Siegel's Lemma and Sum-Distinct Sets

On Zeta Functions and Families of Siegel Modular Forms

Let $p$ be a prime, and let $Γ=\Sp_g(\Z)$ be the Siegel modular group of genus $g$. We study $p$-adic families of zeta functions and Siegel modular forms. $L$-functions of Siegel modular forms are described in terms of motivic $L$-functions attached to $\Sp_g$, and their analytic properties are given. Critical values for the spinor $L$-functions and $p$-adic constructions are discussed. Rankin's lemma of higher genus is established. A general conjecture on a lifting from $ GSp_{2m} \times GSp_{2m}$ to $GSp_{4m}$ (of genus $g=4m$) is formulated. Constructions of $p$-adic families of Siegel modu…

An Efficient Version of the Bombieri-Vaaler Lemma

In their celebrated paper "On Siegel's Lemma", Bombieri and Vaaler found an upper bound on the height of integer solutions of systems of linear Diophantine equations. Calculating the bound directly, however, requires exponential time. In this paper, we present the bound in a different form that can be computed in polynomial time. We also give an elementary (and arguably simpler) proof for the bound.

Hecke eigenvalues of Siegel modular forms of "different weights"

Given two Siegel eigenforms of different weights, we determine explicit sets of Hecke eigenvalues for the two forms that must be distinct. In degree two, and under some additional conditions, we determine explicit sets of Fourier coefficients of the two forms that must be distinct.

Shidlovsky's multiplicity estimate and Irrationality of zeta values

In this paper we follow the approach of Bertrand-Beukers (and of later work of Bertrand), based on differential Galois theory, to prove a very general version of Shidlovsky's lemma that applies to Pad{é} approximation problems at several points, both at functional and numerical levels (i.e., before and after evaluating at a specific point). This allows us to obtain a new proof of the Ball-Rivoal theorem on irrationality of infinitely many values of Riemann zeta function at odd integers, inspired by the proof of the Siegel-Shidlovsky theorem on values of E-functions: Shidlovsky's lemma is used …

Arithmetic behaviour of Hecke eigenvalues of Siegel cusp forms of degree two

Let $F$ and $G$ be Siegel cusp forms for $\Sp_4(\Z)$ and weights $k_1, k_2$ respectively. Also let $F$ and $G$ be Hecke eigenforms lying in distinct eigen spaces. Further suppose that neither $F$ nor $G$ is a Saito-Kurokawa lift. In this article, we study simultaneous arithmetic behaviour of Hecke eigenvalues of these Hecke eigenforms.

On the pre-image of a point under an isogeny and Siegel's theorem

Consider a rational point on an elliptic curve under an isogeny. Suppose that the action of Galois partitions the set of its pre-images into n orbits. It is shown that all such points above a certain height have their denominator divisible by at least n distinct primes. This generalizes Siegel's theorem and more recent results of Everest et al. For multiplication by a prime l, it is shown that if n>1 then either the point is l times a rational point or the elliptic curve emits a rational l-isogeny.

A dynamical Borel-Cantelli lemma via improvements to Dirichlet's theorem

Let $X\cong \operatorname{SL}_2(\mathbb R)/\operatorname{SL}_2(\mathbb Z)$ be the space of unimodular lattices in $\mathbb R^2$, and for any $r\ge 0$ denote by $K_r\subset X$ the set of lattices such that all its nonzero vectors have supremum norm at least $e^{-r}$. These are compact nested subset{s} of $X$, with $K_0 = {\bigcap}_{r}K_r$ being the union of two closed horocycles. We use an explicit second moment formula for the Siegel transform of the indicator functions of squares in $\mathbb R^2$ centered at the origin to derive an asymptotic formula for the volume of sets $K_r$ as $r\to 0$. …

