# Exponential sums over finite fields and the large sieve

## TL;DR

This paper develops advanced zero-density estimates for $ ext{ell}$-adic trace functions over finite fields, leveraging a variant of Kowalski's large sieve, with applications to exponential sums like hyper-Kloosterman sums.

## Contribution

It introduces a new variant of the large sieve for Frobenius that improves zero-density estimates for trace functions with known monodromy groups.

## Key findings

- Zero-density estimates for $ ext{ell}$-adic trace functions over finite fields.
- Application to hyper-Kloosterman sums and Katz's exponential sums.
- Enhanced understanding of the distribution of trace function arguments.

## Abstract

By using a variant of Kowalski's large sieve for Frobenius in compatible systems, we obtain zero-density estimates for arguments of $\ell$-adic trace functions over finite fields with values in some algebraic subsets of the cyclotomic integers, when the monodromy groups are known. This applies in particular to hyper-Kloosterman sums and general exponential sums considered by Katz.