Recent zbMATH articles in MSC 11Ghttps://www.zbmath.org/atom/cc/11G2021-04-16T16:22:00+00:00WerkzeugOn the \(\ell\)-adic cohomology of some \(p\)-adically uniformized Shimura varieties.https://www.zbmath.org/1456.111122021-04-16T16:22:00+00:00"Shen, Xu"https://www.zbmath.org/authors/?q=ai:shen.xuIn the present paper, the author determines the Galois
representations inside the \(\ell\)-adic cohomology of some
unitary Shimura varieties at split places where they admit
uniformization by finite products of Drinfeld upper half
spaces. His results confirm Langlands-Kottwitz's description
of the cohomology of Shimura varieties in new cases. In fact,
\textit{P. Scholze} [J. Am. Math. Soc. 26, No. 1, 227--259 (2013; Zbl 1383.11082)] has developed the Langlands-Kottwitz approach for some
PEL Shimura varieties with arbitrary level at \(p\). The key new
ingredient is to define some test functions by deformation spaces
of \(p\)-divisible groups with some additional structures. The author
defines the test functions by Scholze's method and proves the
vanishing property of these test functions by global method. He
establishes the trace formula as a sum over the set of equivalent
Kottwitz triples and deduces the character identity of the transfers
of test functions. In the end, he gives the local semi-simple zeta
functions of Shimura varieties.
Reviewer: Lei Yang (Beijing)Bounds for the number of points on curves over finite fields.https://www.zbmath.org/1456.111132021-04-16T16:22:00+00:00"Arakelian, Nazar"https://www.zbmath.org/authors/?q=ai:arakelian.nazar"Borges, Herivelto"https://www.zbmath.org/authors/?q=ai:borges.heriveltoSummary: Let \(\mathcal{X}\) be a projective irreducible nonsingular algebraic curve defined over a finite field \(\mathbb{F}_q\). This paper presents a variation of the Stöhr-Voloch theory and sets new bounds to the number of \(\mathbb{F}_{q^r}\)-rational points on \(\mathcal{X}\). In certain cases, where comparison is possible, the results are shown to improve other bounds such as Weil's, Stöhr-Voloch's and Ihara's.On twists of smooth plane curves.https://www.zbmath.org/1456.111172021-04-16T16:22:00+00:00"Badr, Eslam"https://www.zbmath.org/authors/?q=ai:badr.eslam-e"Bars, Francesc"https://www.zbmath.org/authors/?q=ai:bars.francesc"Lorenzo García, Elisa"https://www.zbmath.org/authors/?q=ai:lorenzo-garcia.elisaLet \(C\) be a projective, smooth, non-hyperelliptic, curve and genus \(g \geq 3\) defined over a field \(k\). Denote by \(\bar{k }\) a fixed separable closure of \(k\) and by \(\bar{C}\) the curve \(C \times_k \bar{k}\). A twist of \(C\) over \(k\) is a projective, non-singular \(C^{\prime}\) defined over \(k\) with a \(\bar{k}\)-isomorphism \(\varphi : \overline{C^{\prime}} \rightarrow \bar{C}\). The paper under review deals with the following question: Assuming that \(C\) admits a smooth \(\bar{k}\)-plane model, does it have a smooth plane model over \(k\)? And if the answer is yes, does every twist of C over
\(k\) also have smooth plane model over \(k\)? The answer, in general, is negative. The twists possessing such models are characterized and an example of a twist not admitting any non-singular plane model over \(k\) is given. An interesting consequence is that explicit equations for a non-trivial Brauer-Severi surface are obtained. Furthermore, for smooth plane curves defined over \(k\) with a cyclic automorphism group generated by a diagonal matrix, a general theoretical result to
compute all its twists is presented.
Reviewer: Dimitros Poulakis (Thessaloniki)Elliptic curves arising from the triangular numbers.https://www.zbmath.org/1456.111032021-04-16T16:22:00+00:00"Juyal, Abhishek"https://www.zbmath.org/authors/?q=ai:juyal.abhishek"Kumar, Shiv Datt"https://www.zbmath.org/authors/?q=ai:datt-kumar.shiv"Moody, Dustin"https://www.zbmath.org/authors/?q=ai:moody.dustinGiven a positive integer \(t\), the \(t\)-th triangular number is \(t(t+1)/2\) the sum of the natural numbers up to \(t\). In the paper under review, the authors considered the elliptic curves of the Legendre's form associated to triangular numbers given by \[E_t: y^2 = x (x-1)(x-\frac{t (t+1)}{2}).\] The main results of the paper are about the rank of Mordell-Weil group
of \(E_t\) over \(\mathbb Q(t)\) and \({\bar{\mathbb Q} (t)}\). In Theorem 1 of the paper, they prove
that the elliptic surface associated to \(E_t\) is rational. The ranks of \(E_t\) over \(\mathbb Q(t)\) and \({\bar{\mathbb Q} (t)}\) are \(0\) and \(1\), respectively. Furthermore, the torsion subgroup of \(E_t(\mathbb Q(t))\) is isomorphic to \({\mathbb Z}_2 \times {\mathbb Z}_2\). The proof of these results are included in Section 3. The authors provide infinite families of positive rank ans a subfamilies of tank two in Section 4. Finally, they tried to find high rank elliptic curve over \(\mathbb Q\) by searching in those families. They find only one elliptic curve of rank \(6\) and several of rank \(5\).
