Recent zbMATH articles in MSC 11https://zbmath.org/atom/cc/112023-09-22T14:21:46.120933ZWerkzeugMysteries of sequences in ``Observations cyclometricae'' by Adam Adamandy Kochańskihttps://zbmath.org/1517.010172023-09-22T14:21:46.120933Z"Fukś, Henryk"https://zbmath.org/authors/?q=ai:fuks.henrykFor the entire collection see [Zbl 1258.01002].Number theory in the works of Franciszek Mertenshttps://zbmath.org/1517.010252023-09-22T14:21:46.120933Z"Schinzel, Andrzej"https://zbmath.org/authors/?q=ai:schinzel.andrzejFor the entire collection see [Zbl 1258.01002].Wacław Sierpiński's contribution to number theoryhttps://zbmath.org/1517.010262023-09-22T14:21:46.120933Z"Schinzel, Andrzej"https://zbmath.org/authors/?q=ai:schinzel.andrzejFor the entire collection see [Zbl 1258.01002].Number theory and algebra in the works of Salomon Lubelskihttps://zbmath.org/1517.010272023-09-22T14:21:46.120933Z"Schinzel, Andrzej"https://zbmath.org/authors/?q=ai:schinzel.andrzejFor the entire collection see [Zbl 1258.01002].A short note on polynomials \(f(X)=X+A X^{1 + q^2 (q - 1) / 4}+B X^{1 + 3 q^2 (q - 1) / 4} \in \mathbb{F}_{q^2}[X]\), \(q\) evenhttps://zbmath.org/1517.050052023-09-22T14:21:46.120933Z"Bartoli, Daniele"https://zbmath.org/authors/?q=ai:bartoli.daniele"Bonini, Matteo"https://zbmath.org/authors/?q=ai:bonini.matteoSummary: An alternative proof of the necessary conditions on \(A,B \in \mathbb{F}_{q^2}^\ast\) for \(f(X)=X+A X^{1 + q^2 (q - 1) / 4}+B X^{1 + 3 q^2 (q - 1) / 4}\) to be a permutation polynomial in \(\mathbb{F}_{q^2}\), \(q\) even, is given. This proof involves standard arguments from algebraic geometry over finite fields and fast symbolic computations.\((M,i)\)-multiset Eulerian polynomialshttps://zbmath.org/1517.050112023-09-22T14:21:46.120933Z"Ma, Jun"https://zbmath.org/authors/?q=ai:ma.jun.2|ma.jun"Pan, Kaiying"https://zbmath.org/authors/?q=ai:pan.kaiyingSummary: Let \(M = \{ 1^{p_1}, \ldots, n^{p_n} \}\) be a multiset, \( \mathfrak{S}_{M, i}\) be the set of multipermutations over \(M\) with \(i\) as the first entry and \(A_{M, i}(x)\) be the enumerators of descents over \(\mathfrak{S}_{M,i}\). \(A_{M,i}(x)\) is called the \((M, i)\)-multiset Eulerian polynomial. A Carlitz-type identity of \(A_{M,i}(x)\) is derived. It is proved that \(x A_{M,i}(x)\), \(c_1 A_{M,i}(x)+c_2 A_{M,j}(x)\) and \(c_1 x A_{M,i}(x) + c_2 A_{M,j}(x)\) have only real roots, where \(c_1\) and \(c_2\) are nonnegative real number, \(i, j \in M\) and \(i < j\). For the multiset \(M = \{ 1^k, 2^k, \ldots, n^k \} \), it is shown that \(A_{M,i}(x)\) is reciprocal with \(A_{M, n - i + 1}(x)\), \(A_{M,i}(x) + A_{M,n-i+1}(x)\) and \(x A_{M,i}(x) + A_{M,n - i + 1}(x)\) are \(\gamma \)-positive, and \(A_{M, i}(x)\) is bi-\( \gamma \)-positive for any \(1 \leq i \leq \frac{ n + 1}{ 2} \). For \(M = \{1, 2, \ldots, n \}\) and \(1 \leq i \leq n\), we give a combinatorial interpretation for \(\gamma \)-coefficients of \(A_{M,i}(x) + A_{M,n - i + 1}(x)\).Cycles of even-odd drop permutations and continued fractions of Genocchi numbershttps://zbmath.org/1517.050122023-09-22T14:21:46.120933Z"Pan, Qiongqiong"https://zbmath.org/authors/?q=ai:pan.qiongqiong"Zeng, Jiang"https://zbmath.org/authors/?q=ai:zeng.jiangSummary: Recently \textit{A. Lazar} and \textit{M. L. Wachs} [Comb. Theory 2, No. 1, Paper No. 2, 34 p. (2022; Zbl 1502.52019)] proved two new permutation models, called D-permutations and E-permutations, for Genocchi and median Genocchi numbers. In a follow-up, \textit{S.-P. Eu} et al. [Electron. J. Comb. 29, No. 2, Research Paper P2.15, 23 p. (2022; Zbl 1487.05011)] studied the even-odd descent permutations, which are in bijection with E-permutations. We generalize Eu et al.'s descent polynomials with eight statistics and obtain an explicit J-fraction formula for their ordinary generaing function. The J-fraction permits us to confirm two conjectures of Lazar-Wachs [loc. cit.] about cycles of D and E permutations and obtain a \((p,q)\)-analogue of Eu et al.'s gamma-formula [loc. cit.]. Moreover, the \((p,q)\) gamma-coefficients have the same factorization flavor as the gamma-coefficients of \textit{P. Brändén}'s \((p,q)\)-Eulerian polynomials [in: Handbook of enumerative combinatorics. Boca Raton, FL: CRC Press. 437--483 (2015; Zbl 1327.05051)].A unified approach to generalized Pascal-like matrices: \(q\)-analysishttps://zbmath.org/1517.050132023-09-22T14:21:46.120933Z"Akkus, Ilker"https://zbmath.org/authors/?q=ai:akkus.ilker"Kizilaslan, Gonca"https://zbmath.org/authors/?q=ai:kizilaslan.gonca"Verde-Star, Luis"https://zbmath.org/authors/?q=ai:verde-star.luisSummary: In this paper, we present a general method to construct \(q\)-analogues and other generalizations of Pascal-like matrices. Our matrices are obtained as functions of strictly lower triangular matrices and include several types of generalized Pascal-like matrices and matrices related with modified Hermite polynomials of two variables and other polynomial sequences. We find explicit expressions for products, powers, and inverses of the matrices and also some factorization formulas using this method.Linked partition ideals and Euclidean billiard partitionshttps://zbmath.org/1517.050172023-09-22T14:21:46.120933Z"Chern, Shane"https://zbmath.org/authors/?q=ai:chern.shaneThis paper considers recently introduced Euclidean billiard partitions, that arose in the study of integrable billiards in Euclidean spaces of arbitrary dimensions and represent the winding numbers for each elliptic coordinate corresponding to periodic trajectories. Such partitions have distinct parts such that the adjacent ones are never both odd and the smallest part is even. This work uses linked partition ideals to establish several relevant trivariate generating function identities, which then enable a new way of obtaining the generating functions for the billiard partitions.
Reviewer: Milena Radnović (Sydney)On a generalized basic series and Rogers-Ramanujan type identities. IIhttps://zbmath.org/1517.050222023-09-22T14:21:46.120933Z"Sonik, P."https://zbmath.org/authors/?q=ai:sonik.p"Goyal, Megha"https://zbmath.org/authors/?q=ai:goyal.meghaSummary: This paper is in continuation with our recent paper ``On a generalized basic series and Rogers-Ramanujan type identities'' [the first author et al., Contrib. Discrete Math. 18, No. 1, 15--28 (2023; Zbl 1517.05023)]. Here, we consider two generalized basic series and interpret these basic series as the generating function of some restricted \((n + t)\)-color partitions and restricted weighted lattice paths. The basic series discussed in the aforementioned paper, is now a mere particular case of one of the generalized basic series that are discussed in this paper. Besides, eight particular cases are also discussed which give combinatorial interpretations of eight Rogers-Ramanujan type identities which are combinatorially unexplored till date.On a generalized basic series and Rogers-Ramanujan type identitieshttps://zbmath.org/1517.050232023-09-22T14:21:46.120933Z"Sonik, P."https://zbmath.org/authors/?q=ai:sonik.p"Ranganatha, D."https://zbmath.org/authors/?q=ai:ranganatha.d"Goyal, Megha"https://zbmath.org/authors/?q=ai:goyal.meghaSummary: In this paper, we give the generalization of MacMahon's type combinatorial identities. A generalized \(q\)-series is interpreted as the generating function of two different combinatorial objects, viz., restricted \(n\)-color partitions and weighted lattice paths which give entirely new Rogers-Ramanujan-MacMahon type combinatorial identities. This result yields an infinite class of 2-way combinatorial identities which further extends the work of Agarwal and Goyal. We also discuss the bijective proof of the main result. Forbye, eight particular cases are also discussed which give a combinatorial interpretation of eight entirely new Rogers-Ramanujan type identities.Beck-type companion identities for Franklin's identityhttps://zbmath.org/1517.050252023-09-22T14:21:46.120933Z"Ballantine, Cristina"https://zbmath.org/authors/?q=ai:ballantine.cristina-m"Welch, Amanda"https://zbmath.org/authors/?q=ai:welch.amandaSummary: In 2017, Beck conjectured that the difference in the number of parts in all partitions of \(n\) into odd parts and the number of parts in all strict partitions of \(n\) is equal to the number of partitions of \(n\) whose set of even parts has one element, and also to the number of partitions of \(n\) with exactly one part repeated. Andrews proved the conjecture using generating functions. Beck's identity is a companion identity to Euler's identity. The theorem has been generalized (with a combinatorial proof) by Yang to a companion identity to Glaisher's identity. Franklin generalized Glaisher's identity, and in this article, we provide a Beck-type companion identity to Franklin's identity and prove it both analytically and combinatorially. Andrews' and Yang's respective theorems fit naturally into this very general frame. We also discuss how Franklin's identity and the companion Beck-type identities can be further generalized to Euler pairs of any order.Several properties of differential equation with \((p, q)\)-Genocchi polynomialshttps://zbmath.org/1517.050262023-09-22T14:21:46.120933Z"Kang, J. Y."https://zbmath.org/authors/?q=ai:kang.jiayi|kang.jianying|kang.jung-yoog|kang.ji-yoon|kang.jiayu|kang.jingyu|kang.jung-yup|kang.jiayin|kang.jiyangSummary: We construct several differential equations of which are related to \((p, q)\)-Genocchi polynomials in this paper. From these differential equation, we also investigate some relations which are related to Genocchi, \(q\)-Genocchi, and \((p, q)\)-Genocchi polynomials.Representing the Stirling polynomials \(\sigma_n(x)\) in dependence of \(n\) and an application to polynomial zero identitieshttps://zbmath.org/1517.050272023-09-22T14:21:46.120933Z"Kovačec, Alexander"https://zbmath.org/authors/?q=ai:kovacec.alexander"de Tovar Sá, Pedro Barata"https://zbmath.org/authors/?q=ai:de-tovar-sa.pedro-barataSummary: Denote by \(\sigma_n\) the \(n\)-th Stirling polynomial in the sense of the well-known book [Concrete mathematics. A foundation for computer science. Reading, MA: Addison-Wesley Publishing Company (1989; Zbl 0668.00003)] by \textit{R. L. Graham} et al.. We show that there exist developments \(x{\sigma}_n(x)={\sum}_{j=0}^n{({2}^j j!)}^{-1}{q}_{n-j}(j){x}^j\) with polynomials \(q_j\) of degree j. We deduce from this the polynomial identities
\[
\sum_{a+b+c+d=n}{(-1)}^d\frac{{(x-2a-2b)}^{3n-s-a-c}}{a!b!c!d!(3n-s-a-c)!}=0, \quad \text{for } s\in{{\mathbb{Z}}}_{\ge 1},
\]
found in an attempt to find a simpler formula for the density function in a five-dimensional random flight problem. We point out a probable connection to Riordan arrays.Bivariate extension of the \(r\)-Dowling polynomials and two forms of generalized Spivey's formulahttps://zbmath.org/1517.050282023-09-22T14:21:46.120933Z"Mangontarum, Mahid M."https://zbmath.org/authors/?q=ai:mangontarum.mahid-mSummary: We extend the notion of \(r\)-Dowling polynomials to their bivariate forms and establish several properties that generalize those of the bivariate Bell and \(r\)-Bell polynomials. Lastly, we obtain two forms of generalized Spivey's formula.Properties of \((p, q)\)-differential equations with \((p, q)\)-Euler polynomials as solutionshttps://zbmath.org/1517.050292023-09-22T14:21:46.120933Z"Yu, C. H."https://zbmath.org/authors/?q=ai:yu.chunhui|yu.cheng-he|yu.chunhai|yu.chenghai|yu.changhua|yu.ching-hao|yu.ching-hua|yu.cheng-hao|yu.caihua|yu.chiew-hui|yu.chien-hsien|yu.cihai|yu.chaohang|yu.chao-hua|yu.chonghua|yu.chang-hsi|yu.chenhai|yu.canghai|yu.chung-hyo|yu.chung-hsin|yu.chi-hua|yu.chung-hyun|yu.cunhai|yu.chong-ho|yu.changhui|yu.cheng-han|yu.chunhua|yu.caihui|yu.chenghui|yu.chuanhua"Kang, J. Y."https://zbmath.org/authors/?q=ai:kang.jiayin|kang.jiayu|kang.ji-yoon|kang.jung-yup|kang.jiyang|kang.jingyu|kang.jianying|kang.jung-yoog|kang.jiayiSummary: In this paper, we discuss \((p, q)\)-differential equations which are related to \((p, q)\)-Euler polynomials. Also, we find a basic symmetric property for \((p, q)\)-differential equation using the generating function of \((p, q)\)-Euler polynomials.A simple proof of higher order Turán inequalities for Boros-Moll sequenceshttps://zbmath.org/1517.050302023-09-22T14:21:46.120933Z"Zhao, James Jing Yu"https://zbmath.org/authors/?q=ai:zhao.james-jing-yuThis paper is a different approach to a recent result concerning higher-order Turán inequalities for the Boros-Moll sequence \(\{d_l(m)\}_{l=0}^m\) obtained by \textit{J. J. F. Guo} [J. Number Theory 225, 294--309 (2021; Zbl 1465.05017)]. Here, \(d_l\) is the coefficient of \(x^l\) in the Boros-Moll polynomials
\[
P_m(x)=\sum_{j,k}\binom{2m+1}{2j}\binom{m-j}{k}\binom{2k+2j}{k+j}\frac{(x+1)^j(x-1)^k}{2^{3(k+j)}}.
\]
These polynomials arise in the study of a quartic integral
\[
\int_0^\infty \frac{dt}{{(t^4+2xt^2+1)}^{m+1}}=\frac{\pi}{2^{m+3/2}(x+1)^{m+1/2}}P_m(x).
\]
The paper discusses, among other things, a sharper bound for \(\frac{d_l(m+1)}{d_l(m)}\), and one for \(\frac{d_l(m)^2}{d_{l-1}(m)d_{l+1}(m)}\).
Reviewer: Firdous Ahmad Mala (Srinagar)The projectivization matroid of a \(q\)-matroidhttps://zbmath.org/1517.050342023-09-22T14:21:46.120933Z"Jany, Benjamin"https://zbmath.org/authors/?q=ai:jany.benjaminSummary: In this paper, we investigate the relation between a \(q\)-matroid and its associated matroid called the projectivization matroid. The latter arises by projectivizing the groundspace of the \(q\)-matroid and considering the projective space as the groundset of the associated matroid on which is defined a rank function compatible with that of the \(q\)-matroid. We show that the projectivization map is a functor from categories of \(q\)-matroids to categories of matroids, which allows us to prove new results about maps of \(q\)-matroids. We furthermore show the characteristic polynomial of a \(q\)-matroid is equal to that of the projectivization matroid. We use this relation to establish a recursive formula for the characteristic polynomial of a \(q\)-matroid in terms of the characteristic polynomial of its minors. Finally we use the projectivization matroid to prove a \(q\)-analogue of the Critical Theorem in terms of \(\mathbb{F}_{q^m}\)-linear rank metric codes and \(q\)-matroids.Integer colorings with forbidden rainbow sumshttps://zbmath.org/1517.050522023-09-22T14:21:46.120933Z"Cheng, Yangyang"https://zbmath.org/authors/?q=ai:cheng.yangyang"Jing, Yifan"https://zbmath.org/authors/?q=ai:jing.yifan"Li, Lina"https://zbmath.org/authors/?q=ai:li.lina.2"Wang, Guanghui"https://zbmath.org/authors/?q=ai:wang.guanghui"Zhou, Wenling"https://zbmath.org/authors/?q=ai:zhou.wenlingSummary: For a set of positive integers \(A\subseteq [n]\), an \(r\)-coloring of \(A\) is rainbow sum-free if it contains no rainbow Schur triple. In this paper we initiate the study of the rainbow Erdős-Rothschild problem in the context of sum-free sets, which asks for the subsets of \([n]\) with the maximum number of rainbow sum-free \(r\)-colorings. We show that for \(r=3\), the interval \([n]\) is optimal, while for \(r\geq 8\), the set \([\lfloor n/2\rfloor,n]\) is optimal. We also prove a stability theorem for \(r\geq 4\). The proofs rely on the hypergraph container method, and some ad-hoc stability analysis.On magic distinct labellings of simple graphshttps://zbmath.org/1517.051572023-09-22T14:21:46.120933Z"Xin, Guoce"https://zbmath.org/authors/?q=ai:xin.guoce"Xu, Xinyu"https://zbmath.org/authors/?q=ai:xu.xinyu"Zhang, Chen"https://zbmath.org/authors/?q=ai:zhang.chen"Zhong, Yueming"https://zbmath.org/authors/?q=ai:zhong.yueming\textit{J. A. MacDougall} et al. [Util. Math. 61, 3--21 (2002; Zbl 1008.05135)] have introduced the concept of magic labelling of simple graphs. A magic labelling of a graph $G$ with magic sum $s$ is a labelling of the edges of $G$ by nonnegative integers such that for each vertex $v\in V$, the sum of labels of all edges incident to $v$ is equal to the same number $s$. A magic distinct labelling is a magic labelling whose labels are distinct. It is said to be pure if the labels are $1,2,\dots,n$. The authors consider here the complete construction of all magic labellings of a given graph $G$. They illustrate it in detail by dealing with three regular graphs and a non-regular graph. They give combinatorial proofs. The structure they find here can be used to enumerate the corresponding magic distinct labellings. The idea of using the generating function is highly laudable.
Reviewer: V. Yegnanarayanan (Chennai)Some new results on monochromatic sums and products in the rationalshttps://zbmath.org/1517.051742023-09-22T14:21:46.120933Z"Hindman, Neil"https://zbmath.org/authors/?q=ai:hindman.neil"Ivan, Maria-Romina"https://zbmath.org/authors/?q=ai:ivan.maria-romina"Leader, Imre"https://zbmath.org/authors/?q=ai:leader.imreThe main result of the paper states that there exists a finite colouring of the rational numbers with the following property: There is no infinite set of rationals such that the set of its finite sums and products is monochromatic and the set of primes that divide the denominators of its terms is finite. In other words, given an infinite set of rational numbers whose denominators are divisible only by a finite number of primes, then the set of its finite sums and products it not monochromatic.
Besides this amazing result, they prove several auxiliary results concerning colorings of the natural numbers and the real numbers. For example, they prove that there exists a finite colouring of the natural numbers, such that there is no injective sequence \((x_n)_{n\geq 1}\) of natural numbers with the property that for any \(1\leq n<m\), all numbers \(x_n+x_m\) and \(x_n\cdot x_m\) have the same colour.
Reviewer: Lorenz Halbeisen (Zürich)A note on the action of the Hecke group \(H(2)\) on subsets of the form \(\mathbb{Q}^{\ast}(\sqrt{n})\)https://zbmath.org/1517.051822023-09-22T14:21:46.120933Z"Cimpoeaş, Mircea"https://zbmath.org/authors/?q=ai:cimpoeas.mirceaSummary: Here, we study the action of the groups \(H(\lambda)\) generated by the linear fractional transformations \(x:z\mapsto -\frac{1}{z}\) and \(w:z\mapsto z+\lambda\) and \(\lambda\) is a positive integer, on the subsets \(\mathbb{Q}^{\ast}(\sqrt{n})=\{\frac{a+\sqrt{n}}{c}\mid a,b=\frac{a^2 -n}{c},c\in\mathbb{Z}\}\cup\{0,1,\infty\}\) and \(|n|\) is a square-free positive integer. We prove that this action has a finite number of orbits if and only if \(\lambda =1\) or \(\lambda =2\). Moreover, we give an upper bound for the number of orbits for \(\lambda =2\).Quadrinomial-like versions for Wolstenholme, Morley and Glaisher congruenceshttps://zbmath.org/1517.110012023-09-22T14:21:46.120933Z"Belbachir, Hacène"https://zbmath.org/authors/?q=ai:belbachir.hacene"Otmani, Yassine"https://zbmath.org/authors/?q=ai:otmani.yassineMotivated by the famous congruence by \textit {J. Wolstenholme} (1862), valid for any odd prime \(p \geq 5\),
\[
\binom{2p-1}{p-1}\equiv 1 \pmod {p^3},
\]
and thanks to two further well-known congruences from, respectively, \textit {E. Lehmer} [Ann. Math. (2) 39, 350--360 (1938; Zbl 0019.00505)], and \textit {W. Ljunggren} et al. [11. Skand. Mat.-Kongr., Trondheim 1949, 42--54 (1952; Zbl 0048.27204)], the authors prove, via Fermat's Little Theorem and Taylor expansion, that
\[
\kappa_{p} \equiv 4+8 p^{2} \left(\frac{-1}{p} \right) E_{p-3} \pmod {p^{3}},
\]
where \(E_{n}\) is the \(n\)-th Euler number and \(\kappa_{p}\) is the central quadrinomial coefficient, i.e., the coefficient of \(x^{3p}\) in the polynomial expansion of \((1+x+x^2+x^3)^{2p}\).
Then, after employing three congruences by \textit {Z.-H. Sun} [Fibonacci Q. 40, No. 4, 345--351 (2002; Zbl 1009.11004)], a theorem by \textit {R. Tauraso} [J. Integer Seq. 19, No. 5, Article 16.5.4 (2016; Zbl 1417.11002)], a note by \textit {F. Morley} [Ann. Math. 9, 168--170 (1895; JFM 26.0208.02)], and a lemma by \textit {S. Mattarei} and \textit {R. Tauraso} [J. Number Theory 133, No. 1, 131--157 (2013; Zbl 1300.11020)], the authors establish that for any odd prime \(p \geq 5\),
\[
\kappa_{(p-1) / 2} \equiv \left( \frac{-2}{p} \right)+p \left(q_{p}(2) \left(\frac{7}{2} \left(\frac{-2}{p} \right)-3 \left( \frac{-1}{p} \right) \right)-2 \left(\frac{2}{p} \right) A_{p} \right) \pmod {p^{2}},
\]
where
\[
q_p(x) := \frac{x^{p-1}-1}{p}
\]
is the Fermat quotient, with \(x\) relatively prime to \(p\), and
\[
A_{p} := \frac{(-1)^{(p-1) / 2} P_{p}-(-8)^{(p-1) / 2}}{p},
\]
\( \left( P_n \right)_n \) being the Pell sequence.
Related to \(A_{p}\) is also a congruence provided by \textit {Kh. Hessami Pilehrood} et al. [Int. J. Number Theory 8, No. 7, 1789--1811 (2012; Zbl 1261.11001)] and here used to show that, for any positive integer \(n\) and any odd prime \(p \geq 5\),
\[
\left( \begin{array}{c} np-1 \\
p-1 \end{array} \right)_{3} \equiv \frac{1}{2} \left( \left( \frac{-1}{p} \right)+1 \right)+p q_{p}(2) \frac{n}{4} \left(5 \left( \frac{-1}{p} \right)+3 \right) \pmod {p^{2}},
\]
whose explicit source of inspiration is a result given by \textit {J. W. L. Glaisher} [Q. J. Math. 31, 1--35 (1900; JFM 30.0180.01)].
Reviewer: Enzo Bonacci (Latina)Some supercongruences of arbitrary lengthhttps://zbmath.org/1517.110022023-09-22T14:21:46.120933Z"Beukers, Frits"https://zbmath.org/authors/?q=ai:beukers.frits"Delaygue, Eric"https://zbmath.org/authors/?q=ai:delaygue.ericSummary: We prove supercongruences modulo \(p^2\) for values of truncated hypergeometric series at some special points. The parameters of the hypergeometric series are \(d\) copies of \(1 / 2\) and \(d\) copies of 1 for any integer \(d \geq 2\). In addition we describe their relation to hypergeometric motives.A note on the least nonresidue modulo prime \(p\equiv 7\pmod 8\)https://zbmath.org/1517.110032023-09-22T14:21:46.120933Z"Shivarajkumar"https://zbmath.org/authors/?q=ai:shivarajkumar.ch-srikanth"Srikanth, Ch."https://zbmath.org/authors/?q=ai:cherukupally.srikanth|srikanth.chGiven a prime number \(p\), let us denote by \(n(p)\) the least positive integer that is not a quadratic residue modulo \(p\). Vinogradov's conjecture states that, given any \(\varepsilon>0\), for sufficiently large primes \(p\) it holds that
\[
n(p)\leq p^{\varepsilon}.
\]
It is known that this conjecture is true under the assumption of the Generalised Riemann Hypothesis. It is also know to be true unconditionally provided \(\varepsilon>\frac{1}{4\sqrt{e}}\).
A positive integer \(x\) is \(y\)-smooth if all its prime factors are smaller than or equal to \(y\). Vinogradov's conjecture turns out to be related to the so-called ``smoothness in short intervals'' conjecture which, roughly speaking, states that every interval of length \(x^{\varepsilon}\) close to \(x\) contains an \(x^{\varepsilon}\)-smooth integer.
This conjecture remains open but Granville has shown that this conjecture implies Vinogradov's for \(p\equiv 3\pmod{4}\).
This paper slightly improves Granville's result in the sense that the author proves that, if the interval \(\left[\frac{p-1}{2}-\lfloor\frac{p^{\varepsilon}}{2}\rfloor,\frac{p-1}{2}\right]\) contains a \(p^{\varepsilon}\)-smooth integer, then Vinogradov's conjecture hold for such \(p\equiv 3\pmod{4}\).
Reviewer: Antonio M. Oller Marcén (Zaragoza)Supercongruences involving products of three binomial coefficientshttps://zbmath.org/1517.110042023-09-22T14:21:46.120933Z"Sun, Zhi-Hong"https://zbmath.org/authors/?q=ai:sun.zhihongSummary: Let \(p > 3\) be a prime, and let \(a\) be a rational \(p\)-adic integer. Using the WZ method we establish the congruences for \(\sum_{k=0}^{p-1}\binom{a}{k}\binom{-1-a}{k}\binom{2k}{k}\frac{w(k)}{4^k}\) modulo \(p^3\), where \(w(k)\in\{1, \frac{1}{k+1}, \frac{1}{(k+1)^2}, \frac{1}{2k-1}\}\). Taking \(a = -\frac{1}{2}, -\frac{1}{3}, -\frac{1}{4}, -\frac{1}{6}\) in the congruences confirms some conjectures posed by the author earlier.Generalizing Zeckendorf's theorem to homogeneous linear recurrences. Ihttps://zbmath.org/1517.110052023-09-22T14:21:46.120933Z"Martinez, Thomas C."https://zbmath.org/authors/?q=ai:martinez.thomas-c"Miller, Steven J."https://zbmath.org/authors/?q=ai:miller.steven-j"Mizgerd, Clayton"https://zbmath.org/authors/?q=ai:mizgerd.clayton-m"Sun, Chenyang"https://zbmath.org/authors/?q=ai:sun.chenyangSummary: Zeckendorf's theorem states that every positive integer can be written uniquely as the sum of nonconsecutive shifted Fibonacci numbers \(\{F_n\}\), where we take \(F_1=1\) and \(F_2=2\). This has been generalized for any Positive Linear Recurrence Sequence (PLRS), which informally is a sequence satisfying a homogeneous linear recurrence with a positive leading coefficient and nonnegative integer coefficients. These decompositions are generalizations of base \(B\) decompositions. In this and the followup paper, we provide two approaches to investigate linear recurrences with leading coefficient zero, followed by nonnegative integer coefficients, with differences between indices relatively prime (abbreviated ZLRR).
The first approach involves generalizing the definition of a legal decomposition for a PLRS found in [\textit{M. Koloğlu} et al., Fibonacci Q. 49, No. 2, 116--130 (2011; Zbl 1225.11021)]. We prove that every positive integer \(N\) has a legal decomposition for any ZLRR using the greedy algorithm. We also show that a specific family of ZLRRs loses uniqueness of decompositions.
The second approach converts a ZLRR to a PLRR that has the same growth rate. We develop the Zeroing Algorithm, a powerful helper tool for analyzing the behavior of linear recurrence sequences. We use it to prove a general result that guarantees the possibility of conversion between certain recurrences, and develop a method to quickly determine whether certain sequences diverge to \(+\infty\) or \(-\infty\), given any real initial values.
This paper investigates the first approach.Triforce and cornershttps://zbmath.org/1517.110062023-09-22T14:21:46.120933Z"Fox, Jacob"https://zbmath.org/authors/?q=ai:fox.jacob"Sah, Ashwin"https://zbmath.org/authors/?q=ai:sah.ashwin"Sawhney, Mehtaab"https://zbmath.org/authors/?q=ai:sawhney.mehtaab-s"Stoner, David"https://zbmath.org/authors/?q=ai:stoner.david"Zhao, Yufei"https://zbmath.org/authors/?q=ai:zhao.yufeiThe \textit{triforce} is the \(3\)-graph (\(3\)-uniform hypergraph) with hyperedges \(123'\), \(12'3\) and \(1'23\). Thus the triforce is a \(6\)-vertex \(3\)-graph with three hyperedges. Let \(g(\delta)\) be the minimum density of triforces in a \(3\)-graph with hyperedge density \(\delta\). The first main result in the paper under review is that \(g(\delta)=\delta^{4-o(1)}\) but \(g(\delta)/\delta^4\to\infty\) as \(\delta\to0\). Via results of \textit{M. Mandache} [Math. Proc. Camb. Philos. Soc. 171, No. 3, 607--621 (2021; Zbl 1486.11015)], these estimates on \(g(\delta)\) translate to estimates on the density of so called \textit{popular corners} in \(G\times G\) for abelian groups \(G\) (see [\textit{M. Mandache}, Math. Proc. Camb. Philos. Soc. 171, No. 3, 607--621 (2021; Zbl 1486.11015)]). The authors also note that their proof of their estimates for~\(g(\delta)\) generalize to prove analogous estimates for the minimum density of \textit{\(k\)-forces} in \(k\)-graphs, where the \(k\)-force is the \(k\)-graph on \(2k\) vertices and hyperedges \(1'2\dots k\), \(12'\dots k\), \(\dots\), \(12\dots k'\): if \(g_k(\delta)\) is the corresponding extremal density function, then \(g_k(\delta)=\delta^{k+1-o(1)}\) but
\(g_k(\delta)/\delta^{k+1}\to\infty\) as \(\delta\to\infty\). One might expect these bounds to translate to good, polynomial bounds for popular \((k-1)\)-dimensional corners, but the authors prove a result (Theorem 1.6) that dashes any such hopes: for some absolute constant \(c>0\), given any fixed \(0<\delta<1/2\), for every sufficiently large \(N\), there is \(A\subset[N]^3\) with \(|A|\geq\delta N^3\) such that for every non-zero integer \(d\), there are at most \(\delta^{c\log1/\delta}N^3\) triples \((x,y,z)\) with \((x,y,z)\), \((x+d,y,z)\), \((x,y+d,z)\), and \((x,y,z+d)\) all in \(A\). The construction proving this result is related to a construction of sets of integers lacking popular differences for \(5\)-APs (see [\textit{V. Bergelson} et al., Invent. Math. 160, No. 2, 261--303 (2005; Zbl 1087.28007)]), which is in fact generalized in this paper to all \(5\)-point patterns in \(\mathbb N\) (Theorem 1.7).
Reviewer: Yoshiharu Kohayakawa (São Paulo)Consecutive ratios in second-order linear recurrence sequenceshttps://zbmath.org/1517.110072023-09-22T14:21:46.120933Z"Berend, Daniel"https://zbmath.org/authors/?q=ai:berend.daniel"Kumar, Rishi"https://zbmath.org/authors/?q=ai:kumar.rishiThe authors study a second order difference equations with constant coefficients, namely
\[
a_{n+1} = c a_{n-1} + d a_{n-2}, \quad n \geq 2
\]
with some initial values \(a_0,~ a_1\), where \(c,d~\) are fixed complex numbers (with some restrictions designed to avoid trivialities). In particular, they studied the limit points and asymptotic distribution of the sequence of consecutive ratios \(a_{n+1}/a_n\).
Reviewer: Raghib Abu-Saris (Edmonton)Fence tiling derived identities involving the metallonacci numbers squared or cubedhttps://zbmath.org/1517.110082023-09-22T14:21:46.120933Z"Allen, Michael A."https://zbmath.org/authors/?q=ai:allen.michael-a"Edwards, Kenneth"https://zbmath.org/authors/?q=ai:edwards.kennethSummary: We refer to the generalized Fibonacci sequence \((M^{(c)}_n)_{n\ge 0}\), where \(M^{(c)}_{n+1}=cM^{(c)}_n+M^{(c)}_{n-1}\) for \(n>0\) with \(M^{(c)}_0=0\), \(M^{(c)}_1=1\), for \(c=1,2,\dots\) as the \(c\)-metallonacci numbers. We consider the tiling of an \(n\)-board (an \(n\times 1\) rectangular board) with \(c\) colours of \(1/p\times 1\) tiles (with the shorter sides always aligned horizontally) and \((1/p,1-1/p)\)-fence tiles for \(p\in\mathbb{Z}^+\). A \((w,g)\)-fence tile is composed of two \(w\times 1\) sub-tiles separated by a \(g\times 1\) gap. The number of such tilings equals \((M^{(c)}_{n+1})^p\) and we use this result for the cases \(p=2, 3\) to devise straightforward combinatorial proofs of identities relating the metallonacci numbers squared or cubed to other combinations of metallonacci numbers. Special cases include relations between the Pell numbers cubed and the even Fibonacci numbers. Most of the identities derived here appear to be new.Some combinatorial aspects of bi-periodic incomplete Horadam sequenceshttps://zbmath.org/1517.110092023-09-22T14:21:46.120933Z"Belkhir, Amine"https://zbmath.org/authors/?q=ai:belkhir.amine"Tan, Elif"https://zbmath.org/authors/?q=ai:tan.elif"Dağh, Mehmet"https://zbmath.org/authors/?q=ai:dagh.mehmetSummary: We have recently introduced the bi-periodic incomplete Horadam numbers as a generalization of incomplete Horadam numbers, and studied their properties. In this paper, we provide some combinatorial interpretations of bi-periodic incomplete Horadam numbers by using the weighted tilings approach. We also define bi-periodic hyper Horadam numbers and show that each bi-periodic hyper Horadam number can be written as the difference of a bi-periodic Horadam number and a bi-periodic incomplete Horadam number.The Diophantine equations \(P_n^x+P_{n+1}^y=P_m^x\) or \(P_n^y+P_{n+1}^x=P_m^x\)https://zbmath.org/1517.110102023-09-22T14:21:46.120933Z"Faye, Bernadette"https://zbmath.org/authors/?q=ai:faye.bernadette"Luca, Florian"https://zbmath.org/authors/?q=ai:luca.florian"Rihane, Salah Eddine"https://zbmath.org/authors/?q=ai:rihane.salah-eddine"Togbé, Alain"https://zbmath.org/authors/?q=ai:togbe.alainLet \( (P_n)_{n\ge 0} \) be the Pell sequence defined by the binary recurrence relation: \( P_0=0 \), \( P_1=1 \), and \( P_{n+2}=2P_{n+1}+P_n \) for all \( n\ge 0 \). In the paper under review, the authors prove the following theorem, which is the main result in the paper.
