Bilinear forms with trace functions over arbitrary sets, and applications to Sato-Tate

TL;DR

This paper establishes new bounds for bilinear forms involving trace functions over arbitrary sets, with applications to Sato-Tate distributions, extending previous results and surpassing classical barriers using advanced combinatorial and algebraic techniques.

Contribution

It introduces novel bounds for bilinear forms with trace functions over arbitrary supports, applying deep results to improve classical estimates and analyze Sato-Tate distributions.

Findings

Bounds for bilinear forms with trace functions over arbitrary sets

Application to Sato-Tate distributions of Kloosterman sums and elliptic curves

Beating the Pólya-Vinogradov barrier for hyper-Kloosterman sums

Abstract

We prove non-trivial upper bounds for general bilinear forms with trace functions of bountiful sheaves, where the supports of two variables can be arbitrary subsets in $\mathbf{F}_p$ of suitable sizes. This essentially recovers the P\'olya-Vinogradov range, and also applies to symmetric powers of Kloosterman sums and Frobenius traces of elliptic curves. In the case of hyper-Kloosterman sums, we can beat the P\'olya-Vinogradov barrier by combining additive combinatorics with a deep result of Kowalski, Michel and Sawin on sum-products of Kloosterman sheaves. Two Sato-Tate distributions of Kloosterman sums and Frobenius traces of elliptic curves in sparse families are also concluded.