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    2023 Workshop on Number Theory and Representation Theory

        2023-12-08    to    2023-12-11


    2023-12-08 Check-in & Registration
    2023-12-09 Morning Session Chair: Bin Xu           Conference Room 105 at Experiment Building
    09:00-09:50 Bingrong Huang
    Shandong University    		 Moments of quadratic twisted $L$-functions Abstract
    10:00-10:50 Hengfei Lu
    Beihang University    		 Multiplicity one theorem for symmetric pairs Abstract
    10:50-11:10                      Tea Break
    11:10-12:00 Wei Cao
    Minnan Normal University    		 Zeros of Polynomials over Finite Witt Rings Abstract
    Afternoon Session Chair: Yongxiao Lin           Conference Room 105 at Experiment Building
    14:30-15:20 Zhi Qi
    Zhejiang University    		 Bessel  积分公式在数论中的应用 Abstract
    15:30-16:20 Haowu Wang
    Wuhan University    		 Polynomial rings of modular forms on symmetric domains Abstract
    16:20-16:40                      Tea Break
    16:40-17:30 Shucheng Yu
    University of Science and Technology of China    		 Some quantitative results in Diophantine approximation Abstract
    2023-12-10 Morning Session Chair: Chong Zhang           Conference Room 105 at Experiment Building
    09:00-09:50 Biao Wang
    Yunnan University    		 Modern generalizations and analogues of the prime number theorem in dynamical systems Abstract
    10:00-10:50 Huixi Li
    Nankai University    		 Covering systems and the minimum modulus problem Abstract
    10:50-11:10                      Tea Break
    11:10-12:00 Lian Duan
    ShanghaiTech University    		 On the irreducibility of low-dimensional geometric Galois representations Abstract
    Afternoon Session Free Discussions
    2023-12-11 Departure

    Abstracts

    Bingrong Huang - 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.

    Hengfei Lu - 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.

    Wei Cao - 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.

    Zhi Qi - 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。

    Haowu Wang - 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.

    Shucheng Yu - 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.

    Biao Wang - 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.

    Huixi Li - 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.

    Lian Duan - 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.