z"Jing, Xiaoyan"https://zbmath.org/authors/?q=ai:jing.xiaoyan"Feng, Keqin"https://zbmath.org/authors/?q=ai:feng.keqinSummary: The arithmetic crosscorrelation of binary \(\mathfrak{m}\)-sequences with coprime periods \(2^{n_1} -1\) and \(2^{n_2} -1\) \((\gcd (n_1, n_2) = 1)\) is determined. The result shows that the absolute value of arithmetic crosscorrelation of such binary \(\mathfrak{m}\)-sequences is not greater than \(2^{\min (n_1, n_2)}-1\).Frobenius endomorphisms of binary Hessian curveshttps://zbmath.org/1537.940412024-07-25T18:28:20.333415Z"Sohn, Gyoyong"https://zbmath.org/authors/?q=ai:sohn.gyoyong-ySummary: This paper introduces the Frobenius endomophisms on the binary Hessian curves. It provides an efficient and computable homomorphism for computing point multiplication on binary Hessian curves. As an application, it is possible to construct the GLV method combined with the Frobenius endomorphism to accelerate scalar multiplication over the curve.Classical and quantum 3 and 4-sieves to solve SVP with low memoryhttps://zbmath.org/1537.940482024-07-25T18:28:20.333415Z"Chailloux, André"https://zbmath.org/authors/?q=ai:chailloux.andre"Loyer, Johanna"https://zbmath.org/authors/?q=ai:loyer.johannaSummary: The Shortest Vector Problem (SVP) is at the foundation of lattice-based cryptography. The fastest known method to solve SVP in dimension \(d\) is by lattice sieving, which runs in time \(2^{td+o(d)}\) with \(2^{md+o(d)}\) memory for constants \(t,m \in \varTheta (1)\). Searching reduced vectors in the sieve is a problem reduced to the configuration problem, i.e. searching \(k\) vectors satisfying given constraints on their pairwise scalar products.
In this work, we present a framework for \(k\)-sieve algorithms: we filter the input list of lattice vectors using a code structure modified from [\textit{A. Becker} et al., in: Proceedings of the 27th annual ACM-SIAM symposium on discrete algorithms, SODA 2016, Arlington, VA, USA, January 10--12, 2016. Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM); New York, NY: Association for Computing Machinery (ACM). 10--24 (2016; Zbl 1410.68093)] to get lists centred around \(k\) codewords summing to the null-vector. Then, we solve a simpler instance of the configuration problem in the \(k\) filtered lists. Based on this framework, we describe classical sieves for \(k=3\) and 4 that introduce new time-memory trade-offs. We also use the \(k\)-Lists algorithm [\textit{E. Kirshanova} et al., Lect. Notes Comput. Sci. 11921, 521--551 (2019; Zbl 1456.94093)] inside our framework, and this improves the time for \(k=3\) and gives new trade-offs for \(k=4\).
For the entire collection see [Zbl 1535.94002].Fast enumeration algorithm for multivariate polynomials over general finite fieldshttps://zbmath.org/1537.940542024-07-25T18:28:20.333415Z"Furue, Hiroki"https://zbmath.org/authors/?q=ai:furue.hiroki"Takagi, Tsuyoshi"https://zbmath.org/authors/?q=ai:takagi.tsuyoshiSummary: The enumeration of all outputs of a given multivariate polynomial is a fundamental mathematical problem and is incorporated in some algebraic attacks on multivariate public key cryptosystems. For a degree-\(d\) polynomial in \(n\) variables over the finite field with \(q\) elements, solving the enumeration problem classically requires \(O\left( \binom{n+d}{d} \cdot q^n\right)\) operations. \textit{C. Bouillaguet} et al. [Lect. Notes Comput. Sci. 6225, 203--218 (2010; Zbl 1297.94055)] proposed a fast enumeration algorithm over the binary field \(\mathbb{F}_2\). Their proposed algorithm covers all the inputs of a given polynomial following the order of Gray codes and is completed by \(O(d\cdot 2^n)\) bit operations. However, to the best of our knowledge, a result achieving the equivalent efficiency in general finite fields is yet to be proposed.
In this study, we propose a novel algorithm that enumerates all the outputs of a degree-\(d\) polynomial in \(n\) variables over \(\mathbb{F}_q\) with a prime number \(q\) by \(O(d\cdot q^n)\) operations. The proposed algorithm is constructed by using a lexicographic order instead of Gray codes to cover all the inputs. This result can be seen as an extension of the result of Bouillaguet et al. [loc. cit.] to general finite fields and is almost optimal in terms of time complexity. We can naturally apply the proposed algorithm to the case where \(q\) is a prime power. Notably, our enumeration algorithm differs from the algorithm by Bouillaguet et al. [loc. cit.] even in the case of \(q=2\).
