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    2023年数论与表示论研讨会

    2023-12-08   至   2023-12-11


    2023-12-08 签到
    2023-12-09 上午会场 主持人: 许宾           厦门大学海韵园实验楼105
    09:00-09:50 黄炳荣
    山东大学    		 Moments of quadratic twisted $L$-functions 摘要
    10:00-10:50 吕恒飞
    北京航空航天大学    		 Multiplicity one theorem for symmetric pairs 摘要
    10:50-11:10                      茶歇
    11:10-12:00 曹炜
    闽南师范大学    		 Zeros of Polynomials over Finite Witt Rings 摘要
    下午会场 主持人: 林永晓           厦门大学海韵园实验楼105
    14:30-15:20 齐治
    浙江大学    		 Bessel 积分公式在数论中的应用 摘要
    15:30-16:20 王好武
    武汉大学    		 Polynomial rings of modular forms on symmetric domains 摘要
    16:20-16:40                      茶歇
    16:40-17:30 于树澄
    中国科学技术大学    		 Some quantitative results in Diophantine approximation 摘要
    2023-12-10 上午会场 主持人: 张翀           厦门大学海韵园实验楼105
    09:00-09:50 王标
    云南大学    		 Modern generalizations and analogues of the prime number theorem in dynamical systems 摘要
    10:00-10:50 栗慧曦
    南开大学    		 Covering systems and the minimum modulus problem 摘要
    10:50-11:10                      茶歇
    11:10-12:00 段炼
    上海科技大学    		 On the irreducibility of low-dimensional geometric Galois representations 摘要
    下午会场 自由讨论
    2023-12-11 离会

    摘要

    黄炳荣 - Moments of quadratic twisted $L$-functions

    In this talk, we will discuss some results on moments of quadratic twisted $L$-functions. As applications, we will talk about nonvanishing and extreme values of $L$-functions, lower bounds for higher moments, and determination of cusp forms by central $L$-values. This is based on joint works with Shenghao Hua.

    吕恒飞 - Multiplicity one theorem for symmetric pairs

    It is known that the symmetric pair $(G,H)=(\mathrm{GL}(p+q),\mathrm{GL}(p)\times \mathrm{GL}(q))$ is a Gelfand pair due to Aizenbud-Gourevitch, i.e. for any irreducible representation $\pi$ of $G(F)$, its restriction to $H(F)$ contains the trivial representation as the quotient with multiplicity at most 1. Furthermore, Chen and Sun show that if $p=q$, its restriction to $H(F)$ contains any character with mulitplicity at most 1 except for countable many characters (with only finite exceptions when $F$ is $p$-adic). We will use the theta correspondence to show that if $F$ is $p$-adic, $\pi$ is generic, then it still holds. Then we may talk about the case $(\mathrm{GL}(2n,F),\mathrm{GL}(n,E))$ with any character on $E^*$ where $E/F$ is a quadratic field extension.

    曹炜 - Zeros of Polynomials over Finite Witt Rings

    Let $\mathbb{F}_q$ denote the finite field of characteristic $p$ and order $q$. Let $\mathbb{Z}_q$ denote the unramified extension of the $p$-adic rational integers $\mathbb{Z}_p$ with residue field $\mathbb{F}_q$. Given two positive integers $m,n$, define a box $\mathcal B_m$ to be a subset of $\mathbb{Z}_q^n$ with $q^{nm}$ elements such that $\mathcal B_m$ modulo $p^m$ is equal to $(\mathbb{Z}_q/p^m \mathbb{Z}_q)^n$. For a collection of nonconstant polynomials $f_1,\dots,f_s\in \mathbb{Z}_q[x_1,\ldots,x_n]$ and positive integers $m_1,\dots,m_s$, define the set of common zeros inside the box $\mathcal B_m$ to be $$V=\{X\in \mathcal B_m:\; f_i(X)\equiv 0\mod {p^{m_i}}\mbox{ for all } 1\leq i\leq s\}.$$ It is an interesting problem to give the sharp estimates for the $p$-divisibility of $|V|$. This problem has been partially solved for the three cases: (i) $m=m_1=\cdots=m_s=1$, which is just the Ax-Katz theorem, (ii) $m=m_1=\cdots=m_s>1$, which was solved by Katz, Marshal and Ramage, and (iii) $m=1$, and $ m_1,\dots,m_s\geq 1$, which was recently solved by Cao, Wan and Grynkiewicz. Based on the multi-fold addition and multiplication of the finite Witt rings over $\mathbb{F}_q$, we investigate the remaining unconsidered case of $m>1$ and $m\neq m_j$ for some $1\leq j\leq s$, and finally provide a complete answer to this problem.

    齐治 - Bessel 积分公式在数论中的应用

    首先我会回顾几个经典的 Bessel 积分公式及其在诸如 Waldspurger 公式，Beyond Endoscopy，Motohashi 公式，Kuznetsov 公式等数论问题与公式中的应用，然后我会介绍最近证明的复数域上的 Bessel 积分公式以及 Gauss 数域上的 Bruggeman-Motohashi 和 Kuznetsov-Motohashi 公式。

    王好武 - Polynomial rings of modular forms on symmetric domains

    It is an important problem in the theory of modular forms to determine the structure of rings of modular forms, that is, to find explicit generators and their relations. In this talk, I will introduce the modular Jacobian approach to construct and classify the arithmetic groups acting on type IV symmetric domains and complex balls, for which the rings of modular forms are freely generated. This talk is based on joint work with Brandon Williams.

    于树澄 - Some quantitative results in Diophantine approximation

    Siegel transform is a classical transform in geometry of numbers taking functions on a Euclidean space to functions on the space of lattices. In this talk we describe a general strategy of obtaining some quantitative results in Diophantine approximation (especially some Khintchine-type results) using moment formulas of some generalized Siegel transforms. This is based on several joint works with Mahbub Alam, Anish Ghosh and Dubi Kelmer.

    王标 - Modern generalizations and analogues of the prime number theorem in dynamical systems

    In this talk, we will introduce some recent developments on the prime number theorem (PNT), including Bergelson and Richter's dynamical generalizations of the PNT and Kanigowski-Lemańczyk-Radziwiłł's work on the PNT for analytic skew products.

    栗慧曦 - Covering systems and the minimum modulus problem

    In this presentation, we will first introduce covering systems and the minimum modulus problem, then we will talk about some recent results about them in different settings.

    段炼 - On the irreducibility of low-dimensional geometric Galois representations

    Given an $\ell$-adic Galois representation, a basic question to ask is if this representation is irreducible or not. Usually, this is not easy to answer. However, when this representation arises from a geometric source, we can apply some known results from the study of such kinds of representations to give at least certain criteria to guarantee its irreducibility. In this talk, we will introduce one of such criteria for three-dimensional self-dual geometric representations. We will give an application of our criterion by verifying the Tate conjecture for a specific family of elliptic surfaces of genus three. If time allows, we will roughly introduce an ongoing work on five-dimensional representations. This is a joint work with Xiyuan Wang and Ariel Weiss.