Arithmetic Fourier transforms over finite fields: generic vanishing, convolution, and equidistribution

TL;DR

This paper develops a general framework for arithmetic Fourier transforms over finite fields, establishing vanishing theorems, constructing a tannakian category, and proving equidistribution results for trace functions, with applications and open questions.

Contribution

It introduces a generic vanishing theorem for twisted perverse sheaves, constructs a convolution-based tannakian category, and generalizes equidistribution results for trace functions over finite fields.

Findings

Proved a generic vanishing theorem for twists of perverse sheaves.

Constructed a tannakian category with convolution as tensor operation.

Established a general equidistribution theorem for trace functions.

Abstract

We study the arithmetic Fourier transforms of trace functions on general connected commutative algebraic groups. To do so, we first prove a generic vanishing theorem for twists of perverse sheaves by characters, and using this tool, we construct a tannakian category with convolution as tensor operation. Using Deligne's Riemann Hypothesis, we show how this leads to a general equidistribution theorem for the discrete Fourier transforms of trace functions of perverse sheaves, generalizing the work of Katz in the case of the multiplicative group. We then give some concrete examples of applications of these results and raise a number of questions.