The Arithmetic of Automorphic Forms

数学领域被称为数论，涉及理解多项式方程的整数和有理解。这些解集在几何上与所谓的自守形式（automorphic forms）相连，自守形式是三角函数正弦和余弦函数的高维类似物。像正弦函数和余弦函数一样，自守形式是满足某些微分方程并具有无限多个离散对称的函数，它们本身就是数学研究的对象，而不仅仅是因为它们与多项式方程的联系。PI将研究自守形式的研究主题，特别是那些自守形式的对称系统是“例外”的。“PI还将研究自守形式的L函数中的主题。L函数是Riemann zeta函数的推广，并且在理论上包含大量微妙的算术信息。更详细地说，这个项目有两个不同的重点领域。第一个问题是关于特殊群上一类特殊自守形式的意外算术性。有证据表明，例外群上的（非全纯）“模形式”的行为与经典的全纯模形式惊人地相似，并且具有惊人的算术特征。本项目旨在进一步发展例外群上模形式的理论，例如发展可用于产生例外群G_2上模形式数据库的数学。这个项目的第二个重点关注的工作符合贝林森的猜想的特殊价值的L-功能的动机。在这个方向上的努力涉及获得调节器公式广义贝林森-Flach动机类，并找到一个推广的克罗内克极限公式。该项目涉及的技术和思想包括特殊的θ对应关系、Rankin-Selberg方法和Deligne上同调。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

The area of mathematics known as number theory concerns understanding integer and rational solutions to polynomial equations. These solution sets are conjecturally connected to what are called automorphic forms, high-dimensional analogues of the trigonometric sine and cosine functions. Like the sine and cosine functions, automorphic forms are functions that satisfy certain differential equations and have infinitely many discrete symmetries, and they are objects of intense mathematical study in their own right, not just for their connection to polynomial equations. The PI will investigate topics in the study of automorphic forms, especially those automorphic forms whose system of symmetries is "exceptional." The PI will also investigate topics in the L-functions of automorphic forms. L-functions are generalizations of the Riemann zeta function, and conjecturally contain large amounts of subtle arithmetic information.In more detail, this project has two distinct areas of focus. The first concerns unexpected arithmeticity in a class of special automorphic forms on exceptional groups. There is evidence that (non-holomorphic) "modular forms" on exceptional groups behave surprisingly similarly to classical holomorphic modular forms and possess surprising arithmetic features. This project aims to further develop the theory of these modular forms on exceptional groups, such as developing the mathematics that could be used to produce a database of modular forms on the exceptional group G_2. The second focus of this project concerns work consistent with Beilinson's conjecture about the special values of L-functions of motives. Efforts in this direction involve obtaining regulator formulas for generalized Beilinson-Flach motivic classes and finding a generalization of the Kronecker limit formula. The techniques and ideas involved in the project include exceptional theta correspondences, the Rankin-Selberg method, and Deligne cohomology.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.