Recent zbMATH articles in MSC 11Rhttps://zbmath.org/atom/cc/11R2022-07-25T18:03:43.254055ZWerkzeugReal quadratic fields admitting universal lattices of rank 7https://zbmath.org/1487.110372022-07-25T18:03:43.254055Z"Kim, Byeong Moon"https://zbmath.org/authors/?q=ai:kim.byeong-moon"Kim, Myung-Hwan"https://zbmath.org/authors/?q=ai:kim.myung-hwan"Park, Dayoon"https://zbmath.org/authors/?q=ai:park.dayoonLet \(F\) be a totally real algebraic number field, \(\mathcal O_F\) the ring of algebraic integers of \(F\), and \(\mathcal O_F^+\) the set of totally positive elements of \(\mathcal O_F\). A quadratic \(\mathcal O_F\)-lattice is a finitely generated \(\mathcal O_F\)-module \(L\) equipped with a quadratic map \(Q:L \rightarrow \mathcal O_F\) for which \(B(x,y)=\frac{1}{2}[Q(x+y)-Q(x)-Q(y)]\) is a symmetric bilinear form on \(L\). Such a lattice \(L\) is said to be positive definite if \(Q(x)\in \mathcal O_F^+\) for all \(x\in L\smallsetminus \{0\}\), and a positive definite lattice \(L\) is said to be universal if \(Q(L)=\mathcal O_F^+ \cup \{0\}\).
The first author of the present paper previously proved [Comment. Math. Helv. 75, No. 3, 410--414 (2000; Zbl 1120.11301)] that there exist infinitely many real quadratic fields \(F\) that admit universal positive definite quadratic \(\mathcal O_F\)-lattices of rank \(8\) (in fact, it was shown that there exist free lattices with these properties). In the paper under review, it is shown that \(8\) is the minimal rank with this property. That is, it is proved that for sufficiently large squarefree positive integers \(d\), there do not exist universal positive definite quadratic \(\mathcal O_F\)-lattices of rank \(7\) when \(F=\mathbb Q(\sqrt{d})\). For the special case of diagonal free lattices, this result was established by the first author in [Manuscr. Math. 99, No. 2, 181--184 (1999; Zbl 0961.11016)].
Reviewer: Andrew G. Earnest (Carbondale)On Petersson's partition limit formulahttps://zbmath.org/1487.110442022-07-25T18:03:43.254055Z"Castaño-Bernard, Carlos"https://zbmath.org/authors/?q=ai:castano-bernard.carlos"Luca, Florian"https://zbmath.org/authors/?q=ai:luca.florianAuthors' abstract: For each prime \(p\equiv 1\pmod{4}\) consider the Legendre character \(\chi=\left(\frac{.}{p}\right)\). Let \(p_\pm(n)\) be the number of partitions of \(n\) into parts \(\lambda>0\) such that \(\chi(\lambda)=\pm 1\). Peterson proved a beautiful limit formula for the ratio of \(p_+(n)\) to \(p_-(n)\) as \(n\to\infty\) expressed in terms of important invarians of the real quadratic field \(K=Q(\sqrt{p})\). But his proof is not illuminating and Grosswald conjectured a more natural proof using a Tauberian converse of the Stolz-Cesáro theorem. In this paper, we suggest an approach to address Grosswald's conjecture. We discuss a monotonicity conjecture which looks quite natural in the context of the monotonicity theorems of Bateman-Erdős.
So, using the method described above, the authors of the present paper show that, as \(\nu\to\infty\), the ratiio \((\sum_{n=0}^\nu p_+(n))/(\sum_{n=0}^\nu p_-(n))\) approaches the limit given in Peterson's formula.
Reviewer: Ljuben Mutafchiev (Sofia)On the problem of periodicity of continued fraction expansions of \(\sqrt{f}\) for cubic polynomials \(f\) over algebraic number fieldshttps://zbmath.org/1487.110702022-07-25T18:03:43.254055Z"Platonov, Vladimir P."https://zbmath.org/authors/?q=ai:platonov.vladimir-p"Zhgoon, Vladimir S."https://zbmath.org/authors/?q=ai:zhgoon.vladimir-s"Petrunin, Maksim M."https://zbmath.org/authors/?q=ai:petrunin.maksim-maksimovichMoments of central values of cubic Hecke \(L\)-functions of \(\mathbb{Q}(i)\)https://zbmath.org/1487.110802022-07-25T18:03:43.254055Z"Gao, Peng"https://zbmath.org/authors/?q=ai:gao.peng.1"Zhao, Liangyi"https://zbmath.org/authors/?q=ai:zhao.liangyiThe authors consider \(L\)-functions of cubic characters in two fields: the field \(K={\mathbb Q}(i)\) of Gaussian integers and the field \(F={\mathbb Q}(\zeta_{12})={\mathbb Q}(e(1/12))\).
They estimate (Theorem 1.1) a sum of central values of such \(L\)-functions in \(K\) weighted with a smooth, compactly-supported function \(w\):
\[
\sum_{q\in{\mathcal O}_K/{\mathcal O}_K^\times,(q,6)\sim 1} \sum_{\text {primitive cubic }\chi\text{ mod }q} L(1/2,\chi)w(N_K(q)/Q) = C_0 Q \hat w(0) + O(Q^{37/38+\varepsilon}).
