山大数论讨论班

星期三，15:00-16:00，2022年

线上会议  或者  中心校区明德楼 C704

组织者：山大数论组

讨论班着重于数论中的新进展。报告人为校内外学者和学生。欢迎推荐报告。

报告信息（摘要和往年报告信息见列表下方）

Evening Section: 

• 2022-09-07，  王六权 （武汉大学）

  题目：Sign changes of coefficients of powers of the infinite Borwein product

  摘要：We denote by $c_t^{(m)}(n)$ the coefficient of $q^n$ in the series expansion of $(q;q)_\infty^m(q^t;q^t)_\infty^{-m}$, which is the $m$-th power of the infinite Borwein product. Let $t$ and $m$ be positive integers with $m(t-1)\leq 24$. We  provide asymptotic formula for $c_t^{(m)}(n)$, and give characterizations of $n$ for which $c_t^{(m)}(n)$ is positive, negative or zero. We show that $c_t^{(m)}(n)$ is ultimately periodic in sign and conjecture that this is still true for other positive integer values of $t$ and $m$. Furthermore, we confirm this conjecture in the cases $(t,m)=(2,m),(p,1),(p,3)$ for arbitrary positive integer $m$ and prime $p$.

  地点：腾讯会议ID：358 9878 8585

• 2022-09-14，  沈权利 （University of Lethbridge）

  题目：The eighth moment of the Riemann zeta function

  摘要：The first and second moments of the Riemann zeta function were established asymptotically by Hardy-Littlewood and Ingham, respectively. It is believed that sixth and higher moments are beyond current techniques. Ng recently proved an asymptotic formula for the sixth moment by assuming a conjecture on additive divisor sums. In this talk, we will discuss about how to compute the eighth moment under a conjecture on additive divisor sums and the Riemann hypothesis. It is a joint work with Nathan Ng and Peng-Jie Wong.

  地点：腾讯会议ID：358 9878 8585

• 2022-09-23 (周五), 10:00-11:00,  刘青阳 （中国科学院）

  题目：On Random Multiplicative Functions

  摘要：Random multiplicative functions were first introduced by Wintner as a probabilistic model for the Mobius function and have attracted quite a lot of attention in number theory and probability. In this report, we will introduce the development and recent research of the random multiplicative function theory, and briefly describe the probabilistic methods behind it. Finally, we will present the reporter's contribution in this direction. 

  地点：腾讯会议ID：358 9878 8585

• 2022-09-28，  王梦迪 （KTH Stockholm）

  题目：Linear equations in smooth numbers

  摘要：In this talk, I will introduce an approach to counting system of linear equations with smooth number variables, and this is based on joint work with Lilian Matthiesen. I plan to avoid introducing formal notation in higher-order Fourier analysis, hope this will be accessible to a fairly general audience.

  地点：腾讯会议ID：358 9878 8585

• 2022-10-05， 无报告 （国庆节）

• 2022-10-12， 栗慧曦 （南开大学）

  题目：Two results related to Pan’s work

  摘要：In 1979 Pan proved a new mean value theorem and applied it to prove Chen’s theorem. In the first part of the talk we will apply this theorem to the representation of a large integer as ap + bP_2. In 1982 Pan made a new attempt on the Goldbach conjecture. In the second part of the talk we will discuss how his approach connects the Goldbach conjecture to equidistribution of some arithmetic functions.

  地点：腾讯会议ID：358 9878 8585

• 2022-10-19，  无报告

• 2022-10-26，  吴云辉 （清华大学）    14:00-15:00

  题目：Recent progress on first eigenvalues of hyperbolic surfaces for large genus

  摘要：In this talk we wil discuss several recent results on first eigenvalues of closed hyperbolic surfaces for large genus. For example, we show that a random hyperbolic surface of large genus has first eigenvalue greater than $\frac{3}{16}-\epsilon$, extending Mirzakhani's lower bound $0.0024$. This talk is based on several joint works with Yuhao Xue.

  地点：腾讯会议ID：358 9878 8585

• 2022-10-26，  李加宁 （山东大学）

  题目：Genus理论的一个技巧和应用

  摘要：给数域的循环扩张, Genus理论描述了大域的理想类群Galois不变部分的大小. 它由Gauss发现并用此研究二次域的2理想类群. 本报告将介绍一个构造辅助循环扩张并嵌套使用genus理论的技巧, 并用此证明当素数p =4,7 mod 9时, Q(sqrt[3]{p})的类数恰好被3整除, 这证明了Barrucand-Cohn, Lemmermeyer等人关于纯三次域类群和单位群的猜想. 这是与张神星的合作工作. 