希伯特曲面、模形式与希伯特-维克常数

Hilbert surfaces, modular forms, and Siegel-Veech constants

本文研究了翻译表面中普林姆特征形式定位的希伯特-维克常数，证明其在不同定位间一致，并揭示了相关希伯特模曲面的欧拉特征关系。

Chowla猜想与Landau-Siegel零点

The Chowla conjecture and Landau-Siegel zeroes

本文在Landau-Siegel零点存在的条件下，对Chowla猜想中的相关和式建立了非平凡界，改进了先前结果。

由具有有限不同素因子数的有理数进行内在迪奥菲安逼近

Intrinsic Diophantine approximation by rationals of height with a bounded number of distinct prime factors

本文研究了由具有有限不同素因子数的有理数逼近的自相似集的Hausdorff维数，修正了前一版本中引理3.8的证明错误。

Exponential Sums and Distinct Points on Arcs

Suppose that some harmonic analysis arguments have been invoked to show that the indicator function of a set of residue classes modulo some integer has a large Fourier coefficient. To get information about the structure of the set of residue classes, we then need a certain type of complementary result. A solution to this problem was given by Gregory Freiman in 1961, when he proved a lemma which relates the value of an exponential sum with the distribution of summands in semi-circles of the unit circle in the complex plane. Since then, Freiman's Lemma has been extended by several authors. Rathe…

Étude du cas rationnel de la théorie des formes linéaires de logarithmes. (French) [Study of the rational case of the theory of linear forms in logarithms]

We establish new measures of linear independence of logarithms on commutative algebraic groups in the so-called \emph{rational case}. More precisely, let k be a number field and v_{0} be an arbitrary place of k. Let G be a commutative algebraic group defined over k and H be a connected algebraic subgroup of G. Denote by Lie(H) its Lie algebra at the origin. Let u\in Lie(G(C_{v_{0}})) a logarithm of a point p\in G(k). Assuming (essentially) that p is not a torsion point modulo proper connected algebraic subgroups of G, we obtain lower bounds for the distance from u to Lie(H)\otimes_{k} C_{v_{0}…

Comments on Y. O. Hamidoune's Paper "Adding Distinct Congruence Classes"

The main result in Y.~O.~Hamidoune's paper "Adding Distinct Congruence Classes" ({\em Combin.~Probab.~Comput.}~{\bf 7} (1998) 81-87) is as follows: If $S$ is a generating subset of a cyclic group $G$ such that $0 \not \in S$ and $|S| \geq 5$, then the number of sums of the subsets of $S$ is at least $\min (|G|, 2|S| )$. Unfortunately, argument of the author, who, sadly, passed away in 2011, relies on a lemma whose proof is incorrect; in fact, the lemma is false for all cyclic groups of even order. In this short note we point out this mistake, correct the proof, and discuss why the main result …

用于 ${\cal N}=4$ 计数的 Siegel 模形式的 Rademacher 展开

Rademacher expansion of a Siegel modular form for ${\cal N}= 4$ counting

本文利用 Igusa 尖点 Siegel 模形式 $Φ_{10}$ 的辛对称性，构造了细粒度的 Rademacher 型展开，将杂化弦理论中 $1/4$ BPS 态简并度表示为极点留数的正则化求和，并揭示了其与 $1/η^{24}$ 傅里叶系数的连分数结构关系。

具有移位素数差的集合的多项式进展定量界

Quantitative bounds for sets lacking polynomial progressions with shifted prime difference

该研究证明了移位参数限制在移位素数集合的多项式进展的定量Szemerédi型定理，改进了线性情形结果，不同次数非线性构型的定量界为首创，结合了整数多项式构型与素数Gowers一致性界等方法。

每个准完全数至少有八个不同的素因子

Every quasiperfect number has at least eight distinct prime factors

本文通过三个中学代数引理消除了准完全数下界的阻碍，结合机器计算首次完整证明准完全数若存在则至少有8个不同素因子，填补了此前研究的覆盖缺口。

凸范数的五距离定理

The Five Distance Theorem For An Arbitrary Norm

该论文证明了二维严格凸范数下，幺模格上克罗内克序列的最近邻距离数至多为五，解决了对应ℓₚ范数的猜想界，扩展了相关格论论证。