\]
They also give (Theorem 1.2) an upper bound for the second moment:
\[
\sum_{q\in{\mathcal O}_K/{\mathcal O}_K^\times,(q,6)\sim 1} \sum_{\text {primitive cubic }\chi\text{ mod }q} \lvert L(1/2,\chi)\rvert^2 = O(Q^{11/9+\varepsilon}(1+\lvert t \rvert)^{1+\varepsilon}).
\]
For characters of \(F\) they give a similar upper bound for the sum of squares, however, the cubic characters involved are of a special type, parametrized by elements of \({\mathcal O}_K\).
The authors also mention a lower bound on the number of primitive cubic characters \(\chi\) of \(K\) of a given norm, satisfying \(L(1/2,\chi)\neq 0\), as consequence of these estimates.
Reviewer: Maciej Radziejewski (Poznań)Sextic reciprocal monogenic dihedral polynomialshttps://zbmath.org/1487.110952022-07-25T18:03:43.254055Z"Jones, Lenny"https://zbmath.org/authors/?q=ai:jones.lenny-kA polynomial \(f(x)\in\mathbb{Z}[x]\) with integer coefficients is said to be \textit{reciprocal} (elsewhere \textit{self-reciprocal}, or \textit{palindromic} or \textit{symmetrical})), if \(f(1/x)=f(x)/x^{deg(f)}\). A polynomial \(f\) is called \textit{monogenic}, if \(f\) is irreducible over the rationals and \(\mathbb{Z}_K=\mathbb{Z}[\theta]\), where \(\theta\) is a root of \(f\) and \(K=\mathbb{Q}(\theta)\); equivalently, the respective discriminants coincide: \(\Delta(f)=\Delta(K)\). The central result is as follows:
Let Gal(\(f\)) denote the Galois group over the rationals \(\mathbb{Q}\) of the polynomial \(f(x)\in\mathbb{Z}[x]\). Denote the dihedral group of order \(2n\) by \(D_n\). Then the following hold:
\begin{itemize}
\item[1.] Let \(q\neq 7\) be a prime such that \(q\not\equiv\pm 1\) (mod 7). Then there exist infinitely many primes \(p\) such that the polynomial \(f(x)=x^6+x^5+(pq+1)x^4+(2pq+1)x^3+(pq+1)x^2+x+1 \) is monogenic. Furthermore, if \(q\equiv 3\) (mod 7) or \(q\equiv 5\) (mod 7), then \(\mathrm{Gal}(f)\cong D_6\).
\item[2.] There are infinitely many primes \(p\) such that \(f(x)=x^6+px^3+1\) is monogenic and \(\mathrm{Gal}(f)\cong D_6\).
\item[3.] There exist infinitely many primes \(p\) such that \(f(x)=x^6+3x^5+(p+6)x^4+(2p+7)x^3+(p+6)x^2+3x+1\) is monogenic and \(\mathrm{Gal}(f)\cong D_3\).
\end{itemize}
The emphasis in all three parts of the result is on the existence of infinitely many primes. The appropriately long proof is aided by computations in computer packages MAGMA, Maple, and Sage.
Reviewer: Radoslav M. Dimitrić (New York)Infinite families of reciprocal monogenic polynomials and their Galois groupshttps://zbmath.org/1487.110962022-07-25T18:03:43.254055Z"Jones, Lenny"https://zbmath.org/authors/?q=ai:jones.lenny-kA monic irreducible polynomial \(f(x)\in \mathbb{Z}[x]\) is said to be reciprocal (resp. monogenic) if \(f(x)=x^{\deg (f)}f(1/x)\) (resp. if the ring of integers of the field \(\mathbb{Q}(\alpha ),\) where \(\alpha \) is a zero of \(f,\) is of the form \(\mathbb{Z+\alpha Z}+\cdot \cdot \cdot +\alpha ^{\deg (f)-1}\mathbb{Z}).\)
In the paper under review, the author provides an irreducibility criterion for a certain class of polynomials with integers coefficients, and uses it to construct infinite families of reciprocal monogenic polynomials.
More precisely, he proves that if \(a\geq 0,\) \(b\geq 1\) are integers, \(q\in \{3,5,7\}\) and \(r\geq 3\) is a prime primitive root modulo \(q^{2},\) then there exist infinitely many primes \(p\) such that a polynomial of the form
\[
f_{(a,b,q,r,p)}(x):=\Phi _{2^{a}q^{b}}(x)+4q^{2}rpx^{\frac{\varphi (2^{a}q^{b})}{2}},
\]
is monogenic (\(\Phi _{2^{a}q^{b}}\) is the cyclotomic polynomial of index \(2^{a}q^{b}\) and \(\varphi \) is the Euler function).