  地点：腾讯会议ID：358 9878 8585

• 2022-11-02，  安金鹏 （北京大学）

  题目：丢番图逼近中的奇异性与齐性动力系统

  摘要：对于一个实数，如果Dirichlet逼近定理中的常数可以任意小，则称该实数具有奇异性。容易证明，这样的实数即为有理数，所以这一定义并无实质意义。但是，对向量或矩阵引入类似的定义后，人们发现奇异向量或奇异矩阵的集合具有丰富的分形性质。报告人将探讨具有奇异性的数、向量和矩阵的性质及其与齐性动力系统的关系，并介绍在联立奇异性方面与人合作的一项工作。

  地点：腾讯会议ID：358 9878 8585

• 2022-11-09，  肖梁  （北京大学）

  题目：模形式的p进斜率及相关问题

  摘要：模形式是朗兰兹纲领中的重要研究对象之一，一方面它与椭圆曲线和伽罗华表示的算术性质紧密联系，另一方面它固有的解析性质为研究椭圆曲线提供了巨大的帮助。这个报告将首先介绍模形式相关的背景和结果之后慢慢引入模形式的p进性质的研究，特别是对特征模形式系数被p整除次数（即p进斜率）的性质的研究。这一类研究始于90年代Gouvea和Mazur大量的数值计算，他们对特征模形式的p进斜率提出若干猜想。之后Coleman-Mazur, Buzzard-Calegari, Bergdall-Pollack逐步从理论上完善关于p进斜率的这些猜想。如果时间允许，我会汇报在这个方面与刘若川、万大庆、赵斌、Nha Truong的系列合作，在某些情况下证明关于p进斜率的这些猜想。

  地点：腾讯会议ID：358 9878 8585

• 2022-11-16，  张涵  （清华大学）

  题目：Nondivergence of reductive group actions on homogeneous spaces

  摘要：Let $G/\Gamma$ be the quotient of a semisimple Lie group $G$ by its arithmetic lattice. Let $H$ be a reductive algebraic subgroup of $G$ acting on $G/\Gamma$. The question we are interested in is whether there is a compact set of $G/\Gamma$ that intersects every H-orbit.

  We show that the failure of this can be explained by a single algebraic reason, which generalizes several previous results towards this question. We also obtain a way to find this algebraic obstruction, if there is any. 

  This talk is based on joint work with Runlin Zhang.

  地点：腾讯会议ID：358 9878 8585

• 2022-11-23，   刘若川 （北京大学）

  题目：p进簇的黎曼-希尔伯特问题

  摘要：我们将介绍p进簇的黎曼-希尔伯特问题及一些相关的进展。

  地点：腾讯会议ID：358 9878 8585

• 2022-11-23，  Ritabrata Munshi   (Indian Statistical Institute)   19:00-20:00
  

  Title:  GL(3) L-functions on the critical line

  Abstract:  Traditionally moments of L-functions on the critical line have been used to bound the size of the L-functions on the critical line. For example the Weyl bound for the Riemann Zeta function can be concluded either by obtaining a strong bound for the error term in the second moment or by bounding short fourth moment. The same method works in the case of degree two L-functions, although computing the moment becomes a tricky business. Till recently we had no similar result for degree three (or higher) L-functions. This talk will be about t-aspect sub convexity for degree three L-functions - the delta method approach, its limitations and the recent blending of the moment method and the delta method (a joint work with Aggarwal and Leung). 

  Venue: Zoom: 825 9309 8083, password: 2022

• 2022-11-30，   马蔷   （山东大学）  15:00-16:00

  题目：Exponential sums with the Dirichlet coefficients of Rankin-Selberg L-functions

  摘要：In this talk, we shall introduce our recent work concerning cancellation  inadditively twisted sums  on $\mathrm{GL}(m)\times\mathrm{GL}(m^{\prime})$. We describe a new method to obtain upper bounds for exponential sums with multiplicative coefficients without the Ramanujan conjecture. We verify these hypothesis for (with mild restrictions) the Rankin-Selberg $L$-functions attached to two cuspidal automorphic representations. 

  地点：腾讯会议ID：358 9878 8585

• 2022-11-30，  Péter Maga   (Alfréd Rényi Institute of Mathematics)   20:00-21:00
  

  Title: On the sup-norm problem of automorphic forms: new counting methods

  Abstract: As it was proved by Sarnak, the supnorm of eigenfunctions of the Laplacian on a compact symmetric Riemannian manifold can be estimated from above by an appropriate power (given in terms of some invariants of the space) of their Laplace eigenvalue. Examples show that Sarnak's exponent is sharp in some cases. However, when the space has also arithmetic symmetries (i.e. Hecke operators) and we restrict to joint eigenfunctions of the Laplacian and the Hecke operators, one might expect a better exponent. Such power-savings are known for arithmetic quotients of several symmetric spaces. In the talk, I will give a brief overview of the history of the problem, and discuss some new matrix counting methods which might potentially yield the desired power-saving in some currently unsettled cases. The talk will be based on ongoing research (joint with Gergely Zábrádi).

