An introduction to exponential sums in function fields

Wednesday, 14 June 2023

Abstract

Number theorists have long noticed similarities between the theory of number fields and the theory of function fields over finite fields. One notable aspect of this connection is the stark similarlty between the theories of exponential sums in these two spaces. In this talk we will give a (gentle) introduction to working with exponential sums in rational function fields over finite fields, by using and building upon our intuition for analogous sums over the real numbers. We will explore where exponential sums in these fields offer distinct advantages over their real counterparts, and also where we encounter obstacles inherent in the positive characteristic setting.  We will also discuss some recent results and applications to point counting estimates.