Short sums of trace functions over function fields and their applications

TL;DR

This paper establishes near square-root cancellation for short sums of trace functions over function fields, advancing understanding of their behavior and applications in analytic number theory over $\

Contribution

It introduces new bounds for short sums of trace functions with specific monodromy conditions, extending previous results to a broader class of functions.

Findings

Near square-root cancellation achieved for certain trace functions

Progress on Mordell's problem regarding least residue classes

Improved understanding of variance of short Kloosterman sums

Abstract

For large enough (but fixed) prime powers $q$, and trace functions to squarefree moduli in $\mathbb{F}_q[u]$ with slopes at most $1$ at infinity, and no Artin--Schreier factors in their geometric global monodromy, we come close to square-root cancellation in short sums. A special case is a function field version of Hooley's Hypothesis $R^*$ for short Kloosterman sums. As a result, we are able to make progress on several problems in analytic number theory over $\mathbb{F}_q[u]$ such as Mordell's problem on the least residue class not represented by a polynomial and the variance of short Kloosterman sums.