As an application of this theorem, the author obtains from the class \(\{f_{(a,b,q,r,p)}(x)\}\) infinite families of reciprocal monogenic polynomials with prescribed Galois group. These results are non-trivial extensions of previous results of the author [Ramanujan J. 56, No. 3, 1099--1110 (2021; Zbl 1487.11095)] on reciprocal monogenic polynomials of degree \(6.\)
Reviewer: Toufik Zaïmi (Riyadh)On necessary and sufficient conditions for the monogeneity of a certain class of polynomialshttps://zbmath.org/1487.110972022-07-25T18:03:43.254055Z"Jones, Lenny"https://zbmath.org/authors/?q=ai:jones.lenny-kSummary: Let \(f(x)\in\mathbb{Z}[x]\) be monic and irreducible over \(\mathbb{Q}\), with \(\deg (f)=n\). Let \(K = \mathbb{Q}(\theta)\), where \(f(\theta)=0\), and let \(\mathbb{Z}_K\) denote the ring of integers of \(K\). We say \(f(x)\) is \textit{monogenic} if \(\{1,\theta,\theta^2,\dots,\theta^{n-1}\}\) is a basis for \(\mathbb{Z}_K \). Otherwise, \(f(x)\) is called \textit{non-monogenic}. In this article, we give necessary and sufficient conditions for a certain class of polynomials to be monogenic. Using these conditions allows us to generate infinite families of non-monogenic polynomials. In particular, for quadrinomials our results show that there exist infinitely many primes \(p\geq 3\), and integers \(t\geq 1\) coprime to \(p\), such that \(f(x)=x^p-2ptx^{p -1}+p^2t^2x^{p-2}+1\) is non-monogenic. Finally, we illustrate this situation with an explicit example.Rikuna's generic cyclic polynomial and the monogenityhttps://zbmath.org/1487.110982022-07-25T18:03:43.254055Z"Sekigawa, Ryutaro"https://zbmath.org/authors/?q=ai:sekigawa.ryutaroA number field extension \(L/K\) is said to be monogenic if the ring of integers of \(L\) is of the form \(\mathbb{Z}_{K}+\alpha \mathbb{Z}_{K}+\cdots +\alpha^{\lbrack L:K]-1}\mathbb{Z}_{K}\) where \(\alpha\)
is an element of \(L\) and \(\mathbb{Z}_{K}\) is the ring of integers of \(K\).
According to [\textit{M.-N. Gras}, J. Number Theory 23, 347--353 (1986; Zbl 0564.12008)], an absolute normal extension \(L/\mathbb{Q}\) of prime degree at least \(5\) is monogenic if and only if \(L\) is the maximal real subfield of the cyclotomic field of prime conductor \(2[L:\mathbb{Q}]+1\); thus, for any prime number \(p\geq 5\) the number of monogenic cyclic extensions of degree \(p\) over \(\mathbb{Q}\) is one or zero.
A corollary of the main result of the paper under review gives that this last mentioned property is not always true for a relative extension \(L/K\). More precisely, let \(K:=\mathbb{Q(\zeta +\zeta}^{-1})\), where \(\zeta\) is a primitive \(p\)-th root of unity and \(p\) is again a prime number greater than \(4\), \(s\in \mathbb{Z}_{K}\) and \(\theta _{s}\) a root of the polynomial
\[
\frac{\mathbb{\zeta}^{-1}(X-\zeta )^{p}-\zeta (X-\mathbb{\zeta}^{-1})^{p}}{\mathbb{\zeta}^{-1}-\zeta}-s\frac{(X-\zeta )^{p}-(X-\mathbb{\zeta}^{-1})^{p}}{p(\mathbb{\zeta}^{-1}-\zeta )}\in \mathbb{Z}_{K}[X].
\]
Then, the extension \(K(\theta _{s})/K\) is cyclic of degree \(p\) [\textit{Y. Rikuna}, Proc. Am. Math. Soc. 130, No. 8, 2215--2218 (2002, Zbl 0990.12005)], and the author provides a sufficient condition for the monogenity of such an extension. Also, from this result he obtains that an infinite number of extensions \(K(\theta _{s})/K\) are monogenic. The proof of this last theorem is based on that of a result of [\textit{D. S. Dummit} and \textit{H. Kisilevsky}, Number Theory and Algebra; Collect. Pap. dedic. H.B. Mann, A.E. Ross, O. Taussky-Todd, 29--42 (1977; Zbl 0377.12003)], saying that there are infinitely many monogenic (resp. non-monogenic) normal extensions of degree \(3\) over \(\mathbb{Q}\).
Reviewer: Toufik Zaïmi (Riyadh)On the compositum of orthogonal cyclic fields of the same odd prime degreehttps://zbmath.org/1487.110992022-07-25T18:03:43.254055Z"Greither, Cornelius"https://zbmath.org/authors/?q=ai:greither.cornelius"Kučera, Radan"https://zbmath.org/authors/?q=ai:kucera.radanThe authors study the circular units (a.k.a. cyclotomic units) of a compositum \(\mathbb K\) of \(t\) cyclic extensions of \(\mathbb{Q}\) having all the same degree \(\ell\) over \(\mathbb{Q}\), where \(\ell\) is a fixed odd prime and \(t\geq 2\). We say that two abelian fields are arithmetically orthogonal if each prime that ramifies in one of them splits completely in the other. They assume that these cyclic extension fields are pairwise arithmetically orthogonal and that the number \(s\) of primes which are ramified in the extension \(\mathbb K\) of \(\mathbb{Q}\) is larger than \(t\).
Let \(m\) be the conductor of \(\mathbb K\) and let \(\zeta_m\) be a \(m\)-th root of unity. Let \(\eta= N_{\mathbb{Q}(\zeta_m)/K}(1-\zeta_m)\) be a top generator of the group of circular units of \(\mathbb K\). They construct a non trivial root \(\varepsilon\) of the top generator \(\eta\) and use this unit \(\varepsilon\) to define an enlarged group of circular units of \(\mathbb K\). The arithmetically orthogonal property is used for the construction of \(\varepsilon \). The existence of \(\varepsilon\) allows them to prove that the class number of \(\mathbb K\) is divisible by \(\ell\) to the power \((s-t)\ell^{t-1}\). This is a huge improvement upon the divisibility \(\ell^{s-t}|h_K\) provided by genus theory. It happens that the genus field \(\overline{\mathbb K}\) of \(\mathbb K\) comes into play and the authors make a tour de force by showing that even if the unit \(\varepsilon\) is, in the sense of Sinnott, a circular unit of the genus field \(\overline{\mathbb K}\) of \(\mathbb K\), it is not in the Sinnott group of circular units of \(\mathbb K\).