\textbf{Theorem 1.} The only solution in positive integers \( (n,m,x,y) \) of either the equations
\[
P_n^{x}+P_{n+1}^{y}=P_m^{x}\text{ or }P_n^{y}+P_{n+1}^{x}=P_m^{x},
\]
is \( (n,m,x,y)=(1,3,1,2) \), for which \( 1+2^2=5=P_3 \).
The proof of Theorem 1 follows from a clever combination of techniques in Diophantine number theory, the usual properties of the Pell sequence, Carmichael's theorem on primitive divisors for Lucas sequences, Baker's theory for nonzero lower bounds for linear forms in logarithms of algebraic numbers, both complex and \(p\)-adic, reduction techniques involving the theory of continued fractions, as well as the LLL algorithm. All computations are done with the aid of a computer program in \texttt{Mathematica}.
Reviewer: Mahadi Ddamulira (Kampala)A system of four simultaneous recursions: generalization of the Ledin-Shannon-Ollerton identityhttps://zbmath.org/1517.110112023-09-22T14:21:46.120933Z"Hendel, Russell Jay"https://zbmath.org/authors/?q=ai:hendel.russell-jaySummary: This paper further generalizes the Ledin-Shannon-Ollerton result, the recent result of \textit{A. G. Shannon} and \textit{R. L. Ollerton} [Fibonacci Q. 59, No. 1, 47--56 (2021; Zbl 1486.11026)] who resurrected an old identity due to \textit{G. Ledin jun.} [Fibonacci Q. 5, 45--58 (1967; Zbl 0153.06402)], to all metallic sequences. The results presented in this paper give closed-form formulas for the sum of products of powers of the first \(n\) integers with the first \(n\) members of the metallic sequence. Three key innovations of this paper are i) reducing the proof of the generalization to the solution of a system of 4 simultaneous recursions; ii) use of the shift operation to prove equality of polynomials; and iii) new OEIS sequences arising from the coefficients of the four polynomial families satisfying the 4 simultaneous recursions.\(\mathbb{Q}\)-bonacci words and numbershttps://zbmath.org/1517.110122023-09-22T14:21:46.120933Z"Kirgizov, Sergey"https://zbmath.org/authors/?q=ai:kirgizov.sergeySummary: We present a quite curious generalization of multi-step Fibonacci numbers. For any positive rational \(q\), we enumerate binary words of length \(n\) whose maximal factors of the form \(0^a1^b\) satisfy \(a=0\) or \(aq>b\). When \(q\) is an integer we rediscover classical multistep Fibonacci numbers: Fibonacci, Tribonacci, Tetranacci, etc. When \(q\) is not an integer, obtained recurrence relations are connected to certain restricted integer compositions. We also discuss Gray codes for these words, and a possibly novel generalization of the golden ratio.Sums involving Jacobsthal polynomialshttps://zbmath.org/1517.110132023-09-22T14:21:46.120933Z"Koshy, Thomas"https://zbmath.org/authors/?q=ai:koshy.thomasSummary: We explore the Jacobsthal versions of seven gibonacci sums.Additional sums involving gibonacci polynomialshttps://zbmath.org/1517.110142023-09-22T14:21:46.120933Z"Koshy, Thomas"https://zbmath.org/authors/?q=ai:koshy.thomasSummary: We continue the exploration of sums involving gibonacci polynomials and their numeric versions, and their Pell versions.Additional sums involving gibonacci polynomials: graph-theoretic confirmationshttps://zbmath.org/1517.110152023-09-22T14:21:46.120933Z"Koshy, Thomas"https://zbmath.org/authors/?q=ai:koshy.thomasSummary: Using graph-theoretic techniques, we confirm four sums involving gibonacci polynomials.Additional sums involving gibonacci polynomials revisitedhttps://zbmath.org/1517.110162023-09-22T14:21:46.120933Z"Koshy, Thomas"https://zbmath.org/authors/?q=ai:koshy.thomasSummary: We explore three sums involving gibonacci polynomials and extract their Pell versions.The Narayana sequence in finite groupshttps://zbmath.org/1517.110172023-09-22T14:21:46.120933Z"Kuloğlu, Bahar"https://zbmath.org/authors/?q=ai:kuloglu.bahar"Özkan, Engin"https://zbmath.org/authors/?q=ai:ozkan.engin"Shannon, Anthony G."https://zbmath.org/authors/?q=ai:shannon.anthony-grevilleSummary: In this paper, the Narayana sequence modulo \(m\) is studied. The paper outlines the definition of Narayana numbers and some of their combinatorial links with Eulerian, Catalan and Delannoy numbers and other special functions. From the definition, the Narayana orbit of a 2-generator group for a generating pair \((x,y)\in G\) is defined, so that the lengths of the period of the Narayana orbit can be examined. These yield in turn the Narayana lengths of the polyhedral group and the binary polyhedral group for the generating pair \((x,y)\) and associated properties.Pairwise modular multiplicative inverses and Fibonacci numbershttps://zbmath.org/1517.110182023-09-22T14:21:46.120933Z"Sanna, Carlo"https://zbmath.org/authors/?q=ai:sanna.carloLet \((p,q)\) be a pair of relatively prime integers greater than \(1\). The pairwise modular multiplicative inverse, or PMMI for short, of \((p,q)\) is defined as the unique pair \((p', q')\) of positive integers such that \(p p'\equiv 1\pmod q\), \(p'<q\), \(q q'\equiv 1\pmod p\), \(q'<p\). In other words, \(p'\) is the inverse of \(p\) modulo \(q\), and \(q'\) is the inverse of \(q\) modulo \(p\). Motivated by some results in knot theory, \textit{H.-J. Song} [East Asian Math. J. 35, No. 3, 285--288 (2019; Zbl 1427.11019)] four families of pairs of Fibonacci numbers whose PMMIs are pairs of Fibonacci numbers. In this paper it is proved that these families, together with some isolated pairs, are indeed all the pairs of Fibonacci numbers whose PMMIs are pairs of Fibonacci numbers. Namely, let \(a\), \(b\), \(c\), \(d\) be integers with \(a>b\ge 3\), \(c,d\ge 2\) and \(\gcd(F_a, F_b)=1\). Then the PMMI of \((F_a, F_b)\) is equal to \((F_c, F_d)\) if and only if \((a,b,c,d)\) is equal to \((4,3,2,3)\), \((5,3,2,4)\), \((2 k+1,2 k,2 k-1,2 k-1)\), \((2 k+2,2 k+1,2 k-1,2 k+1)\), \((2 k+2,2 k,2 k-1,2 k)\), or \((2 k+3,2 k+1,2 k-1,2 k+2)\), where \(k\) is a positive integer.
Reviewer: Takao Komatsu (Hangzhou)Application of the BenTaher-Rachidi method in numerical sequenceshttps://zbmath.org/1517.110192023-09-22T14:21:46.120933Z"Vieira, Renata"https://zbmath.org/authors/?q=ai:vieira.renata-passos-machado"Mangueira, Milena"https://zbmath.org/authors/?q=ai:mangueira.milena-carolina-dos-santos"Alves, Francisco Regis"https://zbmath.org/authors/?q=ai:alves.francisco-regis-vieira"Catarino, Paula"https://zbmath.org/authors/?q=ai:catarino.paula-maria-machado-cruzSummary: This article presents the application of the study on the BenTaher-Rachidi method for the solving of linear and recurrent numerical sequences of higher order. Thus, Binet's formula is obtained, using the BenTaher-Rachidi method, in the sequences of Lucas, Pell, Leonardo, Mersenne, Oresme, Jacobsthal, Padovan, Perrin and Narayana.Some identities involving degenerate Stirling numbers associated with several degenerate polynomials and numbershttps://zbmath.org/1517.110202023-09-22T14:21:46.120933Z"Kim, T. K."https://zbmath.org/authors/?q=ai:kim.tong-kuk|kim.tae-keuk|kim.tae-kyu|kim.timur-k|kim.tian-khoon|kim.tae-kyun|kim.taekyung.1|kim.tak-kyeom|kim.taekyun"Kim, D. S."https://zbmath.org/authors/?q=ai:kim.doo-seok|kim.dae-shik|kim.dae-su|kim.dae-sig|kim.david-s|kim.dong-sie|kim.dong-seon|kim.duk-sun|kim.do-sang|kim.dae-seung|kim.dae-san|kim.dong-seong|kim.das-san|kim.dae-seoung|kim.deok-soo|kim.dong-sik|kim.doh-suk|kim.daesoo|kim.dongsoo-s|kim.dong-seoSummary: The aim of this paper is to investigate some properties, recurrence relations and identities involving degenerate Stirling numbers of both kinds associated with degenerate hyperharmonic numbers and also with degenerate Bernoulli, degenerate Euler, degenerate Bell, and degenerate Fubini polynomials.A study on multi-Stirling numbers of the first kindhttps://zbmath.org/1517.110212023-09-22T14:21:46.120933Z"Ma, Yuankui"https://zbmath.org/authors/?q=ai:ma.yuankui"Kim, Dae San"https://zbmath.org/authors/?q=ai:kim.dae-san"Lee, Hyunseok"https://zbmath.org/authors/?q=ai:lee.hyunseok"Park, Seongho"https://zbmath.org/authors/?q=ai:park.seongho"Kim, Taekyun"https://zbmath.org/authors/?q=ai:kim.taekyun(no abstract)Some identities of fully degenerate dowling and fully degenerate Bell polynomials arising from \(\lambda\)-umbral calculushttps://zbmath.org/1517.110222023-09-22T14:21:46.120933Z"Ma, Yuankui"https://zbmath.org/authors/?q=ai:ma.yuankui"Kim, Taekyun"https://zbmath.org/authors/?q=ai:kim.taekyun"Lee, Hyunseok"https://zbmath.org/authors/?q=ai:lee.hyunseok"Kim, Dae San"https://zbmath.org/authors/?q=ai:kim.dae-san(no abstract)Shape of the asymptotic maximum sum-free sets in integer lattice gridshttps://zbmath.org/1517.110232023-09-22T14:21:46.120933Z"Liu, Hong"https://zbmath.org/authors/?q=ai:liu.hong.1"Wang, Guanghui"https://zbmath.org/authors/?q=ai:wang.guanghui"Wilkes, Laurence"https://zbmath.org/authors/?q=ai:wilkes.laurence"Yang, Donglei"https://zbmath.org/authors/?q=ai:yang.dongleiFor an integer \(n\in \mathbb{N}\), a subset \(A\subset [n]\) is sum-free if it has no solution for the equation \(x+y = z\), i.e., for all \(x, y\in A\) we have \(x+y\notin A\). As a generalization of the definition for sum-free sets, the $(p,q)$-sum free sets and problems associated with the extremal structure are investigated. In this paper, the four authors determine the shape of all sum-free sets in \(\{1,2,\ldots,n\}^2\) of size close to the maximum \(\frac{3}{5}n^2\), solving a problem of \textit{C. Elsholtz} and \textit{L. Rackham} [J. Lond. Math. Soc., II. Ser. 95, No. 2, 353--372 (2017; Zbl 1427.11009)]. They show that all such asymptotic maximum sum-free sets lie completely in the stripe \(\frac{4}{5}n-o(n)\leq x+y\leq \frac{5}{8}n+o(n)\). They also determine for any positive integer \(p\) the maximum size of a subset \(A\subseteq \{1,2,\ldots,n\}^2\) which forbids the triple \((x, y, z)\) satisfying \(px+py=z\).
Reviewer: Weidong Gao (Tianjin)Dot-product sets and simplices over finite ringshttps://zbmath.org/1517.110242023-09-22T14:21:46.120933Z"Nguyen Van The"https://zbmath.org/authors/?q=ai:nguyen-van-the.lionel"Vinh, Le Anh"https://zbmath.org/authors/?q=ai:vinh.le-anhLet \(P\) be a subset of \(\mathbb Z_n^d\) and define the dot product of two elements \(p=(p_1, \dots,p_d) \in \mathbb Z_n^d\) and \(q=(q_1,\dots,q_d) \in \mathbb Z_n^d\) to be the value
\[
p_1q_1 + \dots + p_dq_d \in \mathbb Z_n.
\]
Define
\[
\Pi(P)= \{ p \cdot q : p,q \in \Pi \}.
\]
This paper proves that, if \(P\) is sufficiently large then \(\Pi(P)=\mathbb Z_n\). The bound on \(P\) is dependent on the number of divisors of \(n\). The case when \(n\) is a prime number has been considered extensively (see for instance [\textit{D. Hart} and \textit{A. Iosevich}, Contemp. Math. 464, 129--135 (2008; Zbl 1256.11022)]), and can be considered as a particular finite field variant of the Erdős distinct distances problem.
Reviewer: Oliver Roche-Newton (Linz)Proofs of Chappelon and Ramírez Alfonsín conjectures on square Frobenius numbers and their relationship to simultaneous Pell equationshttps://zbmath.org/1517.110252023-09-22T14:21:46.120933Z"Binner, Damanvir Singh"https://zbmath.org/authors/?q=ai:binner.damanvir-singhSummary: Recently, \textit{J. Chappelon} and \textit{J. L. Ramírez Alfonsín} [Semigroup Forum 105, No. 1, 149--171 (2022; Zbl 07570869)] defined the square Frobenius number of coprime numbers \(m\) and \(n\) to be the largest perfect square that cannot be expressed in the form \(mx+ny\) for nonnegative integers \(x\) and \(y\). When \(m\) and \(n\) differ by 1 or 2, they found simple expressions for the square Frobenius number if neither \(m\) nor \(n\) is a perfect square. If either \(m\) or \(n\) is a perfect square, they formulated some interesting conjectures which have an unexpected close connection with a known recursive sequence, related to the denominators of Farey fraction approximations to \(\sqrt{2 }\). In this note, we prove these conjectures. Our methods involve solving Pell equations \(x^2 - 2y^2 = 1\) and \(x^2 - 2y^2 = -1\). Finally, to complete our proofs of these conjectures, we eliminate several cases using several results related to solutions of simultaneous Pell equations.A parametric family of ternary purely exponential Diophantine equation \(A^x + B^y = C^z\)https://zbmath.org/1517.110262023-09-22T14:21:46.120933Z"Fujita, Yasutsugu"https://zbmath.org/authors/?q=ai:fujita.yasutsugu"Le, Maohua"https://zbmath.org/authors/?q=ai:le.maohuaIn the paper under review, the authors study the Diophantine equation
\[
(am^2+1)^x+(bm^2-1)^y=(cm)^z,\tag{1}
\]
in positive integers \( x,y,z \), where \( m>1 \) is any positive integer and \( a,b,c \) are fixed positive integers such that \( a+b=c^2 \), \( 2\not|~c \), and \( b \) is a quadratic nonresidue for every prime divisor \( p \) of \( c \). Their main result is the following.
\textbf{Theorem 1.} If \( m>\max\{10^{8}, c^2\}\), then the Diophantine equation (1) has only one solution \( (x,y,z)=(1,1,2) \).
The proof of Theorem 1 follows from a clever combination of techniques in elementary number theory as well as Baker's theory for nonzero lower bounds for linear forms in logarithms of algebraic numbers.
Reviewer: Mahadi Ddamulira (Kampala)On some arithmetic questions of reductive groups over algebraic extensions of local and global fieldshttps://zbmath.org/1517.110272023-09-22T14:21:46.120933Z"Thắng, Nguyễn Quốc"https://zbmath.org/authors/?q=ai:nguyen-quoc-thang.From the introduction: ``In the present paper, we are interested in answering the following question: which of the classical results in Galois cohomology theory of linear algebraic groups over local or global fields still hold in the case of infinite algebraic extensions of local and global fields? We investigate here some results which are related to the finiteness, the surjectivity (bijectivity) of a coboundary map in Galois cohomology which are important in arithmetic of algebraic groups over field. In particular, we extend Kneser's Theorem on the surjectivitiy of certain coboundary maps in Galois cohomology and Conrad's Theorem (thus partially also Borel-Serre's Theorem) on the finiteness of Galois cohomology of pseudo-reductive groups to the case of infinite algebraic extensions of local and global fields. As an application, we apply the finiteness of Galois cohomology to show the finiteness of the obstruction to weak approximation of connected reductive groups at finite set of places.''
A detailed version of the paper will be published elsewhere.
The following topics are discussed in sections:
\begin{itemize}
\item[--] Tate-Nakayama duality theory for algebraic group schemes of multiplicative type.
\item[--] Surjectivity of a coboundary map \(H^1_{\mathrm{fppf}}(S,\tilde G/Z)\to H^2_{\mathrm{fppf}}(S,Z(\tilde G))\) where \(Z\) is a subgroup \(S\)-scheme of the center \(Z(\tilde G)\) of a semisimple simply connected \(S\)-group scheme \(\tilde G\).
\item[--] Finiteness of Galois (flat) cohomology for pseudo-reductive groups.
\item[--] The obstruction to weak approximation.
\end{itemize}
Reviewer: Wilberd van der Kallen (Utrecht)Spectral zeta function on discrete tori and Epstein-Riemann hypothesishttps://zbmath.org/1517.110282023-09-22T14:21:46.120933Z"Meiners, Alexander"https://zbmath.org/authors/?q=ai:meiners.alexander"Vertman, Boris"https://zbmath.org/authors/?q=ai:vertman.borisThis authors first derive an asymptotic expansion of the spectral zeta function of the combinatorial Laplacian on a sequence of discrete tori which approximate the \(\alpha\)-dimensional torus for \(\alpha=2\). Then they prove that a certain conjecture on the asymptotics gives an equivalence formulation of the Epstein-Riemann Hypothesis for \(\alpha=2\), if they replace the standard discrete Laplacian with the 9-point star discrete Laplacian. Here the Epstein-Riemann Hypothesis asserts that the spectral zeta function of the torus, which is the Epstein zeta function for the identity matrix of size \(\alpha\), satisfies the analog of the Riemann Hypothesis having all nontrivial zeros in the critical strip \(0<\mathrm{Re}(s)<\alpha/2\) on the line \(\mathrm{Re}(s)=\alpha/4\).
These results generalize the theorems of \textit{F. Friedli} and \textit{A. Karlsson} [Tôhoku Math. J. (2) 69, No. 4, 585--610 (2017; Zbl 1429.11153)] who obtained an asymptotic formula for \(\alpha=1\) and proved the equivalence between a certain conjecture on it and the Riemann Hypothesis.
Reviewer: Shin-ya Koyama (Yokohama)Pointwise bounds for Eisenstein series on \(\Gamma_0(q)\backslash \mathrm{SL}_2(\mathbb{R})\)https://zbmath.org/1517.110292023-09-22T14:21:46.120933Z"Musicantov, Evgeny"https://zbmath.org/authors/?q=ai:musicantov.evgeny"Zehavi, Sa'ar"https://zbmath.org/authors/?q=ai:zehavi.saarSummary: We construct pointwise bounds in the weight aspect for Eisenstein series on \(X_0(q)=\Gamma_0(q)\backslash SL_2(\mathbb{R})\), with squarefree level \(q\), using a Sobolev technique. More specifically, we show that for an Eisenstein series \(E\) on \(X_0(q)\) of weight parameter \(n\) and type \(t\), one has for all \(x\in X_0(q)\): \(|E(x,1/2+it)|\ll_{\epsilon}q^{\epsilon}(1+|n|^{1/2+\epsilon}+|t|^{1/2+\epsilon})\sqrt{y(x)+y(x)^{-1}}\), where \(y(x)\) is the Iwasawa \(y\)-coordinate of the point \(x\).Residual finiteness of extensions of arithmetic subgroups of \(\mathrm{SU}(d,1)\) with cuspshttps://zbmath.org/1517.110302023-09-22T14:21:46.120933Z"Hill, Richard M."https://zbmath.org/authors/?q=ai:hill.richard-michaelSummary: Let \(\Gamma\) be an arithmetic subgroup of \(\mathrm{SU}(d,1)\) with cusps, and let \(X_\Gamma\) be the associated locally symmetric space. In this paper we investigate the pre-image of \(\Gamma\) in the covering groups of \(\mathrm{SU}(d,1)\). Let \(H^\bullet_!(X_\Gamma ,\mathbb{C})\) be the inner cohomology, i.e. the image in \(H^\bullet (X_\Gamma ,\mathbb{C})\) of the compactly supported cohomology. We prove that if the first inner cohomology group \(H^1_!(X_\Gamma ,\mathbb{C})\) is non-zero then the pre-image of \(\Gamma\) in each connected cover of \(\mathrm{SU}(d,1)\) is residually finite. At the end of the paper we give an example of an arithmetic subgroup \(\Gamma\) satisfying the criterion \(H^1_!(X_\Gamma ,\mathbb{C}) \ne 0\).Heat operators on modular and quasimodular polynomialshttps://zbmath.org/1517.110312023-09-22T14:21:46.120933Z"Lee, Min Ho"https://zbmath.org/authors/?q=ai:lee.minho.1|lee.minhoJacobi-like forms are generalizations of Jacobi forms motivated by the Taylor expansions of Jacobi forms. In the paper under review, the author considers the action of the heat operator on Jacobi-like forms, and studies their relations with quasi-modular forms, modular forms, quasi-modular polynomials and modular polynomials.
Reviewer: Haowu Wang (Bonn)Ramanujan-type systems of nonlinear ODEs for \(\Gamma_0 (2)\) and \(\Gamma_0 (3)\)https://zbmath.org/1517.110322023-09-22T14:21:46.120933Z"Nikdelan, Younes"https://zbmath.org/authors/?q=ai:nikdelan.younesSummary: This paper aims to introduce two systems of nonlinear ordinary differential equations whose solution components generate the graded algebra of quasi-modular forms on Hecke congruence subgroups \(\Gamma_0 (2)\) and \(\Gamma_0 (3)\). Using these systems, we provide the generated graded algebras with an \(\mathfrak{sl}_2 (\mathbb{C})\)-module structure. As applications, we introduce Ramanujan-type tau functions for \(\Gamma_0 (2)\) and \(\Gamma_0 (3)\), and obtain some interesting and non-trivial recurrence and congruence relations.Modular forms and ellipsoidal \(T\)-designshttps://zbmath.org/1517.110332023-09-22T14:21:46.120933Z"Pandey, Badri Vishal"https://zbmath.org/authors/?q=ai:pandey.badri-vishalThe paper under review is a nice application of the modular forms, especially theta functions. \textit{T. Miezaki} [Discrete Math. 313, No. 4, 375--380 (2013; Zbl 1259.05030)] defines spherical \(T\)-design in \(\mathbb{R}^2\). In the paper under review, the author extends this result to special ellipses and the norm form shells for rings of integers of imaginary quadratic fields with class number \(1\). Here the shell means \(\mathbb{Z}^2\)-lattice points with fixed integer norm. The proof is based on calculations with the help of theta functions.
Reviewer: İlker İnam (Bilecik)Computational arithmetic of modular formshttps://zbmath.org/1517.110342023-09-22T14:21:46.120933Z"Wiese, Gabor"https://zbmath.org/authors/?q=ai:wiese.gaborSummary: These course notes are about computing modular forms and some of their arithmetic properties. Their aim is to explain and prove the modular symbols algorithm in as elementary and as explicit terms as possible, and to enable the devoted student to implement it over any ring (such that a sufficient linear algebra theory is available in the chosen computer algebra system). The chosen approach is based on group cohomology and along the way the needed tools from homological algebra are provided.
For the entire collection see [Zbl 1415.11003].Character analogues of infinite series identities related to generalized non-holomorphic Eisenstein serieshttps://zbmath.org/1517.110352023-09-22T14:21:46.120933Z"Lim, Sung-Geun"https://zbmath.org/authors/?q=ai:lim.sung-geunSummary: In this paper, we derive analogues of a couple of classes of infinite series identities with the confluent hypergeometric functions involving Dirichlet characters.The density of elliptic Dedekind sumshttps://zbmath.org/1517.110362023-09-22T14:21:46.120933Z"Berkopec, Nicolas"https://zbmath.org/authors/?q=ai:berkopec.nicolas"Branch, Jacob"https://zbmath.org/authors/?q=ai:branch.jacob"Heikkinen, Rachel"https://zbmath.org/authors/?q=ai:heikkinen.rachel"Nunn, Caroline"https://zbmath.org/authors/?q=ai:nunn.caroline"Wong, Tian An"https://zbmath.org/authors/?q=ai:wong.tian-anSummary: Elliptic Dedekind sums were introduced by \textit{R. Sczech} [Invent. Math. 76, 523--551 (1984; Zbl 0521.10021)] as generalizations of classical Dedekind sums to complex lattices. We show that for any lattice with real \(j\)-invariant, the values of suitably normalized elliptic Dedekind sums are dense in the real numbers. This extends an earlier result of Ito for Euclidean imaginary quadratic rings. Our proof is an adaptation of the recent work of \textit{W. Kohnen} [Ramanujan J. 45, No. 2, 491--495 (2018; Zbl 1420.11076)], which gives a new proof of the density of values of classical Dedekind sums.A generalization of Iseki's formula and the Jacobi theta functionhttps://zbmath.org/1517.110372023-09-22T14:21:46.120933Z"Me'Meh, Maher"https://zbmath.org/authors/?q=ai:memeh.maher"Saraeb, Ali"https://zbmath.org/authors/?q=ai:saraeb.aliSummary: In this paper we give a generalization of Iseki's formula and use it to prove the transformation law of \(\theta_1(z,\tau)\).A note of a modular equation of Ramanujan-Selberg continued fractionhttps://zbmath.org/1517.110382023-09-22T14:21:46.120933Z"Xi, Gao-Wen"https://zbmath.org/authors/?q=ai:xi.gaowen"Luo, Qiu-Ming"https://zbmath.org/authors/?q=ai:luo.qiumingLet \(z\) and \(q\) be complex numbers such that \(|q| < 1\). The \(q\)-shifted factorial is defined by
\[
(z;q)_\infty = (1=z)(1-zq)\cdots(1-zq^n)\cdots.
\]
Let, for \(|q| < 1\), define the Ramanujan-Selberg continued fraction of the form
\[
T(q)=\cfrac{1}{1+}\; \cfrac{q}{1+} \; \cfrac{q+q^2}{1+}\; \cfrac{q^3}{1+} \; \cfrac{q^2+q^4}{1+} \cdots .
\]
It's known that \textit{H.-C. Chan} [Proc. Am. Math. Soc. 137, No. 9, 2849--2856 (2009; Zbl 1284.05016)] proved the following modular equation. Let \(\chi(q)=\cfrac{1}{2T^2(q)}\). Then
\[
\chi(q)-\cfrac{q^{\tfrac{1}{2}}}{\chi(q)}=\cfrac{(q^{\tfrac{1}{2}};q^{\tfrac{1}{2}})^4_\infty}{2(q^2;q^2)^4_\infty}.
\]
A short proof of Chan's modular equation on the Ramanujan-Selberg continued fraction is given in this paper.
Reviewer: Mykhaylo Pahirya (Uzhhorod)Uniform estimates for sums of coefficients of symmetric power \(L\)-functionshttps://zbmath.org/1517.110392023-09-22T14:21:46.120933Z"Chen, Guohua"https://zbmath.org/authors/?q=ai:chen.guohua"He, Xiaoguang"https://zbmath.org/authors/?q=ai:he.xiaoguangSummary: Let \(f(z)\) be a holomorphic Hecke eigenform of even weight \(k\) for \(SL (2, \mathbb{Z})\), and denote by \(L(s, \mathrm{sym}^m f)\) the corresponding symmetric power \(L\)-function associated with \(f\). Denote by \(\lambda_{\mathrm{sym}^m} {}_f (n)\) the \(n\)th normalized coefficient of \(L(s, \mathrm{sym}^m f)\). In this paper, we investigate the sum \(\Sigma_{n \leqslant x}\lambda_{\mathrm{sym}^m}{}_f (n)\) for \(m \geqslant 2\) and get the uniform upper bound extending the previous results.On the Fourier coefficients of Siegel-Eisenstein series of degree 2 for odd levelhttps://zbmath.org/1517.110402023-09-22T14:21:46.120933Z"Gunji, Keiichi"https://zbmath.org/authors/?q=ai:gunji.keiichiSummary: We calculate the Fourier coefficients of the Siegel-Eisenstein series of degree 2, level \(p^n\) with trivial character. The main result is to compute the Siegel series at \(p\) that is essentially a calculation of exponential sums.A note on the cancellations of sums involving Hecke eigenvalueshttps://zbmath.org/1517.110412023-09-22T14:21:46.120933Z"Hua, Guodong"https://zbmath.org/authors/?q=ai:hua.guodongSummary: Let \(f\) and \(g\) be distinct primitive holomorphic cusp forms of even integral weights \(k_1\) and \(k_2\) for the full modular group \(\Gamma=\mathrm{SL}(2,\mathbb{Z})\), respectively. Denote by \(\lambda_f(n)\) and \(\lambda_g(n)\) the \(n\)-th normalized Fourier coefficients of \(f\) and \(g\), respectively. In this paper, we consider short sums of isotypic trace functions associated to some sheaves modulo primes \(q\) of bounded conductor, twisted by the multiplicative function \(\lambda_f(n^i)\lambda_f(n^j)\) and \(\lambda_f(n^i)\lambda_g(n^j)\) for any integers \(i,j\ge 1\). We are able to establish non-trivial bounds for these algebraic twisted sums with intervals of length of at least \(q^{\frac{1}{2}+\varepsilon}\) for arbitrarily small \(\varepsilon>0\). In the similar manner, We also establish the nontrivial bounds for short sums of isotypic trace functions twisted by the coefficients \(\lambda_{f\times f\times f}(n)\) and \(\lambda_{f\times f\times g}(n)\) of triple product \(L\)-functions \(L(f\times f\times f,s)\) and \(L(f\times f \times g,s)\), respectively.Slopes of modular forms and reducible Galois representations, an oversight in the ghost conjecturehttps://zbmath.org/1517.110422023-09-22T14:21:46.120933Z"Bergdall, John"https://zbmath.org/authors/?q=ai:bergdall.john"Pollack, Robert"https://zbmath.org/authors/?q=ai:pollack.robert.1|pollack.robert.2In an earlier article [Trans. Am. Math. Soc. 372, No. 1, 357--388 (2019; Zbl 1451.11034)], the two authors had stated a conjecture they coined \textit{Ghost Conjecture}, describing the slopes of certain overconvergent \(p\)-adic modular forms where \(p\ge 5\) is a prime. In view of counter examples, they correct the statement of their conjecture in the article under review. The original statement of the conjecture had a global and a local assumption. In the new formulation, the global assumption was replaced by a local one.
The main insight of the present work is that Galois representations that, locally at \(p\), are twists of a representation with unramified semi-simplification must be excluded as well. To state the full conjecture, we let \(\rho: \mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \to \mathrm{GL}_2(\overline{\mathbb{F}}_p)\) be a Galois representation.
Conjecture. Suppose that \(\rho_p\), the restriction of \(\rho\) to a decomposition group at \(p\), is \textit{regular}, that is, reducible and its semi-simplification is not a twist of an unramified representation with Frobenius trace \(0\). Then the Newton polygon of the characteristic power series of the \(U_p\)-operator acting on the space of \(p\)-adic overconvergent modular forms (of fixed weight and tame level) is the same as the Newton polygon of the \textit{ghost series} (for the given weight and level), which is an explicit power series defined by the authors.
The article includes a comprehensive discussion of the new local condition and describes how to detect and verify it. It turns out that the regularity condition is closely related to the twists of the modular form by the cyclotomic character being ordinary (or, in one case, having valuation of \(a_p\) equal to \(1\)). It further contains theoretical and computational support for the correction of the conjecture and also some relation with a classical conjecture by \textit{F. Gouvêa} and \textit{B. Mazur} [Math. Comput. 58, No. 198, 793--805 (1992; Zbl 0773.11030)]. Finally, the authors offer an interesting conjectural relation between non-(half)-integral slopes and the irregularity of \(\overline{V}_{k,a_p}\), which is the semi-simplification of the reduction of the unique two-dimensional irreducible crystalline representation of \(\mathrm{Gal}(\overline{\mathbb{Q}}_p/\mathbb{Q}_p)\) of Hodge-Tage weights \(\{0,k-1\}\).
Reviewer: Gabor Wiese (Luxembourg)On Siegel eigenvarieties at Saito-Kurokawa pointshttps://zbmath.org/1517.110432023-09-22T14:21:46.120933Z"Berger, Tobias"https://zbmath.org/authors/?q=ai:berger.tobias"Betina, Adel"https://zbmath.org/authors/?q=ai:betina.adelSummary: We study the geometry of the \(p\)-adic Siegel eigenvariety \(\mathcal{E}\) of paramodular tame level at certain Saito-Kurokawa points having a critical slope. For \(k\ge 2\) let \(f\) be a cuspidal new eigenform of \(\mathrm{S}_{2k-2}(\Gamma_0(N))\) ordinary at a prime \(p\nmid N\) with sign \(\epsilon_f=-1\) and write \(\alpha\) for the \(p\)-adic unit root of the Hecke polynomial of \(f\) at \(p\). Let \(\pi_\alpha\) be the semi-ordinary \(p\)-stabilization of the Saito-Kurokawa lift of the cusp form \(f\) to \(\mathrm{GSp}(4)\) of weight \((k,k)\) and paramodular tame level. Under the assumption that the dimension of the Selmer group \(\mathrm{H}^1_{f,\mathrm{unr}}(\mathbb{Q},\rho_f(k-1))\) attached to \(f\) is at most one and some mild assumptions on the automorphic representation attached to \(f\), we show that \(\mathcal{E}\) is smooth at the point corresponding to \(\pi_\alpha\), and that the irreducible component of \(\mathcal{E}\) specializing to \(\pi_\alpha\) is not globally endoscopic. Finally we give an application to the Bloch-Kato conjecture, by proving under some mild assumptions that the smoothness failure of \(\mathcal{E}_{\Delta}\) at \(\pi_\alpha\) yields that \(\dim\mathrm{H}^1_{f,\mathrm{unr}}(\mathbb{Q},\rho_f(k-1))\geq 2\).On \(p\)-adic \(L\)-functions for \(\mathrm{GL}(2)\times\mathrm{GL}(3)\) via pullbacks of Saito-Kurokawa liftshttps://zbmath.org/1517.110442023-09-22T14:21:46.120933Z"Casazza, Daniele"https://zbmath.org/authors/?q=ai:casazza.daniele"de Vera-Piquero, Carlos"https://zbmath.org/authors/?q=ai:de-vera-piquero.carlosSummary: We build a one-variable \(p\)-adic \(L\)-function attached to two Hida families of ordinary \(p\)-stabilised newforms \(\mathbf{f},\mathbf{g}\), interpolating the algebraic part of the central values of the complex \(L\)-series \(L(f\otimes\text{Ad}^0(g),s)\) when \(f\) and \(g\) range over the classical specialisations of \(\mathbf{f},\mathbf{g}\) on a suitable line of the weight space. The construction rests on two major results: an explicit formula for the relevant complex central \(L\)-values, and the existence of non-trivial \(\Lambda\)-adic Shintani liftings and Saito-Kurokawa liftings studied in a previous work by the authors. We also illustrate that, under an appropriate sign assumption, this \(p\)-adic \(L\)-function arises as a factor of a triple product \(p\)-adic \(L\)-function attached to \(\mathbf{f},\mathbf{g}\), and \(\mathbf{g}\).Mod-2 dihedral Galois representations of prime conductorhttps://zbmath.org/1517.110452023-09-22T14:21:46.120933Z"Kedlaya, Kiran"https://zbmath.org/authors/?q=ai:kedlaya.kiran-sridhara"Medvedovsky, Anna"https://zbmath.org/authors/?q=ai:medvedovsky.annaSummary: For all odd primes \(N\) up to \(500000\), we compute the action of the Hecke operator \(T_2\) on the space \(S_2(\Gamma_0(N), \mathbb{Q})\) and determine whether or not the reduction mod 2 (with respect to a suitable basis) has 0 and/or 1 as eigenvalues. We then partially explain the results in terms of class field theory and modular mod-2 Galois representations. As a byproduct, we obtain some nonexistence results on elliptic curves and modular forms with certain mod-2 reductions, extending prior results of Setzer, Hadano, and Kida.