  Venue: Zoom: 825 9309 8083, password: 2022
  

• 2022-12-07，   曹炜   （闽南师范大学）

  题目：Divisibility on point counting over finite Witt rings

  摘要：Let F_q denote the finite field of q elements with characteristic p. Let Z_q denote the unramified extension of the p-adic integers Z_p with residue field F_q. In a joint work with Prof. Daqing Wan, we study the q-divisibility for the number of solutions of a polynomial system in n variables over the finite Witt ring Z_q/p^mZ_q, where the n variables of the polynomials are restricted to run through a combinatorial box lifting F_q^n. We prove a q-divisibility theorem for any box of low algebraic complexity, including the simplest Teichmuller box. This extends the classical Ax-Katz theorem over finite field F_q (the case m=1). Taking q=p to be a prime, our result extends and improves a recent combinatorial theorem of Grynkiewicz. Our different approach is based on the addition operation of Witt vectors and is conceptually much more transparent.

  地点：腾讯会议ID：358 9878 8585

• 2022-12-14，   Sheng-Chi Liu  (Washington State University)   13:00-14:00

  题目：A GL(3) Analog of Selberg's Result on S(t)

  摘要：The function S(t) appears in an asymptotic formula for counting the number of nontrivial zeros of the Riemann zeta function with imaginary part less than t. It was shown by Littlewood that the function has a lot of cancellation on the average over t. Later Selberg studied the moments of S(t) and the moments of the analog function associated with a Dirichlet L-function for a primitive Dirichlet character. A GL(2) analog of Selberg's result was proved by Hejhal and Luo. In this talk we will discuss a GL(3) analog of such results. This is joint work with Shenhui Liu.

  地点：Zoom: 825 9309 8083, password: 2022

• 2022-12-14，  Min Lee   (Bristol University)   20:00-21:00
  

  Title:  Non-vanishing of symmetric cube L-functions

  Abstract:  The non-vanishing of L-series at the centre of the critical strip has long been a subject of great interest. An important example of the significance of non-vanishing is in the case of an L-series corresponding to a modular form of weight 2, where the non-vanishing at the central point has been shown to be equivalent to the finiteness of the group of rational points of the associated elliptic curve. In the case of higher rank L-functions whose Euler product has an even degree, such connections between non-vanishing at the central point and the finiteness of certain groups are believed to be true, but the relations remain purely conjectural. The symmetric cube L-series plays a role in one of these conjectures.

  Ginzburg, Jiang and Rallis (2001) proved that the non-vanishing at the central point of the critical strip of the symmetric cube L-series of any GL(2) automorphic form is equivalent to the non-vanishing of a certain triple product integral. The main purpose of this talk is to use this equivalence to prove that there are infinitely many Maass-Hecke cuspforms over the imaginary quadratic field of discriminant -3 such that the central values of their symmetric cube L-functions do not vanish.

  This is joint work with Jeff Hoffstein and Junehyuk Jung. 

  Venue: Zoom: 825 9309 8083, password: 2022

• 2022-12-21，   郑维喆  （中科院）

  题目：平展上同调和超积

  摘要：平展上同调的超积提供了代数簇的一族Weil上同调理论，其性质与ℓ进上同调的ℓ无关性和无挠性密切相关。本报告将介绍超积上同调研究的新进展。

  地点：腾讯会议ID：358 9878 8585

• 2022-12-21， Lior Bary-Soroker   (Tel Aviv University)     20:00-21:00
  

  Title:  Rational points coming from ramified covers

  Abstract:  The talk aims to present recent progress in the area Hilbert's irreducibility theorem.

  The theorem may be formulated as the statement there are "many" rational points on the *line* not coming from rational points on a given cover. If one replaces the line by other varieties, the situation becomes more complicated and, in particular, there are obstructions to the theorem coming from rational points of unramified covers (recall that the line is simply connected).

  I will present recent two results: The first result, j/w Daniele Garzoni, deals with affine groups: We show that a random walk on a finitely generated Zariski dense subgroup almost surely misses rational points coming from a ramified cover. The second result, j/w Arno Fehm and Sebastian Petersen, deals with abelian varieties and with rational points over the maximal cyclotomic field (or more generally, the field obtained by adding the torsion points of an abelian variety).

  Venue: Zoom: 825 9309 8083, password: 2022

• 2022-12-26，  Rizwanur Khan  (University of Mississippi)

  Title: The fourth moment of Dirichlet L-functions and related problems

  Abstract: I will discuss asymptotics for the fourth moment of Dirichlet L-functions and related problems, especially with regards to simplifying existing approaches and sharpening the error terms in these asymptotics. 

  Venue: Zoom: 825 9309 8083, password: 2022

往年报告：

• 2022年春季报告  

• 2021年秋季报告   2021年春季报告

• 2020年报告

Contact us at: brhuang "at" sdu dot edu dot cn,   Office: Mingde C701,   Tel: 883-69786