The Arithmetic of Automorphic Forms

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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.