中文 | 首页

    2025 Workshop on Automorphic Forms and Representations

    2025-12-12   to   2025-12-15


    2025-12-12 Check-in & Registration
    2025-12-13 Morning Session Chair: Zhao Xu           Conference Room S106 at Experiment Building
    09:00-09:50 Yongxiao Lin
    Shandong University    		 Levinson's method with a short mollifier Abstract
    09:50-10:10                      Tea Break
    10:10-11:00 Rufei Ren
    Fudan University    		 The Ghost Conjecture on Irreducible Local Representations Abstract
    11:10-12:00 Jingsong Chai
    Anhui Polytechnic University    		 On parabolic induction and Jacquet module over $p$-adic group Abstract
    Afternoon Session Chair: Yongqi Feng           Conference Room S106 at Experiment Building
    14:30-15:20 Caihua Luo
    The Chinese University of Hong Kong (Shenzhen)    		 Socle Filtration and Intertwining Operator Abstract
    15:20-15:40                      Tea Break
    15:40-16:30 Yisheng Tian
    Harbin Institute of Technology    		 On cohomological invariants of algebraic groups Abstract
    16:40-17:30 Dongming She
    Beijing Yanqi Lake Institute of Applied Mathematics    		 Coperiods and central symmetric cube $L$-values Abstract
    2025-12-14 Morning Session Chair: Jianing Li           Conference Room S106 at Experiment Building
    09:00-09:50 Shanwen Wang
    Renmin University of China    		 The $p$-adic Eichler-Shimura map Abstract
    09:50-10:10                      Tea Break
    10:10-11:00 Yujiao Jiang
    Shandong University    		 On Hypothesis H of Rudnick and Sarnak Abstract
    11:10-12:00 Shenxing Zhang
    Hefei University of Technology    		 含非同余数因子的非同余数 Abstract
    Afternoon Session Free Discussions
    2025-12-15 Departure

    Abstracts

    Yongxiao Lin - Levinson's method with a short mollifier

    This is joint with Brian Conrey, David Farmer, Chung-Hang Kwan, and Caroline Turnage-Butterbaugh. When studying the zeros of Riemann zeta function at a height $T$ up the critical strip one often multiplies zeta by a Dirichlet polynomial, called a mollifier, of length $T^\theta$ before averaging in order to neutralize the irregularities of zeta. Levinson in his 1974 Advances paper famously proved that at least 1/3 of the zeros of zeta are on the critical line, by using a mollifier of length $T^\theta$ with $\theta<1/2$. Significant efforts in the literature have been devoted to refine and optimize Levinson's mollifer. We prove that Levinson's method, as modified by Conrey, will in fact produce a positive proportion of critical zeros, regardless how short the mollifier is.

    Rufei Ren - The Ghost Conjecture on Irreducible Local Representations

    The Ghost Conjecture was jointly proposed by Bergdall and Pollack. When the local representation is reducible and generic, this conjecture has been resolved by Liu Ruochuan, Xiao Liang, Zhao Bin, and Nha Truong. In this talk, I will present our progress on the Ghost Conjecture under the condition of irreducible representations, particularly focusing on the construction of ghost series in this case. This is a joint work with Zhao Bin.

    Jingsong Chai - On parabolic induction and Jacquet module over $p$-adic group

    In this talk, we will use tensor product with appropriate bimodules over Hecke algebras to discuss parabolic induction and Jacquet module. We will also discuss a similar construction in local theta correspondence.

    Caihua Luo - Socle Filtration and Intertwining Operator

    Analogous to the celebrated Birch and Swinnerton-Dyer conjecture (BSD conjecture for short), relating the rank of rational points of an elliptic curve and the vanishing order of the corresponding $L$-functions, we uncover a local BSD-type phenomenon linking the lengths of socle filtrations of an induced representation to the vanishing orders of the associated normalized standard intertwining operator, i.e., local $L$-functions.

    Yisheng Tian - On cohomological invariants of algebraic groups

    In this talk, we first recall definitions and basic properties of cohomological invariants of algebraic groups. Subsequently, we use cohomological invariants to construct a so-called versal torsor. Finally, we introduce a comparison of cohomological obstructions via versal torsor over $p$-adic function fields. This is based on a joint work in progress with CAO Yang (曹阳).

    Dongming She - Coperiods and central symmetric cube $L$-values

    The deep and intrinsic connections between period integrals and special values of automorphic $L$-functions form a central theme in modern automorphic representation theory. In this talk, I will discuss how the central symmetric cube $L$-values of automorphic representations of $\mathrm{GL}(2)$ are related to certain coperiod integrals associated to the cubic Kazhdan-Patterson covers of $\mathrm{GL}(2)$. I will formulate the corresponding GGP-type conjecture and discuss the Ichino-Ikeda type formula for Eisenstein periods. This is a joint work with Li Cai and Yangyu Fan.

    Shanwen Wang - The $p$-adic Eichler-Shimura map

    Using the CdR conjecture for modular forms, we give an explicit description of the $p$-adic Eichler-Shimura map. This talk is based on joint work with Pierre Colmez.

    Yujiao Jiang - On Hypothesis H of Rudnick and Sarnak

    The generalized Ramanujan conjecture (GRC) is one of the foundational problems in modern number theory. Hypothesis H can often substitute for the GRC in analytic applications. In this talk, we will discuss our work that proves this hypothesis, establishing it in its full generality and with a bound that is stronger than its original formulation. This result has a number of immediate applications to other central problems in analytic number theory.

    Shenxing Zhang - 含非同余数因子的非同余数

    我们将回顾非同余数 $n$ 对应的同余椭圆曲线 $E_n:y^2=x^3-n^2x$ 的2-Selmer 秩达到第二小情形的已知结果, 并介绍新的构造此类非同余数的方法.