Bounds on certain Higher-Dimensional Exponential Sums via the Self-Reducibility of the Weil Representation

TL;DR

This paper introduces a novel method leveraging the self-reducibility of the Weil representation to bound higher-dimensional exponential sums related to symplectic tori over finite fields, leading to a sharp quantum ergodicity result.

Contribution

It presents a new approach to bounding exponential sums using the Weil representation's self-reducibility, improving understanding of quantum ergodicity in symplectic dynamics.

Findings

Established sharp bounds on higher-dimensional exponential sums.

Proved a refined quantum unique ergodicity theorem for generic symplectomorphisms.

Demonstrated the effectiveness of the self-reducibility method in this context.

Abstract

We describe a new method to bound certain higher-dimensional exponential sums which are associated with tori in symplectic groups over finite fields. Our method is based on the self-reducibility property of the Weil representation. As a result, we obtain a sharp form of the Hecke quantum unique ergodicity theorem for generic linear symplectomorphisms of the 2N-dimensional torus.