The authors deduce from their new divisibility property of the class number \(h_{\mathbb K}\) some annihilation statement for the ideal class group of \(\mathbb K\). Some numerical examples illustrate their results.
Reviewer: Claude Levesque (Québec)Rubin's conjecture on local units in the anticyclotomic tower at inert primeshttps://zbmath.org/1487.111002022-07-25T18:03:43.254055Z"Burungale, Ashay"https://zbmath.org/authors/?q=ai:burungale.ashay-a"Kobayashi, Shinichi"https://zbmath.org/authors/?q=ai:kobayashi.shinichi"Ota, Kazuto"https://zbmath.org/authors/?q=ai:ota.kazutoLet \(p \geq 5\) be a prime. The authors prove a conjecture of \textit{K. Rubin} [Invent. Math. 88, 405--422 (1987; Zbl 0623.14006)] on the structure of local units along the anticyclotomic \(\mathbb{Z}_p\)-extension of the unramified quadratic extension \(\Phi\) of \(\mathbb{Q}_p\). \par We briefly sketch Rubin's conjecture. Fix a Lubin-Tate formal group \(\mathcal{F}\) over the ring of integers \(\mathcal{O}\) in \(\Phi\) for the uniformizer \(\pi := -p\). Let \(\Phi_n\) be the field obtained from \(\Phi\) by adjoining all \(\pi^{n+1}\)-torsion points of \(\mathcal{F}\) and set \(\Phi_{\infty} := \bigcup_n \Phi_n\). Then there is a natural isomorphism \(\kappa:\mathrm{Gal}(\Phi_{\infty}/\Phi) \simeq \mathcal{O}^{\times}\). Moreover, one has a decomposition
\[
\mathrm{Gal}(\Phi_{\infty}/\Phi_0) \simeq G^+ \times G^-,
\]
where both \(G^{\pm}\) are isomorphic to \(\mathbb{Z}_p\) and \(\mathrm{Gal}(\Phi/\mathbb{Q}_p)\) acts upon \(G^+\) and \(G^-\) by \(+1\) and \(-1\), respectively. We may identify \(G^+\) and \(G^-\) with the Galois groups of the cyclotomic and the anticyclotomic \(\mathbb{Z}_p\)-extension of \(\Phi\), respectively. \par Let \(U_n\) be the group of principal units in \(\Phi_n\) and consider the inverse limit \(\varprojlim_n U_n \otimes_{\mathbb{Z}_p} \mathcal{O}\) with respect to the norm maps. We denote the part of this limit upon which \(\mathrm{Gal}(\Phi_0/\Phi)\) acts via the Teichmüller character by \(U_{\infty}\). Then \(U_{\infty}\) is free of rank \(2\) over the Iwasawa algebra \(\mathcal{O}[[\mathrm{Gal}(\Phi_{\infty}/\Phi_0)]]\). The main object of study is the quotient
\[
V^{\ast}_{\infty} := U_{\infty}^{\ast} / (\sigma-1),
\]
where \(U_{\infty}^{\ast}\) is \(U_{\infty}\) with the Galois action twisted by \(\kappa^{-1}\) and \(\sigma\) is a topological generator of \(G^+\). Then \(V^{\ast}_{\infty}\) is a free \(\Lambda := \mathcal{O}[[G^-]]\)-module of rank \(2\). Rubin defines two subspaces \(V^{\ast, \pm}_{\infty}\) of \(V^{\ast}_{\infty}\) and conjectures that one has a decomposition
\[
V^{\ast}_{\infty} = V^{\ast,+}_{\infty} \oplus V^{\ast,-}_{\infty}.
\]
He showed that both subspaces are free of rank \(1\) and that their intersection is trivial. \par The strategy of proof of Rubin's conjecture is now as follows. Consider the Coates-Wiles derivative
\[
\delta: V^{\ast}_{\infty} / V^{\ast,-}_{\infty} \rightarrow \mathcal{O}.
\]
The authors prove that (i) there is \(\xi \in V^{\ast,+}_{\infty}\) such that \(\delta(\xi) \in \mathcal{O}^{\times}\) and (ii) there is an isomorphism \(V^{\ast}_{\infty}/V^{\ast,-}_{\infty} \simeq \Lambda\). Under this identification, \(\xi\) is not contained in the maximal ideal of \(\Lambda\) (as \(\delta(\xi) \in p \mathcal{O}\) otherwise) and hence is a generator of the quotient \(V^{\ast}_{\infty} / V^{\ast,-}_{\infty}\). Rubin's conjecture follows. \par For (i) the authors construct a certain auxiliary imaginary quadratic field \(K\) and an elliptic curve \(E\) over its Hilbert class field with complex multiplication by \(\mathcal{O}_K\). By a criterion of Rubin it would be sufficient to construct such an \(E\) with good supersingular reduction at \(p\) whose central \(L\)-value is \(p\)-divisible. The authors generalize this approach to allow more general \(L\)-values (they consider twists by certain Hecke characters). The proof of (i) then crucially relies on work of \textit{T. Finis} [Ann. Math. (2) 163, No. 3, 767--807 (2006; Zbl 1111.11047)]. For (ii) the authors make use of the theory of quasi-canonical lifts of \textit{B.H. Gross} [Invent. Math. 84, 321--326 (1986; Zbl 0597.14044)] to construct an optimal system of local points of the formal group. This is the main difficulty of the present work. \par Finally, we note that the main result of this article makes the work of \textit{A. Agboola} and \textit{B. Howard} [Math. Res. Lett. 12, No. 5--6, 611--621 (2005; Zbl 1130.11058)] on a variant of the Iwasawa main conjecture unconditional.
