Exponential sums and equidistribution

Organized by Matthew Hase-Liu and Caleb Ji

This is a learning seminar broadly about the sheaf-theoretic approach to equidistribution results for families of exponential sums over finite fields. For families parametrized by a variety, this is closely related to understanding the geometric monodromy groups of the corresponding sheaves. Katz famously proved an average version of the Sato-Tate law for Kloosterman sums in several variables and developed a framework to study exponential sums/arithmetic Fourier transforms of trace functions of perverse sheaves on $\mathbb{G}_a$. Katz and later Forey, Fresan, and Kowalski extended these ideas to families parametrized by multiplicative characters of the points of any connected commutative algebraic group by exploiting the formalism of Tannakian categories and Sawin's quantitative sheaf theory.

We will roughly follow parts of "Arithmetic Fourier transforms over finite fields" by Forey, Fresan, and Kowalski, as well as "Gauss Sums, Kloosterman Sums, and Monodromy Groups" by Katz. Ideally talks will be relatively self-contained, aside from the first few establishing some technical background.