For the entire collection see [Zbl 1416.11009].Ramanujan-style congruences for prime levelhttps://zbmath.org/1517.110462023-09-22T14:21:46.120933Z"Kumar, Arvind"https://zbmath.org/authors/?q=ai:kumar.arvind"Kumari, Moni"https://zbmath.org/authors/?q=ai:kumari.moni"Moree, Pieter"https://zbmath.org/authors/?q=ai:moree.pieter"Singh, Sujeet Kumar"https://zbmath.org/authors/?q=ai:singh.sujeet-kumar.1The prototype of a Ramanujan congruence asserts that \(\tau(n) \equiv \sigma_{11}(n) \mod 691\) for every positive integer \(n\); it means we have (coefficient-wise) congruence between the unique cusp form \(\Delta(z) = \sum_{n=1}^{\infty}\tau(n)e^{2\pi inz}\) of weight \(12\) and level \(1\) and the normalized Eisenstein series \(E_{12}(z)\). There are several well-known ways to prove, interpret and generalize this [\textit{B. Datskovsky} and \textit{P. Guerzhoy}, Proc. Am. Math. Soc. 124, No. 8, 2283--2291 (1996; Zbl 0864.11023); \textit{N. Billerey} and \textit{R. Menares}, Math. Res. Lett. 23, No. 1, 15--41 (2016; Zbl 1417.11094); \textit{N. Dummigan} and \textit{D. Fretwell}, J. Number Theory 143, 248--261 (2014; Zbl 1304.11027); \textit{R. Gaba} and \textit{A. A. Popa}, J. Number Theory 193, 48--73 (2018; Zbl 1441.11100)].
The authors establish Ramanujan-style congruences modulo certain primes between an Eisenstein series of weight \(k\), prime level \(p\) and a cuspidal newform in the \(\varepsilon\)-eigenspace of the Atkin-Lehner operator inside the space \(S_k(p)\) of cusp forms of weight \(k\) for \(\Gamma_0(p)\) (Theorems 1.2 and 1.3). As an application they give a non-trivial lower bound of the degree of the number field generated by all normalized eigenforms in the space \(S_k(p)\).
Reviewer: Andrzej Dąbrowski (Szczecin)On Motohashi's formulahttps://zbmath.org/1517.110472023-09-22T14:21:46.120933Z"Wu, Han"https://zbmath.org/authors/?q=ai:wu.hanSummary: We complement and offer a new perspective of the proof of a Motohashi-type formula relating the fourth moment of \(L\)-functions for \(\mathrm{GL}_1\) with the third moment of \(L\)-functions for \(\mathrm{GL}_2\) over number fields, studied earlier by Michel-Venkatesh and Nelson. Our main tool is a new type of pre-trace formula with test functions on \(M_2(\mathbb{A})\) instead of \(\mathrm{GL}_2(\mathbb{A})\), on whose spectral side the matrix coefficients are replaced by the standard Godement-Jacquet zeta integrals. This is also a generalization of Bruggeman-Motohashi's other proof of Motohashi's formula. We give a variation of our method in the case of division quaternion algebras instead of \(M_2\), yielding a new spectral reciprocity, for which we are not sure if it is within the period formalism given by Michel-Venkatesh. We also indicate a further possible generalization, which seems to be beyond what the period method can offer.Analytic evaluation of Hecke eigenvalues for Siegel modular forms of degree twohttps://zbmath.org/1517.110482023-09-22T14:21:46.120933Z"Colman, Owen"https://zbmath.org/authors/?q=ai:colman.owen"Ghitza, Alexandru"https://zbmath.org/authors/?q=ai:ghitza.alexandru"Ryan, Nathan"https://zbmath.org/authors/?q=ai:ryan.nathan-cSummary: The standard approach to evaluate Hecke eigenvalues of a Siegel modular eigenform \(F\) is to determine a large number of Fourier coefficients of \(F\) and then compute the Hecke action on those coefficients. We present a new method based on the numerical evaluation of \(F\) at explicit points in the upper half-space and of its image under the Hecke operators. The approach is more efficient than the standard method and has the potential for further optimization by identifying good candidates for the points of evaluation, or finding ways of lowering the truncation bound. A limitation of the algorithm is that it returns floating point numbers for the eigenvalues; however, the working precision can be adjusted at will to yield as close an approximation as needed.
For the entire collection see [Zbl 1416.11009].Modular forms on \(\mathrm{SU}(2,1)\) with weight \(\frac{1}{3}\)https://zbmath.org/1517.110492023-09-22T14:21:46.120933Z"Freitag, Eberhard"https://zbmath.org/authors/?q=ai:freitag.eberhard"Hill, Richard M."https://zbmath.org/authors/?q=ai:hill.richard-michaelSummary: In this note, we describe several new examples of holomorphic modular forms on the group \(\mathrm{SU}(2,1)\). These forms are distinguished by having weight \(\frac{1}{3}\). We also describe a method for determining the levels at which one should expect to find such fractional weight forms.Triple product \(p\)-adic \(L\)-function attached to \(p\)-adic families of modular formshttps://zbmath.org/1517.110502023-09-22T14:21:46.120933Z"Fukunaga, Kengo"https://zbmath.org/authors/?q=ai:fukunaga.kengo\textit{M.-L. Hsieh} constructed the three-variable \(p\)-adic \(L\)-functions attached to Hida families of elliptic newforms and proved interpolation formulas [Am. J. Math. 143, No. 2, 411--532 (2021; Zbl 1470.11115)]. The author generalizes his result in the unbalanced case and constructs a three-variable triple product \(p\)-adic \(L\)-function attached to a primitive Hida family and two more general \(p\)-adic families of modular forms (Coleman families and CM-families).
Reviewer: Andrzej Dąbrowski (Szczecin)The Eisenstein and winding elements of modular symbols for odd square-free levelhttps://zbmath.org/1517.110512023-09-22T14:21:46.120933Z"Krishnamoorthy, Srilakshmi"https://zbmath.org/authors/?q=ai:krishnamoorthy.srilakshmiSummary: We explicitly write down the \textit{Eisenstein elements} inside the space of modular symbols for Eisenstein series with integer coefficients for the congruence subgroups \(\Gamma_0(N)\) with \(N\) odd square-free. We also compute the \textit{winding elements} explicitly for these congruence subgroups. Our results are explicit versions of the Manin-Drinfeld Theorem (Thm. 6). These results are the generalization of the paper [\textit{D. Banerjee} and the author, Pac. J. Math. 281, No. 2, 257--285 (2016; Zbl 1341.11023)] results to odd square-free level.Appearance of the Kashiwara-Saito singularity in the representation theory of \(p\)-adic \(\mathrm{GL}(16)\)https://zbmath.org/1517.110522023-09-22T14:21:46.120933Z"Cunningham, Clifton"https://zbmath.org/authors/?q=ai:cunningham.clifton-l-r"Fiori, Andrew"https://zbmath.org/authors/?q=ai:fiori.andrew"Kitt, Nicole"https://zbmath.org/authors/?q=ai:kitt.nicoleIn the paper under the review, the authors calculate certain ABV-packets for the general linear groups, using a method which should be useful for both algorithmic implementation and symbolic calculation. It is known that A-packets for general linear groups are \(L\)-packets which are singletons. The authors prove that the ABV-packets for general linear groups do not have to be singletons. They obtain an unramified Langlands parameter for the group \(\mathrm{GL}_{16}(F)\), for a \(p\)-adic field \(F\), such that the corresponding ABV-packet contains exactly two representations.
A consequence of this result is the existence of nonsingleton ABV-packets of \(\mathrm{GL}_n(F)\) for all \(n \geq 16\). It is also expected that the obtained example for the group \(\mathrm{GL}_{16}(F)\) is the simplest example of an admissible representation of the general linear group with a corona.
Reviewer: Ivan Matić (Osijek)Derived parabolic inductionhttps://zbmath.org/1517.110532023-09-22T14:21:46.120933Z"Scherotzke, Sarah"https://zbmath.org/authors/?q=ai:scherotzke.sarah"Schneider, Peter"https://zbmath.org/authors/?q=ai:schneider.peter.6|schneider.peter.5|schneider.peter.3|schneider.peter.2|schneider.peter.1Summary: The classical parabolic induction functor is a fundamental tool on the representation theoretic side of the Langlands program. In this article, we study its derived version. It was shown by the second author that the derived category of smooth \(G\)-representations over \(k, G\) a \(p\)-adic reductive group and \(k\) a field of characteristic \(p\), is equivalent to the derived category of a certain differential graded \(k\)-algebra \(H_G^\bullet \), whose zeroth cohomology is a classical Hecke algebra. This equivalence predicts the existence of a derived parabolic induction functor on the dg Hecke algebra side, which we construct in this paper. This relies on the theory of six-functor formalisms for differential graded categories developed by O. Schnürer. We also discuss the adjoint functors of derived parabolic induction.The Barnes-Hurwitz zeta cocycle on \(\operatorname{PGL}_2(\mathbb{Q})\)https://zbmath.org/1517.110542023-09-22T14:21:46.120933Z"Espinoza, Milton"https://zbmath.org/authors/?q=ai:espinoza.miltonThe author uses Barnes' double zeta-function given for positive real \(z,\omega_1,\omega_2\) and complex \(s\) with \(\Re(s) >2\) by
\[
\zeta_2(z,\binom{\omega_1}{\omega_2},s) := \sum_{m=0}^\infty \sum_{n=0}^\infty (z + m\omega_1 + n\omega_2)^{-s},
\]
and meromorphically extended (in \(s\)) to the complex plane in order to define a certain non-vanishing 1-cocycle on \(G=\mathrm{PGL}_2(\mathbb{Q})\) with values in a certain module \(\mathcal{D}\) of distributions.
More precisely, the distributions under consideration are certain maps defined on triples \((x,\omega,s)\) with \(x\in \mathbb{Q}^2 / \mathbb{Z}^2,\) \(\omega = \binom{\omega_1}{\omega_2}\) with incommensurable real \(\omega_1,\omega_2,\) and which depend meromorphically on \(s,\) have values in \(\mathbb{P}^1(\mathbb{C})\) and obey several other conditions. If \(\gamma\) is an integer primitive \(2\times 2\)-matrix with non-zero determinant and class \([\gamma]\in G,\) then for \(\nu\in \mathcal{D}\) a new distribution \([\gamma]\cdot \nu\) is defined by
\[
([\gamma]\cdot \nu)(x,\omega,s) := \operatorname{sgn}(\det(\gamma))\cdot \sum_{\mu\in \mathbb{Z}^2 / \mathbb{Z}^2\gamma^\top} \nu((x+\mu)(\gamma^\top)^{-1}, \gamma\top \omega , s).
\]
This definition gives the \(G\)-module structure on \(\mathcal{D}\).
The author then defines a 1-cocycle \(\mathfrak{Z} \in Z^1(G,\mathcal{D})\) which turns out to be parabolic and to have non-zero cohomology class. The values of the distribution \(\mathfrak{Z}\left[\begin{matrix} 0 & -1 \\ 1 & 0 \end{matrix}\right]\) at \((x,\omega,s)\) are calculated to be (if \(x_1,x_2\not\in \mathbb{Z}\))
\[
\zeta_2(\langle x_1\rangle \omega_1 + \langle x_2\rangle \omega_2, \omega , s),
\]
where \(\langle r\rangle\) denotes the smallest positive element in \(r+\mathbb{Z}.\) If \(x_1\) or \(x_2\) are integers, there is an extra summand involving the Hurwitz zeta function.
Lastly the author is able to show that there are no non-zero \(G\)-invariant elements in \(\mathcal{D},\) using elementary facts about real quadratic fields.
Reviewer: Stefan Kühnlein (Karlsruhe)Patching and the completed homology of locally symmetric spaceshttps://zbmath.org/1517.110552023-09-22T14:21:46.120933Z"Gee, Toby"https://zbmath.org/authors/?q=ai:gee.toby"Newton, James"https://zbmath.org/authors/?q=ai:newton.jamesThis paper generalises and unites two significant recent developments concerning the Taylor-Wiles patching method:
\textit{F. Calegari} and \textit{D. Geraghty} overcame in [Invent. Math. 211, No. 1, 297--433 (2018; Zbl 1476.11078)] the necessity to restrict to discrete series, i.e. cohomology supported only in middle degree, by patching chain complexes rather than homology groups. This construction was subsequently refined by \textit{C. B. Khare} and \textit{J. A. Thorne} [Am. J. Math. 139, No. 5, 1205--1273 (2017; Zbl 1404.11076)].
\textit{A. Caraiani} et al. [Camb. J. Math. 4, No. 2, 197--287 (2016; Zbl 1403.11073)] introduced the idea of patching completed cohomology in order to provide a new approach to the \(p\)-adic Langlands correspondence to more general groups than \(\mathrm{GL}_2/{\mathbb Q}\). This method was refined by \textit{P. Scholze} [Ann. Sci. Éc. Norm. Supér. (4) 51, No. 4, 811--863 (2018; Zbl 1419.14031)] using ultrafilters.
The paper under review has a similar goal and combines the above methods to directly patch complexes computing homology rather than minimal resolutions to obtain natural actions of the Hecke algebras and \(p\)-adic groups at the level of complexes.
More concretely, write \(\mathcal O\) for the ring of integers of a finite extension \(E/{\mathbb Q}_p\) and consider the group \(\mathrm{PGL}_n\) over a general number field \(F\). Some choice of Taylor-Wiles primes determines two power series rings \(\mathcal O_\infty\) and \(R_\infty\) over \(\mathcal O\) and over the completed tensor product of certain local Galois deformation rings for \(v\mid p\) in \(F\).
the new patching procedure produces a perfect complex \(\widetilde{\mathcal C}(\infty)\) of \(\mathcal O_\infty[[\prod_{v\mid p}\mathrm{PGL}_n({\mathcal O}_{F_v})]]\)-modules, equipped with an \(\mathcal O_\infty\)-linear action of \(\prod_{v\mid p}\mathrm{PGL}_n({F_v})\) and an \(\mathcal O_\infty\)-linear action
\[
R_\infty\to\mathrm{End}_{D(\mathcal O_\infty)}(\widetilde{\mathcal C}(\infty)),
\]
commuting with the action of \(\prod_{v\mid p}\mathrm{PGL}_n({F_v})\).
Reducing modulo a suitable ideal \(\mathbf{a}\) of \(\mathcal O_\infty\) the above complex computes the completed comology
\[
\widetilde{H}_\ast(X_{U^p},\mathcal O)_{\mathfrak{m}}:= \varprojlim_{U_p}H_\ast(X_{U_pU^p},\mathcal O)_{\mathfrak{m}}
\]
localized at a non-Eisenstein maximal ideal \(\mathfrak{m}\) of a `big' Hecke algebra \(\mathbb{T}^S(U^p)\).
Assuming a standard non-vanishing conjecture by Calegary and Geraghty on the vanishing of homology with coefficients in the residue field of \(\mathcal O\) localized at \(\mathfrak{m}\) outside the cuspidal range, and a second conjecture of Calegary and Emerton that codimension of completed cohomology is at least the length \(l_0\) of the cuspidal range, it is shown that the cohomology of the complex \(\widetilde{\mathcal C}(\infty)\) vanishes outside the bottom degree \(q_0\) and is Cohen-Macaulay over \(\mathcal O_\infty[[\prod_{v\mid p}\mathrm{PGL}_n({\mathcal O}_{F_v})]]\) and \(R_\infty[[\prod_{v\mid p}\mathrm{PGL}_n({\mathcal O}_{F_v})]]\). Moreover the corresponding projective dimensions are computed.
Moreover, assuming an additional codimension inequality, flatness of the cohomology of \(\widetilde{\mathcal C}(\infty)\) over \(R_\infty\) is established, it is shown that \(R_\infty\mathbf{a}\) is generated by a regular sequence, and that for a suitable `big' Galois deformation ring \(R\), there are canonical isomorphisms
\[
R_\infty/{\mathbf{a}}\to R\to\mathbb{T}^S(U^p)_{\mathfrak{m}},
\]
thus establishing an `\(R=\mathbb{T}\)' theorem. Those rings are local complete intersections with Krull dimension \(1+(\frac{n(n+1)}{2}-1)[F:{\mathbb{Q}}]-l_0\).
Remarkably, freeness of \(\widetilde{H}_{q_0}(X_{U^p},\mathcal O)_{\mathfrak{m}}\) is not needed and not established. The authors prove that this cohomology is faithfully flat over \(\mathbb{T}^S(U^p)_{\mathfrak{m}}\).
Reviewer: Fabian Januszewski (Paderborn)Galois representations with big image in the general symplectic group \(\mathrm{GSp}_4( \mathbb{Z}_p)\)https://zbmath.org/1517.110562023-09-22T14:21:46.120933Z"Maletto, Simone"https://zbmath.org/authors/?q=ai:maletto.simoneThe author is interested in the following question: Given a prime number \(p\), does there exist a continuous Galois representation \(\rho: \mathrm{Gal}(\overline{\mathbb Q}/\mathbb Q) \to \mathrm{GSp}_4(\mathbb Z_p)\) satisfying the following two properties (i) \(\rho\) is unramified at all primes \(l \not= p\), (ii) the image of \(\rho\) contains a finite index subgroup of \(\mathrm{Sp}_4(\mathbb Z_p)\) ? A related question was first studied by \textit{R. Greenberg} [Ann. Math. Qué. 40, No. 1, 83--119 (2016; Zbl 1414.11151)], who showed that if \(p\) is a regular prime such that \(p > 4[n/2] + 1\), then there is a continuous Galois representation \(\rho: \mathrm{Gal}(\overline{\mathbb Q}/\mathbb Q) \to \mathrm{GL}(\mathbb Z_p)\) which is unramified away from \(p\) and has big image.
Let \(e_p\) denote the irregularity index of an odd prime \(p\). If \(4e_p + 8 < \frac{p-1}{2}\), then the author can construct the corresponding examples. The main idea is to start from a residual diagonal representation and deform it, under certain hypothesis, producing a characteristic zero lift (a variant of the strategy used by \textit{A. Ray} [J. Number Theory 222, 168--180 (2021; Zbl 1470.11141)]). An example of how to find such representation in the first nontrivial case of \(p=37\) is given.
Reviewer: Andrzej Dąbrowski (Szczecin)Ranks, 2\-Selmer groups, and Tamagawa numbers of elliptic curves with \(\mathbb{Z} /2\mathbb{Z} \times \mathbb{Z} /8\mathbb{Z}\)-torsionhttps://zbmath.org/1517.110572023-09-22T14:21:46.120933Z"Chan, Stephanie"https://zbmath.org/authors/?q=ai:chan.stephanie"Hanselman, Jeroen"https://zbmath.org/authors/?q=ai:hanselman.jeroen"Li, Wanlin"https://zbmath.org/authors/?q=ai:li.wanlinSummary: In 2016, \textit{J. S. Balakrishnan} et al. [LMS J. Comput. Math. 19A, 351--370 (2016; Zbl 1391.11077)] produced a database of elliptic curves over \(\mathbb{Q}\) ordered by height in which they computed the rank, the size of the \(2\)-Selmer group, and other arithmetic invariants. They observed that after a certain point, the average rank seemed to decrease as the height increased. Here we consider the family of elliptic curves over \(\mathbb{Q}\) whose rational torsion subgroup is isomorphic to \(\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/8\mathbb{Z}\). Conditional on GRH and BSD, we compute the rank of \(92\%\) of the \(202,\!461\) curves with parameter height less than \(10^3\). We also compute the size of the \(2\)-Selmer group and the Tamagawa product, and prove that their averages tend to infinity for this family.
For the entire collection see [Zbl 1416.11009].The Lang-Trotter conjecture for products of non-CM elliptic curveshttps://zbmath.org/1517.110582023-09-22T14:21:46.120933Z"Chen, Hao"https://zbmath.org/authors/?q=ai:chen.hao.7"Jones, Nathan"https://zbmath.org/authors/?q=ai:jones.nathan-a"Serban, Vlad"https://zbmath.org/authors/?q=ai:serban.vladSummary: Inspired by the work of Lang-Trotter on the densities of primes with fixed Frobenius traces for elliptic curves defined over \(\mathbb{Q}\) and by the subsequent generalization of \textit{A. C. Cojocaru} et al. [Int. Math. Res. Not. 2017, No. 12, 3557--3602 (2017; Zbl 1405.11078)] to generic abelian varieties, we study the analogous question for abelian surfaces isogenous to products of non-CM elliptic curves over \(\mathbb{Q}\) that are not \(\overline{\mathbb{Q}}\)-isogenous. We formulate the corresponding conjectural asymptotic, provide upper bounds, and explicitly compute (when the elliptic curves lie outside a thin set) the arithmetically significant constants appearing in the asymptotic. This allows us to provide computational evidence for the conjecture.A group theoretic perspective on entanglements of division fieldshttps://zbmath.org/1517.110592023-09-22T14:21:46.120933Z"Daniels, Harris B."https://zbmath.org/authors/?q=ai:daniels.harris-b"Morrow, Jackson S."https://zbmath.org/authors/?q=ai:morrow.jackson-salvatoreThe aim of the paper under review is to provide an abstract, group-theoretic framework which encompasses the various possible ways in which the family of division fields of an elliptic curve defined over \(\mathbb{Q}\) can be entangled. In particular, the authors give explanations or classifications for all the types of entanglement can can occur at the level of two primes \(p,q \in \mathbb{Z}\) for infinitely many elliptic curves.
Let us be more precise. Let \(E\) be an elliptic curve defined over \(\mathbb{Q}\), and \(a, b \in \mathbb{Z}_{\geq 1}\) be positive integers. Then, the authors say that \(E\) has an \((a,b)\)-entanglement if the intersection of the division fields \(\mathbb{Q}(E[a]), \mathbb{Q}(E[b])\) is bigger than \(\mathbb{Q}(E[d])\) where \(d = \gcd(a,b)\). The type of this entanglement is then defined to be the group \(T = \mathrm{Gal}(\mathbb{Q}(E[a]) \cap \mathbb{Q}(E[b])/\mathbb{Q}(E[d]))\).
The first main new contribution of the paper under review is an abstract group theoretic framework which describes the subgroups \(G \subseteq \mathrm{GL}_2(\mathbb{Z}/n \mathbb{Z})\) that represent entanglements. More precisely, Definition 3.1 of the paper under review says that the subgroup \(G\) represents an \((a,b)\)-entanglement, where \(a,b\) are two divisors of \(n\), if and only if the subgroup \(N_{a,b}(G) \subseteq \mathrm{GL}_2(\mathbb{Z}/c \mathbb{Z})\) generated by \(\ker(\pi_{c,a}) \cap \pi_{n,c}(G)\) and \(\ker(\pi_{c,b}) \cap \pi_{n,c}(G)\) differs from the subgroup generated by \(\ker(\pi_{c,d}) \cap \pi_{n,c}(G)\), where we set \(c = \mathrm{lcm}(a,b)\) and \(d = \gcd(a,b)\), and for every \(M \mid N\) we denote by \(\pi_{N,M} \colon \mathrm{GL}_2(\mathbb{Z}/N \mathbb{Z}) \twoheadrightarrow \mathrm{GL}_2(\mathbb{Z}/M \mathbb{Z})\) the canonical reduction map. Moreover, the authors introduce a notion of primitiveness of triples \(((a,b),G)\) where \(G\) represents an \((a,b)\)-entanglement, and that such primitive entanglements are explained whenever one has the equality of indices \(\lvert (\mathbb{Z}/c \mathbb{Z})^\times \colon \det(N_{a,b}(G)) \rvert = \lvert \pi_{n,c}(G) \colon N_{a,b}(G) \rvert\), where \(\det \colon \mathrm{GL}_2(\mathbb{Z}/c \mathbb{Z}) \twoheadrightarrow (\mathbb{Z}/c \mathbb{Z})^\times\) denotes the determinant map.
The fourth section of the paper under review is then devoted to explain how the group theoretic language introduced above encompasses many properties of entanglement fields of elliptic curves. For instance, the image of the Galois representation
\[
\rho_{E,n} \colon G_\mathbb{Q} \to \mathrm{Aut}(E(\overline{\mathbb{Q}})_\text{tors}) \cong \mathrm{GL}_2(\mathbb{Z}/n \mathbb{Z})
\]
represents an explained \((a,b)\)-entanglement if and only if the intersection \(\mathbb{Q}(E[a]) \cap \mathbb{Q}(E[b])\) is bigger than \(\mathbb{Q}(E[d])\) but equals the compositum of \(\mathbb{Q}(E[a]) \cap \mathbb{Q}(\zeta_b)\) and \(\mathbb{Q}(\zeta_a) \cap \mathbb{Q}(E[b])\). In particular, these entanglements are explained by the inclusions \(\mathbb{Q}(\zeta_N) \subseteq \mathbb{Q}(E[N])\), which hold true for every \(N \in \mathbb{Z}_{\geq 1}\) as a consequence of the existence of the Weil pairing. Moreover, the fourth section of the paper under review contains many examples of unexplained entanglements. In particular, the authors observe in Example 4.8 that it is sufficient in many cases to consider only the maximal subgroups that represent an entanglement of given level and type. This is certainly the case if one wants to study elliptic curves which have an entanglement represented by a subgroup \(G \subseteq \mathrm{GL}_2(\mathbb{Z}/n \mathbb{Z})\) by studying the rational points of the corresponding modular curve \(X_G\), which parametrizes elliptic curves \(E\) such that the image of \(\rho_{E,n}\) is contained in a conjugate of \(G\).
The fifth, sixth and seventh section of the paper under review are then devoted to the study of such modular curves which have an infinite number of rational points, and such that \(a = p\) and \(b = q\) are primes. More precisely, the authors determine that in this case \(n = p q \in \{6, 10, 14, 15, 21, 22, 26, 33, 39 \}\), and one has a total of \(24\) conjugacy classes of such possible subgroups \(G \subseteq \mathrm{GL}_2(\mathbb{Z}/n \mathbb{Z})\) which represent an unexplained \((p,q)\)-entanglement, corresponding to \(22\) modular curves of genus zero, and \(2\) modular curve of genus one having infinitely many rational points. In particular, the authors note that these two modular curves are isomorphic over \(\mathbb{Q}\), a fact which is reminiscent of the exceptional isomorphism proved by \textit{B. Baran} [J. Number Theory 145, 273--300 (2014; Zbl 1300.11055)]. To conclude, the authors compute explicit models and \(j\)-maps for each one of these modular curves, which yields an explicit classification of the \(\overline{\mathbb{Q}}\)-isomorphism classes of all the existing families of elliptic curves defined over \(\mathbb{Q}\) having an unexplained entanglement of level \(p \cdot q\).
To conclude, the paper under review provides a new interesting abstract framework to classify entanglements, which was already generalized and strengthened in a subsequent work by \textit{H. B. Daniels} et al. [``Towards a classification of entanglements of Galois representations attached to elliptic curves '', Preprint, \url{arXiv:2105.02060}].
Reviewer: Riccardo Pengo (Bonn)Unlikely intersections for curves in products of Carlitz moduleshttps://zbmath.org/1517.110602023-09-22T14:21:46.120933Z"Brownawell, W. D."https://zbmath.org/authors/?q=ai:brownawell.w-dale"Masser, D."https://zbmath.org/authors/?q=ai:masser.david-williamThe paper under review continues the research on the Zilber-Pink problem, which generalizes the results of Manin-Mumford and Mordell-Lang. This problem was originally studied over zero characteristic, and the present paper focuses on its study in positive characteristic. The second author of the present paper initiated this research in [\textit{D. Masser}, Q. J. Math. 65, No. 2, 505--515 (2014; Zbl 1317.11067)], focusing on curves in multiplicative groups. The study was further developed by the authors of the current paper in [Proc. Am. Math. Soc. 145, No. 11, 4617--4627 (2017; Zbl 1401.11110)], this time examining curves in additive groups equipped with an ``\(F\)-structure'', where ``\(F\)'' denotes the action of the Frobenius map. The present paper extends this exploration by considering curves in additive groups equipped with the Carlitz action.
Let \(p\) be a fixed prime number, and let \(K\) be an algebraically closed field containing \({\mathbb F}_{p}(t)\). Let \(C\) be an irreducible curve in \({\mathbf G}_{a}^{n}\) defined over \(K.\) The problem is to understand how \(C\) intersects with algebraic subgroups of \({\mathbf G}_{a}^{n}\) with suitable codimensions. Over positive characteristic, there are well-known counterexamples to the naive analogues of the conjectures due to \textit{B. Zilber} [J. Lond. Math. Soc., II. Ser. 65, No. 1, 27--44 (2002; Zbl 1030.11073)] and to \textit{R. Pink} [``A common generalization of the conjectures of André-Oort, Manin-Mumford, and Mordell-Lang'', Preprint, \url{https://people.math.ethz.ch/~pink/ftp/AOMMML.pdf}] for situations over zero characteristic. The results of the two papers mentioned in the first paragraph suggest that extra structures on \({\mathbf G}_{a}^{n}\) are needed to have an analogue parallel to the situation over zero characteristic.
Recall that the additive group \({\mathbf G}_{a}(K)\) has the structure of a Carlitz module under the action of the ring \(R := {\mathbb F}_{p}[{\mathcal C}]\), where \({\mathcal C} x = t x + x^{p}\) for \(x\in {\mathbf G}_{a}(K)\). The action of \(R\) naturally extends to \({\mathbf G}_{a}^{n}(K)\) by acting diagonally, i.e., \(\alpha (x_{1},\ldots, x_{n}) = (\alpha x_{1}, \ldots , \alpha x_{n})\) for \(\alpha \in R\). Here, an algebraic subgroup of \({\mathbf G}_{a}^{n}\) is an \(R\)-submodules defined by several equations of the form \(\alpha_{1} x_{1} + \cdots + \alpha_{n} x_{n} = 0\) where \(\alpha_{1}, \ldots, \alpha_{n}\) are in \(R.\) In this setting, the authors formulate a conjecture that is parallel to the situation over zero characteristic.
Conjecture: assume for any non-zero \((\rho_{1},\ldots, \rho_{n})\) in \(R^{n}\) that the form \(\rho_{1}x_{1} + \cdots + \rho_{n} x_{n}\) is not identically zero on \(C\). Then there are at most finitely many \((\xi_{1},\ldots,\xi_{n})\) in \(C(K)\) for which there exist linearly independent \((\alpha_{1}, \ldots, \alpha_{n}), (\beta_{1}, \ldots, \beta_{n}\)) in \(R^{n}\) such that \(\alpha_{1} \xi_{1} + \cdots + \alpha_{n} \xi_{n} = \alpha_{1} \xi_{1} + \cdots + \alpha_{n} \xi_{n} = 0\).
The main result of this paper is a proof for the case when \(n=3\). It is worth noting that there is nothing to prove if \(n=1\), and the case when \(n=2\) has already been proven by \textit{T. Scanlon} in the general context of Drinfeld modules, using techniques from model theory [J. Number Theory 97, No. 1, 10--25 (2002; Zbl 1055.11037)].
The proof of the main result closely follows the strategy presented in [\textit{E. Bombieri} et al., Int. Math. Res. Not. 1999, No. 20, 1119--1140 (1999; Zbl 0938.11031)]. The main ingredient is a new relative version of Dobrowolski's classical lower bound for canonical heights in the Carlitz context, along with a suitable upper bound. For higher-dimensional cases, the authors suggest possible approaches to tackle the problem.
Reviewer: Liang-Chung Hsia (Taipei)Remarks on rank one Drinfeld modules and their torsion elementshttps://zbmath.org/1517.110612023-09-22T14:21:46.120933Z"El Kati, Mohamed"https://zbmath.org/authors/?q=ai:el-kati.mohamed"Oukhaba, Hassan"https://zbmath.org/authors/?q=ai:oukhaba.hassanThe torsion elements of rank one Drinfeld modules of a global function field \(k\) give an explicit description of abelian class fields, produce units, Euler systems and annihilators of the ideal class group of the abelian extensions of \(k\). In this paper, the authors consider these torsion elements.
Let \(\rho\) be a sgn-normalized Drinfeld \(A\)-module of generic characteristic, where \(A\) is the Dedekind ring of elements of \(k\) regular outside a fixed place \(\infty\) of \(k\). Let \(\Omega\) be the completion of a fixed algebraic closure of \(k_{\infty}\), the completion of \(k\) at \(\infty\). Let \(H_A^*\) be the subfield of \(\Omega\) generated by the coefficients of the polynomials \(\rho_x\), for \(x\in A\). David Hayes called \(H_A^*\) the {\em normalizing field with respect to} sgn.
The first main result of this paper is that if \(E_{\rho}\subseteq H_A^* [\tau]\) is the vector space generated by all the polynomials \(\rho_x\), \(x\in A\), then \(R_{\rho}=H_A^*[\tau]/E_{\rho}\) is a finitely generated vector space of dimension less than or equal to the genus \(g\) of \(k\). Further, if \(\infty\) is of degree \(1\), then \(\dim_{H_A^*} ( R_{\rho})=g\).
The second main result is the following. There exists a suitable ideal \({\mathfrak m}_{\rho}\) such that if \({\mathfrak m}\) is an ideal of \(A\) prime to \({\mathfrak m}_{\rho}\), then the \(B\)-module \(B\Lambda_{ \mathfrak m}\) is free of rank \(\deg({\mathfrak m})-s+1\) if \(q=2\) and \(s\geq 1\) and of rank \(\deg({\mathfrak m})\) if \(q>2\) or (\(q=2\) and \(s=0\)), where \(s\) is the number of prime ideals \({\mathfrak p}\) dividing \({\mathfrak m}\) and such that \(\deg({\mathfrak p})=1\), \(\Lambda_{\mathfrak m}\) is the \(A\)-module of the \({\mathfrak m}\)-torsion elements of \(\Omega\) and \(B\) is the integral closure of \(A\) in \(H_A^*\).
Reviewer: Gabriel D. Villa Salvador (Ciudad de México)CM abelian varieties and large wild finite monodromyhttps://zbmath.org/1517.110622023-09-22T14:21:46.120933Z"Philip, Séverin"https://zbmath.org/authors/?q=ai:philip.severinSummary: In this paper we study the wild part of the finite monodromy groups of abelian varieties over number fields. We solve Grunwald problems for groups of the form \(\mathbb{Z}/p\mathbb{Z} \,\wr\, \mathfrak{S}_n\) over number fields to build CM abelian varieties with maximal wild finite monodromy in the odd prime case. For the even prime case we prove a new bound on the 2-part of the order of the finite monodromy group for CM abelian varieties and build varieties that reach it.Principally polarized squares of elliptic curves with field of moduli equal to \(\mathbb{Q}\)https://zbmath.org/1517.110632023-09-22T14:21:46.120933Z"Gélin, Alexandre"https://zbmath.org/authors/?q=ai:gelin.alexandre"Howe, Everett"https://zbmath.org/authors/?q=ai:howe.everett-w"Ritzenthaler, Christophe"https://zbmath.org/authors/?q=ai:ritzenthaler.christopheSummary: We give equations for \(13\) genus-\(2\) curves over \(\overline{\mathbb{Q}}\), with models over \(\mathbb{Q}\), whose unpolarized Jacobians are isomorphic to the square of an elliptic curve with complex multiplication by a maximal order. If the generalized Riemann hypothesis is true, there are no further examples of such curves. More generally, we prove under the generalized Riemann hypothesis that there exist exactly \(46\) genus-\(2\) curves over \(\overline{\mathbb{Q}}\) with field of moduli \(\mathbb{Q}\) whose Jacobians are isomorphic to the square of an elliptic curve with complex multiplication by a maximal order.
For the entire collection see [Zbl 1416.11009].The derived Hecke algebra for dihedral weight one formshttps://zbmath.org/1517.110642023-09-22T14:21:46.120933Z"Darmon, Henri"https://zbmath.org/authors/?q=ai:darmon.henri"Harris, Michael"https://zbmath.org/authors/?q=ai:harris.michael-howard"Rotger, Victor"https://zbmath.org/authors/?q=ai:rotger.victor"Venkatesh, Akshay"https://zbmath.org/authors/?q=ai:venkatesh.akshayA conjecture proposed by Venkatesh and others posits that when spaces of automorphic forms show up in different degrees of cohomology, they are related by the action of a motivic cohomology group. The paper [\textit{M. Harris} and \textit{A. Venkatesh}, Exp. Math. 28, No. 3, 342--361 (2019; Zbl 1480.11049)] formulates an explicit version of this conjecture in the case of weight \(1\) modular forms. The present paper then proves the Harris-Venkatesh conjecture in the \textit{dihedral} case under additional ramification assumptions.
