The Subconvexity Problem

University Of Wisconsin-Madison, Madison WI

Investigators

Abstract

Prime numbers are of fundamental importance in mathematics, as every number can be uniquely written as a product of primes. A mysterious feature of prime numbers is that the statistics of how they are distributed is similar to what a random sequence of numbers would produce. This is despite the fact that whether a number is prime or not is in no respect random. L-functions are mathematical objects that conjecturally allow us to understand this statistical feature of primes and other objects from the theory of numbers. This project investigates an important conjecture about L-functions, known as the subconvexity problem, which states that the values taken by these L-functions are smaller than expected. If true, it would be an important step in unlocking the statistical information that L- functions contain. The aim of this award is to make progress on the subconvexity problem for L-functions in higher rank, and to prove subconvexity for as wide a range of automorphic forms as possible. The project will focus on the groups U(n+1) x U(n) and GL(n+1) x GL(n). The PI will approach the problem using arithmetic amplification, representation theory, and ideas from microlocal analysis. A key role will be played by microlocal lifts. The methods used will not depend on the rank of the group in an essential way. 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.