Kloosterman sums and Siegel zeros

 Joint IAS/Princeton University Number Theory Seminar Topic: Kloosterman sums and Siegel zeros Speaker: James Maynard Affiliation: Member, School of Mathematics Date: Thursday, September 28 Time/Room: 4:30pm - 5:30pm/Fine 214, Princeton University

Kloosterman sums arise naturally in the study of the distribution of various arithmetic objects in analytic number theory. The 'vertical' Sato-Tate law of Katz describes their distribution over a fixed field $\mathbb F_p$, but the equivalent 'horizontal' distribution as the base field varies over primes remains open. We describe work showing cancellation in the sum over primes if there are exceptional Siegel-Landau zeros. This is joint work with Sary Drappeau, relying on a fun blend of ideas from algebraic geometry, the spectral theory of automorphic forms and sieve theory.