We now describe in slightly more detail the Harris-Venkatesh conjecture and the contents of this paper. The Harris-Venkatesh conjecture computes a pairing \(\langle G, \mathscr{G} \rangle\) arising from Serre duality. The element \(G \in H^0(X^0(N), \Omega^1)\) is a weight \(2\) level \(N\) cusp form given as the trace \(X_0(Nd) \to X_0(N)\) of \(g(z)g^*(NZ) \in S_2(Nd)\), where \(g \in H^0(X_1(d), \omega)\) is a weight \(1\) level \(d\) newform and \(g^*\) is the newform whose Fourier expansion is the complex conjugate of that of \(g\). The element \(\mathscr{G} \in H^1(\overline{X}, \mathcal{O}_{\overline{X}})\) (where \(\overline{X} = X_0(N)_{\mathbb{Z}/p^t\mathbb{Z}}\) and \(p^t\) is the highest power of \(p\) dividing \(N-1\)) is the \emph{Shimura Class}, coming from the class in \(H^1_{\mathrm{et}}(X_0(N), (\mathbb{Z} / N \mathbb{Z})^{\times} \otimes \mathbb{Z}[\frac{1}{6}])\) corresponding to the cover \(X_1(N) \to X_0(N)\) via a fixed discrete logarithm \(\mathrm{log}: (\mathbb{Z}/N\mathbb{Z})^{\times} \to \mathbb{Z} / p^t \mathbb{Z}\). Then the Harris-Venkatesh conjecture says this pairing is equal (up to constant independent of \(N\) and \(p\)) to \(\mathrm{log}(\mathrm{red}_N(u_g))\), the discrete logarithm of the modulo \(N\) reduction of a certain unit \(u_g\) attached to \(g\).
To carry out the computation, they transport the computation to a suitable quaternion algebra \(B\). In particular, there is a theta lift \(\Theta: \text{modular forms on } B \to S_2(N)\) and (up to something pairing trivially with \(\mathscr{G}\)) one has \(G = \Theta(Z_{K,\psi})\) where \(Z_{K,\psi}\) is a certain Heegner cycle, equal to a formal combination of supersingular points in characteristic \(N\). Dually, one has \(\Theta^*(\mathscr{G}) = \mathfrak{U}_N\) where \(\mathfrak{U}_N\) is an explicit \textit{higher Eisenstein element} as in [\textit{E. Lecouturier}, Invent. Math. 223, No. 2, 485--595 (2021; Zbl 1472.11134)]. Hence, one reduces to evaluating \(\langle Z_{K, \psi} , \mathfrak{U}_N \rangle\) which is possible due to explicit constructions of units available in the dihedral case.
Reviewer: Alexander Bertoloni Meli (Ann Arbor)Motivic action on coherent cohomology of Hilbert modular varietieshttps://zbmath.org/1517.110652023-09-22T14:21:46.120933Z"Horawa, Aleksander"https://zbmath.org/authors/?q=ai:horawa.aleksanderSummary: We propose an action of a certain motivic cohomology group on the coherent cohomology of Hilbert modular varieties, extending conjectures of Venkatesh, Prasanna, and Harris. The action is described in two ways: on cohomology modulo \(p\) and over \(\mathbb{C}\), and we conjecture that they both lift to an action on cohomology with integral coefficients. The conjecture is supported by theoretical evidence based on Stark's conjecture on special values of Artin \(L\)-functions and by numerical evidence in base change cases.Optimal curves of genus 3 over finite field with discriminant \(-19\)https://zbmath.org/1517.110662023-09-22T14:21:46.120933Z"Alekseenko, E. S."https://zbmath.org/authors/?q=ai:alekseenko.e-s"Aleshnikov, S. I."https://zbmath.org/authors/?q=ai:aleshnikov.s-i"Zaytsev, A. I."https://zbmath.org/authors/?q=ai:zaytsev.a-iSummary: In the paper, we obtain the equations for optimal curves of genus 3 defined over a finite field of the discriminant \(-19\).Curves of fixed gonality with many rational pointshttps://zbmath.org/1517.110672023-09-22T14:21:46.120933Z"Vermeulen, Floris"https://zbmath.org/authors/?q=ai:vermeulen.florisSummary: Given an integer \(\gamma \geq 2\) and an odd prime power \(q\) we show that for every large genus \(g\) there exists a non-singular curve \(C\) defined over \(\mathbb{F}_q\) of genus \(g\) and gonality \(\gamma\) and with exactly \(\gamma(q+1)\mathbb{F}_q\)-rational points. This is the maximal number of rational points possible. This answers a recent conjecture by Faber-Grantham. Our methods are based on curves on toric surfaces and Poonen's work on squarefree values of polynomials.Geometric quadratic Chabauty over number fieldshttps://zbmath.org/1517.110682023-09-22T14:21:46.120933Z"Čoupek, Pavel"https://zbmath.org/authors/?q=ai:coupek.pavel"Lilienfeldt, David T.-B. G."https://zbmath.org/authors/?q=ai:lilienfeldt.david-t-b-g"Xiao, Luciena X."https://zbmath.org/authors/?q=ai:xiao.luciena-x"Yao, Zijian"https://zbmath.org/authors/?q=ai:yao.zijianLet \(C\) be a smooth, projective, geometrically irreducible curve of genus \(g \geq 2\) defined over a number field \(K\) of degree \(d\), and \(J\) its Jacobian. Falting's theorem asserts that the set \(C(K)\) of rational points on \(C\) over \(K\) is finite. Faltings' proof, however, does not provide a method for determining the set \(C(K)\). Following the pioneering work of Chabauty, methods have been developed to explicitly determine the elements of \(C(K)\) on curves that satisfy certain conditions on rank \(r\) of \(J(K)\). In this paper, the geometric quadratic Chabauty method, initiated over \(\mathbb{Q}\) in [\textit{B. Edixhoven} and \textit{G. Lido}, J. Inst. Math. Jussieu 22, No. 1, 279--333 (2023; Zbl 07662119)], is generalised to curves defined over arbitrary number fields.
Let \(\rho = \mathrm{rank}_{\mathbb{Z}} NS(J)\) be the rank of the Néron-Severi group of \(J\). Further, let \(\mathcal{O}_K\) be the ring of integers of \(K\) and \(p\) a prime satisfying the following assumptions:
\begin{itemize}
\item[(a)] The curve \(C\) has good reduction at each prime \(\wp_1,\ldots,\wp_s\) of \(K\) that lies above \(p\).
\item[(b)] The ramification index of every \(\wp_i\) satisfies \(e(\wp_i/p) < p-1\).
\item[(c)] The prime \(p\) does not divide \(|O_{K,\mathrm{tors}}^* |\).
\end{itemize}
Let \(\delta = \mathrm{rank}_{\mathbb{Z}} O_K^*\) and \( R = \mathbb{Z}_p \langle z_1, \ldots , z_{r+\delta(\rho-1)}\rangle\) be the \(p\)-adically completed polynomial algebra over \(\mathbb{Z}_p\). Suppose that \(C\) satisfies the ``quadratic Chabauty condition'' \(r + \delta(\rho- 1) \leq (g + \rho -2)d\). Then it is proved that there exists an ideal \(I\) of \(R\), which is explicitly computable \(\bmod\ p\), such that if \(\bar{A} = (R/I) \otimes_{\mathbb{Z}_p}\mathbb{F}_p\) is finite dimensional over \(\mathbb{F}_p\), then the set of rational points \(C(K)\) is finite, and its cardinality is bounded by \(\dim_{\mathbb{F}_p}\bar{A}\).
Reviewer: Dimitros Poulakis (Thessaloniki)Optimal strong approximation for quadrics over \(\mathbb{F}_q [t]\)https://zbmath.org/1517.110692023-09-22T14:21:46.120933Z"Sardari, Naser Talebizadeh"https://zbmath.org/authors/?q=ai:talebizadeh-sardari.naser"Zargar, Masoud"https://zbmath.org/authors/?q=ai:zargar.masoudLet \(\mathcal O=\mathbb F_q[t]\) be the polynomial ring over the finite field \(\mathbb F_q\) with \(q\) elements, where \(q\) is a fixed odd prime power, let \(F({\mathbf x})\) be a non-degenerate quadratic form of discriminant \(\Delta\) over \(\mathcal O\) in \(d\geq 4\) variables \(\mathbf{x}=(x_1,\ldots,x_d)\), and let \(f\in \mathcal O\). In this paper, the authors study the optimal strong approximation problem for the quadric \(X_f\) given by the equation \(F(\mathbf{x})=f\). That is, given \(g\in \mathcal O\) and \(\boldsymbol{\lambda} =(\lambda_1,\ldots,\lambda_d)\in \mathcal O^d\), they consider solutions \(\mathbf{x}=(x_1,\ldots,x_d)\in \mathcal O^d\) to the equation \(F(\mathbf{x})=f\) for which \(\mathbf{x}\equiv \boldsymbol{\lambda} \mbox{ mod }g\) (that is, \(x_i \equiv \lambda_i \mbox{ mod }g\) for all \(i=1,\ldots,d\)). Necessary local local conditions for such a solution are that \(X_f(\mathbb F_q((1/t)))\neq \emptyset\) and, for all irreducible \(\varpi \in \mathcal O\), \(F(\mathbf{x})=f\) has a solution \(\mathbf{x}_{\varpi}\in \mathcal O_{\varpi}^d\) such that \(\mathbf{x}_{\varpi} \equiv \boldsymbol{\lambda}\mbox{ mod }\varpi^{\mbox{ord}_{\varpi}(g)}\). The main strong approximation result obtained (Theorem 1.1) is as follows. Let \(\varepsilon >0\) be given, let \(f,g\) be polynomials in \(\mathcal O\) for which the irreducible divisors of \(\Delta\) appear with bounded multiplicity in \(fg\), and let \(\boldsymbol{ \lambda}\in \mathcal O^d\). Suppose that the necessary local conditions for a solution are satisfied. Then there is a constant \(C_{\varepsilon,F}\), independent of \(f\), \(g\) and \(\boldsymbol{\lambda}\), such that if \(d\geq 5\) and \(\mbox{deg }f \geq (4+\varepsilon)\mbox{deg }g+C_{\varepsilon,f}\), then there exists \(\mathbf{x}\in \mathcal O^d\) such that \(F(\mathbf{x})=f\) and \(\mathbf{x}\equiv \boldsymbol{\lambda}\mbox{ mod }g\). In the case \(d=4\), the same result holds with \((4+\varepsilon)\) replaced by \((6+\varepsilon)\).
The method of proof is based on a version of the circle method that was developed by \textit{D. R. Heath-Brown} over the integers [J. Reine Angew. Math. 481, 149--206 (1996; Zbl 0857.11049)] and further developed in a paper of the first author [Duke Math. J. 168, 1887--1927 (2019; Zbl 1443.11030)] to prove an optimal strong approximation result for quadratic forms over the integers. In the present paper, the authors extend the circle method over function fields by proving a stationary phase theorem that allows them to bound certain oscillatory integrals that appear in the circle method. The result obtained is optimal for \(d\geq 5\).
The strong approximation result stated above is used to give a new proof, independent of the Ramanujan conjecture over function fields, that the diameter of a \(k\)-regular Morgenstern Ramanujan graph \(G\) is bounded above by \((2+\varepsilon)\log_{k-1}|G|+O_{\varepsilon}(1)\).
Reviewer: Andrew G. Earnest (Carbondale)On the exceptional zeros of \(p\)-non-ordinary \(p\)-adic \(L\)-functions and a conjecture of Perrin-Riouhttps://zbmath.org/1517.110702023-09-22T14:21:46.120933Z"Benois, Denis"https://zbmath.org/authors/?q=ai:benois.denis"Büyükboduk, Kâzım"https://zbmath.org/authors/?q=ai:buyukboduk.kazimThe aim of the paper under review is to address a conjecture of Mazur-Tate-Teitelbaum for a \(p\)-semistable non-ordinary eigenform \(f\) of arbitrary even weight. To achieve that, they express the second derivative of the \(p\)-adic \(L\)-function in terms of a \(p\)-adic regulator defined on an extended trianguline Selmer group defined by \((\varphi,\Gamma)\)-modules and this can be regard as a form of \(p\)-adic Birch and Swinnerton-Dyer formula for the second derivative.
While their results appear to rely heavily on \textit{R. Venerucci}'s results [Invent. Math. 203, No. 3, 923--972 (2016; Zbl 1406.11060)], a careful task has been done to form a general framework and one expects that it would lead to further research on the exceptional zero phenomenon for non-ordinary \(p\)-adic \(L\)-functions.
Reviewer: Kazuma Morita (Sapporo)On the rank-part of the Mazur-Tate refined conjecture for higher weight modular formshttps://zbmath.org/1517.110712023-09-22T14:21:46.120933Z"Ota, Kazuto"https://zbmath.org/authors/?q=ai:ota.kazutoSummary: Under some assumptions, we prove the rank-part of the Mazur-Tate refined conjecture of BSD type. More concretely, we prove that the rank of the Selmer group of an elliptic modular form is less than or equal to the order of zeros of Mazur-Tate elements, or modular elements, which are elements in certain group rings constructed from special values of the associated \(L\)-function. Our main result is regarded as a generalization of our previous work on elliptic curves.On the leading constant in the Manin-type conjecture for Campana pointshttps://zbmath.org/1517.110722023-09-22T14:21:46.120933Z"Shute, Alec"https://zbmath.org/authors/?q=ai:shute.alecSummary: We compare the Manin-type conjecture for Campana points recently formulated by \textit{M. Pieropan} et al. [Proc. Lond. Math. Soc. (3) 123, No. 1, 57--101 (2021; Zbl 1479.11116)] with an alternative prediction of \textit{T. D. Browning} and \textit{K. Van Valckenborgh} [Exp. Math. 21, No. 2, 204--211 (2012; Zbl 1327.11022)] in the special case of the orbifold \((\mathbb P^1,D)\), where \(D = \frac{1}{2}[0]+\frac{1}{2}[1]+\frac{1}{2}[\infty ]\). We find that the two predicted leading constants do not agree, and we discuss whether thin sets could explain this discrepancy. Motivated by this, we provide a counterexample to the Manin-type conjecture for Campana points, by considering orbifolds corresponding to squarefull values of binary quadratic forms.Integer polynomials and Minkowski's theorem on linear formshttps://zbmath.org/1517.110732023-09-22T14:21:46.120933Z"Bernik, Vasiliĭ Ivanovich"https://zbmath.org/authors/?q=ai:bernik.vasili-i"Korlyukova, Irina Aleksandrovna"https://zbmath.org/authors/?q=ai:korlyukova.irina-aleksandrovna"Kudin, Alekseĭ Sergeevich"https://zbmath.org/authors/?q=ai:kudin.aleksei-sergeevich"Titova, Anastasiya Vladimirovna"https://zbmath.org/authors/?q=ai:titova.anastasiya-vladimirovnaSummary: In paper Minkowski's theorem on linear forms [\textit{J. W. S. Cassels}, An introduction to Diophantine approximation. Translation from the English by A. M. Polosuev. Edited and with an addition by A. O. Gel'fond. Moskau: Verlag für ausländische Literatur (1961; Zbl 0098.26301)] is applied to polynomials with integer coefficients
\[P(x) = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0 \tag{3}\] with degree \(degP = n\) and height \(H(P)=\max_{0 \le i \le n} |a_i|\). Then, for any \(x \in [0,1)\) and a natural number \(Q > 1\), we obtain the inequality
\[|P(x)| < c_1(n) Q^{-n} \tag{4}\]
for some \(P(x), H(P) \leq Q\). Inequality (4) means that the entire interval \([0,1)\) can be covered by intervals \(I_i\), \(i = 1, 2, \ldots\) at all points of which inequality (4) is true. An answer is given to the question about the size of the \(I_i\) intervals. The main result of this paper is proof of the following statement.
For any \(v, 0 \leq v < \frac{n+1}{3} \), there is an interval \(J_k, k = 1, \ldots, K\), such that for all \(x \in J_k\), the inequality (4) holds and, moreover,
\begin{align*} c_2 Q^{-n-1+v} < \mu J_k < c_3 Q^{-n-1+v}. \end{align*}Badly approximable matrices and Diophantine exponentshttps://zbmath.org/1517.110742023-09-22T14:21:46.120933Z"German, O. N."https://zbmath.org/authors/?q=ai:german.oleg-nSummary: This paper is a survey of results concerning different kinds of Diophantine exponents. Special attention is paid to the transference principle and the generalization of the concept of badly approximable numbers to matrices and lattices.Upper bounds and spectrum for approximation exponents for subspaces of \(\mathbb{R}^n\)https://zbmath.org/1517.110752023-09-22T14:21:46.120933Z"Joseph, Elio"https://zbmath.org/authors/?q=ai:joseph.elioFrom the abstract: ``Given two subspaces of \(\mathbb{R}^n\), \(A\) and \(B\) of respective dimensions \(d\) and \(e\), with \(d+e\leqslant n\), the proximity between \(A\) and \(B\) is measured by \(t = \min(d, e)\) canonical angles \(0 \leqslant \theta_1 \leqslant \cdots \leqslant \theta_t \leqslant \pi/2\); we set \(\psi_j(A,B) = \sin\theta_j\). If \(B\) is a rational subspace, its complexity is measured by its height \(H(B) = \mathrm{covol}(B\cap\mathbb{Z}^n)\). We denote by \(\mu_n(A|e)_j\) the exponent of approximation defined as the upper bound (possibly equal to \(+\infty\)) of the set of \(\beta > 0\) such that for infinitely many rational subspaces \(B\) of dimension \(e\), the inequality \(\psi_j(A, B) \leqslant H(B)^{-\beta}\) holds. We are interested in the minimal value \(\mathring{\mu}_n(d|e)_j\) taken by \(\mu_n(A|e)_j\) when \(A\) ranges through the set of subspaces of dimension \(d\) of \(\mathbb{R}^n\) such that for all rational subspaces \(B\) of dimension \(e\) one has \(\dim(A\cap B) < j\). We show that if \(A\) is included in a rational subspace \(F\) of dimension \(k\), its exponent in \(\mathbb{R}^n\) is the same as its exponent in \(\mathbb{R}^k\) via a rational isomorphism \(F\rightarrow\mathbb{R}^k\). This allows us to deduce new upper bounds for \(\mathring{\mu}_n(d|e)_j\). We also study the values taken by \(\mu_n(A|e)_e\) when \(A\) is a subspace of \(\mathbb{R}^n\) satisfying \(\dim(A\cap B) < e\) for all rational subspaces \(B\) of dimension \(e\).''
Reviewer: István Gaál (Debrecen)Diophantine approximation by Piatetski-Shapiro primeshttps://zbmath.org/1517.110762023-09-22T14:21:46.120933Z"Dimitrov, S. I."https://zbmath.org/authors/?q=ai:dimitrov.stoyan-ivanovSummary: Let \([\,\cdot\,]\) be the floor function. In this paper we show that whenever \(\eta\) is real, the constants \(\lambda_i\) satisfy some necessary conditions, then for any fixed \(1<c<38/37\) there exist infinitely many prime triples \(p_1,\, p_2,\, p_3\) satisfying the inequality
\[
|\lambda_1 p_1 +\lambda_2 p_2 +\lambda_3 p_3 +\eta |<(\max p_j)^{\frac{37c-38}{26c}}(\log \max p_j)^{10}
\]
and such that \(p_i =[n_i^c], i=1,\,2,\,3\).Polyadic Liouville numbershttps://zbmath.org/1517.110772023-09-22T14:21:46.120933Z"Chirskii, V. G."https://zbmath.org/authors/?q=ai:chirskii.vladimir-grigorevichSummary: The study of polyadic Liouville numbers has begun relatively recently. They make up an important part of the author's works concerning the infinite linear independence of the polyadic numbers \({{f}_0}(\lambda ) = \sum\limits_{n = 0}^\infty{{(\lambda )}_n}{{\lambda }^n},{{f}_1}(\lambda ) = \sum\limits_{n = 0}^\infty{{(\lambda + 1)}_n}{{\lambda }^n},\) where \(\lambda\) is a polyadic Liouville number. Here, \({{(\gamma )}_n}\) denotes the Pochhammer symbol, i.e., \({{(\gamma )}_0} = 1\) and, for \(n \geqslant 1, {{(\gamma )}_n} = \gamma (\gamma + 1)\dots(\gamma + n - 1).\) The considered series converge in any field \(\mathbb Q_p \). A parameter of the considered Euler-type series is a polyadic Liouville number, and the values of these series are calculated at a polyadic Liouville point. We note E.S. Krupitsyn's works establishing estimates for polynomials in sets of polyadic Liouville numbers and Yudenkova's works in which the values of \(F\)-series are considered at polyadic Liouville points. The canonic expansion of a polyadic number \(\lambda\) is of the form \(\lambda = \sum\limits_{n = 0}^\infty{{a}_n}n!,{{a}_n} \in{\text{Z}},0 \leqslant{{a}_n} \leqslant n.\) This series converges in any field \(\mathbb Q_p\) of \(p\)-adic numbers. A polyadic number \(\lambda\) is called a polyadic Liouville number (or a Liouville polyadic number) if for any \(n\) and \(P\) there exists a positive integer \(A\) such that for all primes \(p\) satisfying \(p \leqslant P\) the inequality \({{\left| {\lambda - A} \right|}_p} < {{A}^{{ - n}}}\) holds. We prove a simple statement that a Liouville polyadic number is transcendental in any field \(\mathbb Q_p\). In other words, a Liouville polyadic number is globally transcendental. Additionally, a theorem is proved about the properties of approximations of a set of \(p\)-adic numbers, and its corollary is established, which is a sufficient condition for the algebraic independence of a set of \(p\)-adic numbers. A theorem on the global algebraic independence of polyadic numbers is obtained.Arithmetic properties of values at polyadic Liouville points of Euler-type series with polyadic Liouville parameterhttps://zbmath.org/1517.110782023-09-22T14:21:46.120933Z"Chirskii, V. G."https://zbmath.org/authors/?q=ai:chirskii.vladimir-grigorevichSummary: We study the infinite linear independence of polyadic numbers \({{f}_0}(\lambda ) = \sum\limits_{n = 0}^\infty{{(\lambda )}_n}{{\lambda }^n}, f_1(\lambda ) = \sum\limits_{n = 0}^\infty{{(\lambda + 1)}_n}{{\lambda }^n},\) where \(\lambda\) is a polyadic Liouville number. Here, \({{(\gamma )}_n}\) denotes the Pochhammer symbol, i.e., \({{(\gamma )}_0} = 1\) and \({{(\gamma )}_n} = \gamma (\gamma + 1)\dots(\gamma + n - 1)\) for \(n \geqslant 1\). The considered series converge in any field \(\mathbb Q_p \). The results presented extends the author's previous results concerning the arithmetic properties of the polyadic numbers \({{f}_0}(1) = \sum\limits_{n = 0}^\infty{{(\lambda )}_n},{{f}_1}(1) = \sum\limits_{n = 0}^\infty{{(\lambda + 1)}_n} \). The values of generalized hypergeometric series have been extensively studied. If the series parameters are rational numbers, then the series belong to the class of \(E\)-functions (if these series are entire functions), to the class of \(G\)-functions (if they have a finite nonzero radius of convergence), or to the class of \(F\)-series (in the case of a zero radius of convergence in the field of complex numbers, but they converge in the fields of \(p\)-adic numbers). In all these cases, the Siegel-Shidlovskii method and its generalizations are applicable. If the parameters of the series contain algebraic irrational numbers, then the study of their arithmetic properties is based on Hermite-Padé approximations. In the considered case, the parameter is a transcendental number. Note that earlier A.I. Galochkin proved the algebraic independence of the values of \(E\)-functions at points that are real Liouville numbers. We also mention E.Yu. Yudenkova's papers (submitted for publication) on the values of \(F\)-series at polyadic Liouville points. It should be emphasized that this paper deals with the values, at polyadic transcendental points, of hypergeometric series with a parameter that is a polyadic transcendental (Liouville) number.Infinite linear independence with constraints on a subset of prime numbers for values of Euler-type series with polyadic Liouville parameterhttps://zbmath.org/1517.110792023-09-22T14:21:46.120933Z"Chirskii, V. G."https://zbmath.org/authors/?q=ai:chirskii.vladimir-grigorevichSummary: The ring of polyadic integers is the direct product of rings of \(p\)-adic integers over all primes \(p\). Thus, elements \(\theta\) of this ring can be treated as infinite-dimensional vectors, whose coordinates in the corresponding ring of \(p\)-adic integers are denoted by \({{\theta }^{{(p)}}}\). The infinite linear independence of polyadic numbers \({{\theta }_1},\dots,{{\theta }_m}\) means that, for any nonzero linear form \({{h}_1}{{x}_1} + \dots + {{h}_m}{{x}_m}\) with integer coefficients \({{h}_1},\dots,{{h}_m}\), there are infinitely many primes \(p\) such that in the field \(\mathbb{Q}_p{{h}_1}\theta_1^{{(p)}} + \dots + {{h}_m}\theta_m^{{(p)}} \ne 0\). At the same time, problems in which primes are considered only from some proper subsets of the set of primes are of interest. In this case, we talk about infinite linear independence with constraints on the specified set. The canonical representation of an element \(\theta\) of the ring of polyadic integers has the form of a series \(\theta = \sum\limits_{n = 0}^\infty{{a}_n}n!,{{a}_n} \in{\text{Z}},0 \leqslant{{a}_n} \leqslant n.\) Of course, a series with integer members that converges in all fields of \(p\)-adic numbers is a polyadic integer. We say that a polyadic number \(\theta\) is a polyadic Liouville number (or a Liouville polyadic number) if for any numbers \(n\) and \(P\) there exists a positive integer \(A\) such that for all primes \(p\) satisfying the inequality \(p \leqslant P\) it is true that \({{\left| {\theta - A} \right|}_p} < A^{-n} \). Here, we prove the infinite linear independence of polyadic numbers \(f_0(1) = \sum\limits_{n = 0}^\infty{{(\lambda )}_n},{{f}_1}(1) = \sum\limits_{n = 0}^\infty{{(\lambda + 1)}_n}\) with constraints on a set of prime numbers in the aggregate of arithmetic progressions. An important apparatus for obtaining this result is Hermite-Padé approximations of generalized hypergeometric functions constructed by \textit{Yu. V. Nesterenko} in [Russ. Acad. Sci., Sb., Math. 83, No. 1, 39--72 (1994; Zbl 0849.11052); translation from Mat. Sb. 185, No. 10, 39--72 (1994)]. Additionally, we use the approach from the work by \textit{A.-M. Ernvall-Hytönen} et al. [J. Integer Seq. 22, No. 2, Article 19.2.2, 10 p. (2019; Zbl 1443.11138)].On polyadic Liouville numbershttps://zbmath.org/1517.110802023-09-22T14:21:46.120933Z"Chirskii, V. G."https://zbmath.org/authors/?q=ai:chirskii.vladimir-grigorevichSummary: The study of polyadic Liouville numbers has begun relatively recently. The canonical expansion of a polyadic number \(\lambda\) is of the form \(\lambda = \sum\limits_{n = 0}^\infty{{a}_n}n!,{{a}_n} \in\mathbb{Z},0 \leqslant{{a}_n} \leqslant n.\) This series converges in any field \(\mathbb{Q}_p\) of \(p\)-adic numbers. A polyadic number \(\lambda\) is called a polyadic Liouville number (or a Liouville polyadic number) if for any \(n\) and \(P\) there exists a positive integer \(A\) such that for all primes \(p\) satisfying \(p \leqslant P\) the inequality \({{\left| {\lambda - A} \right|}_p} < {{A}^{{ - n}}}\) holds. Given a positive integer \(m\), let \(\Phi (k,m) = {{k}^{{{{k}^{{{{{\dots}}^k}}}}}}}\) denote the result of \(k\) raised to the power \(k\) successively \(m\) times. Let \({{n}_m} = \Phi (k,m)\), and let \(\alpha = \sum\limits_{m = 0}^\infty ({{n}_m})!.\) Theorem 1 states that, for any positive integer \(k \geqslant 2\) and any prime number \(p\), the series \(\alpha\) converges to a transcendental element of the ring \(\mathbb{Z}_p\). In other words, the polyadic number \(\alpha\) is globally transcendental.Infinite linear independence with constraints on a subset of prime numbers of values of Eulerian-type series with polyadic Liouville parameterhttps://zbmath.org/1517.110812023-09-22T14:21:46.120933Z"Chirskiĭ, Vladimir Grigor'evich"https://zbmath.org/authors/?q=ai:chirskii.vladimir-grigorevichSummary: A ring of polyadic integers is a direct product of rings of integer \(p\)-adic numbers over all primes \(p\). The elements \(\theta\) of this ring can thus be considered as infinite-dimensional vectors whose coordinates in the corresponding ring of integer \(p\)-adic numbers are denoted by \(\theta^{(p)}\). The infinite linear independence of polyadic numbers \(\theta_1,\ldots,\theta_m\) means that for any nonzero linear form \(h_1x_1+\ldots+h_mx_m\) with integer coefficients \(h_1,\ldots,h_m\) there is an infinite set of primes \(p\) such that in the field \(\mathbb{Q}_p\) the inequality \[h_1\theta_1^{(p)}+\ldots+h_m\theta_m^{(p)}\neq 0\]
holds. At the same time, problems in which primes are considered only from some proper subsets of the set of primes are of interest. In this case, we will talk about infinite linear independence with restrictions on the specified set. Canonical representation of the element \(\theta\) of the ring of polyadic integers has the form of a series
\[\theta= \sum_{n=0}^\infty a_n n!, a_n\in\mathbb{Z}, 0\leq a_n\leq n.\]
Of course, a series whose members are integers converging in all fields of \(p\)-adic numbers is a polyadic integer. We will call a polyadic number \(\theta\) a polyadic Liouville number (or a Liouville polyadic number) if for any numbers \(n\) and \(P\) there exists a natural number \(A\) such that for all primes \(p\) satisfying the inequality \(p\leq P\) the inequality
\[\left|\theta-A\right|_p<A^{-n}.\]
This work continues the development of the basic idea embedded in [the author, Chebyshevskiĭ Sb. 22, No. 5(81), 243--251 (2021; Zbl 1507.11061)]. Here the infinite linear independence with restrictions on the set of prime numbers in the aggregate of arithmetic progressions. of polyadic numbers \[f_0(1)=\sum_{n=0}^\infty (\lambda)_n, f_1(1)=\sum_{n=0}^\infty (\lambda +1)_n.\]
will be proved. An important apparatus for obtaining this result are Hermite-Pade approximations of generalized hypergeometric functions constructed in the work of \textit{Yu. V. Nesterenko} [Russ. Acad. Sci., Sb., Math. 83, No. 1, 39--72 (1994; Zbl 0849.11052); translation from Mat. Sb. 185, No. 10, 39--72 (1994)]. The approach from the work of \textit{A.-M. Ernvall-Hytönen} et al. [J. Integer Seq. 22, No. 2, Article 19.2.2, 10 p. (2019; Zbl 1443.11138)] was used.A \(p\)-adic lower bound for a linear form in logarithmshttps://zbmath.org/1517.110822023-09-22T14:21:46.120933Z"Palojärvi, Neea"https://zbmath.org/authors/?q=ai:palojarvi.neea"Seppälä, Louna"https://zbmath.org/authors/?q=ai:seppala.lounaStarting with the pioneering works of \textit{A. Baker} [Mathematika 14, 220--228 (1967; Zbl 0161.05301)], the effective lower bounds for linear forms in the logarithms of algebraic numbers became one of the most important tools in Diophantine number theory. Since then the bounds have been continuously improved and extended to more general cases. In the present paper, the authors provide explicit \(p\)-adic lower bounds for linear forms in \(p\)-adic logarithms of rational numbers. The proofs are based on Padé approximations of the second kind.
Reviewer: István Gaál (Debrecen)On the parametrization of hyperelliptic fields with \(S\)-units of degrees 7 and 9https://zbmath.org/1517.110832023-09-22T14:21:46.120933Z"Fedorov, G. V."https://zbmath.org/authors/?q=ai:fedorov.gleb-vladimirovich"Zhgoon, V. S."https://zbmath.org/authors/?q=ai:zhgoon.vladimir-s"Petrunin, M. M."https://zbmath.org/authors/?q=ai:petrunin.maksim-maksimovich"Shteinikov, Yu. N."https://zbmath.org/authors/?q=ai:shteinikov.yurii-nikolaevichBased on authors' abstract: This paper shows that if \(k\) is an algebraically closed field with \(\operatorname{char}k=0\), then the set of polynomials \(f\) of degree 5 such that the field \(k(x)(\sqrt{f}\,)\) has a nontrivial \(S\)-unit of degree 7 or 9 and the continued fraction expansion of \(\sqrt{f}/x\) is periodic is a one-parameter set corresponding to a rational curve with finitely many deleted points.
In this paper the authors established two theorems and explain its proofs with the help of different types of proposition and definition.
Reviewer: Vijay Yadav (Virar)Symmetries of the three-gap theoremhttps://zbmath.org/1517.110842023-09-22T14:21:46.120933Z"Dasgupta, Aneesh"https://zbmath.org/authors/?q=ai:dasgupta.aneesh"Roeder, Roland"https://zbmath.org/authors/?q=ai:roeder.roland-k-wThe famous Three Gap Theorem, also known as Steinhaus problem, states that for irrational \(\alpha\), the set of fractional parts \(\{\{n\alpha\}, n = 0, \ldots, N-1\} = \{x_{1,N} < x_{2,N} < \ldots < x_{N,N}\}\) has at most three gaps for every \(N\), that is, \(\left\lvert\{x_{i+1,N} - x_{i,N}: i =1,\ldots,N \}\right\rvert \leq 3\). In this note, the authors study the patterns in which order these gaps occur. They discover the following symmetry in their main result:
For fixed \(N\) and \(\alpha\), let \(a < b < c\) denote the three gap lengths (if only two gap lengths exists, there is no occurrence of the letter \(c\)). The theorem states that around the occurrence of some gap of length \(c\), the gaps of length \(a\) and \(b\) distribute symmetrically until the next occurrence of \(c\).
In formal terms, this can be described in the following way: Let \(W_j = x_{j+1,N} - x_{j,N} \in \{a,b,c\}\) and extend this periodically modulo \(N\). If \(W_j = c\), let \(\ell = \min \{k> 0 : W_{j+k} = c \text{ or } W_{j-k} = c\}\) denote the length until the next time a gap of size \(c\) occurs. Then \(W_{j-k} = W_{j+k}\) for every \(k = 0,\ldots, \ell -1\).
The method of proof is elementary and is supported by exemplary figures. Many of the ideas and statements rely on van Ravenstein's preceding work on the Three Gap Theorem [\textit{T. Van Ravenstein}, J. Aust. Math. Soc., Ser. A 45, No. 3, 360--370 (1988; Zbl 0663.10039)].