Reviewer: Andreas Nickel (Essen)Dirichlet series expansions of \(p\)-adic \(\mathrm{L}\)-functionshttps://zbmath.org/1487.111012022-07-25T18:03:43.254055Z"Knospe, Heiko"https://zbmath.org/authors/?q=ai:knospe.heiko"Washington, Lawrence C."https://zbmath.org/authors/?q=ai:washington.lawrence-cLet \(\chi\) be a Dirichlet character of conductor \(f\). It is well known that the associated complex \(L\)-function \(L(s,\chi)\) is given by an expansion
\[
L(s,\chi) = \sum_{n=1}^{\infty} \frac{\chi(n)}{n^s}
\]
if the real part of \(s\) is greater than \(1\). Recall that in general an expansion of the form \(\sum_{n=1}^{\infty} a_n/n^s\) with \(a_n \in \mathbb{C}\) is called a Dirichlet series. \par
Now assume that \(\chi\) is even and let \(p\) be prime, which we assume to be odd for simplicity. Then there is also a \(p\)-adic \(L\)-function \(L_p(s,\chi)\), which is a \(p\)-adic meromorphic function that interpolates values of complex \(L\)-series at non-positive integers. Starting from a formula for \(L_p(s,\chi)\) in the book of the second named author (Theorem 5.11 in [Introduction to cyclotomic fields. 2nd ed. New York, NY: Springer (1997; Zbl 0966.11047)]), the authors derive at a Dirichlet series expansion for \(L_p(s,\chi)\). More precisely, for any integer \(c>1\), which is coprime to \(pf\), one has the formula
\[
-(1 - \chi(c)\langle c \rangle^{1-s}) \cdot L_p(s,\chi) = \lim_{n \rightarrow \infty} \sum_{\genfrac{}{}{0pt}{}{a=1}{p \nmid a} }^{fp^n} \chi \omega^{-1}(a) \frac{\varepsilon_{a,c,fp^n}}{\langle a \rangle^s},
\]
where \(\omega\) denotes the Teichmüller character and the \(\varepsilon_{a,c,fp^n}\) are certain explicit (half-) integers. A similar formula is obtained for \(S\)-truncated \(p\)-adic \(L\)-series. \par
It is also shown that the numbers \(\varepsilon_{a,c,fp^n}\) are related to the Bernoulli distribution \(E_1\) on \(X := \mathbb{Z}/ f\mathbb{Z} \times \mathbb{Z}_p\) as follows. One first replaces \(E_1\) with a regularization \(E_{1,c}\) which is actually a measure. Then one has
\[
\varepsilon_{a,c,fp^n} = E_{1,c}(a + fp^n X).
\]
In a final section, the authors specialize their formulas to different regularization parameters \(c\). In particular, the case \(c=2\) is considered. In this case the obtained results are similar to those of \textit{D. Delbourgo} [J. Aust. Math. Soc. 81, No. 2, 215--224 (2006; Zbl 1207.11112); Proc. Edinb. Math. Soc., II. Ser. 52, No. 3, 583--606 (2009; Zbl 1243.11104)], where the method of proof is different. Similar expansions for slightly different \(p\)-adic \(L\)-functions are due to \textit{M. S. Kim} and \textit{S. Hu} [J. Number Theory 132, No. 12, 2977--3015 (2012; Zbl 1272.11130)].
Reviewer: Andreas Nickel (Essen)A conjectural improvement for inequalities related to regulators of number fieldshttps://zbmath.org/1487.111022022-07-25T18:03:43.254055Z"Battistoni, Francesco"https://zbmath.org/authors/?q=ai:battistoni.francescoFollowing Remak's remark in [\textit{R. Remak}, Compos. Math. 10, 245--285 (1952; Zbl 0047.27202)] that number fields with small regulators have small discriminants, and Friedman's generalization in [\textit{E. Friedman}, Invent. Math. 98, No. 3, 599--622 (1989; Zbl 0694.12006)], one can search all the number fields with a fixed signature and whose regulator is less than a prescribed bound.
Exhaustive and partial results about this problem can be find in [\textit{S. Astudillo} et al., J. Number Theory 167, 232--258 (2016; Zbl 1415.11158)], where the authors give a list of fields with minimal or small regulators for degrees less than 7 -- except for the signature \((5,1)\) -- and a partial list for degrees 8 and 9.
This paper completes those previous results in degree 8, and refines them in degree 5 and 7, following Friedman and Ramirez-Raposo's ideas found in [\textit{E. Friedman} and \textit{G. Ramirez-Raposo}, J. Number Theory 198, 381--385 (2019; Zbl 1429.11200)], where the upper bound in Remak-Friedman's inequality have been improved, allowing the two authors to solve the missing \((5,1)\)-signature case in degree 7. However, the list of the fields with small regulators given in the paper under review is based on conjectural upper bounds, the validity of those conjectures being extensively discussed by the author of the present paper, and corroborated by numerical experiments conducted on/with PARI/GP and the MATLAB Optimization Toolbox.