Reviewer: Sajad Salami (Rio de Janeiro)Equations and rational points of the modular curves \(X_0^+(p)\).https://www.zbmath.org/1456.111112021-04-16T16:22:00+00:00"Mercuri, Pietro"https://www.zbmath.org/authors/?q=ai:mercuri.pietroSummary: Let \(p\) be an odd prime number and let \(X_0^+(p)\) be the quotient of the classical modular curve \(X_0(p)\) by the action of the Atkin-Lehner operator \(w_p\). In this paper, we show how to compute explicit equations for the canonical model of \(X_0^+(p)\). Then we show how to compute the modular parametrization, when it exists, from \(X_0^+(p)\) to an isogeny factor \(E\) of dimension 1 of its Jacobian \(J_0^+(p)\). Finally, we show how to use this map to determine the rational points on \(X_0^+(p)\) up to a large fixed height.Hyperelliptic curves with maximal Galois action on the torsion points of their Jacobians.https://www.zbmath.org/1456.111182021-04-16T16:22:00+00:00"Landesman, Aaron"https://www.zbmath.org/authors/?q=ai:landesman.aaron"Swaminathan, Ashvin"https://www.zbmath.org/authors/?q=ai:swaminathan.ashvin-anand"Tao, James"https://www.zbmath.org/authors/?q=ai:tao.james"Xu, Yujie"https://www.zbmath.org/authors/?q=ai:xu.yujieSummary: In this article, we show that in each of four standard families of hyperelliptic curves, there is a density-\(1\) subset of members with the property that their Jacobians have adelic Galois representation with image as large as possible. This result constitutes an explicit application of a general theorem on arbitrary rational families of abelian varieties to the case of families of Jacobians of hyperelliptic curves. Furthermore, we provide explicit examples of hyperelliptic curves of genus \(2\) and \(3\) over \(\mathbb{Q}\) whose Jacobians have such maximal adelic Galois representations.New examples of maximal curves with low genus.https://www.zbmath.org/1456.111152021-04-16T16:22:00+00:00"Bartoli, Daniele"https://www.zbmath.org/authors/?q=ai:bartoli.daniele"Giulietti, Massimo"https://www.zbmath.org/authors/?q=ai:giulietti.massimo"Kawakita, Motoko"https://www.zbmath.org/authors/?q=ai:kawakita.motoko-qiu"Montanucci, Maria"https://www.zbmath.org/authors/?q=ai:montanucci.mariaSummary: In this paper, explicit equations for algebraic curves with genus 4, 5, and 10 already studied in characteristic zero, are analyzed in positive characteristic \(p\). We show that these curves have an interesting behaviour on the number of their rational places. Namely, they are either maximal or minimal over the finite field with \(p^2\) elements for infinitely many \(p\)'s. The key tool is the investigation of their Jacobian decomposition. Lists of small \(p\)'s for which maximality holds are provided. In some cases we also describe the automorphism group of the curve.Codimension two cycles in Iwasawa theory and elliptic curves with supersingular reduction.https://www.zbmath.org/1456.112122021-04-16T16:22:00+00:00"Lei, Antonio"https://www.zbmath.org/authors/?q=ai:lei.antonio"Palvannan, Bharathwaj"https://www.zbmath.org/authors/?q=ai:palvannan.bharathwajLet us fix an odd prime \(p\) and \(\mathcal{R}\) be a Noetherian, complete, integrally closed, local domain of characteristic zero with Krull dimension \(n+1\) and whose residue field has characteristic \(p\). To a continuous Galois representation \(\rho_{d, n}:\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\longrightarrow\mathrm{Gal}_d(\mathcal{R})\) satisfying the `Panchishkin condition', which is a type of `ordinariness' assumption for Galois deformations introduced by Greenberg in Section 4 of [Proc. Symp. Pure Math. 55, 193--223 (1994; Zbl 0819.11046)], \textit{R. Greenberg} has formulated a main conjecture in Iwasawa theory. The Iwasawa main conjecture provides a relation involving codimension one cycles in the divisor group of the ring \(\mathcal{R}\), relating a \(p\)-adic \(L\)-function, satisfying suitable interpolation properties, to a Selmer group. The divisor group, denoted by \(Z^1(\mathcal{R})\), is the free abelian group on the set of height \(1\) prime ideals of the ring \(\mathcal{R}\).
One could consider \(Z^2(\mathcal{R})\), the free abelian group on the set of height \(2\) prime ideals of the ring \(\mathcal{R}\). Many standard conjectures in Iwasawa theory expect that pseudonull modules are ubiquitous. For example, see Conjecture 3.5 in \textit{R. Greenberg}'s article [Adv. Stud. Pure Math. 30, 335--385 (2001; Zbl 0998.11054)] and Conjecture B in the work of \textit{J. Coates} and \textit{R. Sujatha} [Math. Ann. 331, No. 4, 809--839 (2005; Zbl 1197.11142)]. These pseudonull \(\mathcal{R}\)-modules are supported in codimension at least two. One desirable extension of the Iwasawa main conjecture is an answer to the following question: Can we use codimension two cycles from \(Z^2(\mathcal{R})\) to associate analytic invariants to pseudonull modules in Iwasawa theory ?
A result of \textit{F. M. Bleher} et al. [Am. J. Math. 142, No. 2, 627--682 (2020; Zbl 07208784)] has initiated the topic of higher codimension Iwasawa theory. As a generalization of the classical Iwasawa main conjecture, they prove a relationship between analytic objects (a pair of Katz's 2-variable \(p\)-adic \(L\)-functions) and algebraic objects (two `everywhere unramified' Iwasawa modules) involving codimension two cycles in a \(2\)-variable Iwasawa algebra. In the current work under review, the authors prove a result by considering the restriction to an imaginary quadratic field \(K\) (where an odd prime \(p\) splits) of an elliptic curve \(E\), defined over \(\mathbb{Q}\), with good supersingular reduction at \(p\). On the analytic side, we consider eight pairs of \(2\)-variable \(p\)-adic \(L\)-functions in this setup. On the algebraic side, they consider modifications of fine Selmer groups over the \(\mathbb{Z}^2_p\)-extension of \(K\). The authors also provide numerical evidence, using algorithms of Pollack [\url{http://math.bu.edu/people/rpollack/Data/data.html}], towards a pseudonullity conjecture of \textit{J. Coates} and \textit{R. Sujatha} [Math. Ann. 331, No. 4, 809--839 (2005; Zbl 1197.11142)].
Reviewer: Wei Feng (Beijing)On distribution formulas for complex and \(l\)-adic polylogarithms.https://www.zbmath.org/1456.111212021-04-16T16:22:00+00:00"Nakamura, Hiroaki"https://www.zbmath.org/authors/?q=ai:nakamura.hiroaki.1"Wojtkowiak, Zdzisław"https://www.zbmath.org/authors/?q=ai:wojtkowiak.zdzislawSummary: We study an \(l\)-adic Galois analogue of the distribution formulas for polylogarithms with special emphasis on path dependency and arithmetic behaviors. As a goal, we obtain a notion of certain universal Kummer-Heisenberg measures that enable interpolating the \(l\)-adic polylogarithmic distribution relations for all degrees.
For the entire collection see [Zbl 1446.81002].Evaluating generating functions for periodic multiple polylogarithms via rational Chen-Fliess series.https://www.zbmath.org/1456.111612021-04-16T16:22:00+00:00"Ebrahimi-Fard, Kurusch"https://www.zbmath.org/authors/?q=ai:ebrahimi-fard.kurusch"Gray, W. Steven"https://www.zbmath.org/authors/?q=ai:gray.w-steven"Manchon, Dominique"https://www.zbmath.org/authors/?q=ai:manchon.dominiqueSummary: The goal of the paper is to give a systematic way to numerically evaluate the generating function of a periodic multiple polylogarithm using a Chen-Fliess series with a rational generating series. The idea is to realize the corresponding Chen-Fliess series as a bilinear dynamical system. A standard form for such a realization is given. The method is also generalized to the case where the multiple polylogarithm has non-periodic components. This allows one, for instance, to numerically validate the Hoffman conjecture. Finally, a setting in terms of dendriform algebras is provided.