Reviewer: Manuel Hauke (Graz)On the distribution of \(\alpha p\) modulo one over Piatetski-Shapiro primeshttps://zbmath.org/1517.110852023-09-22T14:21:46.120933Z"Dimitrov, Stoyan"https://zbmath.org/authors/?q=ai:dimitrov.stoyan-ivanovSummary: Let \([\cdot]\) be the floor function and \(\|x\|\) denotes the distance from \(x\) to the nearest integer. In this paper we show that whenever \(\alpha\) is irrational and \(\beta\) is real then for any fixed \(1<c<12/11\) there exist infinitely many prime numbers \(p\) satisfying the inequality
\[
\|\alpha p + \beta\| \ll p^{\frac{11c-12}{26c}}\log^6p
\]
and such that \(p = [n^c]\).Polyadic estimates for \(F\)-serieshttps://zbmath.org/1517.110862023-09-22T14:21:46.120933Z"Chirskii, V. G."https://zbmath.org/authors/?q=ai:chirskii.vladimir-grigorevichSummary: Polyadic estimates for linear forms in some \(F\)-series are obtained.Arithmetic properties of polyadic integershttps://zbmath.org/1517.110872023-09-22T14:21:46.120933Z"Chirskii, V. G."https://zbmath.org/authors/?q=ai:chirskii.vladimir-grigorevichSummary: Arithmetic properties of series of the form \(\sum\limits_{n = 0}^\infty{{a}_n} \cdot n!\) with \({{a}_n} \in \mathbb{Z}\) are studied. The concept of infinite algebraic independence of polyadic numbers is discussed. A theorem is proved concerning the infinite algebraic independence of polyadic numbers of the class \(F\left( {\mathbb{Q},{{C}_1},{{C}_2},{{C}_3},{{d}_0}} \right)\) that are connected by a system of linear differential equations of a certain type.About minimal polynomial of residual fractions for algebraic irrationalitieshttps://zbmath.org/1517.110882023-09-22T14:21:46.120933Z"Dobrovol'skii, N. M."https://zbmath.org/authors/?q=ai:dobrovolskii.n-m"Dobrovol'skii, N. N."https://zbmath.org/authors/?q=ai:dobrovolskii.n-nSummary: We study the form and properties of minimal polynomials of residual fractions in continued fraction expansions of algebraic numbers. For purely real algebraic irrationalities \(\alpha\) of degree \(n \geqslant 2\), it is shown that, starting at some index \({{m}_0} = {{m}_0}(\alpha )\), the sequence of residual fractions \(\alpha_m\) is a sequence of reduced algebraic irrationalities. The definition of generalized Pisot numbers is given, which differs from the definition of Pisot numbers in that the former are not required to be integer. For an arbitrary real algebraic irrationality \(\alpha\) of degree \(n \geqslant 2\), it is shown that, starting at some index \({{m}_0} = {{m}_0}(\alpha )\), the sequence of residual fractions \(\alpha_m\) is a sequence of generalized Pisot numbers. An asymptotic formula for the conjugates of residual fractions of generalized Pisot numbers is found. This formula implies that the conjugates of the residual fraction \(\alpha_m\) are concentrated about the fractions \(- \frac{{{{Q}_{{m - 2}}}}}{{{{Q}_{{m - 1}}}}}\) either in an interval of radius \(O\left( {\frac{1}{{Q_{{m - 1}}^2}}} \right)\) in the case of purely real algebraic irrationalities or in circles of the same radius in the general case of real algebraic irrationalities having complex conjugates. It is established that, starting at some number \({{m}_0} = {{m}_0}(\alpha )\), a recurrence formula for incomplete private \(q_m\) expansions of real algebraic irrationals \(\alpha \), Express \(q_m\) using the values of the minimal polynomial \({{f}_{{m - 1}}}(x)\) for residual fractions \({{\alpha }_{{m - 1}}}\) and its derivative at the point \({{q}_{{m - 1}}}\). A recurrence formula using linear fractional transformations is found for minimal polynomials of residual fractions. A composition of such linear fractional transformations is a linear fractional transformation that takes the conjugates of an algebraic irrationality \(\alpha\) to the conjugates of the residual fraction, the latter exhibiting a pronounced effect of concentration about the rational fraction \(- \frac{{{{Q}_{{m - 2}}}}}{{{{Q}_{{m - 1}}}}} \). It is established that the sequence of minimal polynomials for residual fractions is a sequence of polynomials with equal discriminants. In conclusion, for a real algebraic irrational number \(\alpha \), the problem of the structure of its rational conjugate spectrum and its limit points is posed.Estimates of linear forms and polynomials in polyadic numbershttps://zbmath.org/1517.110892023-09-22T14:21:46.120933Z"Chirskii, V. G."https://zbmath.org/authors/?q=ai:chirskii.vladimir-grigorevichSummary: The paper presents lower estimates of the polyadic distance from zero for linear forms and polynomials with integer coefficients in certain polyadic numbers.Distinction of measures of Haar cylinders in the Dirichlet theorem for the field of \(p\)-adic numbershttps://zbmath.org/1517.110902023-09-22T14:21:46.120933Z"Bernik, V. I."https://zbmath.org/authors/?q=ai:bernik.vasili-i"Kudin, A. S."https://zbmath.org/authors/?q=ai:kudin.aleksei-sergeevich"Titova, A. V."https://zbmath.org/authors/?q=ai:titova.anastasiya-vladimirovnaSummary: The Dirichlet box principle gives surprisingly accurate results in problems of approximation of real numbers by rational numbers, transcendental numbers by real algebraic numbers. Every polynomial taking small values at a given point \(x\) also takes small values in its neighborhood. A problem of studying such neighborhoods and obtaining possible Lebesgue measure values arises frequently. In this paper we solve the problem in the p-adic case using recent results of the metric theory of Diophantine approximations.The generalised Hausdorff measure of sets of Dirichlet non-improvable numbershttps://zbmath.org/1517.110912023-09-22T14:21:46.120933Z"Bos, Philip"https://zbmath.org/authors/?q=ai:bos.philip"Hussain, Mumtaz"https://zbmath.org/authors/?q=ai:hussain.mumtaz"Simmons, David"https://zbmath.org/authors/?q=ai:simmons.davidOne can begin with authors' abstract:
``Let \(\psi: \mathbb R_{+} \to \mathbb R_{+}\) be a non-increasing function. A real number \(x\) is said to be \(\psi\)-Dirichlet improvable if the system
\[
|qx-p|<\psi (t)\text{ and }|q| < t
\]
has a non-trivial integer solution for all large enough \(t\). Denote the collection of such points by \(D(\psi)\). In this paper, we prove a zero-infinity law valid for all dimension functions under natural non-restrictive conditions. Some of the consequences are zero-infinity laws, for all essentially sublinear dimension functions proved by \textit{M. Hussain} et al. [Mathematika 64, No. 2, 502--518 (2018; Zbl 1412.11082)], for some non-essentially sublinear dimension functions, and for all dimension functions but with a growth condition on the approximating function.''
In this paper, special attention is given to basics of the theory of Diophantine approximation and to comprehensive metrical theory of some sets, as well as to explanations and descriptions of auxiliary notions. It also includes the consideration of some properties of continued fractions, as well as the Hausdorff measure and dimension.
The proofs of the main statements are given with explanations. Connections between known and presented results are described.
Reviewer: Symon Serbenyuk (Kyjiw)A note on well distributed sequenceshttps://zbmath.org/1517.110922023-09-22T14:21:46.120933Z"Moshohevitin, Nikolay"https://zbmath.org/authors/?q=ai:moshohevitin.nikolaySummary: We prove an easy statement about inhomogeneous approximation for non-singular vectors in metric theory of Diophantine approximation.Padovan or Perrin numbers that are concatenations of two distinct base \(b\) repdigitshttps://zbmath.org/1517.110932023-09-22T14:21:46.120933Z"Adédji, Kouèssi N."https://zbmath.org/authors/?q=ai:adedji.kouessi-norbert"Dossou-Yovo, Virgile"https://zbmath.org/authors/?q=ai:dossou-yovo.virgile"Rihane, Salah E."https://zbmath.org/authors/?q=ai:rihane.salah-eddine"Togbé, Alain"https://zbmath.org/authors/?q=ai:togbe.alainRecently special structures of numbers in classical recurrence sequences are intensively investigated. In this paper all Padovan and Perrin numbers are determined which are concatenations of two base \(b\) repdigits with \(2\le b\le 9\). The largest such numbers are \(P_{26}=816=\overline{2244}_7\) and \(E_{24}=853=\overline{31111}_4\). The proofs are based on estimates for linear forms in the logarithms for algebraic numbers and Baker-Davenport reduction.
Reviewer: István Gaál (Debrecen)On transformations of periodic sequenceshttps://zbmath.org/1517.110942023-09-22T14:21:46.120933Z"Chirskii, V. G."https://zbmath.org/authors/?q=ai:chirskii.vladimir-grigorevichSummary: In the study of pseudorandom number generators, a problem is whether the generated sequence is periodic. Some generators yield, in principle, a periodic sequence. To get rid of periodicity or to increase the period length, various methods are used, for example, filtering or combining generators. However, they can lead to a reduced total length of the generated sequence. The general idea underlying another approach (proposed in this paper) is to find an easy way of introducing chaos in an initially ordered set. Suppose that a given periodic sequence consists of digits in some positional notation. These digits are assigned a new number obtained by applying a fairly simple transformation to the initial sequence of digits. If the new number is irrational, then the sequence of its digits is not periodic. For example, if positive integers \({{a}_1}, \ldots ,{{a}_T}\) are treated as elements of a periodic continued fraction, then, by Lagrange's theorem, the resulting number is a quadratic irrationality. Note that this number is a badly approximable rational number. Another method relying on the same baseline idea is to consider series of the form \(\sum\limits_{n = 0}^\infty \frac{{{{a}_n}}}{{n!}}\) with a periodic sequence of integers \(\{ {{a}_n}\} ,{{a}_{{n + T}}} = {{a}_n} \). The irrationality of these numbers is proved if at least one of the numbers \({{a}_1}, \ldots ,{{a}_T}\) is different from 0. Additionally, the order of approximation of these numbers can be estimated using the Siegel-Shidlovskii method. However, the computation of digits in a number of this form requires numerous division operations. It is possible to consider series of the form \(\sum\limits_{n = 0}^\infty{{a}_n}n!\) with a periodic sequence of positive integers \(\{ {{a}_n}\} ,{{a}_{{n + T}}} = {{a}_n} \). Some properties of such series are described.An effective Schmidt's subspace theorem for arbitrary hypersurfaces over function fieldshttps://zbmath.org/1517.110952023-09-22T14:21:46.120933Z"Le, Giang"https://zbmath.org/authors/?q=ai:le.giangThe celebrated theorem of Wolfgang Schmidt, the subspace theorem became one of the most important tools of diophantine approximation. Using techniques from Nevanlinna theory it is possible to obtain an effective versions of it over algebraic function fields.
Such an effective version of the subspace theorem is formulated by the author on a smooth projective variety over function fields of characteristic zero for an arbitrary family of hypersurfaces.
This result is applied to a far reaching generalized form of the classical Thue equation over function fields.
The exact formulation of the results is not possible within thos review.
Reviewer: István Gaál (Debrecen)Continued \(A_2\)-fractions and singular functionshttps://zbmath.org/1517.110962023-09-22T14:21:46.120933Z"Pratsiovytyi, M. V."https://zbmath.org/authors/?q=ai:pratsiovytyi.mykola"Goncharenko, Ya. V."https://zbmath.org/authors/?q=ai:goncharenko.ya-v"Lysenko, I. M."https://zbmath.org/authors/?q=ai:lysenko.i-m"Ratushniak, S. P."https://zbmath.org/authors/?q=ai:ratushniak.s-pThis article is devoted to a new singular function defined in terms of \(A_2\)-continued fractions \( \frac{1}{a_1+\frac{1}{a_2+\frac{1}{\ddots}}}\equiv [0;a_1, a_2,\dots,a_n,\dots]\), where \(a_n\in A_2:=\{1/2,1\}\). It is proved that almost all (in the sense of the Lebesgue measure) numbers of segment \(I=[1/2,1]\) in their \(A_2\)-representations use each of the tuples of elements of the alphabet of arbitrary length as consecutive digits of the representation infinitely many times. Consider quasi-exponential functions \(f\) related to the \(A_2\)-representation of numbers, defined by equality \(f(x=[0;a_1,a_2,\dots,a_n,\dots])=\exp(\sum_{n=1}^\infty(2 a_n-1)v_n)\), where \(v_1+v_2+\cdots+v_n+\cdots\) is a given absolutely convergent series. For a function \(f\), structural and functional relationships are indicated as well as necessary and sufficient conditions for continuity (which are: \(v_n=v_1(-1/2)^{n-1}\), \(v_1\in \mathbb R\)) and monotonicity are found. In the case of the continuity of the function \(f\), we give the expression of its derivative and prove the singularity (the equality of the derivative to zero almost everywhere in the sense of the Lebesgue measure) using the above-mentioned normal property of numbers in terms of their \(A_2\)-representation. The relation between this new strictly monotonic singular function and the classical strictly increasing Minkowski question-mark function is indicated.
Reviewer: Takao Komatsu (Hangzhou)Hausdorff dimension estimates applied to Lagrange and Markov spectra, Zaremba theory, and limit sets of Fuchsian groupshttps://zbmath.org/1517.110972023-09-22T14:21:46.120933Z"Pollicott, M."https://zbmath.org/authors/?q=ai:pollicott.mark"Vytnova, P."https://zbmath.org/authors/?q=ai:vytnova.polinaThis paper is devoted to the exact value of the Hausdorff dimension and to limits sets for some dimensional Markov iterated function schemes. Also, the present research deals with the following topics: Diophantine approximations, the difference between the Markov and Lagrange spectra, and denominators of finite continued fractions and the Zaremba conjecture, as well as the spectrum of the Laplacian on certain Riemann surfaces.
Special attention is given to auxiliary notions, explanations, and examples. Such notions as the Hausdorff dimension, continued fractions, and the Markov spectrum, as well as the Markov iterated function schemes and the transfer operator, etc., are recalled. In addition, the pressure function is considered.
Approaches for estimating the Hausdorff dimension of dynamically defined sets given by Markov iterated function schemes, are discussed. The authors present the other approach which ``is based on combining elements of the methods of \textit{K. I. Babenko} and \textit{S. P. Yur'ev} [Sov. Math., Dokl. 19, 731--735 (1978; Zbl 0416.10040); translation from Dokl. Akad. Nauk SSSR 240, 1273--1276 (1978)] and \textit{E. Wirsing} [Acta Arith. 24, 507--528 (1974; Zbl 0283.10032)] originally developed for the Gauss map''.
Ingredients for effective estimates of the Hausdorff dimension are described. The used techniques and related peculiarities and known results are explained.
Finally, one can note the following text from authors' abstract:
``\dots we are particularly concerned with the role of the value of the Hausdorff dimension in solving conjectures and problems in other areas of mathematics. As our first application we confirm, and often strengthen, conjectures on the difference of the Lagrange and Markov spectra in Diophantine analysis, which appear in the work of \textit{C. Matheus} and \textit{C. G. Moreira} [Comment. Math. Helv. 95, No. 3, 593--633 (2020; Zbl 1465.11165)]. As a second application we (re-)validate and improve estimates connected with the Zaremba conjecture in number theory, used in the work of \textit{J. Bourgain} and \textit{A. Kontorovich} [Ann. Math. (2) 180, No. 1, 137--196 (2014; Zbl 1370.11083)], \textit{S. Huang} [Geom. Funct. Anal. 25, No. 3, 860--914 (2015; Zbl 1333.11078)] and \textit{I. D. Kan} [Sb. Math. 210, No. 3, 364--416 (2019; Zbl 1437.11010); translation from Mat. Sb. 210, No. 3, 75--130 (2019)]. As a third more geometric application, we rigorously bound the bottom of the spectrum of the Laplacian for infinite area surfaces, as illustrated by an example studied by \textit{C. T. McMullen} [Am. J. Math. 120, No. 4, 691--721 (1998; Zbl 0953.30026)]\dots''
Reviewer: Symon Serbenyuk (Kyjiw)Estimation of weighted Kloosterman sums with the help of additive shifthttps://zbmath.org/1517.110982023-09-22T14:21:46.120933Z"Korolev, M. A."https://zbmath.org/authors/?q=ai:korolev.maxim-aSummary: Additive shift is a widely used tool for estimating exponential sums and character sums. According to it, the summation variable \(n\) is replaced by an expression of the type \(n + x\) with the subsequent summation over the artificially introduced variable \(x\). The transformation of a simple sum into a multiple one gives additional opportunities for obtaining a nontrivial bound for the initial sum. This technique was widely used by I.G. van der Corput, I.M. Vinogradov, D.A. Burgess, A.A. Karatsuba, and many other researchers. It became a useful tool for dealing with character sums in finite fields and with multiple exponential sums. E. Fouvry and P. Michel (1998) and J. Bourgain (2005) successfully used this shift to estimate Kloosterman sums. Fouvry and Michel combined additive shift with profound results from algebraic geometry. On the contrary, the method of J. Bourgain is completely elementary. For example, it allowed the author to give an elementary proof of an estimate for the Kloosterman sum modulo a prime \(q\) with primes in the case when its length \(N\) exceeds \({{q}^{{{\kern 1pt} 1/2 + \varepsilon }}}\). In this paper, we give some new elementary applications of additive shift to weighted Kloosterman sums of the type \(\sum\limits_{n \leqslant N} f(n)\exp \left( {\frac{{2\pi ia}}{q}{\kern 1pt} (n + b){\text{*}}} \right),(ab,q) = 1,mm{\text{*}} \equiv 1\;({\text{mod}}q),\) where \(q\) is a prime and the weight function \(f(n)\) is equal to \(\tau (n)\), which is the number of divisors of \(n\), or to \(r(n)\), which is the number of representations of \(n\) by a sum of two squares of integers. The bounds obtained for these sums are nontrivial for \(N \geqslant{{q}^{{{\kern 1pt} 2/3 + \varepsilon }}} \). As a corollary of such bounds, we obtain some new results concerning the distribution of fractional parts of the type \(\left\{ {\frac{a}{q}{\kern 1pt} (u{v} + b){\text{*}}} \right\},\left\{ {\frac{a}{q}{\kern 1pt} ({{u}^2} + {{{v}}^2} + b){\text{*}}} \right\},\) where \(u\) and \({v}\) are integers running through hyperbolic \((u{v} \leqslant N)\) and circular \(({{u}^2} + {{{v}}^2} \leqslant N)\) domains, respectively.Methods of estimating short Kloosterman sumshttps://zbmath.org/1517.110992023-09-22T14:21:46.120933Z"Korolev, M. A."https://zbmath.org/authors/?q=ai:korolev.maxim-aSummary: This survey is an extended version of the mini-course read by the author in November 2015 during the Chinese-Russian workshop on exponential sums and sumsets. This workshop was organized by Professor Chaohua Jia (Institute of Mathematics, Academia Sinica) and Professor Ke Gong (Henan University) at the Academy of Mathematics and System Science, CAS (Beijing). The author is warmly grateful to them for support and hospitality. The survey consists of the Introduction, three sections, and the Conclusion. The basic definitions and results concerning complete Kloosterman sums are given in the Introduction. A method for estimating incomplete Kloosterman sums modulo a growing power of a fixed prime is described in Section 1. This method is based on an idea of A.G. Postnikov, according to which the estimation of such sums reduces to estimating exponential sums with a polynomial in the exponent by applying I.M. Vinogradov's mean value theorem. A.A. Karatsuba's method for estimating incomplete sums to an arbitrary modulus is described in Section 2. This method is based on a fairly accurate estimate for the number of solutions of a symmetric congruence involving inverse residues to a given modulus. This estimate plays the same role in the problems under consideration as Vinogradov's mean value theorem in estimating corresponding exponential sums. The method of J. Bourgain and M.Z. Garaev is described in Section 3. This method is based on a profound sum-product estimate and on an improvement of Karatsuba's bound for the number of solutions of a symmetric congruence. The Conclusion contains a number of recent results concerning estimates of short Kloosterman sums without proofs.On nonlinear Kloosterman sumshttps://zbmath.org/1517.111002023-09-22T14:21:46.120933Z"Korolev, M. A."https://zbmath.org/authors/?q=ai:korolev.maxim-aSummary: Exponential sums of a special type -- the so-called Kloosterman sums -- play a key role in numerous number-theoretic problems concerning the distribution of inverse residues in residue rings modulo a given \(q\). In many cases, estimates of such sums are based on A. Weil's bound for complete Kloosterman sums over primes. This bound allows one to estimate Kloosterman sums of length \(N \geqslant{{q}^{{0.5 + \varepsilon }}}\) for any fixed \(\varepsilon > 0\) with a power-saving factor. Weil's bound was originally proved using methods of algebraic geometry. Later, S.A. Stepanov gave an elementary proof of this bound, but this proof was also complicated enough. The aim of this paper is to give an elementary proof of an estimate for the Kloosterman sum of length \(N \geqslant{{q}^{{0.5 + \varepsilon }}} \), which also leads to a power-saving factor. This proof is based on the trick of additive shift of the summation variable, which is widely used in various number theory problems.Speed of convergence of Weyl sums over Kronecker sequenceshttps://zbmath.org/1517.111012023-09-22T14:21:46.120933Z"Colzani, Leonardo"https://zbmath.org/authors/?q=ai:colzani.leonardoAuthor's abstract: We study the speed of convergence in the numerical integration with Weyl sums over Kronecker sequences in the torus,
\[
\frac{1}{N} \sum_{n=1}^N f( x+n\alpha) -\int_{\mathbb{T}^d }f(y)dy.
\]
Reviewer: Giovanni Coppola (Napoli)New identities involving certain Hardy sums and two-term exponential sumshttps://zbmath.org/1517.111022023-09-22T14:21:46.120933Z"Dağlı, Muhammet Cihat"https://zbmath.org/authors/?q=ai:dagli.muhammet-cihatSummary: In this paper, we present a new reciprocity formula for certain Hardy sums and study a computational problem of one type mean value containing certain Hardy sums and the two-term exponential sum with the help of the properties of Gauss sums and Dirichlet \(L\)-function.Large oscillations of the argument of the Riemann zeta-functionhttps://zbmath.org/1517.111032023-09-22T14:21:46.120933Z"Chirre, Andrés"https://zbmath.org/authors/?q=ai:chirre.andres"Mahatab, Kamalakshya"https://zbmath.org/authors/?q=ai:mahatab.kamalakshyaSummary: Let \(S(t)\) denote the argument of the Riemann zeta-function, defined as
\[
S(t) = \frac{1}{\pi} \mathrm{Im} \log \zeta (1/2+it).
\]
Assuming the Riemann hypothesis, we prove that
\[
S(t) = \Omega_\pm \left(\frac{\log t \log \log \log t}{\log \log t}\right).
\]
This improves the classical \(\Omega\)-results of
\textit{H. L. Montgomery} [Comment. Math. Helv. 52, 511--518 (1977; Zbl 0373.10024), Theorem 2] and matches with the \(\Omega\)-result obtained by \textit{A. Bondarenko} and \textit{K. Seip} [Math. Ann. 372, No. 3--4, 999--1015 (2018; Zbl 1446.11153)].A new Ramanujan-type identity for \(L(2k+1, \chi_1)\)https://zbmath.org/1517.111042023-09-22T14:21:46.120933Z"Chourasiya, Shashi"https://zbmath.org/authors/?q=ai:chourasiya.shashi"Jamal, Md Kashif"https://zbmath.org/authors/?q=ai:jamal.md-kashif"Maji, Bibekananda"https://zbmath.org/authors/?q=ai:maji.bibekanandaThe paper deals with Ramanujan-type identities for \(L(2k+1, \chi_1)\) where \(\chi_1\) is the principal character modulo a prime \(p\).
Theorem 1. Let \(\alpha\) and \(\beta\) be positive real numbers such that \(\alpha\beta = \pi^2/4\). For any integer \(k\geq 1\), we have
\begin{multline*}
(4\alpha)^{-k}\left(\frac{1}{2}\zeta(2k+1)\left(1-2^{-2k-1} \right)-\sum_{n=0}^\infty\frac{(2n+1)^{-2k-1}}{e^{2(2n+1)\alpha}+1} \right)\\
-(-4\beta)^{-k}\left(\frac{1}{2}\zeta(2k+1)\left(1-2^{-2k-1} \right)-\sum_{n=0}^\infty\frac{(2n+1)^{-2k-1}}{e^{2(2n+1)\beta}+1} \right)\\
=\sum_{j=1}^k(-1)^{j-1}(2^{2j}-1)(2^{2k+2-2j}-1)\frac{B_{2j}}{(2j)!}\frac{B_{2k+2-2j}}{(2k+2-2j)!}\alpha^{k+1-j}\beta^j.
\end{multline*}
Here and in the next results the \(B_{2i}\)s are the Bernoulli numbers.
Theorem 2. Let \(p\) be a prime number and \(\alpha\) and \(\beta\) be positive real numbers such that \(\alpha\beta = \pi^2/p^2\). For any integer \(k\geq 1\), we have
\begin{multline*}
(4\alpha)^{-k}\left(\frac{p-1}{2}L(2k+1, \chi_1)-\sum_{n=1}^\infty a_n\left(\sum_{d|n}\frac{\chi_1(d)}{d^{2k+1}}\right)e^{-2n\alpha}\right)\\
-(-4\beta)^{-k}\left(\frac{p-1}{2}L(2k+1, \chi_1)-\sum_{n=1}^\infty a_n\left(\sum_{d|n}\frac{\chi_1(d)}{d^{2k+1}}\right)e^{-2n\beta}\right)\\
=\sum_{j=1}^k(-1)^{j-1}(p^{2j}-1)(p^{2k+2-2j}-1)\frac{B_{2j}}{(2j)!}\frac{B_{2k+2-2j}}{(2k+2-2j)!}\alpha^{k+1-j}\beta^j,
\end{multline*}
where \( L(s,\chi)\) is the Dirichlet \(L\)-function, \(a_n=1\) if \(\gcd(n, p)=1\) and \(a_n= 1-p\), otherwise.
Theorem 3. Using the notations from the previous theorem, we have
\[
\sum_{n=1}^\infty a_n\left(\sum_{d|n}\frac{\chi_1(d)}{d}e^{-2n\alpha}\right) - \sum_{n=1}^\infty a_n\left(\sum_{d|n}\frac{\chi_1(d)}{d}e^{-2n\beta}\right) = \frac{(p-1)^2}{2p}\log\left(\frac{\pi}{p\alpha}\right).
\]
Theorem 4. Let \(p\) be a prime number an define \(\sigma_{-k, \chi}(n):=\sum_{d|n}\chi(d)d^{-k}\). For any \(k\in \mathbb{N}\) and \(z\in \mathbb{H}:=\{\mathrm{Im}(z)\geq 0\}\), we have
\begin{multline*}
(pz)^{2k} \mathfrak{F}_{2k+1, \chi_1}\left(-\frac{1}{p^2z}\right)-\mathfrak{F}_{2k+1, \chi_1}(z) = \frac{p-1}{2}L(2k+1, \chi_1)\left[(pz)^{2k}-1\right]\\
+\frac{(2\pi i)^{2k+1}}{2zp^{2k+2}}\sum_{j=1}^k(p^{2j}-1)(p^{2k+2-2j}-1)\frac{B_{2j}}{(2j)!}\frac{B_{2k+2-2j}}{(2k+2-2j)!}(pz)^{2k+2-2j},
\end{multline*}
where
\[
\mathfrak{F}_{k, \chi_1}(z):=\sum_{n=1}^\infty a_n\sigma_{-k, \chi_1}(n)\exp(2\pi inz).
\]
Some conjectures and their consequences are stated, as well.
Reviewer: Stelian Mihalas (Timişoara)A model zeta function of the monoid of natural numbershttps://zbmath.org/1517.111052023-09-22T14:21:46.120933Z"Dobrovol'skii, N. N."https://zbmath.org/authors/?q=ai:dobrovolskii.n-nSummary: The paper studies the zeta function \(\zeta (M({{p}_1},{{p}_2})|\alpha )\) of the monoid \(M({{p}_1},{{p}_2})\) generated by prime numbers \({{p}_1} < {{p}_2}\) of the form \(3n + 2\). The main monoid \({{M}_{{3,1}}}({{p}_1},{{p}_2}) \subset M({{p}_1},{{p}_2})\) and the main set \({{A}_{{3,1}}}({{p}_1},{{p}_2}) = M({{p}_1},{{p}_2}){{\backslash }}{{M}_{{3,1}}}({{p}_1},{{p}_2})\) are defined. For the corresponding zeta functions, explicit finite formulas are found that give their analytic continuations to the entire complex plane, except for a countable set of poles. Inverse series for these zeta functions are found, and functional equations are derived. Three new types of monoids of natural numbers with a unique prime factorization are defined, namely, monoids of powers, Euler monoids modulo \(q\), and unit monoids modulo \(q\). Expressions for their zeta functions in terms of the Euler product are given. The Davenport-Heilbronn effect for zeta functions of monoids of natural numbers is discussed, which means the appearance of zeros of the zeta functions for terms obtained by decomposition into residue classes to some modulus. For monoids with an exponential sequence of primes, the barrier series hypothesis is proved and it is shown that the holomorphic domain of the zeta function of such a monoid is the complex half-plane to the right of the imaginary axis. In conclusion, topical problems with zeta functions of monoids of natural numbers that require further investigation are considered.Inverse problem for a monoid with an exponential sequence of primeshttps://zbmath.org/1517.111062023-09-22T14:21:46.120933Z"Dobrovol'skii, N. N."https://zbmath.org/authors/?q=ai:dobrovolskii.n-n"Rebrova, I. Yu."https://zbmath.org/authors/?q=ai:rebrova.irina-yurevna"Dobrovol'skii, N. M."https://zbmath.org/authors/?q=ai:dobrovolskii.n-mSummary: In this paper, for an arbitrary monoid of natural numbers, the foundations of the Dirichlet series algebra are constructed over either a number field or the ring of integers of an algebraic number field. For any number field \(\mathbb{K} \), it is shown that the set \(\mathbb{D}{\text{*}}{{(M)}_{\mathbb{K}}}\) of all invertible Dirichlet series of \(\mathbb{D}{{(M)}_{\mathbb{K}}}\) is an infinite Abelian group consisting of series whose first coefficient is nonzero. We introduce the notion of an entire Dirichlet series of a monoid of natural numbers that form an algebra over the ring of algebraic integers \({{\mathbb{Z}}_{\mathbb{K}}}\) of \(\mathbb{K} \). For the group \({{\mathbb{U}}_{\mathbb{K}}}\) of algebraic units of the ring of algebraic integers \({{\mathbb{Z}}_{\mathbb{K}}}\) of \(\mathbb{K} \), it is shown that the set \(\mathbb{D}{{(M)}_{{{{\mathbb{U}}_{\mathbb{K}}}}}}\) of entire Dirichlet series with \(a(1) \in{{\mathbb{U}}_{\mathbb{K}}}\) is a multiplicative group. For any Dirichlet series from the Dirichlet series algebra of a monoid of natural numbers, the reduced series, the noninvertible part, and the additional series are defined. A formula for decomposition of an arbitrary Dirichlet series into the product of a reduced series and a construction of a noninvertible part and an additional series is found. For any monoid of natural numbers, the algebra of Dirichlet series convergent in the whole complex domain is defined. The Dirichlet series algebra with a given half-plane of absolute convergence is constructed. It is shown that, for any nontrivial monoid \(M\) and any real \({{\sigma }_0} \), there is an infinite set of Dirichlet series from \(\mathbb{D}(M)\) such that the domain of their holomorphy is the \(\alpha \)-half-plane \(\sigma > {{\sigma }_0} \). With the help of the universality theorem of S.M. Voronin, a weak form of the universality theorem is proved for a wide class of zeta functions of monoids of natural numbers. In conclusion, topical problems with zeta functions of monoids of natural numbers that require further research are described. In particular, if the Linnik-Ibragimov conjecture is true, then the strong universality theorem should be valid for them.Bounds for moments of cubic and quartic Dirichlet \(L\)-functionshttps://zbmath.org/1517.111072023-09-22T14:21:46.120933Z"Gao, Peng"https://zbmath.org/authors/?q=ai:gao.peng.1"Zhao, Liangyi"https://zbmath.org/authors/?q=ai:zhao.liangyiIn the line of works by \textit{W. Heap} et al. [Q. J. Math. 70, No. 4, 1387--1396 (2019; Zbl 1469.11290)] and \textit{W. Heap} and \textit{K. Soundararajan} [``Lower bounds for moments of zeta and \(L\)-functions revisited'', Mathematika 68, No. 1, 1--14 (2022; \url{doi:10.1112/mtk.12115})] on moments of the Riemann zeta-function on the critical line, the first author recently gave upper and lower bounds for moments of central values \(L(1/2,\chi)\), where \(\chi\) runs over a family of quadratic Dirichlet characters. In this paper, cubic and quartic characters are considered; upper and lower bounds of the same order of magnitude are obtained.
The results are conditional to the generalized Lindelöf or Riemann hypothesis, except in the case of the sharp unconditional lower bound for \(2k\)-th moments of \(\lvert L(1/2,\chi)\rvert\) with \(\chi\) cubic and \(k \ge 1/2\).
Reviewer: Michel Balazard (Marseille)Godement-Jacquet \(L\)-function, some conjectures and some consequenceshttps://zbmath.org/1517.111082023-09-22T14:21:46.120933Z"Kaur, Amrinder"https://zbmath.org/authors/?q=ai:kaur.amrinder"Sankaranarayanan, Ayyadurai"https://zbmath.org/authors/?q=ai:sankaranarayanan.ayyaduraiSummary: In this paper, we investigate the mean square estimate for the logarithmic derivative of the Godement-Jacquet \(L\)-function \(L_f(s)\) assuming the Riemann hypothesis for \(L_f(s)\) and Rudnick-Sarnak conjecture.Asymptotic structure of eigenvalues and eigenvectors of certain triangular Hankel matriceshttps://zbmath.org/1517.111092023-09-22T14:21:46.120933Z"Matiyasevich, Yu. V."https://zbmath.org/authors/?q=ai:matiyasevich.yuri-vSummary: The Hankel matrices considered in this article arose in one reformulation of the Riemann hypothesis proposed earlier by the author. Computer calculations showed that, in the case of the Riemann zeta function, the eigenvalues and the eigenvectors of such matrices have an interesting structure. The article studies a model situation when the zeta function is replaced by a function having a single zero. For this case, we indicate the first terms of the asymptotic expansions of the smallest and largest (in absolute value) eigenvalues and the corresponding eigenvectors.A generalization of the Riemann-Siegel formulahttps://zbmath.org/1517.111102023-09-22T14:21:46.120933Z"O'Sullivan, Cormac"https://zbmath.org/authors/?q=ai:osullivan.cormacAs the author summarizes, the Riemann-Siegel formula compares the Riemann zeta function on the critical line with its partial sums, expressing the difference between them as an expansion in terms of decreasing powers of the imaginary variable \(t\). Siegel anticipated that this formula could be generalized to include the Hardy-Littlewood approximate functional equation, which is valid in any vertical strip.
In the paper, the author provides such generalization, the asymptotics containing Mordell integrals and an interesting new family of polynomials. To reach this result, the author proceeds by showing that the Riemann-Siegel formula and the Hardy-Littlewood approximate functional equation are indeed special cases of a shared natural generalization, provided in Theorem 1.5 of the paper. To finish, the author proposes, as a follow up, to develop in a natural extension of the techniques in his paper is to Dirichlet \(L\)-functions, \(L(s,\chi)\).
Reviewer: Emilio Elizalde (Bellaterra)The first moment of quadratic twists of modular \(L\)-functionshttps://zbmath.org/1517.111112023-09-22T14:21:46.120933Z"Shen, Quanli"https://zbmath.org/authors/?q=ai:shen.quanliAssume \(f\) is a modular form of weight \(\kappa\) for the full modular group \(\mathrm{SL}_2(\mathbb{Z})\), such that \(f\) is an eigenfunction of all Hecke operators and has the Fourier expansion at infinity
\[
f(z) = \sum_{n=1}^\infty\lambda_f(n)n^{(\kappa-1)/2}e^{2\pi inz}.
\]
The twisted modular \(L\)-function is defined by
\[
L(s, f\otimes \chi_d):=\sum_{n=1}^\infty\frac{\lambda_f(n)\chi_d(n)}{n^s} = \prod_{p\not | d}\left(1-\frac{\lambda_f(p)\chi_d(p)}{p^s}+\frac{1}{p^{2s}}\right)^{-1}
\]
for \(\mathrm{Re}(s)>1\), extended to the entire complex plane.
The symmetric square \(L\)-function is defined by
\[
L(s,\mathrm{sym}^2f):=\zeta(2s)\sum_{n=1}^\infty\frac{\lambda_f(n^2)}{n^s}=\prod_p\left(1-\frac{\alpha_f(p)^2}{p^s}\right)^{-1}\left(1-\frac{\alpha_f(p)\beta_f(p)}{p^s}\right)^{-1}\left(1-\frac{\beta_f(p)^2}{p^s}\right)^{-1},
\]
where \(\mathrm{Re}(s)>1\), \(\alpha_f(p)+\beta_f(p)=\lambda_f(p)\) and \(\alpha_f(p)\beta_f(p)=1\). Also, \(\sum^*\) will denote summation over square-free integers.
Theorem 1. Let \(\kappa\equiv 0\ (\mathrm{mod}\ 4)\) and \(\kappa \not =0\). Let \(\Phi : (0, \infty) \to \mathbb{R}\) be a smooth, compactly supported function. We have
\[
\sum^*_{(d, 2)=1} L(1/2, f\otimes\chi_{8d})\Phi\left(\frac{d}{X}\right)=\frac{8\tilde{\Phi}(1)}{\pi^2}L(1, \mathrm{sym}^2f)Z^*(0)X+O(X^{1/2+\varepsilon}),
\]
where \(\tilde{\Phi}\) is the Mellin transform of \(\Phi\), \(Z^*(\alpha):=Z(1/2+\alpha, 1)\), while \(Z\), for \(\mathrm{Re}(\gamma) >0\), can be inferred from
\[
L(1+2\gamma, \mathrm{sym}^2f)Z(1/2+\gamma, l):=\prod_{(p, 2)=1}Z_p(1/2+\gamma, l).
