New Bounds for Automorphic L-Functions

University Of Texas At Austin, Austin TX

Investigators

Abstract

The principal investigator intends to prove new subconvex bounds for automorphic L-functions or improve on existing subconvex bounds. Such bounds reflect the arithmetic nature of their source objects as they cannot be derived from simple analytic principles. In return, they provide the key to the solution of several deep diophantine problems addressing equidistribution phenomena. Proving these bounds unconditionally also sheds light on the Generalized Riemann Hypothesis as they are consequences of it. In the focus of the proposal are families of GL(2) x GL(1) and GL(2) x GL(2) type. The techniques leading to subconvex bounds for these families have so far been restricted to the field of rational numbers or holomorphic forms. The principal investigator will try to extend these techniques to totally real number fields and non-holomorphic forms. This proposal belongs to the theory of integers. Because of their fundamental character, integers are at the source of many theoretical and practical constructions including algorithms for secure communication through the internet or efficient communication through a noisy channel. Automorphic forms and their L-functions are among the mathematical objects that make the hidden symmetries of integers visible. It has been observed, but not proved rigorously in a single instance, that the zeros of every automorphic L-function are distributed in a very special way. This observation is the Generalized Riemann Hypothesis which implies many otherwise unknown properties of the integers. For various important questions a weaker hypothesis, concerning the size of L-functions, suffices. The principal investigator intends to prove new instances of this weaker hypothesis.