For the entire collection see [Zbl 1446.81002].A constructive proof of Masser's theorem.https://www.zbmath.org/1456.110992021-04-16T16:22:00+00:00"Barrios, Alexander J."https://www.zbmath.org/authors/?q=ai:barrios.alexander-jSummary: The modified Szpiro conjecture, equivalent to the \(abc\) Conjecture, states that for each \(\epsilon >0\), there are finitely many rational elliptic curves satisfying \(N_E^{6+\epsilon}<\max\!\left\{\left\vert c_4^3\right\vert,c_6^2\right\}\) where \(c_4\) and \(c_6\) are the invariants associated to a minimal model of \(E\) and \(N_E\) is the conductor of \(E\). We say \(E\) is a good elliptic curve if \(N_E^6<\max\!\left\{\left\vert c_4^3\right\vert,c_6^2\right\}\). \textit{D. W. Masser} [Astérisque 183, 19--23 (1990; Zbl 0742.14027)] showed that there are infinitely many good Frey curves. Here we give a constructive proof of this assertion.
For the entire collection see [Zbl 1455.00030].Arithmetic Chern-Simons theory. I.https://www.zbmath.org/1456.140292021-04-16T16:22:00+00:00"Kim, Minhyong"https://www.zbmath.org/authors/?q=ai:kim.minhyongSummary: In this paper, we apply ideas of \textit{R. Dijkgraaf} and \textit{E. Witten} [Commun. Math. Phys. 129, No. 2, 393--429 (1990; Zbl 0703.58011)] and \textit{E. Witten} [Commun. Math. Phys. 121, No. 3, 351--399 (1989; Zbl 0667.57005)] on \(2+1\) dimensional topological quantum field theory to arithmetic curves, that is, the spectra of rings of integers in algebraic number fields. In the first three sections, we define classical Chern-Simons functionals on spaces of Galois representations. In the highly speculative Sect. 6, we consider the far-fetched possibility of using Chern-Simons theory to construct \(L\)-functions.
For the entire collection see [Zbl 07237934].Second descent and rational points on Kummer varieties.https://www.zbmath.org/1456.111072021-04-16T16:22:00+00:00"Harpaz, Yonatan"https://www.zbmath.org/authors/?q=ai:harpaz.yonatanSummary: A powerful method, pioneered by Swinnerton-Dyer, allows one to study rational points on pencils of curves of genus 1 by combining the fibration method with a sophisticated form of descent. A variant of this method, first used by \textit{A. Skorobogatov} and \textit{P. Swinnerton-Dyer} in [Adv. Math. 198, No. 2, 448--483 (2005; Zbl 1085.14021)], can be applied to the study of rational points on Kummer varieties. In this paper we extend the method to include an additional step of second descent. Assuming finiteness of the relevant Tate-Shafarevich groups, we use the extended method to show that the Brauer-Manin obstruction is the only obstruction to the Hasse principle on Kummer varieties associated to abelian varieties with all rational 2-torsion, under relatively mild additional hypotheses.Exceptional zero formulae for anticyclotomic \(p\)-adic \(L\)-functions of elliptic curves in the ramified case.https://www.zbmath.org/1456.112132021-04-16T16:22:00+00:00"Longo, Matteo"https://www.zbmath.org/authors/?q=ai:longo.matteo"Pati, Maria Rosaria"https://www.zbmath.org/authors/?q=ai:pati.maria-rosariaLet \(E\) be an elliptic curve over \(\mathbb Q\) of conductor \(N\) and let \(p\) be a prime of multiplicative reduction for \(E\). Let \(K\) be an imaginary quadratic field, not equal to \(\mathbb Q(\sqrt{-1})\) or \(\mathbb Q(\sqrt{-3})\), in which \(p\) is ramified and all other primes dividing \(N\) are unramified.
Assume \(E\) has multiplicative reduction at the primes dividing \(N\) that are inert in \(K\) and assume that the number of such primes is odd. Using the eigenform of weight 2 on \(\Gamma_0(N)\) attached to \(E\), the authors, following the work of \textit{M. Bertolini} et al. [Am. J. Math. 124, No. 2, 411--449 (2002; Zbl 1079.11036)], construct
a \(p\)-adic \(L\)-function \(L_p(E/K, \chi, s)\) for all finite order ramified characters \(\chi\) of ring class fields of \(K\) of \(p\)-power conductor. Let \(H\) be the Hilbert class field of \(K\) and let \(H_p\) be the maximal subextension in which the prime of \(K\) above \(p\)
splits completely. The main result of the paper is that for characters \(\chi\) that factor through \(H_p\),
\[
L_p'(E/K,\chi, 1)=\frac{2}{[H : H_p]} \log_E(y_{\chi}-\overline{y}_{\chi}),
\]
where \(\log_E\) is the logarithm of the formal group of \(E\) and where \(y_{\chi}\) is a twisted Heegner point constructed from a uniformization of \(E\) by a Shimura curve and \(\overline{y}_{\chi}\) is its complex conjugate.
This result is an analogue of the work of \textit{M. Bertolini} and \textit{H. Darmon} [Invent. Math. 131 No. 2, 453--491 (1998; Zbl 0899.11029)] ], who considered the case where \(\chi\) is trivial and \(p\) is inert in \(K\).