Reviewer: Isabelle Dubois (Metz)On class numbers of pure quartic fieldshttps://zbmath.org/1487.111032022-07-25T18:03:43.254055Z"Li, Jianing"https://zbmath.org/authors/?q=ai:li.jianing"Xu, Yue"https://zbmath.org/authors/?q=ai:xu.yueAuthors' abstract: Let \(p\) be a prime. The 2-primary part of the class group of the pure quartic field \(\mathbb{Q} (\sqrt[4]{p})\) has been determined by \textit{C. J. Parry} [J. Reine Angew. Math. 314, 40--71 (1980; Zbl 0421.12006)] and \textit{F. Lemmermeyer} [J. Ramanujan Math. Soc. 28, No. 4, 415--421 (2013; Zbl 1402.11133)] when \(p \not\equiv \pm\, 1\bmod 16\). In this paper, we improve the known results in the case \(p\equiv \pm\, 1\bmod 16\). In particular, we determine all primes \(p\) such that 4 does not divide the class number of \(\mathbb{Q}(\sqrt[4]{p})\).We also conjecture a relation between the class numbers of \(\mathbb{Q}(\sqrt[4]{p})\) and \(\mathbb{Q}(\sqrt{-2p})\). We show that this conjecture implies a distribution result of the 2-class numbers of \(\mathbb{Q}(\sqrt[4]{p})\).
Reviewer: Bouchaïb Sodaïgui (Valenciennes)How far is an extension of \(p\)-adic fields from having a normal integral basis?https://zbmath.org/1487.111042022-07-25T18:03:43.254055Z"Del Corso, Ilaria"https://zbmath.org/authors/?q=ai:del-corso.ilaria"Ferri, Fabio"https://zbmath.org/authors/?q=ai:ferri.fabio"Lombardo, Davide"https://zbmath.org/authors/?q=ai:lombardo.davide-mLet \(L/K\) be a finite Galois extension of \(p\)-adic fields with Galois group \(G\), and denote the valuation rings of these fields with \(\mathcal O_L\) and \(\mathcal O_K\), resp.. The authors investigate the minimal index \(m(L/K)\) of a free \(\mathcal O_K[G]\)-module contained in \(\mathcal O_L\). It is well-known that for \(L/K\) at most tamely ramified one has \(m(L/K)=1\), i.e., \(L\) has a normal integral basis.
The authors show that \(m(L/K)\) can be determined with a finite, effective procedure, and in Theorem 1.4 give an explicit upper bound for (the \(p\)-exponent of) \(m(L/K)\) in the general case.
Supposing that \(L\) is an abelian extension of \(\mathbb Q_p\) with \(p\) odd, the explicit value for \(m(L/K)\) is given in Theorem 1.6. This is obtained from the known results on the Galois module structure of \(\mathcal O_L\) (local version of Leopoldt's Theorem). If \(p=2\), the authors indicate how the value of \(m(L/K)\) can be obtained in an analogous way.
For the case \([L:K]=p\), Theorem 1.7 gives the explicit value of \(m(L/K)\), which depends on whether the ramification jump is divisible by \(p\) or not. The proof of the latter case uses an explicit \(\mathcal O_K\)-basis of \(\mathcal O_L\) and detailed calculations with the valuations of appropriate elements (Section 6 of the paper). Finally, the authors use these tools to give another proof for a theorem of \textit{F. Bertrandias} and \textit{M.-J. Ferton} [C. R. Acad. Sci., Paris, Sér. A 274, 1330--1333 (1972; Zbl 0235.12007)], stated in Theorem 1.8.
Reviewer: Günter Lettl (Graz)Galois symbol maps for abelian varieties over a \(p\)-adic fieldhttps://zbmath.org/1487.111052022-07-25T18:03:43.254055Z"Hiranouchi, Toshiro"https://zbmath.org/authors/?q=ai:hiranouchi.toshiroLet \(k\) be a finite extension of \(\mathbb{Q}_p\) with resident field \(\mathbb{F}\). Let \(X\) be a projective smooth geometrically connected curve over \(k\). The reciprocity maps \(k(x)^{\times} \to \pi_1^{\mathrm{ab}}(x)\) of the local class field theory of \(k(x)\) for some closed point \(x\) of \(X\) induce the reciprocity map \(\rho\) from \(SK_1(X):= \mathrm{Coker}(\delta: K_2(k(X)) \to \oplus_{x \in X_0}k(x)^{\times})\) to the abelian fundamental group \(\pi_1^{\mathrm{ab}}(X)\) of \(X\). Let \(V(x) = \mathrm{Ker}(\delta: SK_1(X) \to k^{\times})\). Then there is a direct sum decomposition \(V(X) = V(X)_{\mathrm{div}} \oplus V(X)_{\mathrm{fin}}\) where \(V(X)_{\mathrm{div}}\) is the maximal divisible subgroup of \(V(X)\) and \(V(X)_{\mathrm{fin}}\) is a finite group. The main goal of this paper is to study the structure of \(V(X)_{\mathrm{fin}}\).
Let \(\mathscr{X}\) be the regular model of \(X\) over \(O_k\) with \(\mathscr{X} \otimes_{O_k} k \simeq X\). The Jacobian variety \(\mathrm{Jac}(\mathscr{X})\) has generic fiber \(J = \mathrm{Jac}(X)\) and special fiber \(\overline{J} = \mathrm{Jac}(\mathscr{X} \otimes_{O_k} \mathbb{F})\). The following is the main result of this paper.
