Semiclassical Analysis, Amplification, and Subconvexity

University Of Wisconsin-Madison, Madison WI

Investigators

Abstract

L-functions are functions of one complex variable whose study is central to modern number theory. They were introduced in order to study deep questions about the distribution of prime numbers. Today, we know that the values of L-functions at special points are linked to many areas of mathematics, from solving equations in integers to mathematical physics. When these special values are small, it tells us that certain highly structured mathematical objects are in fact behaving randomly. This research project project attempts to prove that this happens by using methods from quantum physics. The principal investigator will use techniques from semiclassical analysis to approach the subconvexity problem for L-functions. The link between these two areas is provided by period integral formulae such as those of Rankin-Selberg, Waldspurger, and Ichino-Ikeda. The PI hopes that the geometric structure of arithmetic manifolds will make it easier to study the oscillatory integrals and equidistribution problems that will arise in this work. The PI also aims to prove chaoticity results for automorphic forms, by showing that their Lp norms are smaller than those of general wave packets.