Reviewer: Lawrence C. Washington (College Park)Patterns of primes in the Sato-Tate conjecture.https://www.zbmath.org/1456.111742021-04-16T16:22:00+00:00"Gillman, Nate"https://www.zbmath.org/authors/?q=ai:gillman.nate"Kural, Michael"https://www.zbmath.org/authors/?q=ai:kural.michael"Pascadi, Alexandru"https://www.zbmath.org/authors/?q=ai:pascadi.alexandru"Peng, Junyao"https://www.zbmath.org/authors/?q=ai:peng.junyao"Sah, Ashwin"https://www.zbmath.org/authors/?q=ai:sah.ashwinSummary: Fix a non-CM elliptic curve \(E/\mathbb{Q} \), and let \(a_E(p) = p + 1 - \#E(\mathbb{F}_p)\) denote the trace of Frobenius at \(p\). The Sato-Tate conjecture gives the limiting distribution \(\mu_{ST}\) of \(a_E(p)/(2\sqrt{p})\) within \([-1, 1]\). We establish bounded gaps for primes in the context of this distribution. More precisely, given an interval \(I\subseteq [-1, 1]\), let \(p_{I,n}\) denote the \(n\)th prime such that \(a_E(p)/(2\sqrt{p})\in I\). We show \(\liminf_{n\rightarrow \infty }(p_{I,n+m}-p_{I,n}) < \infty\) for all \(m\ge 1\) for ``most'' intervals, and in particular, for all \(I\) with \(\mu_{ST}(I)\ge 0.36\). Furthermore, we prove a common generalization of our bounded gap result with the Green-Tao theorem. To obtain these results, we demonstrate a Bombieri-Vinogradov type theorem for Sato-Tate primes.Examples of genuine QM abelian surfaces which are modular.https://www.zbmath.org/1456.110692021-04-16T16:22:00+00:00"Schembri, Ciaran"https://www.zbmath.org/authors/?q=ai:schembri.ciaranSummary: Let \(K\) be an imaginary quadratic field. Modular forms for GL(2) over \(K\) are known as Bianchi modular forms. Standard modularity conjectures assert that every weight 2 rational Bianchi newform has either an associated elliptic curve over \(K\) or an associated abelian surface with quaternionic multiplication over \(K\). We give explicit evidence in the way of examples to support this conjecture in the latter case. Furthermore, the quaternionic surfaces given correspond to \textit{genuine} Bianchi newforms, which answers a question posed by \textit{J. E. Cremona} [J. Lond. Math. Soc., II. Ser. 45, No. 3, 404--416 (1992; Zbl 0773.14023)] as to whether this phenomenon can happen.Weierstrass coefficients of the canonical lifting.https://www.zbmath.org/1456.111052021-04-16T16:22:00+00:00"Finotti, Luís R. A."https://www.zbmath.org/authors/?q=ai:finotti.luis-r-aModular polynomials on Hilbert surfaces.https://www.zbmath.org/1456.110782021-04-16T16:22:00+00:00"Milio, Enea"https://www.zbmath.org/authors/?q=ai:milio.enea"Robert, Damien"https://www.zbmath.org/authors/?q=ai:robert.damienSummary: We describe an evaluation/interpolation approach to compute modular polynomials on a Hilbert surface, which parametrizes abelian surfaces with maximal real multiplication. Under some heuristics we obtain a quasi-linear algorithm. The corresponding modular polynomials are much smaller than the ones on the Siegel threefold. We explain how to compute even smaller polynomials by using pullbacks of theta functions to the Hilbert surface.On the elliptic Stark conjecture in higher weight.https://www.zbmath.org/1456.111202021-04-16T16:22:00+00:00"Gatti, Francesca"https://www.zbmath.org/authors/?q=ai:gatti.francesca"Guitart, Xavier"https://www.zbmath.org/authors/?q=ai:guitart.xavierSummary: We study the special values of the triple product \(p\)-adic \(L\)-function constructed by \textit{H. Darmon} and \textit{V. Rotger} [Ann. Sci. Éc. Norm. Supér. (4) 47, No. 4, 779--832 (2014; Zbl 1356.11039)] at all classical points outside the region of interpolation. We propose conjectural formulas for these values that can be seen as extending the Elliptic Stark Conjecture, and we provide theoretical evidence for them by proving some particular cases.Betti numbers of Shimura curves and arithmetic three-orbifolds.https://www.zbmath.org/1456.111102021-04-16T16:22:00+00:00"Frączyk, Mikołaj"https://www.zbmath.org/authors/?q=ai:fraczyk.mikolaj"Raimbault, Jean"https://www.zbmath.org/authors/?q=ai:raimbault.jeanSummary: We show that asymptotically the first Betti number \(b_1\) of a Shimura curve satisfies the Gauss-Bonnet equality \(2\pi(b_1-2)=\operatorname{vol}\) where \(\operatorname{vol}\) is hyperbolic volume; equivalently \(2g-2=(1+o(1))\operatorname{vol}\) where \(g\) is the arithmetic genus. We also show that the first Betti number of a congruence hyperbolic 3-orbifold asymptotically vanishes relatively to hyperbolic volume, that is \(b_1/\operatorname{vol}\to 0\). This generalizes previous results obtained by the first author [``Strong Limit Multiplicity for arithmetic hyperbolic surfaces and 3-manifolds'', Preprint, \url{arXiv:1612.05354}], on which we rely, and uses the same main tool, namely Benjamini-Schramm convergence.Non-isotrivial elliptic surfaces with non-zero average root number.https://www.zbmath.org/1456.111002021-04-16T16:22:00+00:00"Bettin, Sandro"https://www.zbmath.org/authors/?q=ai:bettin.sandro"David, Chantal"https://www.zbmath.org/authors/?q=ai:david.chantal"Delaunay, Christophe"https://www.zbmath.org/authors/?q=ai:delaunay.christopheSummary: We consider the problem of finding non-isotrivial 1-parameter families of elliptic curves whose root number does not average to zero as the parameter varies in \(\mathbb{Z}\). We classify all such families when the degree of the coefficients (in the parameter \(t\)) is less than or equal to 2 and we compute the rank over \(\mathbb{Q}(t)\) of all these families. Also, we compute explicitly the average of the root numbers for some of these families highlighting some special cases. Finally, we prove some results on the possible values average root numbers can take, showing for example that all rational number in \([- 1, 1]\) are average root numbers for some non-isotrivial 1-parameter family.Bounds