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

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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.