Theorem: Suppose that \(J[p] \subset J(k)\), \(\overline{J}\) has ordinary reduction, and \(k(\mu_{p^{N+1}})/k\) is a nontrivial totally ramified extension where \(\mu_{p^{N+1}}\) is the group of \(p^{N+1}\)-th roots of unity, and \(N = \mathrm{max}\{n \mid J[p^n] \subset J(k)\}\). Then we have \(V(X)_{\mathrm{fin}} \simeq (\mathbb{Z}/p^N)^{\oplus g} \oplus \overline{J}(\mathbb{F})\) where \(g = \mathrm{dim}(J)\).
Reviewer: Xiao Xiao (Utica)Isomorphisms of discriminant algebrashttps://zbmath.org/1487.130182022-07-25T18:03:43.254055Z"Biesel, Owen"https://zbmath.org/authors/?q=ai:biesel.owen"Gioia, Alberto"https://zbmath.org/authors/?q=ai:gioia.albertoMass formula and Oort's conjecture for supersingular abelian threefoldshttps://zbmath.org/1487.140622022-07-25T18:03:43.254055Z"Karemaker, Valentijn"https://zbmath.org/authors/?q=ai:karemaker.valentijn"Yobuko, Fuetaro"https://zbmath.org/authors/?q=ai:yobuko.fuetaro"Yu, Chia-Fu"https://zbmath.org/authors/?q=ai:yu.chia-fuThe paper deals with mass stratification and conjecture by \textit{S. J. Edixhoven} et al. [Bull. Sci. Math. 125, No. 1, 1--22 (2001; Zbl 1009.11002)] on the automorphism groups of generic (supersingular) abelian threefolds for supersingular locus of the Siegel modular variety of degree 3. The authors of the paper under review give the number of strata and obtaine the explicit mass formula for each stratum. The classification of possible automorphism groups on each strata of \(\alpha\)-number one is also given. These give the Oort conjecture for investigated polarized abelian threefolds in the case \(p > 2.\) For supersingular abelian surfaces for any odd prime \(p\) the Oort conjecture is proved by \textit{T. Ibukiyama} [J. Math. Soc. Japan 72, No. 4, 1161--1180 (2020; Zbl 1471.14091)]. Let \((X, \lambda)\) be the \(p\)-power degree polarized abelian variety over an algebraically closed field \(k\) of characteristic \(p\) and \(x = (X_0, \lambda_0)\) be polarized supersingular abelian variety of \(p\)-power degree over \(k\). An abelian variety defined over \(k\) is supersingular if it is isogenous to a product of supersingular elliptic curves and is superspecial if it is isomorphic to a product of supersingular elliptic curves. The authors of the paper under review first define for any integer \(d \ge 1\) the (coarse) moduli space \({\mathcal A}_{g,d}\) over \(\overline{\mathbb F}_p\) of \(g\)-dimensional abelian varieties \((X, \lambda)\) with polarization degree \(\deg \lambda = d^2\) and for any \(m \ge 1\) the supersingular locus \({\mathcal G}_{g,p^m}\) of supersingular abelian varieties in \({\mathcal A}_{g,p^m}\). For abelian variety \((X,\lambda)\) let \(X^\bot\) be its dual and respectively let \(G\) and \(G^\bot\) their \(p\)-divicible groups. A polarization \(\lambda\) is an isogeny which is symmetric \((\lambda: X \to X^\bot)^\bot = \lambda\) with the identification \(X = X^{\bot \bot}\) by \textit{D. Mumford} [Abelian varieties. London: Oxford University Press (1970; Zbl 0223.14022)] and by \textit{T. Oda} and \textit{F. Oort} [in: Proc. int. Symp. on algebraic geometry, Kyoto. 595--621 (1977; Zbl 0402.14016)]. A quasi-polarization \(\lambda: G \to G^\bot\) of a \(p\)-divisible group \(G\) is a symmetric isogeny of \(p\)-divisible groups such that \((\lambda: G \to G^\bot)^\bot = \lambda\) by \textit{F. Oort} [Ann. Math. (2) 152, No. 1, 183--206 (2000; Zbl 0991.14016)]. Then the authors define the set \(\Lambda_x\) of isomorphim classes of \(p\)-power polarization degree polarized abelian varieties \((X, \lambda)\) over \(k\). The set \(\Lambda_x\) consists of those polarized abelian varieties whose assosiated quasi-polarized \(p\)-divisible groups satisfy \((X, \lambda)[p^\infty] = (X_0, \lambda_0)[p^\infty] .\) The set \(\Lambda_x\) is finite by \textit{C.-F. Yu} [J. Aust. Math. Soc. 78, No. 3, 373--392 (2005; Zbl 1137.11323)] The mass of \(\Lambda_x\) is defined by \(\mathrm{Mass}(\Lambda_x) = \sum_{(X, \lambda) \in \Lambda_x} \frac{1}{\#\mathrm{Aut}(X,\lambda)}.\) Sections 4 and 5 are concerned with computing the mass for principally polarized supersingular abelian threefolds respectivly for \(\alpha\)-number \(\ge 2\) (Theorem 4.3) and for \(\alpha\)-number 1 (Theorem 5.21). Section 6 deals with ``the automorphism groups of principally polarised abelian threefolds \((X, \lambda)\) over an algebraically closed field \(k \supseteq {\mathbb F}_p\) with \(\alpha(X) = 1\).'' The main result of the section is Theorem 6.4. The section includes also the discussion of arithmetic properties of definite quaternion algebras over rational numbers and the superspecial case. The section is ended with some open problems. The case of a set-theoretic intersection of the Fermat curve and a curve \(\Delta\) defined by authors in Section 5 is treated in the Appendix.