on 2-torsion in class groups of number fields and integral points on elliptic curves.https://www.zbmath.org/1456.111012021-04-16T16:22:00+00:00"Bhargava, M."https://www.zbmath.org/authors/?q=ai:bhargava.manjul"Shankar, A."https://www.zbmath.org/authors/?q=ai:shankar.arul|shankar.ananth-n"Taniguchi, T."https://www.zbmath.org/authors/?q=ai:taniguchi.takashi.1"Thorne, F."https://www.zbmath.org/authors/?q=ai:thorne.frank"Tsimerman, J."https://www.zbmath.org/authors/?q=ai:tsimerman.jacob"Zhao, Y."https://www.zbmath.org/authors/?q=ai:zhao.yueqing|zhao.yude|zhao.yongan|zhao.yingmin|zhao.yanxing|zhao.yongling|zhao.yannan|zhao.yanliang|zhao.yongqiang|zhao.yuxiao|zhao.yanying|zhao.yaru|zhao.yunping|zhao.youjie|zhao.yane|zhao.yaowu|zhao.yongwang|zhao.yaqin|zhao.yuesheng|zhao.yiyuan|zhao.yibin|zhao.yixing|zhao.yu|zhao.yingshuai|zhao.yongxiang|zhao.yongchen|zhao.yafan|zhao.yuhua|zhao.yanxiang|zhao.yunpeng|zhao.yonglong|zhao.yuyuan|zhao.yongchun|zhao.yuqi|zhao.yuejing|zhao.yuanjun|zhao.yajuan|zhao.yanzhu|zhao.yuzhe|zhao.yongda|zhao.yiwen|zhao.yidong|zhao.yanqi|zhao.yanyong|zhao.yongcai|zhao.yuqing|zhao.yuzhou|zhao.yanyun|zhao.yuane|zhao.yongzhi|zhao.yaming|zhao.yuchun|zhao.yapu|zhao.yongkai|zhao.yingqi|zhao.yaobing|zhao.yihong|zhao.yihui|zhao.yanda|zhao.yougang|zhao.yize|zhao.yaomin|zhao.yuling|zhao.yan|zhao.yili|zhao.yunlong|zhao.yixuan|zhao.yanfang|zhao.ye|zhao.yanjia|zhao.yani|zhao.yanxia|zhao.yongxia|zhao.yunsong|zhao.yizhen|zhao.yanmeng|zhao.yuansong|zhao.yibing|zhao.yinlong|zhao.yijin|zhao.yicai|zhao.yuelong|zhao.yajun|zhao.yuli|zhao.yanxin|zhao.youhui|zhao.yingfeng|zhao.yanju|zhao.yiqiang|zhao.younan|zhao.yaqing|zhao.yongbo|zhao.yongkang|zhao.yuxin|zhao.yuzhuang|zhao.yuhuai|zhao.yonghui|zhao.yanli|zhao.yangyang|zhao.yufan|zhao.yingying|zhao.yifei|zhao.yancai|zhao.yilin|zhao.yulin|zhao.yuhan|zhao.yanjuan|zhao.yang|zhao.yaping|zhao.yongchao|zhao.youqun|zhao.yudong|zhao.yuanxiang|zhao.yinghai|zhao.yongzhe|zhao.yunbo|zhao.yuefei|zhao.yizhi|zhao.yuwen|zhao.yimin|zhao.yongxin|zhao.yuliang|zhao.yadong|zhao.yianhe|zhao.yigong|zhao.yong|zhao.yuzhang|zhao.yongqian|zhao.yueyuan|zhao.yishu|zhao.yaozong|zhao.yuying|zhao.yuqiu|zhao.ying|zhao.yuhai|zhao.yanchun|zhao.you|zhao.yongyi|zhao.yingliang|zhao.yonghua|zhao.yicheng|zhao.yuehua|zhao.yarlwen|zhao.yuxiang|zhao.yingxue|zhao.youxuan|zhao.yinglu|zhao.yijia|zhao.yumeng|zhao.yunmei|zhao.yongdong|zhao.yihan|zhao.yongping|zhao.yinchao|zhao.yongtao|zhao.yuzhong|zhao.yuntao|zhao.yitian|zhao.yangsheng|zhao.yuanlu|zhao.yingzi|zhao.yanyu|zhao.yanguang|zhao.yinghui|zhao.yunfan|zhao.yumin|zhao.yanan|zhao.yucan|zhao.yunhong|zhao.yongchang|zhao.yanwei|zhao.yanjie|zhao.yanhua|zhao.yanqing|zhao.yehua|zhao.yukun|zhao.yuandi|zhao.yingtao|zhao.yaowen|zhao.yiming|zhao.yiwu|zhao.yanping|zhao.yuanying|zhao.yunge|zhao.yongjie|zhao.yanbin|zhao.yanchang|zhao.yaohong|zhao.yajing|zhao.yunwei|zhao.yongshen|zhao.yuchen|zhao.yanwen|zhao.yanhong|zhao.yonghong|zhao.yingchao|zhao.yuemin|zhao.yanqin|zhao.yeye|zhao.yayun|zhao.yueyu|zhao.yueqiang|zhao.yifen|zhao.yushu|zhao.yuanhe|zhao.yuejen|zhao.yizheng|zhao.yadi|zhao.yunyuan|zhao.yuna|zhao.youjian|zhao.yile|zhao.yuxia|zhao.yanyan|zhao.yunzhuan|zhao.yuexu|zhao.yian|zhao.yipeng|zhao.ya|zhao.yibao|zhao.yongjuan|zhao.yanfen|zhao.yanlu|zhao.yingbo|zhao.yuwei|zhao.yingnan|zhao.yinglin|zhao.yunfeng|zhao.yexi|zhao.yunxin|zhao.yalun|zhao.yingxiu|zhao.yanfeng|zhao.yurong|zhao.yunlei|zhao.yuhang|zhao.yanhui|zhao.yisi|zhao.yuning|zhao.yuanyuan|zhao.yigeng|zhao.yubo|zhao.yongwei|zhao.youxing|zhao.yibo|zhao.yuqin|zhao.yuge|zhao.yecheng|zhao.yuyun|zhao.yongfang|zhao.yongcheng|zhao.yueling|zhao.yufei|zhao.yiyi|zhao.yanjun|zhao.yinglei|zhao.yuanshan|zhao.yuchao|zhao.yuanzhang|zhao.yanmin|zhao.yaling|zhao.yangzhang|zhao.yijun|zhao.yuandong|zhao.yijiu|zhao.yanbo|zhao.yao|zhao.yusheng|zhao.yingdong|zhao.yueying|zhao.yingxin|zhao.yafei|zhao.yun|zhao.yunsheng|zhao.yumei|zhao.yuefeng|zhao.yunjie|zhao.yi|zhao.yanlei|zhao.yanzheng|zhao.yongshun|zhao.yage|zhao.yating|zhao.yeqing|zhao.yi.1|zhao.yujie|zhao.yanfei|zhao.yuoli|zhao.yanling|zhao.yuzhen|zhao.yinchuan|zhao.yongye|zhao.yuhuan|zhao.yuping|zhao.yufu|zhao.yinshan|zhao.yaonan|zhao.yige|zhao.yufang|zhao.yujuan|zhao.yinan|zhao.yufeng|zhao.yunfei|zhao.yuda|zhao.yanzhong|zhao.yaxi|zhao.youyi|zhao.yunbin|zhao.yinghong|zhao.yunhe|zhao.yagu|zhao.yueqin|zhao.yichun|zhao.yaqun|zhao.yushan|zhao.yaohua|zhao.yiming.1|zhao.yuejuan|zhao.yakun|zhao.yanlong|zhao.yongrui|zhao.yue|zhao.yongliang|zhao.yinyu|zhao.yusong|zhao.yanfa|zhao.yinliang|zhao.yingjie|zhao.yueru|zhao.yuming|zhao.yuhong|zhao.yongjun|zhao.yiping|zhao.yongyao|zhao.yonggan|zhao.yupeng|zhao.yin|zhao.yanru|zhao.yifan|zhao.yanting|zhao.yaoqing|zhao.yuhui|zhao.yahong|zhao.yulong|zhao.yanlin|zhao.yanbing|zhao.yuting|zhao.yucheng|zhao.yongsheng|zhao.yihua|zhao.yichuan|zhao.yingcui|zhao.yingchun|zhao.yunwang|zhao.yonggang|zhao.yaoshun|zhao.yazhou|zhao.yuee|zhao.yujia|zhao.yunliang|zhao.yuzhuo|zhao.yongchi|zhao.youjun|zhao.yixin|zhao.yuan|zhao.yandong|zhao.yuqianGiven a number field \(K\) of degree \(n\) and any positive integer \(m\), it is conjectured that \(h_m(K) = O_{\varepsilon, n, m}(|\mathrm{Disc}(K)|^{\varepsilon})\), where \(h_m(K)\) denotes the size of the \(m\)-torsion subgroup of the class group of \(K\), and \(\mathrm{Disc}(K)\) is the discriminant of \(K\). There are several works concerning with finding upper bounds for \(h_m(K)\) in the literature, say [\textit{J. S. Ellenberg} and \textit{A. Venkatesh}, Int. Math. Res. Not. 2007, No. 1, Article ID rnm002, 18 p. (2007; Zbl 1130.11060); \textit{H. A. Helfgott} and \textit{A. Venkatesh}, J. Am. Math. Soc. 19, No. 3, 527--550 (2006; Zbl 1127.14029); \textit{E. Landau}, Nachr. Ges. Wiss. Göttingen, Math.-Phys. Kl. 1918, 79--97 (1918; JFM 46.0266.02); \textit{L. B. Pierce}, Forum Math. 18, No. 4, 677--698 (2006; Zbl 1138.11049)].
