P-adic Aspects of L-Values, Congruences Between Automorphic Forms, and Arithmetic Applications

在研究数论中的许多基本问题时，例如素数的分布和一些多项式方程的有理解，复平面上的一个函数，称为l函数，一直与所研究的算术对象联系在一起。我们相信l函数的各种性质，并在许多特殊情况下证明了它们用于编码深度算术信息。本课题主要研究l函数的特殊值之间的同余，以及具有许多对称性的多变量函数自同构形式之间的同余。利用这些同余来研究l函数的特殊值与某些特殊情况下算术上同群的大小之间的关系。在这个项目中，一个方面是研究与对应相关的自同构形式之间p的模幂的同余。这包括利用l函数的合适的积分表示和l值的自同构表示来研究某些提升的p进性质，构造某些p进l函数和构造提升的Hida族。Iwasawa- Greenberg主要猜想预测了p进l函数和附在伽罗瓦表示的p进变形上的Selmer群之间的联系。本课题的另一个方面是利用同余的结果，在权值相同宇称的模形式对的Rankin—Selberg积的特殊情况下研究这一猜想。该奖项反映了美国国家科学基金会的法定使命，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。

In the study of many fundamental problems in number theory, for example distribution of prime numbers and rational solutions to some polynomial equations, a function on the complex plane, called the L-function, has been associated with the arithmetic object under study. Various properties of the L-functions are believed and have been proved in many special cases to encode deep arithmetic information. This research project is mainly concerned with congruences between special values of L-functions, as well as congruences between automorphic forms, which are multi-variable functions with many symmetries. This project also aims at applying these congruences to study the connection between the special values of L-functions and the sizes of arithmetic cohomology groups in some special cases. In this project, one aspect is the study of the congruences modulo powers of p between automorphic forms related to theta correspondence. This includes studying the p-adic properties of certain theta lifts by using suitable integral representations of L-functions and L-values for automorphic representations, constructing certain p-adic L-functions and constructing Hida families of theta lifts. The Iwasawa--Greenberg Main Conjecture predicts a connection between the p-adic L-functions and the Selmer groups attached to p-adic deformations of Galois representations. Another aspect of this project is to exploit the results on the congruences to study this conjecture in the special case of the Rankin--Selberg product of a pair of modular forms whose weights are of the same parity.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.