\]
The formulas for \(Z_p(1/2+\gamma, l)\) are quite complex and, therefore, omitted.
Theorem 2. Let \(\kappa\equiv 2\ (\mathrm{mod}\ 4)\) and \(\kappa \not =0\). Let \(\Phi : (0, \infty) \to \mathbb{R}\) be a smooth, compactly supported function. We have
\begin{multline*}
\sum_{(d, 2)=1}^* L'(1/2, f\otimes\chi_{8d})\Phi\left(\frac{d}{X}\right)=\frac{8\tilde{\Phi}(1)}{\pi^2}L(1, \mathrm{sym}^2 f)Z^*(0)X \\
\times\left[\log X+2\frac{L'(1, \mathrm{sym}^2 f)}{L(1, \mathrm{sym}^2 f)} + \frac{{Z^*}'(0)}{Z^*(0)}+\log\frac{8}{2\pi}+\frac{\Gamma' (\kappa/2)}{\Gamma(\kappa/2)}+\frac{\tilde{\Phi}'(1)}{\tilde{\Phi}(1)}\right] + O(X^{1/2+\varepsilon}).
\end{multline*}
Reviewer: Stelian Mihalas (Timişoara)The Riemann hypothesis as the parity of special binomial coefficientshttps://zbmath.org/1517.111122023-09-22T14:21:46.120933Z"Matiyasevich, Yu. V."https://zbmath.org/authors/?q=ai:matiyasevich.yuri-vSummary: The Riemann hypothesis has many equivalent reformulations. Some of them are arithmetical, that is, they are statements about properties of integers or natural numbers. Among them the reformulations with the simplest logical structure are those from the class \(\Pi_1^0\) of the arithmetical hierarchy, that is, having the form ``for every \({{x}_1}, \ldots ,{{x}_m}\) relation \(A({{x}_1}, \ldots ,{{x}_m})\) holds,'' where \(A\) is decidable. An example is the reformulation of the Riemann hypothesis as the assertion that a certain Diophantine equation has no solution (such a particular equation can be given explicitly). Although the logical structure of this reformulation is very simple, all known methods for constructing such a Diophantine equation produce equations occupying several pages. On the other hand, there are other reformulations that also belong to \(\Pi_1^0\), but have rather short wording. Examples are the criteria for the validity of the Riemann hypothesis proposed by J.-L. Nicolas, G. Robin, and J. Lagarias. A shortcoming of these reformulations (as compared to Diophantine equations) is that they involve constants and functions which are ``more complicated'' than natural numbers and addition and multiplication sufficient for constructing Diophantine equations. This paper presents a system of nine conditions imposed on nine variables. To state these conditions, one needs only addition, multiplication, exponentiation (unary, with fixed base 2), remainders, inequalities, congruences, and binomial coefficients. The whole system can be written explicitly on a single sheet of paper. It is proved that the system is inconsistent if and only if the Riemann hypothesis is true.A generalized regularization theorem and Kawashima's relation for multiple zeta valueshttps://zbmath.org/1517.111132023-09-22T14:21:46.120933Z"Kaneko, Masanobu"https://zbmath.org/authors/?q=ai:kaneko.masanobu"Xu, Ce"https://zbmath.org/authors/?q=ai:xu.ce"Yamamoto, Shuji"https://zbmath.org/authors/?q=ai:yamamoto.shujiSummary: Kawashima's relation is conjecturally one of the largest classes of relations among multiple zeta values.
\textit{G. Kawashima} [J. Number Theory 129, No. 4, 755--788 (2009; Zbl 1220.11103)] introduced and studied a certain Newton series, which we call the Kawashima function, and deduced his relation by establishing several properties of this function. We present a new approach to the Kawashima function without using Newton series. We first establish a generalization of the theory of regularizations of divergent multiple zeta values to Hurwitz type multiple zeta values, and then relate it to the Kawashima function. Via this connection, we can prove a key property of the Kawashima function to obtain Kawashima's relation.A continuous version of multiple zeta functions and multiple zeta valueshttps://zbmath.org/1517.111142023-09-22T14:21:46.120933Z"Li, Jiangtao"https://zbmath.org/authors/?q=ai:li.jiangtao.2|li.jiangtao.1|li.jiangtao|li.jiangtao.3|li.jiangtao.4In this paper, the author defines a continuous version of multiple zeta functions by
\[
\zeta^{\mathscr C}(s_1,\ldots,s_r)=\int_1^\infty\cdots\int_1^\infty\frac{dx_1\cdots dx_r}{x_1^{s_1}(x_1+x_2)^{s_2}\cdot(x_1+\cdot+x_r)^{s_r}}
\]
and proves that these functions can be analytically continued to meromorphic functions on \(\mathbb C^r\) with only simple poles at some special hyperplanes. As in the classical multiple zeta values, it is shown that the values of these functions at positive integers, called \textit{continuous multiple zeta values}, satisfy the shuffle product and the sum formulas.
The author also considers the depth defect phenomena in the algebra of continuous multiple zeta values, and the relation between continuous multiple zeta values and multiple polylogarithms.
Reviewer: Wataru Takeda (Tokyo)On Ecalle's and Brown's polar solutions to the double shuffle equations modulo productshttps://zbmath.org/1517.111152023-09-22T14:21:46.120933Z"Matthes, Nils"https://zbmath.org/authors/?q=ai:matthes.nils"Tasaka, Koji"https://zbmath.org/authors/?q=ai:tasaka.kojiFor a multiple zeta value, (MZV),
\[ \zeta(m_1,\dots,m_r)=\sum_{0<n_1<\dots<n_r}\frac1{n_1^{m_1}\cdots n_r^{m_r}}
\]
the \textit{weight} is \(k=m_1+\dots+m_r\). For \(k\in\mathbb{N}\) let \(\mathcal Z_k\) denote the \(\mathbb{Q}\)-span of the finitely many MZVs of weight \(k\). \textit{D. Zagier} [Prog. Math. 120, 497--512 (1994; Zbl 0822.11001)] conjectured, that \(\dim_{\mathbb{Q}}\mathcal{Z}_k=d_k\), where \(d_k\) is defined by \(\sum_{k=0}^\infty d_k t^k=\frac1{1-t^2-t^3}\). \textit{A. B. Goncharov} [``Multiple polylogarithms and mixed Tate motives'', Preprint, \url{arXiv:Math/0103059}] and \textit{T. Terasoma} [Invent. Math. 149, No. 2, 339--369 (2002; Zbl 1042.11043)] independently proved, that \(\dim_{\mathbb{Q}}\mathcal{Z}_k\le d_k\). This means that there are many relations among the MZV. Several sets of relations are known. For instance in ascending order we have the sets of double shuffle relations, of associator relations and of motivic relations. The latter are due to \textit{F. Brown} [Ann. Math. (2) 175, No. 2, 949--976 (2012; Zbl 1278.19008)], who defined certain elements \(\sigma_{2k+1}\) of a motivic Lie algebra which define relations. It is expected, that these three classes actually coincide, which, among other things, would mean that the relators \(\sigma_{2k+1}\) should be expressible in terms of the double shuffle relators (DSR). The paper under review deals with the question for explicit expressions of (some of) the \(\sigma_{2k+1}\) relators in terms of the DSR. In the literature, there are two approaches, one by \textit{F. Brown} [``Anatomy of an associator'', Preprint, \url{arXiv:1709.02765}], and the other by \textit{J. Ecalle} [J. Théor. Nombres Bordx. 15, No. 2, 411--478 (2003; Zbl 1094.11032)]. In the paper, these two are put in a common algebraic context and similarities and differences are given. For instance, they show that the relators agree in depth \(\le 4\), but not in depth 5.
Reviewer: Anton Deitmar (Tübingen)Sun's three conjectures on Apéry-like sums involving harmonic numbershttps://zbmath.org/1517.111162023-09-22T14:21:46.120933Z"Xu, Ce"https://zbmath.org/authors/?q=ai:xu.ce"Zhao, Jianqiang"https://zbmath.org/authors/?q=ai:zhao.jianqiang|zhao.jianqiang.1Summary: In this paper, we will give another proof of Zhi-Wei Sun's three conjectures on Apéry-like sums involving harmonic numbers by proving some identities among special values of multiple polylogarithms.Generalized Dirichlet problem for a two-dimensional lattice of Dirichlet approximationshttps://zbmath.org/1517.111172023-09-22T14:21:46.120933Z"Dobrovol'skiĭ, Nikolaĭ Nikolaevich"https://zbmath.org/authors/?q=ai:dobrovolskii.n-n"Dobrovol'skiĭ, Mikhail Nikolaevich"https://zbmath.org/authors/?q=ai:dobrovolskii.m-n"Chubarikov, Vladimir Nikolaevich"https://zbmath.org/authors/?q=ai:chubarikov.vladimir-nikolaevich"Rebrova, Irina Yur'evna"https://zbmath.org/authors/?q=ai:rebrova.irina-yurevna"Dobrovol'skiĭ, Nikolaĭ Mikhaĭlovich"https://zbmath.org/authors/?q=ai:dobrovolskii.n-mSummary: The paper studies the relationship between the problem of determining the number of points of a two-dimensional lattice of Dirichlet approximations in a hyperbolic cross and the integral representation of the hyperbolic zeta function of a two-dimensional lattice of Dirichlet approximations. The concept of components of hyperbolic zeta-functions of a two-dimensional lattice of Dirichlet approximations is introduced. A representation is found for the first component of the hyperbolic zeta function of a two-dimensional lattice of Dirichlet approximations via the Riemann zeta function. With respect to the first component, the paradoxical fact is established that it is continuous for any irrational \(\beta\) and discontinuous at all rational points of \(\beta \). This refers to the dependency only on the \(\beta\) parameter.
For the second component of the hyperbolic zeta-function of the two-dimensional lattice of Dirichlet approximations in the case of a rational value \(\beta=\frac{a}{b} \), an asymptotic formula is obtained for the number of points of the second component of the two-dimensional lattice of Dirichlet approximations in the hyperbolic cross. The resulting formula gives an integral representation in the half-plane \(\sigma>\frac{1}{2} \).
The main research tool was the Euler summation formula. For the purposes of the work, it was necessary to obtain explicit expressions of the residual terms in asymptotic formulas for the number of points of residue classes of a two-dimensional lattice of Dirichlet approximations over a stretched fundamental lattice \(b\mathbb{Z} \times \mathbb{Z} \). Both Theorem 1 and Theorem 2, proved in the paper, show the dependence of the second term of the asymptotic formula and the deduction of the hyperbolic zeta function of the lattice \(\Lambda\left(\frac{a}{b}\right)\) depends on the magnitude of the denominator \(b\) and independence from the numerator \(a\). Earlier, similar effects were discovered by A. L. Roscheney for other generalizations of the Dirichlet problem. The paper sets the task of clarifying the order of the residual term in asymptotic formulas by studying the quantities
\[R_1^*(T,b,\delta)=\sum_{q=1}^{\frac{\sqrt{T}}{b}}\left\{\frac{T}{bq}-\delta\right\}-\frac{\sqrt{T}}{2b}, R_2^*(T,b,\delta)=\sum_{p=1}^{\sqrt{T}-\delta}\left\{\frac{T}{bp+b\delta}\right\}-\frac{\sqrt{T}}{2}.\]
It is proposed to first study the possibilities of the elementary method of I. M. Vinogradov, and then to obtain the most accurate estimates using the method of trigonometric sums. The paper outlines the directions of further research on this topic.On some series with gapshttps://zbmath.org/1517.111182023-09-22T14:21:46.120933Z"Somu, Sai Teja"https://zbmath.org/authors/?q=ai:somu.sai-teja"Haw, Johnathan"https://zbmath.org/authors/?q=ai:haw.johnathan"Nguyen, Vincent"https://zbmath.org/authors/?q=ai:nguyen.vincent-phuc"Tran, Duc Van Khanh"https://zbmath.org/authors/?q=ai:tran.duc-van-khanhSummary: Let \(k\) be any positive integer. In this paper, we derive results concerning series with gaps motivated by the open problem about the evaluation of
\[
\sum_{n = 1}^\infty \bigg(\frac{1}{n^3} + \frac{1}{(n + k)^3} + \frac{1}{(n + 2 k)^3} + \cdots \bigg) = \sum_{n = 1}^\infty \sum_{m = 0}^\infty \frac{1}{(n + k m)^3}.
\]
We later prove that there are infinitely many prime numbers using series with gaps.Computing zeta functions of cyclic covers in large characteristichttps://zbmath.org/1517.111192023-09-22T14:21:46.120933Z"Arul, Vishal"https://zbmath.org/authors/?q=ai:arul.vishal"Best, Alex J."https://zbmath.org/authors/?q=ai:best.alex-j"Costa, Edgar"https://zbmath.org/authors/?q=ai:costa.edgar"Magner, Richard"https://zbmath.org/authors/?q=ai:magner.richard"Triantafillou, Nicholas"https://zbmath.org/authors/?q=ai:triantafillou.nicholas-georgeSummary: We describe an algorithm to compute the zeta function of a cyclic cover of the projective line over a finite field of characteristic \(p\) that runs in time \(p^{1/2 + o(1)}\). We confirm its practicality and effectiveness by reporting on the performance of our SageMath implementation on a range of examples. The algorithm relies on Gonçalves's generalization of Kedlaya's algorithm for cyclic covers, and Harvey's work on Kedlaya's algorithm for large characteristic.
For the entire collection see [Zbl 1416.11009].The exponential-type generating function of the Riemann zeta-function revisitedhttps://zbmath.org/1517.111202023-09-22T14:21:46.120933Z"Noda, Takumi"https://zbmath.org/authors/?q=ai:noda.takumiIn the paper under review, the author introduces Dirichlet series associated with the Poincaré series attached to \(\mathrm{SL}(2, \mathbb{Z})\). In this work, certain integral representations and transformation formulas are presented, that describe the Voronoï-type summation formula for the exponential-type generating function of the Riemann zeta-function. In the form of application, the author establishes a new proof of the Fourier series expansion of holomorphic Poincaré series.
Reviewer: Michael Th. Rassias (Zürich)Bounded gaps between product of two primes in imaginary quadratic number fieldshttps://zbmath.org/1517.111212023-09-22T14:21:46.120933Z"Darbar, Pranendu"https://zbmath.org/authors/?q=ai:darbar.pranendu"Mukhopadhyay, Anirban"https://zbmath.org/authors/?q=ai:mukhopadhyay.anirban"Viswanadham, G. K."https://zbmath.org/authors/?q=ai:viswanadham.g-kAuthors' abstract: We study the gaps between products of two primes in imaginary quadratic number fields using a combination of the methods of \textit{D. A. Goldston} et al. [Proc. Lond. Math. Soc. (3) 98, No. 3, 741--774 (2009; Zbl 1213.11171)], and \textit{J. Maynard} [Ann. Math. (2) 181, No. 1, 383--413 (2015; Zbl 1306.11073)]. An important consequence of our main theorem is existence of infinitely many pairs \(\alpha_1, \alpha_2\) which are product of two primes in the imaginary quadratic field \(K\) such that \(|\sigma(\alpha_1- \alpha_2)| \le 2\) for all embeddings \(\sigma\) of \(K\) if the class number of \(K\) is one and \(|\sigma(\alpha_1 - \alpha_2)| \le 8\) for all embeddings \(\sigma\) of \(K\) if the class number of \(K\) is two.
Reviewer: László Tóth (Pécs)An inequality for the distribution of numbers free of small prime factorshttps://zbmath.org/1517.111222023-09-22T14:21:46.120933Z"Fan, Kai (Steve)"https://zbmath.org/authors/?q=ai:fan.kaiLet \(1<z \le x\) be real numbers, and denote by \(\Phi(x,z)\) the number of integers up to \(x\) whose prime divisors are greater than \(z\). By improving older estimates, the author proves that for \(1<z \le x\) one has the sharp inequality \(\Phi(x,z)< x/\log z\). Under stronger conditions, improved estimates are also proved, for example: \(\Phi(x,z) <(2/3)(x/\log z)\) for \(\max(2,x^{2/5}) \le z \le \sqrt x\). The methods are based on application of the arithmetic large sieve, combined with estimates for the prime counting function \(\pi(x)\), as well as Mathematica computations.
Reviewer: József Sándor (Cluj-Napoca)On the consecutive square-free values of the polynomials \(x_1^2 + \cdots + x_k^2+1\), \(x_1^2 + \cdots + x_k^2+2\)https://zbmath.org/1517.111232023-09-22T14:21:46.120933Z"Chen, Bo"https://zbmath.org/authors/?q=ai:chen.bo.6|chen.bo.4|chen.bo.3|chen.bo.2|chen.boSummary: The problem of evaluating square-free values of polynomials is a classical problem that has attracted many authors, including Eetermann, Carlitz, Tolev and Zhou. Recently, for \(1 \leq x, y \leq H\),
\textit{S. Dimitrov} [Czech. Math. J. 71, No. 4, 991--1009 (2021; Zbl 07442468)] established an asymptotic formula for the number of the square-free values attained by the polynomial \(f(x, y) = (x^2 + y^2 + 1)(x^2+y^2+2)\). Motived by the work of Dimitrov, in this paper, we give an asymptotic formula for the consecutive square-free numbers \(x_1^2 + \cdots + x_k^2+1\), \(x_1^2 + \cdots + x_k^2+2\).Error term concerning number of subgroups of group \(\mathbb{Z}_m \times \mathbb{Z}_n\) with \(m^2 + n^2 \le x\)https://zbmath.org/1517.111242023-09-22T14:21:46.120933Z"Sui, Yankun"https://zbmath.org/authors/?q=ai:sui.yankun"Liu, Dan"https://zbmath.org/authors/?q=ai:liu.danLet \(\mathbb{Z}_n\) be the additive group of residue classes modulo \(n\) (the cyclic group of order \(n\)). Let \(s(m, n)\) denote the number of subgroups of the group \(\mathbb{Z}_m \times \mathbb{Z}_n\), where \(m\) and \(n\) are arbitrary positive integers. Asymptotic formulas for the sum \(\sum_{m,n\le x} s(m,n)\) have been obtained by \textit{W. G. Nowak} and \textit{L. Tóth} [Int. J. Number Theory 10, No. 2, 363--374 (2014; Zbl 1307.11098)], \textit{L. Tóth} and \textit{W. Zhai} [Acta Arith. 183, No. 3, 285--299 (2018; Zbl 1435.11130)].
In this paper the authors deduce an asymptotic formula for the sum \(\sum_{m^2+n^2\le x} s(m,n)\) by using analytic arguments.
Reviewer: László Tóth (Pécs)Level of distribution of unbalanced convolutionshttps://zbmath.org/1517.111252023-09-22T14:21:46.120933Z"Fouvry, Etienne"https://zbmath.org/authors/?q=ai:fouvry.etienne"Radziwill, Maksym"https://zbmath.org/authors/?q=ai:radziwill.maksymThis paper studies the \emph{level of distribution} (in a weak sense) of arithmetic functions \(g\), those are exponents \(1/2 + \delta\) for which for any integer \(a \neq 0\) and for any \(A > 0\) holds that
\[
\sum_{\substack{q \leq x^{1/2 + \delta} \\
(q,a) = 1}} \left| \sum_{\substack{n \leq x \\
n \equiv a \: (\operatorname*{mod} q)}} g(n) - \frac{1}{\varphi(q)} \sum_{\substack{n \leq x \\
(n,q) =1 }} g(n)\right| \ll_{a,A} \frac{x}{\log^{A} x}.
\]
The main innovation in this paper is the establishment of levels of distribution for functions that originate as the multiplicative convolution of essentially an arbitrary sequence and a possibly very short sequence (of length \(\geq \exp((\log x)^\varepsilon)\)) that is well-distributed in arithmetic progressions of small moduli. A convolution of sequences having drastically different sizes is referred to as an \textit{unbalanced convolution}.
The proof of the main result starts by applying the classical dispersion method of Linnik which transforms the problem into finding estimates of a certain trilinear sum. The analysis of this sum in this work is then different from the proofs of previous dispersion estimates as one appeals here to estimates of \textit{S. Bettin} and \textit{V. Chandee} [Adv. Math. 328, 1234--1262 (2018; Zbl 1448.11158)] for trilinear forms of Kloosterman fractions and estimates of Duke-Friedlander-Iwaniec for bilinear forms in Kloosterman fractions
[\textit{W. Duke} et al., Invent. Math. 128, No. 1, 23--43 (1997; Zbl 0873.11050)].
The levels of distribution of unbalanced convolutions have several interesting arithmetical applications relying on the observation that most integers \(n\) can be factored as \(n = pm\) with \(p\) being a small prime (in the range \((\exp((\log x)^\varepsilon), x^\varepsilon)\). The authors investigate the consequences on the distribution of the \(k\)th divisor function, almost-prime counting functions, sieve weights and the distribution of sums of two squares.
Reviewer: Gregory Debruyne (Gent)On arithmetic properties of Cantor setshttps://zbmath.org/1517.111262023-09-22T14:21:46.120933Z"Cui, Lu"https://zbmath.org/authors/?q=ai:cui.lu"Ma, Minghui"https://zbmath.org/authors/?q=ai:ma.minghuiSummary: In this paper, we study three types of Cantor sets. For any integer \(m \geqslant 4\), we show that every real number in \([0, k]\) is the sum of at most \textit{k\, m}-th powers of elements in the Cantor ternary set \(C\) for some positive integer \(k\), and the smallest such \(k\) is \(2^m\). Moreover, we generalize this result to the middle-\(\frac{1}{\alpha}\) Cantor set for \(1 < \alpha < 2 + \sqrt{5}\) and \(m\) sufficiently large. For the naturally embedded image \(W\) of the Cantor dust \(C \times C\) into the complex plane \(\mathbb{C}\), we prove that for any integer \(m \geqslant 3\), every element in the closed unit disk in \(\mathbb{C}\) can be written as the sum of at most \(2^{m+8}\; m\)-th powers of elements in \(W\). At last, some similar results on \(p\)-adic Cantor sets are also obtained.Lattice points close to the Heisenberg sphereshttps://zbmath.org/1517.111272023-09-22T14:21:46.120933Z"Campolongo, Elizabeth G."https://zbmath.org/authors/?q=ai:campolongo.elizabeth-g"Taylor, Krystal"https://zbmath.org/authors/?q=ai:taylor.krystalThe authors give an upper bound on the number of lattice points near the surfaces of the Heisenberg norm balls, specifically they give an upper bound on the number of points on and near large dilates of the unit spheres generated by the anisotropic norms \(\|(z,t) \|_\alpha = ( |z|^\alpha + |t|^{\frac{\alpha}{2}})^{\frac{1}{\alpha}}\) where \(\alpha \geq 2\). They establish a bound on the Fourier transform of the surface measures coming from these norms and give a estimate for the number of lattice points in the intersection of two such surfaces.
Reviewer: Steven T. Dougherty (Scranton)On some mock theta functions of order 2 and 3https://zbmath.org/1517.111282023-09-22T14:21:46.120933Z"Kaur, H."https://zbmath.org/authors/?q=ai:kaur.harshil|kaur.harwinder|kaur.harman|kaur.harneet|kaur.harveen|kaur.hargeet|kaur.harmanpreet|kaur.harvinder|kaur.hardish|kaur.harleen|kaur.harpreet|kaur.harsimran|kaur.harvir|kaur.harjeet"Rana, M."https://zbmath.org/authors/?q=ai:rana.mishal|rana.masud|rana.mehwish|rana.meenakshi|rana.mansiAn \((n+t)\)-color partition is a partition of a positive integer \(\nu\), in which a part of size \(n\), \(n\ge0\), can come in \((n+t)\) different colors denoted by \(n_1,n_2,\ldots, n_{n+t}\). Here, zeros are permissible if and only if \(t>0\), and only \(0_t\) is allowed.
In this paper, the authors interpret the following mock theta functions and their generalized version in terms of \((n+t)\)-color partitions:
\begin{align*}
\omega(q) &= \sum_{n=0}^{\infty}\frac{q^{2(n^2+n)}}{(q;q^2)_{n+1}^2},\\
f(q) &= \sum_{n=0}^{\infty}\frac{q^{n^2}}{(-q;q)_{n}^2},\\
A(q) &= \sum_{n=0}^{\infty}\frac{q^{{(n+1)}^2}(-q;q^2)_n}{(q;q^2)_{n+1}^2},\\
B(q) &= \sum_{n=0}^{\infty}\frac{q^{n(n+1)}(-q^2;q^2)_n}{(q;q^2)_{n+1}^2},\\
\mu(q) &= \sum_{n=0}^{\infty}\frac{(-1)^nq^{n^2}(q;q^2)_n}{(-q^2;q^2)_{n}^2}.
\end{align*}
Applying a similar technique they also provide a combinatorial interpretation of two \(q\)-series.
Reviewer: Pankaj Jyoti Mahanta (Lakhimpur)On restricted partitions of numbershttps://zbmath.org/1517.111292023-09-22T14:21:46.120933Z"Mattson, H. F. Jr."https://zbmath.org/authors/?q=ai:mattson.harold-f-junSummary: This paper finds new quasi-polynomials over \(\mathbb{Z}\) for the number \(p_k (n)\) of partitions of \(n\) with parts at most \(k\). Methods throughout are elementary. We derive a small number of polynomials (e.g., one for \(k=3\), two for \(k = 4\) or 5, six for \(k=6)\) that, after addition of appropriate constant terms, take the value \(p_k (n)\). For example, for \(0\leq r<6\) and for all \(q \geq 0, p_3 (6q+r) = p_3 (r)+\pi_0 (q,r)\), a polynomial of total degree 2 in \(q\) and \(r\). In general there are \(M_{\lfloor k/2 \rfloor} = \text{\textsc{lcm}}\{1,2,\ldots,\lfloor k/2 \rfloor\}\) such polynomials. In two variables \(q\) and \(s\), they take the form \(\sum a_{i,j}\left(\begin{smallmatrix} q \\ i \end{smallmatrix}\right) \left(\begin{smallmatrix} s \\ j \end{smallmatrix}\right)\) with \(a_{i,j} \in\mathbb{Z}\), which we call the \textit{proper form} for an integer-valued polynomial. They constitute a quasi-polynomial of period \(M_{\lfloor k/2 \rfloor}\) for the sequence \((p_k (n)-p_k (r))\) with \(n \equiv r \pmod{M_k}\). For each \(k\) the terms of highest total degree are the same in all the polynomials and have coefficients dependent only on \(k\). A second theorem, combining partial fractions and the above approach, finds hybrid polynomials over \(\mathbb{Q}\) for \(p_k (n)\) that are easier to determine than those above. We compare our results to those of Cayley, MacMahon, and Arkin, whose classical results, as recast here, stand up well. We also discuss recent results of Munagi and conclude that circulators in some form are inevitable. At \(k=6\) we find serious errors in Sylvester's calculation of his ``waves''. \textit{J.J. Sylvester} [``On the partition of numbers'', Q. J. Pure Appl. Math. 1, 141-152 (1855)]. The results are generalized to the (not very different) problem called ``making change,'' where significant improvements to existing approaches are found. We find an infinitude of new congruences for \(p_k (n)\) for \(k= 3, 4\), and one new one for \(k=5\). Reduced modulo \(m\) the periodic sequence \((p_k (n))\) is investigated for periodicity and zeros: we find, from scratch, a simple proof of a known result in a special case.A note on Rogers-Ramanujan-Slater type theta function identityhttps://zbmath.org/1517.111302023-09-22T14:21:46.120933Z"Cao, Jian"https://zbmath.org/authors/?q=ai:cao.jian"Arjika, Sama"https://zbmath.org/authors/?q=ai:arjika.sama"Chaudhary, M. P."https://zbmath.org/authors/?q=ai:chaudhary.mahendra-palSummary: In this paper, we research theta function identity involving Rogers-Ramanujan identity and establish a Rogers-Ramanujan-Slater type theta function identity related to \(G(q)\) and \(\varphi(q)\).Weighted words at degree two. I: Bressoud's algorithm as an energy transferhttps://zbmath.org/1517.111312023-09-22T14:21:46.120933Z"Konan, Isaac"https://zbmath.org/authors/?q=ai:konan.isaac\textit{J. Dousse} [Proc. Am. Math. Soc. 145, No. 5, 1997--2009 (2017; Zbl 1357.05010)] introduced a refinement of Siladić's theorem on partitions, where parts occur in two primary and three secondary colors. The proof used the method of weighted words and difference equations. In [Eur. J. Comb. 87, Article ID 103101, 18 p. (2020; Zbl 1439.05025)] the author provided a bijective proof of a generalization of Dousse's theorem from two primary colors to an arbitrary number of primary colors. In this paper, the author gives a generalization of the result given in [loc. cit.], by using the statistic mechanical viewpoint of the integer partitions.
Reviewer: Mircea Merca (Cornu de Jos)Refinement for sequences in partitionshttps://zbmath.org/1517.111322023-09-22T14:21:46.120933Z"Lin, Bernard L. S."https://zbmath.org/authors/?q=ai:lin.bernard-l-s"Lin, Xiaowei"https://zbmath.org/authors/?q=ai:lin.xiaoweiThe \(k\)-measure of a partition \(\lambda\) denoted by \(\mu_k(\lambda)\) is the length of the largest subsequence of parts of \(\lambda\) in which the difference between any two consecutive parts of the subsequence is at least \(k\). It is a new statistic of integer partitions introduced recently by \textit{G. E. Andrews} et al. [Integers 22, Paper A32, 9 p. (2022; Zbl 1494.05007)]. They established a surprising identity that the number of partitions of \(n\) with \(2\)-measure \(m\) is equal to the number of partitions of \(n\) with Durfee square of side \(m\).
In the paper under review, the authors obtain the refinement of the result of Andrews et al., which involves two statistics, the smallest part and the length of partitions. Explicitly, they establish a trivariate generating function identity for partitions counting both the length and the \(2\)-measure, with restriction on the smallest parts and the largest parts. For \(s\geq 0\) and \(N\geq 1\), let \(\mathcal{P}_{s,N}\) be the set of all partitions of \(n\) whose smallest parts are at least \(s+1\) and whose largest parts are at most \(s+N\). Let \(d_N^s(t,z,q)=\sum_{\lambda\in \mathcal{P}_{s,N}}t^{\ell(\lambda)}z^{\mu_2(\lambda)}q^{|\lambda|}\), the identity is:
\[
d_N^s(t,z,q)=\sum_{j\geq 0}\frac{t^jz^jq^{j^2+js}}{(tq^{s+1};q)_j(tq^{s+N-j+1};q)_j} \times\left(\begin{bmatrix}N-j+1 \\ j\end{bmatrix}-tq^{s+N-j+1}\begin{bmatrix}N-j \\ j-1\end{bmatrix}\right).
\]
Letting \(N\rightarrow\infty\) in the above identity, the authors obtain an interesting result which involves a new concept called Durfee rectangle, which is a generalization of the famous Durfee square of partitions. For a nonnegative integer \(d\), the \(d\)-Durfee rectangle of a partition means the largest \(n\times (n+d)\) rectangle that fits in the Ferrers graph, the side of this \(d\)-rectangle is defined as the above largest \(n\).
The interesting result states that the number of partitions of \(n\) with \(\ell\) parts, \(2\)-measure equal to \(j\) and smallest part no less than \(s+1\) is equal to the number of partitions of \(n\) with \(\ell\) parts, \(s\)-Durfee rectangle of side \(j\) and smallest part no less than \(s+1\).
As a corollary of their result, they get the following result which was also established by \textit{G. E. Andrews} et al. [Algebr. Comb. 5, No. 6, 1353--1361 (2022; Zbl 1508.11101)]. That is,
the number of partitions of \(n\) with \(\ell\) parts, \(2\)-measure equal to \(j\) is equal to the number of partitions of \(n\) with \(\ell\) parts, and \(0\)-Durfee square of side \(j\).
Reviewer: Donna Q. J. Dou (Changchun)Neighborly partitions and the numerators of Rogers-Ramanujan identitieshttps://zbmath.org/1517.111332023-09-22T14:21:46.120933Z"Mohsen, Zahraa"https://zbmath.org/authors/?q=ai:mohsen.zahraa"Mourtada, Hussein"https://zbmath.org/authors/?q=ai:mourtada.husseinAmong the most famous and ubiquitous formulas involving \(q\)-series, we find the following two Rogers-Ramanujan identities:
\begin{align*}
\sum_{k=0}^\infty\dfrac{q^{k^2}}{(1-q)(1-q^2)\cdots(1-q^k)} &=\prod_{j=0}^\infty\dfrac{1}{(1-q^{5j+1})(1-q^{5j+4})},\\
\sum_{k=0}^\infty\dfrac{q^{k(k+1)}}{(1-q)(1-q^2)\cdots(1-q^k)} &=\prod_{j=0}^\infty\dfrac{1}{(1-q^{5j+2})(1-q^{5j+3})}.
\end{align*}
In this paper under review, the authors prove two identities which are in some sense dual to the Rogers-Ramanujan identities. These identities are inspired by a correspondence between three kinds of objects, namely, a new type of partitions (neighborly partitions), monomial ideals and some infinite graphs.
Reviewer: Dazhao Tang (Chongqing)Fast multiquadratic S-unit computation and application to the calculation of class groupshttps://zbmath.org/1517.111342023-09-22T14:21:46.120933Z"Biasse, Jean-François"https://zbmath.org/authors/?q=ai:biasse.jean-francois"Van Vredendaal, Christine"https://zbmath.org/authors/?q=ai:van-vredendaal.christineSummary: Let \(L = \mathbb{Q}(\sqrt{d_1},\dots,\sqrt{d_n})\) be a real multiquadratic field and \(S\) be a set of prime ideals of \(L\). In this paper, we present a heuristic algorithm for the computation of the \(S\)-class group and the \(S\)-unit group that runs in time \(\operatorname{Poly}(\log(\Delta) ,\operatorname{Size}(S))e^{\widetilde O(\sqrt{\ln d})}\) where \(d = \prod_{i\leq n} d_i\) and \(\Delta\) is the discriminant of \(L\). We use this method to compute the ideal class group of the maximal order \(\mathcal O_L\) of \(L\) in time \(\operatorname{Poly}(\log(\Delta))e^{\widetilde O(\sqrt{\log d})}\). When \(\log(d)\leq \log(\log(\Delta))^c\) for some constant \(c < 2\), these methods run in polynomial time. We implemented our algorithm using Sage 7.5.1.
For the entire collection see [Zbl 1416.11009].The shape of cyclic number fieldshttps://zbmath.org/1517.111352023-09-22T14:21:46.120933Z"Bolaños, Wilmar"https://zbmath.org/authors/?q=ai:bolanos.wilmar"Mantilla-Soler, Guillermo"https://zbmath.org/authors/?q=ai:mantilla-soler.guillermoSummary: Let \(m>1\) and \(\mathfrak{d} \neq 0\) be integers such that \(v_p(\mathfrak{d}) \neq m\) for any prime \(p\). We construct a matrix \(A(\mathfrak{d})\) of size \((m-1) \times (m-1)\) depending on only of \(\mathfrak{d}\) with the following property: For any tame \(\mathbb{Z}/m \mathbb{Z}\)-number field \(K\) of discriminant \(\mathfrak{d}\), the matrix \(A(\mathfrak{d})\) represents the Gram matrix of the integral trace-zero form of \(K\). In particular, we have that the integral trace-zero form of tame cyclic number fields is determined by the degree and discriminant of the field. Furthermore, if in addition to the above hypotheses, we consider real number fields, then the shape is also determined by the degree and the discriminant.On analogues of Mazur-Tate type conjectures in the Rankin-Selberg settinghttps://zbmath.org/1517.111362023-09-22T14:21:46.120933Z"Cauchi, Antonio"https://zbmath.org/authors/?q=ai:cauchi.antonio"Lei, Antonio"https://zbmath.org/authors/?q=ai:lei.antonioThe authors generalize (under certain technical hypotheses) a recent work of \textit{C.-H. Kim} and \textit{M. Kurihara} on elliptic curves [Int. Math. Res. Not. 2021, No. 14, 10559--10599 (2021; Zbl 1497.11270)] to prove a result very close to the weak main conjecture of Mazur and Tate for Rankin-Selberg convolutions. The main results are stated in Theorem A (in the ordinary setting) and Theorem B (in the supersingular setting).