For the entire collection see [Zbl 1535.94002].NTRU in quaternion algebras of bounded discriminanthttps://zbmath.org/1537.940602024-07-25T18:28:20.333415Z"Ling, Cong"https://zbmath.org/authors/?q=ai:ling.cong"Mendelsohn, Andrew"https://zbmath.org/authors/?q=ai:mendelsohn.andrewSummary: The NTRU assumption provides one of the most prominent problems on which to base post-quantum cryptography. Because of the efficiency and security of NTRU-style schemes, structured variants have been proposed, using modules. In this work, we create a structured form of NTRU using lattices obtained from orders in cyclic division algebras of index 2, that is, from quaternion algebras. We present a public-key encryption scheme, and show that its public keys are statistically close to uniform. We then prove IND-CPA security of a variant of our scheme when the discriminant of the quaternion algebra is not too large, assuming the hardness of Learning with Errors in cyclic division algebras.
For the entire collection see [Zbl 1535.94002].Five infinite families of binary cyclic codes and their related codes with good parametershttps://zbmath.org/1537.940942024-07-25T18:28:20.333415Z"Liu, Hai"https://zbmath.org/authors/?q=ai:liu.hai"Li, Chengju"https://zbmath.org/authors/?q=ai:li.chengju"Ding, Cunsheng"https://zbmath.org/authors/?q=ai:ding.cunshengCyclic codes are interesting and important families of codes as mathematically they are closely related to quite a number of areas of mathematics such as algebra, algebraic number theory, number theory, combinatorics and finite geometry. Cyclic codes are also important in practice due to their efficient encoding and decoding algorithms. However, it is theoretically hard to design cyclic codes of length \(n\) with good parameters if \(n\) has a small divisor \(n_0 > 1\) due to some general theory developed in [\textit{M. Xiong}, IEEE Trans. Inf. Theory 64, No. 9, 6305--6314 (2018; Zbl 1401.94231); \textit{M. Xiong} and \textit{A. Zhang}, IEEE Trans. Inf. Theory 67, No. 8, 5097--5103 (2021; Zbl 1486.94179)]. This fact is also confirmed by the tables of best binary cyclic codes in Appendix A.2 of [\textit{C. Ding}, Codes from difference sets. Hackensack, NJ: World Scientific (2014; Zbl 1367.94005)]. It is harder to design binary cyclic codes with good parameters as the alphabet size is too small. It is a much more difficult problem to design an infinite family of binary cyclic codes such that each code in the family has good parameters. It is an interesting problem to design an infinite family of binary cyclic codes with good parameters such that their duals also have good parameters. For convenience, we call such an infinite family of binary cyclic codes a dually-good infinite family of binary cyclic codes. Dually-good infinite families of binary cyclic codes with small or large dimensions relative to their lengths are relatively easy to construct. The binary Hamming codes and the punctured binary second-order Reed-Muller codes are two dually-good infinite families of binary cyclic codes with a large and small dimension, respectively. However, only a small number of dually-good infinite families of binary cyclic codes with parameters \([n, k]\) and \((n - 6)/3 \leq k \leq 2(n + 6)/3\) are known in the literature. The following is a list of such binary cyclic codes known to the authors.
\begin{itemize}
\item[1.] The family of binary quadratic-residue codes.
\item[2.] The punctured binary Reed-Muller codes of length \(2m-1\) and order \((m-1)/2,\) where \(m\) is odd.
\item[3.] Two families of cyclic codes presented in [\textit{C. Tang} and \textit{C. Ding}, IEEE Trans. Inf. Theory 68, No. 12, 7842--7849 (2022; Zbl 1534.94124)].
