Topics in automorphic Forms and Algebraic Cycles

该奖项涉及数论方面的研究。求解有理数的多项式方程可以追溯到3世纪的丢番图，几代人以来一直是数学的中心课题。丢番图方程的现代研究融合了黎曼的革命性思想，他使用了一类特殊的函数，称为“Zeta函数”或“L函数”。这种特殊的函数建立在计算多项式方程的解的个数的基础上，模运算的设置要简单得多。在20世纪60年代，Birch和Swinnerton-Dyer提出了一个引人注目的猜想，揭示了L函数的零点与一类特殊的有理多项式方程的解之间的关系。后来，Beilinson和Bloch猜想，对于一般的有理多项式方程，L函数的零点与“多项式方程的参数解”的代数圈之间总有一种关系。本课题将通过模簇上的自同构形式和特殊循环来研究L函数的零点。自同构型理论为L函数的零点求法提供了一条卓有成效的途径。模簇要么是数域上的Shimura簇，要么是函数域上的Shtukas模空间。它们在现代数论和算术几何中扮演着核心角色，而且它们经常伴随着大量的代数循环。该项目旨在证明关于模簇的L函数的零点和代数圈的结果，包括关于与酉群相关的Shimura簇的算术Gan-Gross-Prasad猜想的新情况，函数域上的某些高阶Gross-Zagier公式，以及Kudla程序的函数域模拟，重点是生成一系列特殊圈的模性和算术Siegel-Weil公式。该项目还将开发新的相对迹公式，这是一个连接光谱信息和几何结构的强大方程，用于研究包括酉Friedberg-Jacquet周期在内的一般自同构周期积分。该项目的更广泛的影响包括对研究生的指导和研讨会的组织。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

This awards concern research in Number Theory. Solving polynomial equations in rational numbers dates back to Diophantus in the 3rd century and has been a central subject in mathematics for generations. The modern study of Diophantine equations has incorporated the revolutionary idea of Riemann from his use of a class of special functions called "zeta functions” or "L-functions". Such special functions are built up on counting the numbers of solutions of polynomial equations in the much simpler setting of modular arithmetic. In the 1960s, Birch and Swinnerton-Dyer came up with a remarkable conjecture revealing a relation between the zeros of L-functions and the solutions to a special class of polynomial equations in the rationals. Later Beilinson and Bloch conjectured that, for general polynomial equations in the rationals, there should always be a relation between the zeros of L-functions and algebraic cycles which are “parameter solutions to polynomial equations”. The project will study the zeros of L-functions through automorphic forms and special cycles on modular varieties. The theory of automorphic form provides a fruitful way to access the zeros of L-functions. The modular varieties are either Shimura varieties over number fields or moduli spaces of Shtukas over function fields. They play a central role in modern number theory and arithmetic geometry, and they often come with a great supply of algebraic cycles. The project aims to prove results relating zeros of L-functions and algebraic cycles on modular varieties, including new cases of the arithmetic Gan–Gross–Prasad conjecture for Shimura varieties associated to unitary groups, certain Higher Gross–Zagier formula over function fields, and the function field analog of Kudla’s program with an emphasis on the modularity of generating series of special cycles and the arithmetic Siegel—Weil formula. The project will also develop new relative trace formula, a powerful equation connecting spectral information and geometric structure, to study general automorphic period integral including the unitary Friedberg–Jacquet period. The broader impacts of this project include mentoring of graduate students and seminar organization.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.