They study the Fitting ideals over the finite layers of the cyclotomic \(\mathbb Z_p\)-extension of \(\mathbb Q\) of Selmer groups attached to the Rankin-Selberg convolution of two modular forms \(f\) and \(g\). Inspired by the theta elements for modular forms defined by \textit{B. Mazur} and \textit{J. Tate} [Duke Math. J. 54, 711--750 (1987; Zbl 0636.14004)], they define new theta elements for Rankin-Selberg convolutions of \(f\) and \(g\) using Loeffler-Zerbes' geometric \(p\)-adic \(L\)-functions attached to \(f\) and \(g\). Special emphasis is given to the case where \(f\) corresponds to an elliptic curve \(E\) and \(g\) to a two-dimensional odd irreducible Artin representation \(\rho\) with splitting field \(F\).
As an application, they give an upper bound of the dimension of the \(\rho\)-isotypic component of the Mordell-Weil group of \(E\) over the finite layers of the cyclotomic \(\mathbb Z_p\)-extension of \(F\) in terms of the order of vanishing of theta elements.
Reviewer: Andrzej Dąbrowski (Szczecin)Fine Selmer groups of congruent \(p\)-adic Galois representationshttps://zbmath.org/1517.111372023-09-22T14:21:46.120933Z"Kleine, Sören"https://zbmath.org/authors/?q=ai:kleine.soren"Müller, Katharina"https://zbmath.org/authors/?q=ai:muller.katharinaSummary: We compare the Pontryagin duals of fine Selmer groups of two congruent \(p\)-adic Galois representations over admissible pro-\(p\), \(p\)-adic Lie extensions \(K_\infty\) of number fields \(K\). We prove that in several natural settings the \(\pi\)-primary submodules of the Pontryagin duals are pseudo-isomorphic over the Iwasawa algebra; if the coranks of the fine Selmer groups are not equal, then we can still prove inequalities between the \(\mu\)-invariants. In the special case of a \(\mathbb{Z}_p\)-extension \(K_\infty /K\), we also compare the Iwasawa \(\lambda\)-invariants of the fine Selmer groups, even in situations where the \(\mu\)-invariants are nonzero. Finally, we prove similar results for certain abelian non-\(p\)-extensions.On the growth of even \(K\)-groups of rings of integers in \(p\)-adic Lie extensionshttps://zbmath.org/1517.111382023-09-22T14:21:46.120933Z"Lim, Meng Fai"https://zbmath.org/authors/?q=ai:lim.meng-faiLet \(p\) be an odd prime. The author studies the growth of the Sylow \(p\)-subgroups of the even \(K\)-groups of rings of integers in a \(p\)-adic Lie extension.
A main result of the present paper is the following: Let \(i\ge 2\) be given. Let \(F_{\infty}\) be a \(\mathbb Z_p\)-extension of a number field \(F\) and let \(F_n\) be the intermediate field with \(\mathrm{Gal}(F_n/F)\) cyclic of order \(p^n\). Then we have
\[
\mathrm{ord}_p\left(K_{2i-2}({\mathcal O}_{F_n})[p^{\infty}]\right) = \mu p^n + \lambda^{(i)} n + O(1),
\]
for some \(\lambda^{(i)}\) independent of \(n\) and where \(\mu\) depends on \(H_{Iw, S_{p^{\infty}}}(F_{\infty}/F, \mathbb Z_p(i))\).
Another main result is the following: Let \(i\ge 2\) be given. Suppose that \(F_{\infty}\) is a \(p\)-adic Lie extension of a number field \(F\) and \(F_{\infty}/F\) is unramified outside a finite set of primes. Moreover, suppose \(G=\mathrm{Gal}(F_{\infty}/F)\) is a uniform pro-\(p\)-group of dimension \(d\). There is a naturally defined sequence \(\{F_n\}\) of subfields of \(F_{\infty}\) and we have
\[
\mathrm{ord}_p\left(K_{2i-2}({\mathcal O}_{F_n})[p^n]\right) = \mu p^{dn} + O(np^{(d-1)n}).
\]
The work of \textit{V. Voevodsky} [Ann. Math. 174, 401--438 (2011; Zbl 1236.14026)] and others allows the use of Iwasawa cohomology groups with coefficients in \(\mathbb Z_p(i)\) for \(i\ge 2\), and this approach applies to arbitrary \(\mathbb Z_p\)-extensions, rather than only to the case of cyclotomic \(\mathbb Z_p\)-extensions studied by
\textit{J. Coates} [Ann. Math. (2) 95, 99--116 (1972; Zbl 0245.12005)] and by \textit{Q. Ji} and \textit{H. Qin} [J. \(K\)-Theory 12, No. 1, 115--123 (2013; Zbl 1288.11104)].
Reviewer: Lawrence C. Washington (College Park)Residual supersingular Iwasawa theory and signed Iwasawa invariantshttps://zbmath.org/1517.111392023-09-22T14:21:46.120933Z"Nuccio Mortarino Majno di Capriglio, Filippo A. E."https://zbmath.org/authors/?q=ai:nuccio.filippo-alberto-edoardo"Ramdorai, Sujatha"https://zbmath.org/authors/?q=ai:sujatha.ramdoraiSummary: For an odd prime \(p\) and a supersingular elliptic curve over a number field, this article introduces a multi-signed residual Selmer group, under certain hypotheses on the base field. This group depends purely on the residual representation at \(p\), yet captures information about the Iwasawa theoretic invariants of the signed \(p^{\infty}\)-Selmer group that arise in supersingular Iwasawa theory. Working in this residual setting provides a natural framework for studying congruences modulo \(p\) in Iwasawa theory.The distribution of values of \(\frac{L^\prime}{L}(1 / 2 + \epsilon, \chi_D)\)https://zbmath.org/1517.111402023-09-22T14:21:46.120933Z"Hamieh, Alia"https://zbmath.org/authors/?q=ai:hamieh.alia"McClenagan, Rory"https://zbmath.org/authors/?q=ai:mcclenagan.roryLet \(\chi_{D}(n)\) be the Kronecker symbol and \(0<\varepsilon<1/2\). The authors write:
``We determine the limiting distribution of the family of values \(\frac{L'}{L}(1/2+\varepsilon,\chi_{D})\) as \(D\) varies over fundamental discriminants\dots Moreover, we also establish an upper bound for the rate of convergence of this family to its limiting distribution.''
This result strengthens a theorem of \textit{M. Mourtada} and \textit{V. K. Murty} [Mosc. Math. J. 15, No. 3, 497--509 (2015; Zbl 1382.11058)].
For \(N\in\mathbb{N}\), let \(F(N)\) be the set of fundamental discriminants in the interval \([-N, N]\) and let
\[
m(N):=\min\left\{\left|\frac{L'}{L}(1/2+\varepsilon,\chi_{D})\right|: D\in F(N)\right\};
\]
as a consequence of their results, the authors obtain the upper bound
\[
m(N)<< \left(\frac{\log\log n}{\log n}\right)^{1/2+\varepsilon}\ \text{ as }N\rightarrow\infty.
\]
Reviewer: B. Z. Moroz (Bonn)Counting ideals in ray classeshttps://zbmath.org/1517.111412023-09-22T14:21:46.120933Z"Gun, Sanoli"https://zbmath.org/authors/?q=ai:gun.sanoli"Ramaré, Olivier"https://zbmath.org/authors/?q=ai:ramare.olivier"Sivaraman, Jyothsnaa"https://zbmath.org/authors/?q=ai:sivaraman.jyothsnaaLet \(I\) stand for the monoid of the integral ideals of a number field \(k\) of degree \(n:=[k:\mathbb Q]\), let \(B\) be a ray class of \(k\), and let
\[
\mathcal{N} (B, x):= \{{\mathfrak a}\mid {\mathfrak a}\in B\cap I, N{\mathfrak a}\leq x\}.
\]
The authors prove that
\[
|\mathcal{N} (B, x) - \alpha (B) x| \leq x^{(n-1)/n}f(B) + g(B)\text{ for }x\geq 1,
\]
where \(\alpha (B), f(B)\), and \(g(B)\) are expressed in terms of \(k\) and \(B\). According to the authors, they follow the approach developed by \textit{K. Debaene} [Int. J. Number Theory 15, No. 5, 883--905 (2019; Zbl 1456.11218)]. To obtain their asymptotic formula, the authors employ technique of geometry of numbers, citing, in particular, a recent theorem of \textit{M. Widmer} [Trans. Am. Math. Soc. 362, No. 9, 4793--4829 (2010; Zbl 1270.11064)] on the number of lattice points in a bounded subset of \(\mathbb R^{m}, m\geq 2\), of Lipschitz class.
Reviewer: B. Z. Moroz (Bonn)Periodicity criterion for continued fractions of key elements in hyperelliptic fieldshttps://zbmath.org/1517.111422023-09-22T14:21:46.120933Z"Platonov, V. P."https://zbmath.org/authors/?q=ai:platonov.vladimir-p"Fedorov, G. V."https://zbmath.org/authors/?q=ai:fedorov.gleb-vladimirovichSummary: The periodicity and quasi-periodicity of functional continued fractions in the hyperelliptic field \(L = \mathbb{Q}(x)(\sqrt f )\) has a more complex nature than the periodicity of numerical continued fractions of elements of quadratic fields. It is known that the periodicity of a continued fraction of \(\sqrt f {\text{/}}{{h}^{{g + 1}}}\) constructed using the valuation associated with a first-degree polynomial \(h\) is equivalent to the existence of nontrivial \(S\)-units in a field \(L\) of genus \(g\) and is equivalent to the existence of nontrivial torsion in the divisor class group. In this article, we find an exact interval of values of \(s \in \mathbb{Z}\) such that the elements \(\sqrt f {\text{/}}{{h}^s}\) have a periodic continued fraction expansion, where \(f \in \mathbb{Q}[x]\) is a square-free polynomial of even degree. For polynomials \(f\) of odd degree, the periodicity problem for continued fractions of elements of the form \(\sqrt f {\text{/}}{{h}^s}\) was discussed in [the authors, Sb. Math. 209, No. 4, 519--559 (2018; Zbl 1445.11135); translation from Mat. Sb. 209, No. 4, 54--94 (2018)], where it was proved that the length of the quasi-period does not exceed the degree of the fundamental \(S\)-unit of \(L\). For polynomials \(f\) of even degree, the periodicity of continued fractions of elements of the form \(\sqrt f {\text{/}}{{h}^s}\) is a more complicated problem. This is underlined by an example we have found, namely, a polynomial \(f\) of degree 4 for which the corresponding continued fraction has an abnormally long period. Earlier in [loc. cit.], for elements of a hyperelliptic field \(L\), we found examples of continued fractions with a quasi-period length significantly exceeding the degree of the fundamental \(S\)-unit of \(L\).Cogalois theory and Drinfeld moduleshttps://zbmath.org/1517.111432023-09-22T14:21:46.120933Z"Sánchez-Mirafuentes, Marco Antonio"https://zbmath.org/authors/?q=ai:sanchez-mirafuentes.marco-antonio"Salas-Torres, Julio Cesar"https://zbmath.org/authors/?q=ai:salas-torres.julio-cesar"Villa-Salvador, Gabriel"https://zbmath.org/authors/?q=ai:villa-salvador.gabriel-danielGiven a field extension \(L/K\), let \(\mathrm{Cog}(L/K)\) be the cogalois group defined as the torsion subgroup of the quotient group \(L^{*}/K^{*}\) of the multiplicative groups of \(L\) and \(K\). The analogue of the cogalois group for Drinfeld modules is obtained by replacing the multiplicative structure of the field by the one given by the Drinfeld module structure. \textit{M. Sánchez-Mirafuentes} and \textit{G. Villa-Salvador} [J. Algebra 398, 284--302 (2014; Zbl 1308.12003)] considered the case of Carlitz-Hayes action and obtained several results for cyclotomic function fields.
The authors generalize the results of the cited paper to rank one Drinfeld modules with class number one. The also show that, in the present form, there does not exist a cogalois theory for Drinfeld modules of rank or class number larger than one.
Reviewer: Andrzej Dąbrowski (Szczecin)Crystalline interpretation of the Deligne-Illusie isomorphismhttps://zbmath.org/1517.111442023-09-22T14:21:46.120933Z"Huyghe, Christine"https://zbmath.org/authors/?q=ai:huyghe.christine"Wach, Nathalie"https://zbmath.org/authors/?q=ai:wach.nathalieSummary: Let \(k\) be a finite field of characteristic \(p>0\), \(W(k)\) the ring of Witt vectors of \(k,X\) a smooth scheme over \(\operatorname{spec}W(k)\) of dimension \(<p-1\) and \(X_0\) the special fiber of \(X\). In 1987, \textit{P. Deligne} and \textit{L. Illusie} [Invent. Math. 89, 247--270 (1987; Zbl 0632.14017)] proved the degeneration of the spectral sequence ``de Hodge vers de Rham'' in a purely algebraic way, by constructing a quasi-isomorphism at the level of derived categories between the de Rham complex of \(X_0\) with a complex with \(0\) differentials. Simultaneously \textit{J.-M. Fontaine} and \textit{W. Messing} [Contemp. Math. 67, 179--207 (1987; Zbl 0632.14016)] constructed a divided Frobenius map on the crystalline complexes associated with \(X_0\). We show that both morphisms of derived categories are compatible mod \(p>0\). We use this compatibility to compute the filtered \(\varphi\)-module \(\bmod p\) and the \((\varphi,\Gamma)\)-module \(\bmod p\) associated to the Drinfeld curve.L-functions of certain exponential sums over finite fields. IIhttps://zbmath.org/1517.111452023-09-22T14:21:46.120933Z"Lin, Xin"https://zbmath.org/authors/?q=ai:lin.xin"Chen, Chao"https://zbmath.org/authors/?q=ai:chen.chaoLet \(n=\sum_{i=1}^r n_i\) be a partition of \(n\), \(\mathbb{F}_q\) be the finite field of \(q\) elements and \(p\) its characteristic. For \(\mathbf{a}=(a_1,\ldots,a_r)\in {\mathbb{F}_q^*}^r\) and positive integers \(b_{ij}\), \(1\le j\le n_i\), \(1\le i\le r\), the authors consider the character sums
\[
S_k(\mathbf{a})=\sum_{\sum_{i=1}^r \frac{a_i}{\prod_{j=1}^{n_i}x_{ij}^{b_{ij}}}=1} \psi\left(\mathrm{Tr}_k\left(\sum_{i=1}^r\sum_{j=1}^{n_i} x_{ij}\right)\right),
\]
where \(\psi\) is a nontrivial additive character of \(\mathbb{F}_p\), Tr\(_k\) denotes the absolute trace of \(\mathbb{F}_{q^k}\) and \(x_{ij}\) run through \(\mathbb{F}_{q^k}^*\). They prove that the \(L\)-function
\[
L(\mathbf{a},T)=\exp\left(\sum_{k=1}^\infty S_k(\mathbf{a})\frac{T^k}{k}\right)
\]
is a polynomial of degree
\[
d=\prod_{i=1}^r \left(1+\sum_{j=1}^{n_i}b_{ij}\right)
\]
and the number of its zeros with a certain slope can be expressed with a certain generating function.
This paper generalizes the author's previous paper [Math. Z. 300, No. 2, 1851--1871 (2022; Zbl 1501.11103)], which deals with \(n=4\), \(n_1=n_2=2\) and \(b_{ij}=1\).
The main tools are Adolphson-Sperber's result on toric exponential sums and Wan's decomposition theorem.
Reviewer: Arne Winterhof (Linz)On the index of the Diffie-Hellman mappinghttps://zbmath.org/1517.111462023-09-22T14:21:46.120933Z"Işık, Leyla"https://zbmath.org/authors/?q=ai:isik.leyla"Winterhof, Arne"https://zbmath.org/authors/?q=ai:winterhof.arneIn this paper, the authors determine the index of the univariate Diffie-Hellman mapping \(d(\gamma^a) = \gamma^{a^2}\), \(0 \le a \le n-1\), and \(\gamma\) is a generator of a cyclic group \(G\) of order \(n\). They show that any mapping of small index coincides with \(d\) only on a small subset of \(G\). After that, the authors introduce the index pair of a bivariate function over \(G\) and obtain similar results for the bivariate Diffie-Hellman mapping. Also, in the special case that \(G\) is a subgroup of the multiplicative subgroup \(\mathbb{F}^*_q\) of the finite field \(\mathbb{F}_q\) they obtain some improvements.
Reviewer: Zlatko Varbanov (Veliko Tarnovo)Exponential sums and flowershttps://zbmath.org/1517.111472023-09-22T14:21:46.120933Z"Proskurin, N. V."https://zbmath.org/authors/?q=ai:proskurin.nikolai-vThis short article shows the curious fact that the values of some exponential sums portray a flower in polar coordinates inside the unit complex disk. The exponential sums used to produce such pictures are of the form
\[
E_p(\psi)=\frac{1}{2\sqrt{p}}\sum_{x\in\mathbb{F}_p}\psi(x)e_p(x^2),
\]
where \(\psi\) is a cubic multiplicative character of \(\mathbb{F}_p\) and \(e_p\) denotes the standard additive character. The flowers are obtained by evaluating the sums for \(p\leq 360000\).
An interesting conjecture about the size of the set of primes for which \(|E_p(\psi)|\subset [0,1]\) is also given in Section 7.
Reviewer: Edwin León Cardenal (Zacatecas)Dwork hypersurfaces of degree six and Greene's hypergeometric functionhttps://zbmath.org/1517.111482023-09-22T14:21:46.120933Z"Kumabe, Satoshi"https://zbmath.org/authors/?q=ai:kumabe.satoshiSummary: In this paper, we give a formula for the number of rational points on the Dwork hypersurfaces of degree six over finite fields by using Greene's finite-field hypergeometric function, which is a generalization of Goodson's formula for the Dwork hypersurfaces of degree four. Our formula is also a higher-dimensional and a finite field analogue of Matsumoto-Terasoma-Yamazaki's formula. Furthermore, we also explain the relation between our formula and Miyatani's formula.Certain towers of ramified minimal ring extensions of commutative rings. IIhttps://zbmath.org/1517.130082023-09-22T14:21:46.120933Z"Dobbs, David E."https://zbmath.org/authors/?q=ai:dobbs.david-earlIn previous works, the author of this paper classified the (commutative) rings with a unique minimal subring, as well as the rings of zero characteristic with exactly two minimal subrings. He also reduced the case of positive characteristic \(n\) to classifying the rings \(\mathcal{E}\) for which there exists a tower of ramified extensions \(\mathbb{Z}/n\mathbb{Z}\subseteq\mathcal{B}\subseteq \mathcal{E}\) such that \(\mathcal{B}\) is the only ring properly contained between \(\mathbb{Z}/n\mathbb{Z}\) and \(\mathcal{E}\). Moreover, \(n\) can be assumed to be a power of an odd prime number \(p\): \(n=p^{\alpha}\). The author has already solved the case \(\alpha=1\), so in this paper he solves also the \(\alpha=2\). He classifies the rings \(R\) for which there exists a tower \( A \subseteq \mathcal{B} \subseteq R\) of ramified ring extensions such that \(\mathcal{B}\) is the only ring properly contained between \(A\) and \(R\). Moreover, \(\mathcal{B}\) is isomorphic to the idealization \(A(+)\mathbb{F}_p\). As a result of this classification, the author obtains for each integer \(n>1\) with prime factorization \(n=\prod_{i=1}^k p_{i}^{e_{i}}\), where \(e_{i}\le 4\) for all \(i\), a classification up to isomorphism of the rings of \(n\) elements that have exactly two proper subrings. Here is a concluding remark of the author: ``We will close by remarking that this work has only strengthened our belief that for \(\alpha\ge3\), future research on related questions should be computer-aided, at least to some extent.''
Reviewer: Moshe Roitman (Haifa)Controlling distribution of prime sequences in discretely ordered principal ideal subrings of \(\mathbb{Q}[x]\)https://zbmath.org/1517.130132023-09-22T14:21:46.120933Z"Glivická, Jana"https://zbmath.org/authors/?q=ai:glivicka.jana"Sgallová, Ester"https://zbmath.org/authors/?q=ai:sgallova.ester"Šaroch, Jan"https://zbmath.org/authors/?q=ai:saroch.janThe goal of the paper under review is the study of discretely ordered PIDs, as sub-rings of \(\mathbb{Q}[x]\), with special attention to the behaviour of finite progressions of primes. The authors discover a way to construct a profinite integer \(\tau\) such that for each primes \(f\not=g\in R_{\tau}\setminus\mathbb{Z}\), the difference \(f-g\not\in\mathbb{Z}\). Many applications are given.
Reviewer: Ali Benhissi (Monastir)Cluster duality between Calkin-Wilf tree and Stern-Brocot treehttps://zbmath.org/1517.130182023-09-22T14:21:46.120933Z"Gyoda, Yasuaki"https://zbmath.org/authors/?q=ai:gyoda.yasuakiSummary: We find a duality between two well-known trees, the Calkin-Wilf tree and the Stern-Brocot tree, derived from cluster algebra theory. The vertex sets of these trees are the set of positive rational numbers, and they have cluster structures induced by a one-punctured torus. In particular, the Calkin-Wilf tree is an example of the structure given by initial-seed mutations.
For the entire collection see [Zbl 1516.14004].Open problems in the wild McKay correspondence and related fieldshttps://zbmath.org/1517.140152023-09-22T14:21:46.120933Z"Yasuda, Takehiko"https://zbmath.org/authors/?q=ai:yasuda.takehikoSummary: The wild McKay correspondence is a form of McKay correspondence in terms of stringy invariants that is generalized to arbitrary characteristics. It gives rise to an interesting connection between the geometry of wild quotient varieties and arithmetic on extensions of local fields. The principal purpose of this article is to collect open problems on the wild McKay correspondence, as well as those in related fields that the author believes are interesting or important. It also serves as a survey on the present state of these fields.
For the entire collection see [Zbl 1516.14004].Algebraic points on the hyperelliptic curves \(y^2 = x^5 + n^2\)https://zbmath.org/1517.140212023-09-22T14:21:46.120933Z"Camara, Moustapha"https://zbmath.org/authors/?q=ai:camara.moustapha"Fall, Moussa"https://zbmath.org/authors/?q=ai:fall.moussa"Sall, Oumar"https://zbmath.org/authors/?q=ai:sall.oumarSummary: We give an algebraic description of the set of algebraic points of degree at most \(d\) over \(\mathbb{Q}\) on hyperelliptic curves \(y^2 = x^5 + n^2\).On centres and direct sum decompositions of higher degree formshttps://zbmath.org/1517.150172023-09-22T14:21:46.120933Z"Huang, Hua-Lin"https://zbmath.org/authors/?q=ai:huang.hua-lin"Lu, Huajun"https://zbmath.org/authors/?q=ai:lu.huajun"Ye, Yu"https://zbmath.org/authors/?q=ai:ye.yu"Zhang, Chi"https://zbmath.org/authors/?q=ai:zhang.chi.4The authors show that almost all direct sum decompositions of higher degree forms have trivial centre, i.e., isomorphic to the ground field. This means that they are a priori absolutely indecomposable. They also prove that the centre of the algebra of a higher-degree form is semisimple iff the form is not a limit of direct sums forms, as discussed in [\textit{W. Buczyńska} et al., Mich. Math. J. 64, No. 4, 675--719 (2015; Zbl 1339.13012)]. Furthermore, for forms with semisimple centre the authors develop an elementary criterion for the direct sum decomposability, which is equivalent to computing the rank of a finite set of vectors.
In Theorem 3.2, it is shown that a generic higher degree form is central; with the help of Theorems 3.6 and 3.7, a central form may be expressed as an indecomposable non-LDS (here LDS means limit of direct sums). In Theorem 3.11, an algorithm is discussed, purely in terms of linear algebra, for direct sum decompositions of any higher degree form. Such an algorithm can be obtained by slightly extending the algorithm for forms with semisimple centres again using the Jordan decomposition theorem.
The authors discuss several related results and examples.
Reviewer: M. P. Chaudhary (New Delhi)Centers of multilinear forms and applicationshttps://zbmath.org/1517.150182023-09-22T14:21:46.120933Z"Huang, Hua-Lin"https://zbmath.org/authors/?q=ai:huang.hua-lin"Lu, Huajun"https://zbmath.org/authors/?q=ai:lu.huajun"Ye, Yu"https://zbmath.org/authors/?q=ai:ye.yu"Zhang, Chi"https://zbmath.org/authors/?q=ai:zhang.chi.4Summary: The center algebra of a general multilinear form is defined and investigated. We show that the center of a nondegenerate multilinear form is a finite dimensional commutative algebra, and center algebras can be effectively applied to direct sum decompositions of multilinear forms. As an application of the algebraic structure of centers, we show that almost all multilinear forms are absolutely indecomposable. The theory of centers can also be applied to symmetric equivalence of multilinear forms. Moreover, with a help of the results of symmetric equivalence, we are able to provide a linear algebraic proof for a well known Torelli type result which says that two complex homogeneous polynomials with the same Jacobian ideal are linearly equivalent.Counting non-isomorphic generalized Hamilton quaternionshttps://zbmath.org/1517.160152023-09-22T14:21:46.120933Z"Grau, José María"https://zbmath.org/authors/?q=ai:grau.jose-maria"Miguel, Celino"https://zbmath.org/authors/?q=ai:miguel.celino"Oller-Marcén, Antonio M."https://zbmath.org/authors/?q=ai:oller-marcen.antonio-mQuaternion algebras over finite unital commutative rings \(R\) are studied. Recall that the quaternion algebra \(H=\left(\frac{a,b}{R}\right)\) are studied. Recall that \(H\) has \(R\)-basis \(\{1,i,j,k\}\) with multiplication rules \(i^2=a\), \(j^2=b\), \(ij=-ji=k\) extended linearly. The aim of the paper is to describe when two quaternion algebras over a finite ring of odd characteristic are isomorphic. It is important to remark that the authors consider the case where \(a,b \in R\) are not necessarily invertible. Otherwise the problem is trivial, since the Brauer group of a finite ring is trivial, implying all quaternion algebras given by invertible elements are \(2\times 2\)-matrix algebras. The authors reprove this in Lemma 4.1. As a finite ring is the direct sum of finite local rings, it suffices to study the case where \(R\) is local. All the results are summarized in the isomorphism Theorem 4.10. In the last Section, the authors prove that the number of isomorphism classes, in the case where the maximal ideal \(\mathfrak m\) is principal, is equal to \(2(k^2+1)\), where \(k\) is the index of nilpotence of \(\mathfrak m\).
Reviewer: Stefaan Caenepeel (Brussels)Local properties of Virasoro-like algebrahttps://zbmath.org/1517.170192023-09-22T14:21:46.120933Z"Tang, Xiaomin"https://zbmath.org/authors/?q=ai:tang.xiaomin.1|tang.xiaomin"Xiao, Mingyue"https://zbmath.org/authors/?q=ai:xiao.mingyue"Wang, Peng"https://zbmath.org/authors/?q=ai:wang.peng.2|wang.peng.25|wang.peng.4|wang.peng.7|wang.peng.5Summary: This paper studies local properties of the Virasoro-like algebra. Namely, the 2-local derivations and 2-local automorphisms of the Virasoro-like algebra are considered. It is proved that every 2-local derivation of this Lie algebra is a derivation, and every 2-local automorphism of this Lie algebra is an automorphism.Inverse Satake isomorphism and change of weighthttps://zbmath.org/1517.200092023-09-22T14:21:46.120933Z"Abe, N."https://zbmath.org/authors/?q=ai:abe.norihiro|abe.nobukado|abe.natsuroh|abe.narishige|abe.noriyuki|abe.naoto|abe.naohito|abe.naohiko|abe.naoki|abe.nobuhisa"Herzig, F."https://zbmath.org/authors/?q=ai:herzig.florian"Vignéras, M. F."https://zbmath.org/authors/?q=ai:vigneras.marie-franceLet \(F\) be a local non-Archimedean field with finite residue field \(k\) of characteristic \(p\). Let \(\mathbf{G}\) be a connected reductive \(F\)-group, and \(C\) be an algebraically closed field of characteristic \(p\). In a previous paper of the authors and \textit{G. Henniart} [J. Am. Math. Soc. 30, No. 2, 495--559 (2017; Zbl 1372.22014)], a classification of irreducible admissible smooth \(C\)-representations of \(G=\mathbf{G}(F)\) was given in terms of supercuspidal representations of Levi subgroups of \(G\). A fundamental ingredient of that paper is the so-called ``change of weight theorem'', which we deduced from the existence of certain elements in the image of the \(\bmod\ {p}\) Satake transform. This follows from the existence of certain elements in the image of the \(\bmod\ {p}\) Satake transform.
In the paper under review, the authors make an intensive study of the \(\bmod\ {p}\) Satake transform. They determine the image of the \(\bmod\ {p}\) Satake transform and give an explicit give an explicit formula for the inverse of the \(\bmod\ {p}\) Satake transform, which they call the inverse Satake theorem, from which the change of weight is an immediate consequence.
Reviewer: Enrico Jabara (Venezia)Aritmethic lattices of \(\operatorname{SO}(1,n)\) and units of group ringshttps://zbmath.org/1517.200462023-09-22T14:21:46.120933Z"Chagas, Sheila"https://zbmath.org/authors/?q=ai:chagas.sheila-c"del Rio, Ángel"https://zbmath.org/authors/?q=ai:del-rio.angel"Zalesskii, Pavel A."https://zbmath.org/authors/?q=ai:zalesskij.pavel-aLet \(G\) be a group. \(G\) is conjugacy separable if given any two elements \(x\) and \(y\) that are non-conjugate in \(G\), there exists some finite quotient of \(G\) in which the images of \(x\) and \(y\) are not conjugate. \(G\) is virtually compact special if it has a finite index subgroup isomorphic to the fundamental group of a compact special cube complex and \(G\) is toral relatively hyperbolic if \(G\) is torsion-free, and hyperbolic relative to a finite set of finitely generated abelian subgroups.
The main result of the paper under review is Theorem 1.1: Let \(G\) be a group having compact special and toral relatively hyperbolic subgroup of finite index. Then \(G\) is conjugacy separable.
As a consequence of Theorem 1.1, the authors also prove Theorem 1.2: Standard arithmetic lattices of special orthogonal groups \(\mathrm{SO}(1, n)\) are conjugacy separable.
The results above allow the authors to prove conjugacy separability for the group of units \(\mathcal{U}(\mathbb{Z}(G)\) of the integral group rings of some finite groups \(G\). These groups of units have nice residual properties that allows to approach the question to what extent a finitely generated group is determined by its finite quotients or equivalently by its profinite completion for \(U(\mathbb{Z}G)\).
Reviewer: Enrico Jabara (Venezia)Salem numbers, spectral radii and growth rates of hyperbolic Coxeter groupshttps://zbmath.org/1517.200642023-09-22T14:21:46.120933Z"Kellerhals, R."https://zbmath.org/authors/?q=ai:kellerhals.ruth"Liechti, L."https://zbmath.org/authors/?q=ai:liechti.livioA Coxeter system \((W, S)\) is a pair with \(W\) a group and \(S\) a set of generators that are reflections and the only relations arise from the angles of reflecting hyperplanes. A Coxeter group admits a faithful representation where the generators act as reflections on \(\mathbb{R}^N\). A product of all the generators of \(W\) is called a Coxeter element of \(W\), and the corresponding element in \(\mathrm{GL}_N(\mathbb{R})\) is called a Coxeter transformation. For a Coxeter group, the growth series is defined as
\[
f_S(t) = 1 + \sum_{k\ge 1}a_kt^k
\]
where \(a_k\) is the number of elements of \(S\)-length \(k\) and it is known to be a rational function that depends of the finite subgroups of \(W\). The inverse \(\tau = 1/R = \limsup_{k\to \infty}\sqrt[n]{a_k} \) of the radius of convergence of the series is called the growth rate of \(W\).
A Coxeter polyhedron \(P \subset \mathbb{H}^n\) is a convex polyhedron where all the dihedral angles are \({\pi}/m\) where \(m \in \{2, \dots \infty\}\). It is assumed that the polyhedron is of finite volume and thus it is bounded by finitely many hyperplanes. For \(n = 2\), \(P = (p_1, \dots p_k)\) is a hyperbolic polygon where \(p_1, \dots p_k \ge 2\) are integers and \(1/p_1 + \dots + 1/p_k < k - 2\). Reflections on \(\mathbb{H}^n\) with respect to the hyperplanes define a Coxeter group which is called a hyperbolic Coxeter group. It acts on \(\mathbb{H}^n\) with hyperbolic isometries. If \(n\) is 2 or 3, the growth rate is either a quadratic unit or a Salem number, that is a real algebraic integer \(\tau > 1\), whose conjugates have absolute value \(\le 1\) and at least one of them has absolute value 1.
The first main result of the paper states that not all Salem numbers appear as growth rates of hyperbolic Coxeter groups. For \(n = 2\), it is proved that the smallest growth rate is \(\tau_{[3, 7]}\) which is Lehmer's number \(\alpha_L\) and after that is \(\tau_{[3,8]}\) which is the seventh smallest number in the known list of Salem numbers. So the numbers in between in the list can not be realised as growth rates of hyperbolic Coxeter groups. For \(n = 3\), it is known that the smallest growth rate is the Salem number \(\tau_{[3,4,3]}\). So the first 47 known numbers are not realised. For \(n \ge 4\), it is proved that the growth rate is either larger that \(\tau_{[3,8]}\) or it is not a Salem number. There is no example of a hyperbolic Coxeter group with \(\tau > \tau_{[3,8]}\).
Let \((p_1, \dots p_k)\) be a sequence of integers, \(p_i \ge 2\). We construct the star graph \(\mathrm{Star}(p_1, \dots p_k)\) to be a tree with a vertex with degree \(k\) and \(k\) paths emanating from the vertex of length \(p_i -1\), respectively. Let \((p_1, \dots p_k)\) denote a convex polygon in \(\mathbb{H}^2\) and \(W\) the Coxeter group with Coxeter graph \(\mathrm{Star}(p_1, \dots p_k)\), i.e. with generators corresponding to the vertices of \(\mathrm{Star}(p_1, \dots p_k)\) and relations \((st)^3 = 1\) if two vertices are joined by an edge and \((st)^2 = 1\) otherwise. Then the growth rate of \(W\) equals to the spectral radius of the Coxeter transformation of \(W\).
The tree \(H(i,j,k)\) is defined as follows: we start with two vertices \(v_1\) and \(V_2\) of degree 3 and a path of length \(j\) joining them. To \(v_1\) we attach two paths, one having length \(1\) and the other of length \(i - 1\). Similarly, to \(v_2\) we attach a vertex and a path of length \(k-1\). To this graph, we associate a Coxeter group as before. The authors prove that to Coxeter group \([4, 3, 5]\) we associate the graph \(H(2, 8, 3)\) and the Coxeter group \(W\). They prove that \(\tau_{[4, 3, 5]}\) (which is a Salem number) equal to the spectral radius of the Coxeter transformation of \(W\). Also, they prove that \(\tau_{[3,5,3]}\) is not the spectral radius of a Coxeter element of a Coxeter group associated to a graph as above.
Reviewer: Stratos Prassidis (Karlovasi)Arf numerical semigroups with prime multiplicityhttps://zbmath.org/1517.200852023-09-22T14:21:46.120933Z"Karakaş, Halil İbrahim"https://zbmath.org/authors/?q=ai:karakas.halil-ibrahimAn \textit{Arf numerical semigroup} \(S\) is a numerical semigroup satisfying additionally
\[
x, y, z\in S;\ x\geq y\geq z\Rightarrow x+y-z\in S.
\]
Let \(\mathcal{S}_{\mathrm{ARF}}(m, c)\) denote the set of all Arf numerical semigroups with multiplicity \(m\) (the smallest positive element of \(S\)) and conductor \(c\) (the smallest element of \(S\) for which all subsequent natural numbers belong to \(S\)). It is proved that if \(p\) is prime and \(c > 2p\), then \(\mathcal{S}_{\mathrm{ARF}}(p, c+p)=\{(p+S)\cup\{0\}\colon S\in \mathcal{S}_{\mathrm{ARF}}(p, c)\}\) (and, consequently, \(\vert \mathcal{S}_{\mathrm{ARF}}(p, c+p)\vert=\vert\mathcal{S}_{\mathrm{ARF}}(p, c)\vert\)).