\end{itemize}
It is in general very hard to determine the minimum distance of a cyclic code with parameters \([n, k]\) and \((n - 6)/3 \leq k \leq 2(n + 6)/3\) as the dimension \(k\) is neither small nor large compared with a large length \(n\) [\textit{P. Charpin}, in: Handbook of coding theory. Vol. 1. Part 1: Algebraic coding. Vol. 2. Part 2: Connections, Part 3: Applications. Amsterdam: Elsevier. 963--1063 (1998; Zbl 0927.94017); \textit{S. Li}, SIAM J. Discrete Math. 31, No. 4, 2530--2569 (2017; Zbl 1420.94112); \textit{S. Noguchi} et al., SIAM J. Discrete Math. 35, No. 1, 179--193 (2021; Zbl 1478.94139)]. If it is impossible to determine the minimum distance of such a code, the best one can do is to develop a good lower bound on the minimum distance of the code. This is the only way to show that such a code has a good error-correcting capability. However, this is not easy either. It is more difficult to develop good lower bounds on the minimum distances of both \(C\) and \(C^\perp\) [\textit{B. Gong} et al., IEEE Trans. Inf. Theory 68, No. 2, 953--964 (2022; Zbl 1489.94160); \textit{X. Shi} et al., Des. Codes Cryptography 87, No. 9, 2165--2183 (2019; Zbl 1419.94079)]. This explains why it is very difficult to find a dually-good infinite family of binary cyclic codes with parameters \([n, k]\) and \((n - 6)/3 \leq k \leq 2(n + 6)/3.\) Inspired by the works in [\textit{C. Tang} and \textit{C. Ding}, IEEE Trans. Inf. Theory 68, No. 12, 7842--7849 (2022; Zbl 1534.94124)], in the paper under review, the authors construct and analyse five dually-good infinite families of binary cyclic codes with parameters \([n, k]\) and \((n - 6)/3 \leq k \leq 2(n +6)/3.\) Three of the five infinite families of binary cyclic codes and their duals have a very good lower bound on their minimum distances and contain distance-optimal codes. The other two families of binary cyclic codes are composed of binary duadic codes with a square-root-like lower bound on their minimum distances. As a by-product, two families of self-dual binary codes with a square-root-like lower bound on their minimum distances are obtained.
Reviewer: Djoko Suprijanto (Bandung)Perfect codes in two-dimensional algebraic latticeshttps://zbmath.org/1537.940952024-07-25T18:28:20.333415Z"Strapasson, J. E."https://zbmath.org/authors/?q=ai:strapasson.joao-eloir"Strey, G."https://zbmath.org/authors/?q=ai:strey.giselleSummary: In the present paper, we investigate the existence of lattice perfect codes in two families of two-dimensional lattices built from the Minkowski embedding of the ring of integers of a quadratic field \(\mathbb{K} = \mathbb{Q} [\sqrt{ l}]\), where \(l\) is a square-free integer. We analyzed the forms and quantities of perfect codes for two of these families lattices, one Orthogonal and the other Non-Orthogonal and we verified that we always managed to find a perfect code in these two families of lattices. The greater the value of \(l\) the more perfect codes we get.On BCH split metacyclic codeshttps://zbmath.org/1537.940972024-07-25T18:28:20.333415Z"Behajaina, Angelot"https://zbmath.org/authors/?q=ai:behajaina.angelotThe author investigates metacyclic codes, which means group codes over a semidirect product of cyclic groups. After recalling some basics in Section 2, the author gives some code constructions as submodules in Lemma 3.1. In Theorem 3.5 the author gives a generator polynomial for a principal code which is generated as a submodule. Further, some properties of these principal codes are given. In Section 4, the author investigates dual codes as submodules and gives their generator polynomials (Proposition 2 and Corollary 3). In the last section the author establishes an analogue of BCH bound for the introduced principal codes. It turns out (Theorem 5.1) that the minimum distance of these codes can be determined by the generator polynomial.
Reviewer: Carolin Hannusch (Debrecen)Weight enumerators of all cubic-primitive irreducible cyclic codes of odd prime power lengthhttps://zbmath.org/1537.940982024-07-25T18:28:20.333415Z"Bishnoi, Monika"https://zbmath.org/authors/?q=ai:bishnoi.monika"Kumar, Pankaj"https://zbmath.org/authors/?q=ai:kumar.pankaj.2In the present paper, the authors compute the weight enumerator of any cubic primitive irreducible cyclic code \(C\) of length \(p^u\) over \(\mathbb{F}_q\), where \(p\) and \(q\) are odd primes, and \(q\) is a cubic primitive modulo \(p^u\). Furthermore, they give simple proofs of Theorems \(4\), \(5\), \(6\), \(7\), and \(8\) of the paper [\textit{M. Bishnoi} and \textit{P. Kumar}, Cryptogr. Commun. 15, No. 4, 795--809 (2023; Zbl 07720659)]. They also find solutions of some Diophantine equations to obtain \(k\)-weight codes for \(k = 1\) or \(3\). As a consequence, they obtain all \(1\)-weight cubic primitive irreducible cyclic codes of length \(\ell\), where \(\ell\) is an odd positive integer and they characterize some \(3\)-weight cubic primitive irreducible cyclic codes. Finally, the authors provide bounds on the minimum distances of all cubic primitive irreducible cyclic codes of length \(\ell\) over \(\mathbb{F}_q\).
Reviewer: Sami Omar (Sukhair)