数论与表示论会议IX, Symposium on Number Theory and Representation Theory, IX

Symposium on Number Theory and Representation Theory, IX

数论与表示论会议, IX

2025年6月14日至15日

题目及摘要

时间：2025年6月14日，9：00-17:30；6月15日，9：00-12：00

地点：杭州西轩酒店

报告人：吴涵（中国科学技术大学）

题  目：On a Generalization of Motohashi's formula

摘  要：Spectral reciprocities are equalities between moments of automorphic $L$-functions in different families. They are powerful tools for the study of the moment problem and the subconvexity problem. The first spectral reciprocity formula is Motohashi's formula, which relates the cubic moment of $L$-functions for $\GL_2$ with the fourth moment of $L$-functions for $\GL_1$. The exploitation of this formula (over $\mathbb{Q}$) has led Conrey-Iwaniec (Ann. of Math. 2000) and Petrow-Young (Ann. of Math. 2020, Duke Math. J. 2022) to the uniform Weyl bound for all Dirichlet $L$-functions, a celebrated recent result.

 In a talk with the same title from last year, we presented an adelic version of a generalization of Motohashi's formula relating $\GL_3 \times \GL_2$ with $\GL_3 \times \GL_1$ and $\GL_1$ moments of $L$-functions, and announced a local non-archimedean weight estimation on the fourth moment side. In this talk, we report some progress towards this estimation with details.

报告人：Andreas Mihatsch（浙江大学）

题  目：Construction of Gaussian test functions

摘  要：In this talk, I will explain a method to construct special functions on real Lie groups: In the setting of the adjoint representation, this method produces explicit Euler-Poincaré function. In the setting of the Jacquet--Rallis relative trace formula, it produces explicit Gaussian test functions. The construction is based on Kudla--Millson theory and stems from joint work with Siddarth Sankaran and Tonghai Yang.

报告人：贺乔（哥伦比亚大学）

题  目：Height pairing on Shimura curve revisited and a general conjecture for GSpin Shimura varieties

摘  要：In their paper "Height pairings on Shimura curves and p-adic uniformization" (Invent, 2000), Kudla and Rapoport studied intersections of special cycles on Shimura curves and related it with derivative of Eisenstein series, which is one of the key ingredient to prove arithmetic inner product formula for Shimura curves (a variant/generalization of Gross-Zagier formula). In this talk, we will revisit Kudla and Rapoport's formula by incorporating it into a general conjecture for the GSpin Shimura variety. As evidence of the conjecture, we also discuss the proof for the self product of Shimura curves case. This is a joint work with Baiqing Zhu.

报告人: 罗渝（威斯康星大学麦迪逊分校）

题  目：A new proof of the arithmetic Siegel-Weil formula

摘  要：The arithmetic Siegel–Weil formula establishes a profound connection between intersection numbers in Shimura varieties and the Fourier coefficients of central derivatives of Eisenstein series. This result was proven by C. Li and W. Zhang in 2021 using local methods. In this talk, I will present a new proof of the formula that uses the local-global compatibility and the modularity of generating series of special divisors.

报告人：李兆林（明尼苏达大学）

题  目：Beyond Endoscopy: Kuznetsov Trace Formula and Standard L-functions of GL_2

摘  要：In this talk, we will apply non-standard test functions to the Kuznetsov trace formula, together with a direct proof of a Poisson summation formula on the level of orbital integrals that is responsible for the local Hankel transform calculated by H. Jacquet, to get the analytic continuation and functional equations of standard L-functions of GL_2.

报告人：王好武（武汉大学）

题  目：Hyperbolization of affine Lie algebras

摘  要：In 1983, Feingold and Frenkel discovered a relation between genus-two Siegel modular forms and a rank-three hyperbolic Kac--Moody algebra extending the affine Lie algebra of type A_1. It inspires a problem to explore more general relations between affine Lie algebras, hyperbolic Kac--Moody algebras and modular forms. In this talk we give an automorphic answer to this problem. We classify hyperbolic Borcherds--Kac--Moody superalgebras whose super-denominators define reflective automorphic products of singular weight on certain orthogonal groups of signature (n,2). As a consequence, we prove that there are exactly 81 affine Lie algebras g which have extensions to hyperbolic BKM superalgebras for which the leading Fourier--Jacobi coefficients of super-denominators coincide with the denominators of g. We find that 69 of them appear in Schellekens' list of semi-simple V_1 structures of holomorphic CFT of central charge 24, while 8 of them correspond to the N=1 structures of holomorphic SCFT of central charge 12 composed of 24 chiral fermions. The last 4 cases are related to exceptional modular invariants from nontrivial automorphisms of fusion algebras. This clarifies the relationship of affine Lie algebras, vertex algebras and hyperbolic BKM superalgebras at the level of modular forms. This is based on a joint paper with Kaiwen Sun and Brandon Williams.

报告人：方金辉（南京师范大学）

题  目：On Sidon sets

摘  要：A Sidon set is a set ξAξ of positive integers with the property that all the sums ξa+a'ξ with ξa,a'\in Aξ, ξa\le a'ξ are distinct. In this talk, we will present our recent results on Sidon sets.

报告人：魏志宁（布朗大学）

题  目：The random matrix theory, the weighted moment conjectures and applciaitons

摘  要：In this talk, I will present our recent work on the weighted moment problems, which has applications in the nonvanishing problems and subconvexity problems. I will further discuss its connections with the random matrix theory and the weighted low-lying zero problems.This talk is part of several joint projects with Liyang Yang and Shifan Zhao.