Reviewer: Peeter Normak (Tallinn)Inequalities for Lerch transcendent functionhttps://zbmath.org/1517.260172023-09-22T14:21:46.120933Z"Cerone, Pietro"https://zbmath.org/authors/?q=ai:cerone.pietro"Dragomir, Silvestru Sever"https://zbmath.org/authors/?q=ai:dragomir.sever-silvestruSummary: Some fundamental inequalities for Lerch transcendent function with positive terms by utilising certain classical results due to Hölder, Čebyšev, Grüss and others, are established. Some particular cases of interest for Polylogarithm function, Hurwitz zeta function and Legendre chi function are also given.Finding determinant and integral forms of the 2-iterated \(2D\) Appell polynomialshttps://zbmath.org/1517.330092023-09-22T14:21:46.120933Z"Wani, Shahid Ahmad"https://zbmath.org/authors/?q=ai:wani.shahid-ahmad"Nahid, Tabinda"https://zbmath.org/authors/?q=ai:nahid.tabindaSummary: This article deals with the derivation of determinant and integral forms for the 2-iterated \(2D\) Appell polynomials. The summation formulae and operational rule for these polynomials are derived. Also, certain identities for these polynomials are established by using operational formalism. The exponential operational definitions and integral representations are combined to introduce the extended form of these 2-iterated \(2D\) Appell polynomials. Further, by using computer-aided programs (Mathematica or Matlab), we draw graphs of some particular cases of the 2-iterated \(2D\) Appell polynomials, mainly in order to observe in several angles how zeros of these polynomials are distributed and located.Fractional Laplacians on the sphere, the Minakshisundaram zeta function and semigroupshttps://zbmath.org/1517.351012023-09-22T14:21:46.120933Z"De Nápoli, Pablo Luis"https://zbmath.org/authors/?q=ai:de-napoli.pablo-luis"Stinga, Pablo Raúl"https://zbmath.org/authors/?q=ai:stinga.pablo-raulSummary: In this paper we show novel underlying connections between fractional powers of the Laplacian on the unit sphere and functions from analytic number theory and differential geometry, like the Hurwitz zeta function and the Minakshisundaram zeta function. Inspired by \textit{S. Minakshisundaram}'s ideas [J. Indian Math. Soc., New Ser. 13, 41--48 (1949; Zbl 0033.11605)], we find a precise pointwise description of \((-\Delta _{\mathbb{S}^{n-1}})^su(x)\) in terms of fractional powers of the Dirichlet-to-Neumann map on the sphere. The Poisson kernel for the unit ball will be essential for this part of the analysis. On the other hand, by using the heat semigroup on the sphere, additional pointwise integro-differential formulas are obtained. Finally, we prove a characterization with a local extension problem and the interior Harnack inequality.
For the entire collection see [Zbl 1416.35007].Ergodicity of Iwasawa continued fractions via markable hyperbolic geodesicshttps://zbmath.org/1517.370112023-09-22T14:21:46.120933Z"Lukyanenko, Anton"https://zbmath.org/authors/?q=ai:lukyanenko.anton"Vandehey, Joseph"https://zbmath.org/authors/?q=ai:vandehey.josephSummary: We prove the convergence and ergodicity of a wide class of real and higher-dimensional continued fraction algorithms, including folded and \(\alpha\)-type variants of complex, quaternionic, octonionic, and Heisenberg continued fractions, which we combine under the framework of Iwasawa continued fractions. The proof is based on the interplay of continued fractions and hyperbolic geometry, the ergodicity of geodesic flow in associated modular manifolds, and a variation on the notion of geodesic coding that we refer to as geodesic marking. As a corollary of our study of markable geodesics, we obtain a generalization of Serret's tail-equivalence theorem for almost all points. The results are new even in the case of some real and complex continued fractions.Some counterexamples to the central limit theorem for random rotationshttps://zbmath.org/1517.370122023-09-22T14:21:46.120933Z"Czudek, Klaudiusz"https://zbmath.org/authors/?q=ai:czudek.klaudiuszWhen we talk about Central Limit Theorems (CLTs) in probability theory, we, often quite understandably, focus on the success stories and provide sufficient conditions that ensure a Gaussian limit. The present paper by is a refreshing take on the subject as it provides a number of counterexamples to celebrated CLTs.
The primary object of study is a discrete-time Random Walk (RW) \((Y_n^\alpha)_{n\ge 1}\) on the circle \(\mathbb{S}^{1}\). At each time step, the random walk moves to either \(x+\alpha\) or \(x-\alpha\) with equal probability, where \(\alpha \in \mathbb{R}\). One should think of \(\alpha\) as the angle of rotation. The author is able to connect properties of the angle with the validity of the CLT for additive functionals of the random walk, i.e., if \(n^{-1/2}(\psi(Y_1^\alpha) + \psi(Y_2^\alpha) + \ldots + \psi(Y_n^\alpha))\) converges to a Gaussian distribution. In particular, the angle \(\alpha\) is said to be Diophantine of type \((c, \gamma) \in (0, \infty)\times[2, \infty)\) if \(|\alpha - p/q | \ge c q^{-\gamma}\) for all integers \(p, q \ne 0\). It is called Liouville if it is not Diophantine.
The authors prove the following:
(1) There exists a Liouville angle and an analytic functional for which the CLT does not hold;
(2) If the angle \(\alpha\) is irrational and satisfies \(|\alpha - p/q| \le cq^{-\gamma}\) for infinitely many integers \(p, q\ne 0\) and some \(c>0, \gamma >5\), then there exists a continuous function with first \(r\) derivatives, where \(r\) is the largest positive integer less than \((\gamma - 3)/2\);
(3) For a given Liouville angle, there exist smooth functions \(\psi\) such that the CLT does not hold.
The results obtained in this paper are reminiscent of conjugacy results for circle diffeomorphisms. The proof goes by solving the Poisson equation, which is a functional analytic approach, but has also become standard in probability theory.
The results presented in this paper are refreshing and leave you with uncanny satisfaction!
Reviewer: Wasiur Rahman Khuda Bukhsh (Nottingham)Heights and arithmetic dynamicshttps://zbmath.org/1517.370962023-09-22T14:21:46.120933Z"Yasufuku, Yu"https://zbmath.org/authors/?q=ai:yasufuku.yuSummary: We survey some recent results in the field of arithmetic dynamics. We especially focus on topics where the height functions play important roles, namely integral points in orbits and Kawaguchi-Silverman conjecture relating arithmetic degrees with dynamical degrees.Unlikely intersection problems for restricted lifts of a \(p\)-th powerhttps://zbmath.org/1517.370972023-09-22T14:21:46.120933Z"Peng, Wayne"https://zbmath.org/authors/?q=ai:peng.wayneSummary: We applied the theory of perfectoid spaces to prove dynamical versions of the Manin-Mumford conjecture, Mordell-Lang conjecture, and Tate-Voloch conjecture for lifts of a \(p\)-th power, generalizing the work of \textit{J. Xie} in [Algebra Number Theory 12, No. 7, 1715--1748 (2018; Zbl 1415.37116)] for lifts of Frobenius.Variants of \(q\)-hypergeometric equationhttps://zbmath.org/1517.390022023-09-22T14:21:46.120933Z"Hatano, Naoya"https://zbmath.org/authors/?q=ai:hatano.naoya"Matsunawa, Ryuya"https://zbmath.org/authors/?q=ai:matsunawa.ryuya"Sato, Tomoki"https://zbmath.org/authors/?q=ai:sato.tomoki"Takemura, Kouichi"https://zbmath.org/authors/?q=ai:takemura.kouichiSummary: We introduce two variants of the \(q\)-hypergeometric equation. We obtain several explicit solutions of the variants of the \(q\)-hypergeometric equation. We show that a variant of the \(q\)-hypergeometric equation can be obtained by a restriction of the \(q\)-Appell equation of two variables.Twisted Blanchfield pairings and twisted signatures. I: Algebraic backgroundhttps://zbmath.org/1517.570042023-09-22T14:21:46.120933Z"Borodzik, Maciej"https://zbmath.org/authors/?q=ai:borodzik.maciej"Conway, Anthony"https://zbmath.org/authors/?q=ai:conway.anthony"Politarczyk, Wojciech"https://zbmath.org/authors/?q=ai:politarczyk.wojciechIn this article, the authors lay the algebraic foundations for forthcoming papers devoted to the study of twisted linking forms of knots and three-manifolds. Their goal is to describe how to define and calculate signature invariants associated to a linking form \(M \times M \rightarrow \mathbb{F} (t)/ \mathbb{F} [t^{\pm1}]\) for \(\mathbb{F} = \mathbb{R}, \mathbb{C}\), where \(M\) is a torsion \(\mathbb{F} [t^{\pm1}]\)-module.
They begin by stating and proving a classification result for \(\mathbb{F} [t ^{\pm1}]\)-linking forms. Then they apply these methods to linking forms over local rings and study their representability and their classification up to isometry and Witt equivalence. Finally they define signature jumps and the signature function of a \(\mathbb{F} [t ^{\pm1}]\)-linking form.
Reviewer: Leila Ben Abdelghani (Monastir)Fuglede-Kadison determinants over free groups and Lehmer's constantshttps://zbmath.org/1517.570142023-09-22T14:21:46.120933Z"Ben Aribi, Fathi"https://zbmath.org/authors/?q=ai:ben-aribi.fathiSummary: Lehmer's famous problem asks whether the set of Mahler measures of polynomials with integer coefficients admits a gap at \(1\). In [J. Topol. Anal. 14, No. 4, 901--932 (2022; Zbl 07635915)], \textit{W. Lück} extended this question to Fuglede-Kadison determinants of a general group, and he defined the Lehmer's constants of the group to measure such a gap.
In this paper, we compute new values for Fuglede-Kadison determinants over non-cyclic free groups, which yields the new upper bound \(\frac{2}{\sqrt{3}}\) for Lehmer's constants of all torsion-free groups which have non-cyclic free subgroups.
Our proofs use relations between Fuglede-Kadison determinants and random walks on Cayley graphs, as well as works of \textit{L. Bartholdi} [Enseign. Math. (2) 45, No. 1--2, 83--131 (1999; Zbl 0961.05032)] and \textit{O. T. Dasbach} and \textit{M. N. Lalin} [Forum Math. 21, No. 4, 621--637 (2009; Zbl 1225.11131)].
Furthermore, via the gluing formula for \(L^2\)-torsions, we show that the Lehmer's constants of an infinite number of fundamental groups of hyperbolic 3-manifolds are bounded above by even smaller values than \(\frac{2}{\sqrt{3}}\).A two term Kuznecov sum formulahttps://zbmath.org/1517.580102023-09-22T14:21:46.120933Z"Wyman, Emmett L."https://zbmath.org/authors/?q=ai:wyman.emmett-l"Xi, Yakun"https://zbmath.org/authors/?q=ai:xi.yakunSummary: The Kuznecov sum formula, proved by \textit{S. Zelditch} in the Riemannian setting [Commun. Partial Differ. Equations 17, No. 1--2, 221--260 (1992; Zbl 0749.58062)], is an asymptotic sum formula
\[
N(\lambda ):= \sum_{\lambda_j \le \lambda } \left| \int_H e_j \, dV_H \right|^2 = C_{H,M} \lambda^{{\text{codim}}H} + O(\lambda^{{\text{codim}}H - 1})
\]
where \(e_j\) constitute a Hilbert basis of Laplace-Beltrami eigenfunctions on a Riemannian manifold \(M\) with \(\Delta_g e_j = -\lambda_j^2 e_j\), and \(H\) is an embedded submanifold. Assuming that the looping time set is countable, we show for some suitable definition of `\( \sim\)',
\[
N(\lambda ) \sim C_{H,M} \lambda^{{\text{codim}}H} + Q(\lambda ) \lambda^{{\text{codim}}H - 1} + o(\lambda^{{\text{codim}}H - 1})
\]
where \(Q\) is a bounded oscillating term and is expressed in terms of the geodesics which depart and arrive \(H\) in the normal directions. Our result generalizes a theorem of \textit{Yu. G. Safarov} on the pointwise Weyl law [Funct. Anal. Appl. 22, No. 3, 213--223 (1988; Zbl 0679.35074); translation from Funkts. Anal. Prilozh. 22, No. 3, 53--65 (1988)] in this case. In [\textit{Y. Canzani} et al., Commun. Math. Phys. 360, No. 2, 619--637 (2018; Zbl 1393.53031)] and [\textit{Y. Canzani} and \textit{J. Galkowski}, Duke Math. J. 168, No. 16, 2991--3055 (2019; Zbl 1471.35213)], the authors established (as a corollary to a stronger result involving defect measures) that if the set of recurrent directions of geodesics normal to \(H\) has measure zero, then we obtain improved bounds on the individual terms in the sum -- the period integrals. We are able to give a dynamical condition such that \(Q\) is uniformly continuous and `\( \sim \)' can be replaced with `\(=\)'. This implies improved bounds on period integrals, and this condition holds if the recurrent directions have measure zero. Moreover, our result implies improved bounds for period integrals if there is no \(L^1\) measure on \(SN^*H\) that is invariant under the first return map. This generalizes a theorem of \textit{C. D. Sogge} and \textit{S. Zelditch} [Rev. Mat. Iberoam. 32, No. 3, 971--994 (2016; Zbl 1361.32011)] and \textit{Y. Canzani} and \textit{J. Galkowski} [Duke Math. J. 168, No. 16, 2991--3055 (2019; Zbl 1471.35213)].Minimal automaton for multiplying and translating the Thue-Morse sethttps://zbmath.org/1517.681792023-09-22T14:21:46.120933Z"Charlier, Émilie"https://zbmath.org/authors/?q=ai:charlier.emilie"Cisternino, Célia"https://zbmath.org/authors/?q=ai:cisternino.celia"Massuir, Adeline"https://zbmath.org/authors/?q=ai:massuir.adelineSummary: The Thue-Morse set \(\mathcal{T}\) is the set of those non-negative integers whose binary expansions have an even number of \(1\)'s. The name of this set comes from the fact that its characteristic sequence is given by the famous Thue-Morse word \[\mathtt{0110100110010110}\cdots,\] which is the fixed point starting with \(0\) of the word morphism \(\mathtt{0\mapsto 01}\), \(\mathtt{1\mapsto 10}\). The numbers in \(\mathcal{T}\) are commonly called the evil numbers. We obtain an exact formula for the state complexity of the set \(m\mathcal{T}+r\) (i.e. the number of states of its minimal automaton) with respect to any base \(b\) which is a power of \(2\). Our proof is constructive and we are able to explicitly provide the minimal automaton of the language of all \(2^p\)-expansions of the set of integers \(m\mathcal{T}+r\) for any positive integers \(p\) and \(m\) and any remainder \(r\in\{0,\ldots,m{-}1\} \). The proposed method is general for any \(b\)-recognizable set of integers.Efficient multiplication of somewhat small integers using number-theoretic transformshttps://zbmath.org/1517.684242023-09-22T14:21:46.120933Z"Becker, Hanno"https://zbmath.org/authors/?q=ai:becker.hanno"Hwang, Vincent"https://zbmath.org/authors/?q=ai:hwang.vincent"Kannwischer, Matthias J."https://zbmath.org/authors/?q=ai:kannwischer.matthias-j"Panny, Lorenz"https://zbmath.org/authors/?q=ai:panny.lorenz"Yang, Bo-Yin"https://zbmath.org/authors/?q=ai:yang.bo-yinSummary: Conventional wisdom purports that FFT-based integer multiplication methods (such as the Schönhage-Strassen algorithm) begin to compete with Karatsuba and Toom-Cook only for integers of several tens of thousands of bits. In this work, we challenge this belief, leveraging recent advances in the implementation of number-theoretic transforms (NTT) stimulated by their use in post-quantum cryptography. We report on implementations of NTT-based integer arithmetic on two Arm Cortex-M CPUs on opposite ends of the performance spectrum: Cortex-M3 and Cortex-M55. Our results indicate that NTT-based multiplication is capable of outperforming the big-number arithmetic implementations of popular embedded cryptography libraries for integers as small as 2048 bits. To provide a realistic case study, we benchmark implementations of the RSA encryption and decryption operations. Our cycle counts on Cortex-M55 are about \(10\times\) lower than on Cortex-M3.
For the entire collection see [Zbl 1503.68013].Dirac series of \(E_{7 (- 5)}\)https://zbmath.org/1517.810672023-09-22T14:21:46.120933Z"Ding, Yi-Hao"https://zbmath.org/authors/?q=ai:ding.yihao"Dong, Chao-Ping"https://zbmath.org/authors/?q=ai:dong.chao-ping"Li, Ping-Yuan"https://zbmath.org/authors/?q=ai:li.pingyuanSummary: Using the sharpened Helgason-Johnson bound, this paper classifies all the irreducible unitary representations with non-zero Dirac cohomology of \(E_{7 (- 5)}\). As an application, we find that the cancellation between the even part and the odd part of the Dirac cohomology continues to happen for certain unitary representations of \(E_{7 (- 5)}\). Assuming the infinitesimal character being integral, we further improve the Helgason-Johnson bound for \(E_{7 (- 5)}\). This should help people to understand (part of) the unitary dual of this group.Noncommutative tensor triangular geometryhttps://zbmath.org/1517.810712023-09-22T14:21:46.120933Z"Nakano, Daniel K."https://zbmath.org/authors/?q=ai:nakano.daniel-k"Vashaw, Kent B."https://zbmath.org/authors/?q=ai:vashaw.kent-b"Yakimov, Milen T."https://zbmath.org/authors/?q=ai:yakimov.milen-tSummary: We develop a general noncommutative version of s tensor triangular geometry that is applicable to arbitrary monoidal triangulated categories M\(\Delta\)Cs). Insight from noncommutative ring theory is used to obtain a framework for prime, semiprime, and completely prime (thick) ideals of an M\(\Delta\)C, \textbf{K}, and then to associate to \textbf{K} a topological space-the Balmer spectrum \(\operatorname{Spc} \mathbf{K}\). We develop a general framework for (noncommutative) support data, coming in three different flavors, and show that \(\operatorname{Spc} \mathbf{K}\) is a universal terminal object for the first two notions (support and weak support). The first two types of support data are then used in a theorem that gives a method for the explicit classification of the thick (two-sided) ideals and the Balmer spectrum of an M\(\Delta\)C. The third type (quasi support) is used in another theorem that provides a method for the explicit classification of the thick right ideals of \textbf{K}, which in turn can be applied to classify the thick two-sided ideals and \(\operatorname{Spc} \mathbf{K}\).
As a special case, our approach can be applied to the stable module categories of arbitrary finite dimensional Hopf algebras that are not necessarily cocommutative (or quasitriangular). We illustrate the general theorems with classifications of the Balmer spectra and thick two-sided/right ideals for the stable module categories of all small quantum groups for Borel subalgebras, and classifications of the Balmer spectra and thick two-sided ideals of Hopf algebras studied by \textit{D. Benson} and \textit{S. Witherspoon} [Arch. Math. 102, No. 6, 513--520 (2014; Zbl 1310.16025)].Automorphic scalar fields in two-dimensional de Sitter spacehttps://zbmath.org/1517.830782023-09-22T14:21:46.120933Z"Higuchi, Atsushi"https://zbmath.org/authors/?q=ai:higuchi.atsushi"Schmieding, Lasse"https://zbmath.org/authors/?q=ai:schmieding.lasse"Serrano Blanco, David"https://zbmath.org/authors/?q=ai:serrano-blanco.davidSummary: We study non-interacting automorphic quantum scalar fields with positive mass in two-dimensional de Sitter space. We find that there are no Hadamard states which are de Sitter invariant except in the periodic case, extending the result of Epstein and Moschella for the anti-periodic case. We construct the two-point Wightman functions for the non-Hadamard de Sitter-invariant states by exploiting the fact that they are functions of the geodesic distance between the two points satisfying an ordinary differential equation. We then examine a certain Hadamard state, which is not de Sitter invariant, and show that it is approximately a thermal state with the Gibbons-Hawking temperature when restricted to a static region of the spacetime.Applying iterated mapping to the no-three-in-a-line problemhttps://zbmath.org/1517.901102023-09-22T14:21:46.120933Z"Brower, Cole"https://zbmath.org/authors/?q=ai:brower.cole-rubin"Ponomarenko, Vadim"https://zbmath.org/authors/?q=ai:ponomarenko.vadimSummary: Iterated mapping has seen a lot of success lately in many problems such as bit retrieval, diffraction signal reconstruction, and graph coloring. We add another application of iterated mapping, namely finding solutions to the no-three-in-a-line problem. Given an \(n \times n\) grid, we utilize iterated mapping to find \(2n\) points such that any straight line (of any slope) drawn will not intersect three of the selected points.Block coordinate type methods for optimization and learninghttps://zbmath.org/1517.901162023-09-22T14:21:46.120933Z"Yu, Zhan"https://zbmath.org/authors/?q=ai:yu.zhanSummary: We study nonconvex (composite) optimization and learning problems where the decision variables can be split into blocks of variables. Random block projection is a popular technique to handle this kind of problem for its remarkable reduction of the computational cost from the projection. This powerful method has not been well proposed for the situation that first-order information is prohibited and only zeroth-order information is available. In this paper, we propose to develop different classes of zeroth-order stochastic block coordinate type methods. Zeroth-order block coordinate descent (ZS-BCD) is proposed for solving unconstrained nonconvex optimization problem. For composite optimization, we establish the zeroth-order stochastic block mirror descent (ZS-BMD) and its associated two-phase method to achieve the complexity bound for the composite optimization problem. Furthermore, we also establish zeroth-order stochastic block coordinate conditional gradient (ZS-BCCG) method for nonconvex (composite) optimization. By implementing ZS-BCCG method, in each iteration, only (approximate) linear programming subproblem needs to be solved on a random block instead of a rather costly projection subproblem on the whole decision space, in contrast to the existing traditional stochastic approximation methods. In what follows, an approximate ZS-BCCG method and corresponding two-phase ZS-BCCG method are proposed. This is also the first time that a two-phase BCCG method has been developed to carry out the complexity analysis of nonconvex composite optimization problem.Adventures in supersingularlandhttps://zbmath.org/1517.940572023-09-22T14:21:46.120933Z"Arpin, Sarah"https://zbmath.org/authors/?q=ai:arpin.sarah"Camacho-Navarro, Catalina"https://zbmath.org/authors/?q=ai:camacho-navarro.catalina"Lauter, Kristin"https://zbmath.org/authors/?q=ai:lauter.kristin-e"Lim, Joelle"https://zbmath.org/authors/?q=ai:lim.joelle"Nelson, Kristina"https://zbmath.org/authors/?q=ai:nelson.kristina"Scholl, Travis"https://zbmath.org/authors/?q=ai:scholl.travis"Sotáková, Jana"https://zbmath.org/authors/?q=ai:sotakova.janaSummary: Supersingular Isogeny Graphs were introduced as a source of hard problems in cryptography by \textit{D. X. Charles} et al. [J. Cryptology 22, No. 1, 93--113 (2009; Zbl 1166.94006)] for the construction of cryptographic hash functions and have been used for key exchange SIKE. The security of such systems depends on the difficulty of finding a path between two random vertices. In this article, we study several aspects of the structure of these graphs. First, we study the subgraph given by \(j\)-invariants in \(\mathbb{F}_p\), using the related isogeny graph consisting of only \(\mathbb{F}_p\)-rational curves and isogenies. We prove theorems on how the connected components thereof attach, stack, and fold when mapped into the full graph. The \(\mathbb{F}_p\)-rational vertices are fixed by the Frobenius involution on the graph, and we call the induced graph the spine. Finding paths to the spine is relevant in cryptanalysis. Second, we present numerous computational experiments and heuristics relating to the position of the spine within the whole graph. These include studying the distance of random vertices to the spine, estimates of the diameter of the graph, how often paths are preserved under the Frobenius involution, and what proportion of vertices are conjugate. We compare some of the heuristics with known results on other Ramanujan graphs.Formal security proof for a scheme on a topological networkhttps://zbmath.org/1517.940842023-09-22T14:21:46.120933Z"Civino, Roberto"https://zbmath.org/authors/?q=ai:civino.roberto"Longo, Riccardo"https://zbmath.org/authors/?q=ai:longo.riccardoSummary: Key assignment and key maintenance in encrypted networks of resource-limited devices may be a challenging task, due to the permanent need of replacing out-of-service devices with new ones and to the consequent need of updating the key information. Recently, \textit{R. Aragona} et al. [J. Discrete Math. Sci. Cryptography 25, No. 8, 2429--2448 (2022; Zbl 1504.94094)] proposed a new cryptographic scheme, ECTAKS, which provides a solution to this design problem by means of a Diffie-Hellman-like key establishment protocol based on elliptic curves and on a prime field. Even if the authors proved some results related to the security of the scheme, the latter still lacks a formal security analysis. In this paper, we address this issue by providing a security proof for ECTAKS in the setting of computational security, assuming that no adversary can solve the underlying discrete logarithm problems with non-negligible success probability.Accelerating the Delfs-Galbraith algorithm with fast subfield root detectionhttps://zbmath.org/1517.940862023-09-22T14:21:46.120933Z"Corte-Real Santos, Maria"https://zbmath.org/authors/?q=ai:corte-real-santos.maria"Costello, Craig"https://zbmath.org/authors/?q=ai:costello.craig"Shi, Jia"https://zbmath.org/authors/?q=ai:shi.jiaSummary: We give a new algorithm for finding an isogeny from a given supersingular elliptic curve \(E/\mathbb{F}_{p^2}\) to a subfield elliptic curve \(E'/\mathbb{F}_p\), which is the bottleneck step of the Delfs-Galbraith algorithm for the general supersingular isogeny problem. Our core ingredient is a novel method of rapidly determining whether a polynomial \(f \in L[X]\) has any roots in a subfield \(K \subset L\), while avoiding expensive root-finding algorithms. In the special case when \(f=\Phi_{\ell ,p}(X,j) \in \mathbb{F}_{p^2}[X]\), i.e., when \(f\) is the \(\ell \)-th modular polynomial evaluated at a supersingular \(j\)-invariant, this provides a means of efficiently determining whether there is an \(\ell \)-isogeny connecting the corresponding elliptic curve to a subfield curve. Together with the traditional Delfs-Galbraith walk, inspecting many \(\ell \)-isogenous neighbours in this way allows us to search through a larger proportion of the supersingular set per unit of time. Though the asymptotic \(\tilde{O}(p^{1/2})\) complexity of our improved algorithm remains unchanged from that of the original Delfs-Galbraith algorithm, our theoretical analysis and practical implementation both show a significant reduction in the runtime of the subfield search. This sheds new light on the concrete hardness of the general supersingular isogeny problem (i.e. the foundational problem underlying isogeny-based cryptography), and has immediate implications on the bit-security of schemes like B-SIDH and SQISign for which Delfs-Galbraith is the best known classical attack.
For the entire collection see [Zbl 1514.94003].Constructing the classes of Boolean functions with guaranteed cryptographic properties on the base of coordinate functions of the finite field power mappingshttps://zbmath.org/1517.941102023-09-22T14:21:46.120933Z"Ivanov, A. V."https://zbmath.org/authors/?q=ai:ivanov.alexey-v|ivanov.aleksei-valerevich|ivanov.anton-valerevich|ivanov.aleksandr-vladimirovich|ivanov.aleksei-valerevich.1|ivanov.aleksei-vladimirovich|ivanov.andrey-v"Romanov, V. N."https://zbmath.org/authors/?q=ai:romanov.vyacheslav-nSummary: In the paper an approach to constructing the nonlinear approximations of Boolean functions is offered. The approximations are constructed by using the coordinate functions of the finite field power mapping. The effectiveness of such approximations for bent functions is shown.A survey on functional encryptionhttps://zbmath.org/1517.941312023-09-22T14:21:46.120933Z"Mascia, Carla"https://zbmath.org/authors/?q=ai:mascia.carla"Sala, Massimiliano"https://zbmath.org/authors/?q=ai:sala.massimiliano"Villa, Irene"https://zbmath.org/authors/?q=ai:villa.ireneSummary: Functional Encryption (FE) expands traditional public-key encryption in two different ways: it supports fine-grained access control and allows learning a function of the encrypted data. In this paper, we review all FE classes, describing their functionalities and main characteristics. In particular, we mention several schemes for each class, providing their security assumptions and comparing their properties. To our knowledge, this is the first survey that encompasses the entire FE family.Cryptographic systems based on an algebraic structurehttps://zbmath.org/1517.941322023-09-22T14:21:46.120933Z"Matysiak, Łukasz"https://zbmath.org/authors/?q=ai:matysiak.lukasz"Chrzaniuk, Monika"https://zbmath.org/authors/?q=ai:chrzaniuk.monika"Duda, Maximilian"https://zbmath.org/authors/?q=ai:duda.maximilian"Hanc, Marta"https://zbmath.org/authors/?q=ai:hanc.marta"Kowalski, Sebastian"https://zbmath.org/authors/?q=ai:kowalski.sebastian"Skotnicka, Zoja"https://zbmath.org/authors/?q=ai:skotnicka.zoja"Waldoch, Martin"https://zbmath.org/authors/?q=ai:waldoch.martinSummary: In this paper cryptographic systems based on the Dedekind and Galois structures are considered. We supplement the created cryptosystems based on the Dedekind structure with programs written in C++ and discuss the inner structure of Galois in cryptography. It is well-known that such a structure is based on finite fields only. Our results reveal something more internal. The final section contains additional information about square-free and radical factorizations in monoids consisting in searching for a minimal list of counterexamples. As an open problem, we leave creating a program that would generate such a list and how to use such a list to create a cryptosystem.A note on the \(c\)-differential spectrum of an AP\(c\)N functionhttps://zbmath.org/1517.941372023-09-22T14:21:46.120933Z"Pang, Tingting"https://zbmath.org/authors/?q=ai:pang.tingting"Li, Nian"https://zbmath.org/authors/?q=ai:li.nian"Zeng, Xiangyong"https://zbmath.org/authors/?q=ai:zeng.xiangyong"Zhu, Haiying"https://zbmath.org/authors/?q=ai:zhu.haiyingSummary: Motivated by a recent work of \textit{H. Yan} and \textit{K. Zhang} [Des. Codes Cryptography 90, No. 10, 2385--2405 (2022; Zbl 1506.14057)] on the \(c \)-differential spectrum of some power functions over finite fields, we further study an AP\(c\)N function and express its \(c \)-differential spectrum in terms of \((i, j, k)_2 \), i.e., the cardinality of the intersection \((\mathcal{C}^{(2)}_i+1)\cap\mathcal{C}^{(2)}_j\cap(\mathcal{C}^{(2)}_k-1)\) for \(i, j, k\in\{0, 1\} \), where \(\mathcal{C}^{(2)}_0, \mathcal{C}^{(2)}_1\) are the cyclotomic classes of order two over the finite field \(\mathbb{F}_{p^n} , p\) is an odd prime and \(n\) is a positive integer. By virtue of the cyclotomic numbers of orders two and four, we determine the values of \((i, j, k)_2\) for \(i, j, k\in\{0, 1\} \), which may be of independent interest. As an application, we give another proof of the \(c \)-differential spectrum of an AP\(c\)N function over finite fields with characteristic \(5 \). Further, we refine the result of Zhang and Yan [loc. cit.] in the sense that we completely characterize the conditions when the \(c \)-differential equation of the AP\(c\)N function has one solution and two solutions, respectively.An attack on \(N = p^2q\) with partially known bits on the multiple of the prime factorshttps://zbmath.org/1517.941492023-09-22T14:21:46.120933Z"Ruzai, W. N. A."https://zbmath.org/authors/?q=ai:ruzai.wan-nur-aqlili-wan-mohd"Adenan, N. N. H."https://zbmath.org/authors/?q=ai:adenan.nurul-nur-hanisah"Ariffin, M. R. K."https://zbmath.org/authors/?q=ai:ariffin.mahammad-rezal-kamel|ariffin.muhamad-rezal-kamel"Ghaffar, A. H. A."https://zbmath.org/authors/?q=ai:ghaffar.a-h-a"Johari, M. A. M."https://zbmath.org/authors/?q=ai:johari.m-a-mohamatSummary: This paper presents a cryptanalytic study upon the modulus \(N=p^2q\) consisting of two large primes that are in the same-bit size. In this work, we show that the modulus \(N\) is factorable if \(e\) satisfies the Diophantine equation of the form \(ed - k(N - (ap)^2 - apbq+ap) = 1\) where \(\frac{a}{b}\) is an unknown approximation of \(\frac{q}{p}\). Our attack is feasible when some amount of Least Significant Bits (LSBs) of \(ap\) and \(bq\) is known. By utilising the Jochemsz-May strategy as our main method, we manage to prove that the modulus \(N\) can be factored in polynomial time under certain specified conditions.A Wiener-type attack on an RSA-like cryptosystem constructed from cubic Pell equationshttps://zbmath.org/1517.941592023-09-22T14:21:46.120933Z"Susilo, Willy"https://zbmath.org/authors/?q=ai:susilo.willy"Tonien, Joseph"https://zbmath.org/authors/?q=ai:tonien.josephSummary: This paper investigates a novel RSA-like cryptosystem proposed by \textit{N. Murru} and \textit{F. M. Saettone} [Lect. Notes Comput. Sci. 10737, 91--103 (2018; Zbl 1423.94091)]. This cryptosystem is constructed from a cubic field connected to the cubic Pell equation and Redei rational functions. The scheme is claimed to be secure against the Wiener-type attack. However, in this paper, we show a Wiener-type attack that can recover the secret key from the continued fraction constructed from public information.Two classes of power mappings with boomerang uniformity 2https://zbmath.org/1517.941652023-09-22T14:21:46.120933Z"Yan, Haode"https://zbmath.org/authors/?q=ai:yan.haode"Li, Zhen"https://zbmath.org/authors/?q=ai:li.zhen.2"Song, Zhitian"https://zbmath.org/authors/?q=ai:song.zhitian"Feng, Rongquan"https://zbmath.org/authors/?q=ai:feng.rongquanSummary: Let \(q\) be an odd prime power. Let \(F_1(x) = x^{d_1}\) and \(F_2(x) = x^{d_2}\) be power mappings over \(\mathrm{GF}(q^2) \), where \(d_1 = q-1\) and \(d_2 = d_1+\frac{q^2-1}{2} = \frac{(q-1)(q+3)}{2} \). In this paper, we study the boomerang uniformity of \(F_1\) and \(F_2\) via their differential properties. It is shown that the boomerang uniformity of \(F_i\) \(( i = 1,2 )\) is 2 with some conditions on \(q \).An efficient publicly verifiable and proactive secret sharing schemehttps://zbmath.org/1517.941762023-09-22T14:21:46.120933Z"Bagherpour, Bagher"https://zbmath.org/authors/?q=ai:bagherpour.bagherSummary: A verifiable proactive secret sharing (VPSS) scheme is a verifiable secret sharing scheme with the property that the shareholders can renew their shares without reconstructing the secret and interacting with the dealer. In this paper, we propose a new VPSS scheme using homogeneous linear recursions and prove its security in a standard model. Our scheme is more efficient than the previous VPSS schemes and is secure against mobile and active adversaries. Furthermore, anyone, not only the shareholders of the scheme, can verify the correctness of the produced shares and sub-shares without observing them and interacting with the dealer and shareholders.Goppa codes over the \(p\)-adic integers and integers modulo \(p^e\)https://zbmath.org/1517.942072023-09-22T14:21:46.120933Z"Epelde, Markel"https://zbmath.org/authors/?q=ai:epelde.markelIn this paper, Goppa codes over the ring of \(p\)-adic integers and the ring \(\mathbb{Z}_{p^e}\) are defined, based on the original idea from Goppa. Their basic properties are studied, the creation of chains of Goppa codes over different rings and the relations between their parity check matrices are described. Furthermore, it is shown how to get isomorphic Goppa codes over different rings by changing one of the parameters of the code. Moreover, the McEliece and Niederreiter cryptosystems which are based on Goppa codes are generalised to the above rings, and it is proved that the distinguishability problems for Goppa codes over \(\mathbb{Z}_{p}\) and \(\mathbb{Z}_{p^e}\) are equivalent.
Reviewer: Dimitros Poulakis (Thessaloniki)