In the paper under review, the authors develop a uniform method to provide nontrivial bounds for the case \(m=2\) and all degrees \(n \geq 3\). Indeed, in Theorem 1.1, the authors prove that \[h_2(K) < O_\varepsilon (|\mathrm{Disc}(K)|)^{\frac{1}{2} - \delta_n + \epsilon}\] with \(\delta_n= .2215 \cdots,\) for \(n=3, 4\) and \(\delta_n = 1/2n\) for \(n\geq 5\).
Using their result in the case \(n=3\), in Theorem 1.2, they give upper bounds on the size of \(2\)-Selmer groups, ranks, and the number of integral points on the elliptic curves (in terms of their discriminants) improving the known ones in [\textit{A. Brumer} and \textit{K. Kramer}, Duke Math. J. 44, 715--743 (1977; Zbl 0376.14011); \textit{H. A. Helfgott} and \textit{A. Venkatesh}, J. Am. Math. Soc. 19, No. 3, 527--550 (2006; Zbl 1127.14029)]. Considering their results in the case \(n>4\), in Theorem 1.3, they obtain upper bounds on the size of the \(2\)-Selmer groups, ranks of the Jacobian of the hyperelliptic curves \(C: y^2=f(x)\) over \(\mathbb Q\) in terms of \(|\mathrm{Disc}(K)|\), where \(f\) is a separable polynomial of degree \(n\) and \(K\) is the étale \(\mathbb Q\)-algebra \({\mathbb Q}[x]/f(x)\), which improve the previous result in [\textit{A. Brumer} and \textit{K. Kramer}, Duke Math. J. 44, 715--743 (1977; Zbl 0376.14011)]. As an other consequence of the case \(n=3\) of Theorem 1.1, the authors obtain an upper bound for the number of isomorphism classes of quartic fields having Galois group \(A_4\) and discriminant less than a given \(X>0\). Their results improve those of the previous works [\textit{A. M. Baily}, J. Reine Angew. Math. 315, 190--210 (1980; Zbl 0421.12007); \textit{S. Wong}, Proc. Am. Math. Soc. 133, No. 10, 2873--2881 (2005; Zbl 1106.11041)]. The key tools for proving Theorem 1.1 are given in Theorems 1.5 and 1.6.
Finally, in Theorem 1.7, the authors prove an analogous of Theorems 1.5 and 1.6 to the case of the function fields to obtain a nontrivial upper bound on the \(2\)-torsion points in \(\mathrm{Pic}^0(C)(k)\), where \(C\) ia an algebraic curve of genus \(g\) on a finite field \(k\), in terms of \(g\) and the cardinal number of \(k\).
Reviewer: Sajad Salami (Rio de Janeiro)Some explicit computations in Arakelov geometry of abelian varieties.https://www.zbmath.org/1456.111062021-04-16T16:22:00+00:00"Gaudron, Éric"https://www.zbmath.org/authors/?q=ai:gaudron.ericSummary: Given a polarized complex abelian variety \((\mathsf{A}, \mathsf{L})\), a Gromov lemma makes a comparison between the sup and \(L^2\) norms of a global section of \(\mathsf{L}\). We give here an explicit bound which depends on the dimension, degree and injectivity diameter of \((\mathsf{A}, \mathsf{L})\). It rests on a more general estimate for the jet of a global section of \(\mathsf{L}\). As an application we deduce some estimates of the maximal slope of the tangent and cotangent spaces of a polarized abelian variety defined over a number field. These results are effective versions of previous works by Masser and Wüstholz on one hand and
\textit{J. B. Bost} [Duke Math. J. 82, No. 1, 21--70 (1996; Zbl 0867.14010)] on the other. They also improve some similar statements established by \textit{P. Graftieaux} in [Duke Math. J. 106, No. 1, 81--121 (2001; Zbl 1064.14045)].Rational points on Shimura curves and the Manin obstruction.https://www.zbmath.org/1456.111092021-04-16T16:22:00+00:00"Arai, Keisuke"https://www.zbmath.org/authors/?q=ai:arai.keisukeSummary: In a previous article, we proved that Shimura curves have no points rational over number fields under a certain assumption. In this article, we give another criterion of the nonexistence of rational points on Shimura curves and obtain new counterexamples to the Hasse principle for Shimura curves. We also prove that such counterexamples obtained from the above results are accounted for by the Manin obstruction.The finite subgroups of \(\mathrm{SL}(3,\overline{F})\).https://www.zbmath.org/1456.200562021-04-16T16:22:00+00:00"Flicker, Yuval Z."https://www.zbmath.org/authors/?q=ai:flicker.yuval-zThe \textit{special linear group} \(\mathrm{SL}(n,F)\) of degree \(n\) over a field \(F\) is the set of all \(n\times n\) matrices with determinant \(1\), with the group operations of ordinary matrix multiplication and matrix inversion. In the paper under review, the author gives a complete exposition of the finite subgroups of \(\mathrm{SL}(3,\bar{F})\), where \(\bar{F}\) is a separably closed field of characteristic not dividing the order of the finite subgroup. This completes the earlier work of \textit{H. F. Blichfeldt} [Math. Ann. 63, 552--572 (1907; JFM 38.0192.03)]. Several elementary examples are illustrated by the author in the exposition.
Reviewer: Mahadi Ddamulira (Kampala)Elliptic curves over finite fields with Fibonacci numbers of points.https://www.zbmath.org/1456.111162021-04-16T16:22:00+00:00"Bilu, Yuri"https://www.zbmath.org/authors/?q=ai:bilu.yuri-f"Gómez, Carlos A."https://www.zbmath.org/authors/?q=ai:gomez.carlos-alexis"Gómez, Jhonny C."https://www.zbmath.org/authors/?q=ai:gomez.jhonny-c"Luca, Florian"https://www.zbmath.org/authors/?q=ai:luca.florianThe paper studies and solves a curious problem: let \(E\)\, be an elliptic curve defined over a finite field \(\mathbb{F}_q\)\, with \(\sharp(E)=q+1-a\)\, and let us denote \(E_m(q,a);\,\, m\ge 1\)\, the number of points of \(E\)\, over \(\mathbb{F}_{q^m}\); let \(\{F_n\}_{n\ge 1}\)\, be the Fibonacci sequence. When the intersection of \(\{E_m(q,a)\}_{m\ge 1}\)\, and \(\{F_n\}_{n\ge 1}\)\, has two elements at least?, i.e. to find the solutions of \(\sharp E_{m_1} = F_{n_1}\),\, \(\sharp E_{m_2} = F_{n_2}\), etc.
