Analytic Properties of L-functions

Princeton University, Princeton NJ

Investigators

Abstract

The principal investigator intends primarily to pursue the goal of attacking some of the fundamental problems of the Langlands program by means of analytic methods. He is most interested in the case of base change for holomorphic forms on GL(2) where a non-soluble base change is involved, but the proposed methods should work much more generally. At present the intention is to permit the use of the Riemann Hypothesis (in an appropriate setting) to achieve this goal. A second goal of the proposal is to study some of the fundamental analytic questions on L-functions. This proposal is ultimately concerned with finding methods to solve equations. Since the nineteenth century it has been known that certain equations have formulas (like the quadratic equation and the cubic equation) but that most do not. The cases that interest me are the equations with ordinary integer coefficients. It has become apparent in the last fifty years that there should be a completely different way of studying these general equations by using methods from analysis (i.e. derived from calculus) rather than by using methods from algebra. The idea is to describe and prove many properties of the solutions (which still exist even though there may be no formulas for them).