On character sums with determinants

TL;DR

This paper derives nontrivial bounds for weighted character sums involving determinants of 2x2 matrices modulo a prime, achieving power savings for N as small as p^{1/8+ε}, advancing understanding in character sum estimates.

Contribution

The paper introduces new bounds for character sums with determinants, improving the range of N for which nontrivial estimates are known, approaching optimality under current conjectures.

Findings

Achieved power-saving bounds for N ≥ p^{1/8+ε}

Extended techniques to more general sums with larger N

Provided evidence that bounds are close to optimal without Burgess improvements

Abstract

We estimate weighted character sums with determinants $ad-bc $ of $2\times 2$ matrices modulo a prime $p$ with entries $a,b,c,d $ varying over the interval $ [1,N]$. Our goal is to obtain nontrivial bounds for values of $N$ as small as possible. In particular, we achieve this goal, with a power saving, for $N \ge p^{1/8+\varepsilon}\ $ with any fixed $\varepsilon>0$, which is very likely to be the best possible unless the celebrated Burgess bound is improved. By other techniques, we also treat more general sums but sometimes for larger values of $N$.