Section 1 discusses the problem and formulates the obtained result (Theorem 1.1): the cardinal of the intersection is two for four couples \((q,a)\)\, and three in only a case (for \(q=a=2\)). The rest of the paper is devoted to the proof of that theorem.
Section 2 studies the equation \(E_m(q,a)=F_n\)\, in terms of linear forms in logarithms and discusses how the problem can be reduced to three cases: (i) \(q\)\, small (\(q\le 10.000\)), (ii) \(n_1\)\, small and (iii) \(m_2\)\, small. This provides a list of the possible values for \(q\).
Section 3 gathers some necessary tools regarding linear forms in logarithms and continued fractions. Assuming proved that \(n_2\le 1.000\)\, Section 4 first studies the problem for \(q\le 10.000\)\, and, using the computational package \texttt{Mathematica} finds the five solutions of Theorem 1.1. Then, for \(q> 10.000\),\, also using Mathematica, proves that there is no solution.
The rest of the paper assumes \(n_2> 1.000\)\, and Section 5 to 10 deals with the three cases (i), (ii) and (iii) and, always with the computational help of Mathematica (the paper says that ``the total calculation time for the Mathematica software for this paper was 20 days on 25 parallel desktop computers'') finishes the proof of Theorem 1.1.
Reviewer: Juan Tena Ayuso (Valladolid)Canonical heights and preperiodic points for certain weighted homogeneous families of polynomials.https://www.zbmath.org/1456.371082021-04-16T16:22:00+00:00"Ingram, Patrick"https://www.zbmath.org/authors/?q=ai:ingram.patrickSummary: A family \(f\) of polynomials over a number field \(K \) will be called weighted homogeneous if and only if \(f_t(z) = F(z^e, t)\) for some binary homogeneous form \(F(X, Y)\) and some integer \(e \geq 2\). For example, the family \(z^d + t\) is weighted homogeneous. We prove a lower bound on the canonical height, of the form
\[
\hat{h}_{f_t}(z)\geq \varepsilon \max \left\{h_{\mathsf{M}_d}(f_t), \log|\operatorname{Norm}\mathfrak{R}_{f_t}|\right\},
\]
for values \(z \in K\) which are not preperiodic for \(f_t\). Here \(\varepsilon\) depends only on the number field \(K\), the family \(f\), and the number of places at which \(f_t\) has bad reduction. For suitably generic morphisms \( \varphi :\mathbb{P}^1 \to \mathbb{P}^1\), we also prove an absolute bound of this form for \(t\) in the image of \(\varphi\) over \(K\) (assuming the \(abc\) Conjecture), as well as uniform bounds on the number of preperiodic points (unconditionally).From the monster to Thompson to O'Nan.https://www.zbmath.org/1456.110632021-04-16T16:22:00+00:00"Duncan, John F. R."https://www.zbmath.org/authors/?q=ai:duncan.john-f-rSummary: The commencement of monstrous moonshine is a connection between the largest sporadic simple group-the monster-and complex elliptic curves. Here we explain how a closer look at this connection leads, via the Thompson group, to recently observed relationships between the non-monstrous sporadic simple group of O'Nan and certain families of elliptic curves defined over the rationals. We also describe umbral moonshine from this perspective.
For the entire collection see [Zbl 1452.17002].Equations of hyperelliptic Shimura curves.https://www.zbmath.org/1456.110452021-04-16T16:22:00+00:00"Guo, Jia-Wei"https://www.zbmath.org/authors/?q=ai:guo.jiawei"Yang, Yifan"https://www.zbmath.org/authors/?q=ai:yang.yifanSummary: By constructing suitable Borcherds forms on Shimura curves and using Schofer's formula for norms of values of Borcherds forms at CM points, we determine all of the equations of hyperelliptic Shimura curves \(X_{0}^{D}(N)\). As a byproduct, we also address the problem of whether a modular form on Shimura curves \(X_{0}^{D}(N)/W_{D,N}\) with a divisor supported on CM divisors can be realized as a Borcherds form, where \(X_{0}^{D}(N)/W_{D,N}\) denotes the quotient of \(X_{0}^{D}(N)\) by all of the Atkin-Lehner involutions. The construction of Borcherds forms is done by solving certain integer programming problems.On integral points on isotrivial elliptic curves over function fields.https://www.zbmath.org/1456.111022021-04-16T16:22:00+00:00"Conceição, Ricardo"https://www.zbmath.org/authors/?q=ai:conceicao.ricardo-pLet \(L\) be a number field and \(S\) a finite set of places of \(L\) containing the Archimedean places. The well-known Lang's conjecture states that the number of \(S\)-integral points on an elliptic curve \(E\) defined over \(L\) is bounded by a quantity depending only on \(L\), \(S\) and the rank of the group \(E(L)\). Let \(K\) be the function field of a curve defined over a field of characteristic \(0\). Then for an elliptic curve defined over \(K\) with non-constant \(j\)-invariant, a version of Lang's conjecture has been proved in [\textit{M. Hindry} and \textit{J.Silverman}, Invent. Math. 93, No. 2, 419--450 (1988; Zbl 0657.14018)].
Let \(C\) be a curve over a finite field \(k\) of genus \(g\ge 1\) and \(L\) its function field. Let \(S\) be a finite non-empty set of points on \(C\). Then \(S\) is a set of places of \(L\). Let \(E\) be an elliptic curve defined over \(k\). Suppose that \(E\) is defined over \(L\). The group of \(L\)-rational points \(E(L)\) of \(E\) is isomorphic to the group of \(k\)-morphisms of \(C\) to \(E\), and \(S\)-integral points on \(E\) correspond to \(k\)-morphisms \(\psi:C\rightarrow E\) satisfying \(\psi^{-1}(0_E)\subset S\). Such morphisms \(\psi\) are called \(S\)-integral. Since the composition of any power of the Frobenius endomorphism of \(E\) and a \(S\)-integral morphism is also \(S\)-integral, for the finiteness, it is necessary to consider only separable \(S\)-integral morphisms. In this article, the author shows a version of Lang's conjecture for the number of non-constant separable \(S\)-integral morphisms. Thus he shows that the number of non-constant separable \(S\)-integral morphisms is bounded by \((2\sqrt{|S|+4(g-1)}+1)^{4g}\). This bound is dependent only on \(|S|\) and the genus of \(C\). Let \(A(t)\in k[t]\) be a square-free polynomial of odd order \(d>1\). Assume that characteristic of \(k\) is odd. Let \(\infty\) be the point at infinity of the curve \(C_A:y^2=A(t)\). Let \(f(x)\in k[x]\) be a polynomial of degree \(3\) defining an elliptic curve \(E:y^2=f(x)\). Let \(E_A\) be twist over \(k(t)\) of \(E\) defined by \(A(t)y^2=f(x)\). Proving that non-constant separable \(\{\infty\}\)-integral morphisms of \(C_A\) to \(E\) correspond bijectively to non-constant integral points \((F,G)\) on \(E_A(k[t])\) with \(F'\ne 0\), the author shows that the number of non-constant separable \(\{\infty\}\)-morphisms of \(C_A\) to \(E\) is bounded by \(|k|^{2d-3}\).
