Recent zbMATH articles in MSC 11G05https://zbmath.org/atom/cc/11G052024-03-13T18:33:02.981707ZWerkzeugFormulas for moments of class numbers in arithmetic progressionshttps://zbmath.org/1528.110222024-03-13T18:33:02.981707Z"Bringmann, Kathrin"https://zbmath.org/authors/?q=ai:bringmann.kathrin"Kane, Ben"https://zbmath.org/authors/?q=ai:kane.ben"Pujahari, Sudhir"https://zbmath.org/authors/?q=ai:pujahari.sudhirThe well-known identity \(\sum_{t\in\mathbb{Z}} H(4_p-t^2)= 2p\) where \(p\) is a prime and \(H\) denotes the Hurwitz class number is generalized to
\[
H_{2,m,3}(n) := \sum_{\substack{t\in\mathbb{Z}\\ t\equiv m\bmod 3}} t^2 H(4n-t^2)\quad\text{for }m\in\{0,1,2\}.
\]
A special result for \(p\equiv2\bmod 3\) is \(H_{2,1,3}(p)=\frac{p(p+1)}{2}\). The proof uses non-holomorphic modular forms and their growth towards the cusps and evaluations of generalized quadratic Gauss sums.
Reviewer: Meinhard Peters (Münster)Cube sum problem for integers having exactly two distinct prime factorshttps://zbmath.org/1528.110422024-03-13T18:33:02.981707Z"Majumdar, Dipramit"https://zbmath.org/authors/?q=ai:majumdar.dipramit"Shingavekar, Pratiksha"https://zbmath.org/authors/?q=ai:shingavekar.pratikshaSummary: Given an integer \(n>1\), it is a classical Diophantine problem that whether \(n\) can be written as a sum of two rational cubes. The study of this problem, considering several special cases of \(n\), has a copious history that can be traced back to the works of
\textit{J. J. Sylvester} [Am. J. Math. 2, 280--285 (1879; JFM 11.0141.01)],
\textit{P. Satgé} [Invent. Math. 87, 425--439 (1987; Zbl 0616.14023)],
\textit{E. S. Selmer} [Acta Math. 85, 203--362 (1951; Zbl 0042.26905)] etc., and up to the recent works of \textit{L. Alpöge} et al. [``Integers expressible as the sum of two rational cubes'', Preprint, arXiv:2210.10730]. In this article, we consider the cube sum problem for cube-free integers \(n\) which are coprime to 3 and have exactly two distinct prime factors.On the surjectivity of \(\bmod\, \ell\) representations associated to elliptic curveshttps://zbmath.org/1528.110432024-03-13T18:33:02.981707Z"Zywina, David"https://zbmath.org/authors/?q=ai:zywina.davidSummary: Let \(E\) be an elliptic curve over the rationals that does not have complex multiplication. For each prime \(\ell \), the action of the absolute Galois group on the \(\ell \)-torsion points of \(E\) can be given in terms of a Galois representation \(\rho_{E,\ell }\colon \operatorname{Gal}({\overline{\mathbb{Q}}}/\mathbb{Q}) \rightarrow \operatorname{GL}_2(\mathbb{F}_\ell )\). An important theorem of Serre says that \(\rho_{E,\ell }\) is surjective for all sufficiently large \(\ell \). In this paper, we describe a simple algorithm based on Serre's proof that can quickly determine the finite set of primes \(\ell >13\) for which \(\rho_{E,\ell }\) is not surjective. We will also give some improved bounds for Serre's theorem.
{{\copyright} 2022 The Authors. The publishing rights in this article are licensed to the London Mathematical Society under an exclusive licence.}Moments of Kloosterman sums, supercharacters, and elliptic curveshttps://zbmath.org/1528.110712024-03-13T18:33:02.981707Z"Sayed, Fahim"https://zbmath.org/authors/?q=ai:sayed.fahim"Kalita, Gautam"https://zbmath.org/authors/?q=ai:kalita.gautamAuthors' abstract: In this paper, we use a supercharacter theory for \(\mathbb{F}_p^2\) induced by the action of a subgroup of \(\mathrm{GL}_2\left(\mathbb{F}_p\right)\) to express the fifth and seventh power moments of Kloosterman sums in terms of traces of Frobenius endomorphism for certain families of elliptic curves.
Reviewer: Alexey Ustinov (Khabarovsk)Residual supersingular Iwasawa theory over quadratic imaginary fieldshttps://zbmath.org/1528.111142024-03-13T18:33:02.981707Z"Hamidi, Parham"https://zbmath.org/authors/?q=ai:hamidi.parhamSummary: Let \(p\) be an odd prime. Let \(E\) be an elliptic curve defined over a quadratic imaginary field, where \(p\) splits completely. Suppose \(E\) has supersingular reduction at primes above \(p\). Under appropriate hypotheses, we extend the results of [\textit{F. A. E. Nuccio Mortarino Majno di Capriglio} and \textit{S. Ramdorai}, Rend. Semin. Mat. Univ. Padova 149, 83--129 (2023; Zbl 1517.11139)] to \(\mathbb{Z}_p^2\)-extensions. We define and study the fine double-signed residual Selmer groups in these settings. We prove that for two residually isomorphic elliptic curves, the vanishing of the signed \(\mu\)-invariants of one elliptic curve implies the vanishing of the signed \(\mu\)-invariants of the other. Finally, we show that the Pontryagin dual of the Selmer group and the double-signed Selmer groups have no non-trivial pseudo-null submodules for these extensions.