Reviewer: Nikolaj M. Glazunov (Kyïv)On \(\ell\)-adic Galois polylogarithms and triple \(\ell\)th power residue symbolshttps://zbmath.org/1487.140712022-07-25T18:03:43.254055Z"Shiraishi, Densuke"https://zbmath.org/authors/?q=ai:shiraishi.densukeFor \(\ell=2\) (resp. \(3\)), let \(\mu_\ell=\{\pm 1\}\) (resp. \(\{1,\zeta_3^{\pm 1}\}\)) denote the set of \(\ell\)-th roots of unity. When three primes \(\mathfrak{p}_i\) (\(i=1,2,3\)) of \(\mathbb{Q}(\mu_\ell)\) have pairwise trivial power residue symbols, \textit{M. Morishita} [Knots and primes. An introduction to arithmetic topology. Based on the Japanese original (Springer, 2009). Berlin: Springer (2012; Zbl 1267.57001)] and \textit{F. Amano} et al. [Res. Number Theory 4, No. 1, Paper No. 7, 29 p. (2018; Zbl 1444.11222)] introduced the triple \(\ell\)-th power residue symbol \([\mathfrak{p}_1,\mathfrak{p}_2,\mathfrak{p}_3]_\ell\in\mu_\ell\) as an analog of the Milnor invariant of three knots in the space that are linked as total but pairwise unlinked. Based on a preceding work by \textit{H. Hirano} and \textit{M. Morishita} [J. Number Theory 198, 211--238 (2019; Zbl 1456.11216)], the paper under review shows a formula that expresses \([\mathfrak{p}_1,\mathfrak{p}_2,\mathfrak{p}_3]_\ell\in\mu_\ell\) with a special value of the \(\ell\)-adic Galois dilogarithmic function studied by \textit{H. Nakamura} and \textit{Z. Wojtkowiak} [Proc. Symp. Pure Math. 70, 285--294 (2002; Zbl 1191.11022)] that enables one to interpret a reciprocity law of type \([\mathfrak{p}_1,\mathfrak{p}_2,\mathfrak{p}_3]_\ell [\mathfrak{p}_2,\mathfrak{p}_1,\mathfrak{p}_3]_\ell=1\) (due to Rédei, Amano-Mizusawa-Morishita) as a consequence of the \(\ell\)-adic dilogarithmic functional equation between \(Li_2(z)\) and \(Li_2(1-z)\) shown in [\textit{H. Nakamura} and \textit{Z. Wojtkowiak}, Lond. Math. Soc. Lect. Note Ser. 393, 258--310 (2012; Zbl 1271.11068)]. In the Appendix, computational examples of \([\mathfrak{p}_1,\mathfrak{p}_2,\mathfrak{p}_3]_3\) for various specific primes \(\mathfrak{p}_i\) (\(i=1,2,3\)) are presented.
Reviewer: Hiroaki Nakamura (Osaka)Equivalence between K-functionals and modulus of smoothness on the quaternion algebrahttps://zbmath.org/1487.420092022-07-25T18:03:43.254055Z"Bouhlal, A."https://zbmath.org/authors/?q=ai:bouhlal.aziz"Safouane, N."https://zbmath.org/authors/?q=ai:safouane.najat"Belkhadir, A."https://zbmath.org/authors/?q=ai:belkhadir.a"Daher, R."https://zbmath.org/authors/?q=ai:daher.radouanSummary: In the space \(L^2({\mathbb{R}}^2,{\mathcal{H}})\), using the analog of the Steklov operator, we construct the generalized modulus of smoothness, and also using the Laplacian operator we define the K-functional. The main result of our article is the proof of the equivalence between K-functionals and modulus of smoothness.Renormalisation for inflation tilings. II: Connections to number theoryhttps://zbmath.org/1487.520262022-07-25T18:03:43.254055Z"Mañibo, Neil"https://zbmath.org/authors/?q=ai:manibo.neilThe paper is a two-pages expository (without proofs) of some results previously obtained by the author with Baake, Coons, Gaehler, and Grimm about some connections of the Lyapunov exponent of the matrix cocycles induced by Fourier matrices of inflation rules for aperiodic tilings and Mahler measures of certain polynomials.
For the entire collection see [Zbl 1459.37002].
Reviewer: Anton Shutov (Vladimir)Average electron number in two-island systemhttps://zbmath.org/1487.811242022-07-25T18:03:43.254055Z"Harata, Pipat"https://zbmath.org/authors/?q=ai:harata.pipat"Srivilai, Prathan"https://zbmath.org/authors/?q=ai:srivilai.prathanSummary: We studied the charge fluctuation in a two-metallic island system using a quantum Monte Carlo simulation. The imaginary-time path integral approach was used to express the system's grand canonical partition function. The average excess charge number on the islands was calculated using the distribution of the winding numbers. In the absence of a source-drain voltage, we found that an average electron number exhibited a Coulomb staircase phenomena and the function of the two gate voltages. Furthermore, as the temperature increased, the sharp step of the Coulomb staircase was smeared out. As a result, the system behaved as a single-electron box containing two coupled islands. We also suggest constructing a quantum stability diagram that accounts for the tunneling effect and temperature dependency from the average electron number. The calculation can be used to investigate the effect of the tunneling conductance on a honeycomb shape.