Further, he shows that for a separable integral point \(P=(F,G)\) on \(E_A\), \(\text{deg}F<\text{deg }A-1\) and that if \(\text{deg }A'(t)=0\) and \(E_A\) has a separable integral point \((F,G)\) satisfying \(\text{deg }F\le(\text{deg }A-1)/2\) then \(j(E)=1728\).
Reviewer: Noburo Ishii (Kyoto)On the degeneracy of integral points and entire curves in the complement of nef effective divisors.https://www.zbmath.org/1456.111192021-04-16T16:22:00+00:00"Heier, Gordon"https://www.zbmath.org/authors/?q=ai:heier.gordon"Levin, Aaron"https://www.zbmath.org/authors/?q=ai:levin.aaronSummary: As a consequence of the divisorial case of our recently established generalization of Schmidt's subspace theorem, we prove a degeneracy theorem for integral points on the complement of a union of nef effective divisors. A novel aspect of our result is the attainment of a strong degeneracy conclusion (arithmetic quasi-hyperbolicity) under weak positivity assumptions on the divisors. The proof hinges on applying our recent theorem with a well-situated ample divisor realizing a certain lexicographical minimax. We also explore the connections with earlier work by other authors and make a conjecture regarding bounds for the numbers of divisors necessary, including consideration of the question of arithmetic hyperbolicity. Under the standard correspondence between statements in Diophantine approximation and Nevanlinna theory, one obtains analogous degeneration statements for entire curves.The Lind-Lehmer constant for \(\mathbb{Z}_2^r\times\mathbb{Z}_4^s\).https://www.zbmath.org/1456.112042021-04-16T16:22:00+00:00"Mossinghoff, Michael J."https://www.zbmath.org/authors/?q=ai:mossinghoff.michael-j"Pigno, Vincent"https://www.zbmath.org/authors/?q=ai:pigno.vincent"Pinner, Christopher"https://www.zbmath.org/authors/?q=ai:pinner.christopher-gFor a finite abelian group \(G=\mathbb{Z}_{n_{1}}\times \cdots \times \mathbb{Z}_{n_{k}}\), where \(\mathbb{Z}_{n_{j}}\) \((1\leq j\leq k)\) denotes the cyclic group with order \(n_{j}\), define
\[
\lambda (G)=\min \left( \left\{ \prod_{j_{1}=1}^{n_{1}}\dots\prod_{j_{k}=1}^{n_{k}}\left\vert F(e^{i2\pi j_{1}/n_{1}},\dots,e^{i2\pi j_{k}/n_{k}})\right\vert \mid F\in \mathbb{Z}[x_{1},\dots,x_{k}]\right\} \cap \lbrack 2,\infty )\right) .
\]
According to [\textit{D. Lind} et al., Proc. Am. Math. Soc. 133, No. 5, 1411--1416 (2005; Zbl 1056.43005); \textit{D. Desilva} and \textit{C. Pinner}, Proc. Am. Math. Soc. 142, No. 6, 1935--1941 (2014; Zbl 1294.11185)], if \(G\neq \mathbb{Z}_{2}\), then
\[
\lambda (G)\leq \operatorname{card}(G)-1, \tag{*}
\]
\((\lambda (\mathbb{Z}_{p^{n}}),\lambda (\mathbb{Z}_{2^{n}}))=(2,3)\) for any natural number \(n\) and any odd prime \(p\), and (*) is sharp when \(G=\mathbb{Z}_{3}^{n}\), or when \(G=\mathbb{Z}_{2}^{n}\) and \(n\geq 2\).
In the paper under review, the authors continue to investigate the values of \(\lambda (G)\) for \(G\) running through certain families of \(p\)-groups, where \(p\in \{2,3\}\). Mainly, they show that \(\lambda (\mathbb{Z}_{3}\times \mathbb{Z}_{3^{n}})=8\), \(n\geq 3\Rightarrow \lambda (\mathbb{Z}_{2}\times \mathbb{Z}_{2^{n}})=9\), and equality occurs (again) in (*) whenever \(G\neq \mathbb{Z}_{2}\) and the factors of \(G\) are all \(\mathbb{Z}_{2}\) or \(\mathbb{Z}_{4}\).
The proofs of these results are based on a generalization of Lemma 2.1 of the last mentioned reference about a congruence satisfied by the rational integers defining \(\lambda (G)\), when \(G\) is a \(p\)-group.
Reviewer: Toufik Zaïmi (Riyadh)On CM-types of Galois CM-fields without proper CM-subfields.https://www.zbmath.org/1456.111082021-04-16T16:22:00+00:00"Kida, Masanari"https://www.zbmath.org/authors/?q=ai:kida.masanariSummary: We show that all CM-types of Galois CM-fields without proper CM-subfields are nondegenerate. As a consequence, the Hodge conjecture is true for abelian varieties with complex multiplication by such CM-fields.Statistics for products of traces of high powers of the Frobenius class of hyperelliptic curves in even characteristic.https://www.zbmath.org/1456.111142021-04-16T16:22:00+00:00"Bae, Sunghan"https://www.zbmath.org/authors/?q=ai:bae.sunghan"Jung, Hwanyup"https://www.zbmath.org/authors/?q=ai:jung.hwanyupAn asymptotic for the average number of amicable pairs for elliptic curves.https://www.zbmath.org/1456.111042021-04-16T16:22:00+00:00"Parks, James"https://www.zbmath.org/authors/?q=ai:parks.james-mSummary: Amicable pairs for a fixed elliptic curve defined over \(\mathbb{Q}\) were first considered by \textit{J. H. Silverman} and \textit{K. E. Stange} [Exp. Math. 20, No. 3, 329--357 (2011; Zbl 1269.11056)] where they conjectured an order of magnitude for the function that counts such amicable pairs. This was later refined by Jones to give a precise asymptotic constant. The author previously proved an upper bound for the average number of amicable pairs over the family of all elliptic curves. In this paper we improve this result to an asymptotic for the average number of amicable pairs for a family of elliptic curves.