Character sums to prime power moduli evaluated at binary quadratic forms

TL;DR

This paper develops new estimates for short character sums evaluated at binary quadratic forms with prime power moduli, extending previous results to a broader class of moduli using $p$-adic analysis.

Contribution

It introduces a $p$-adic analytical approach to estimate character sums at prime power moduli, complementing prior work on squarefree moduli.

Findings

Estimates for character sums at prime power moduli are established.

The approach uses $p$-adic exponential sum theory.

Results extend previous squarefree modulus estimates.

Abstract

We establish estimates for short character sums to prime power moduli evaluated at binary quadratic forms. This complements estimates established by Heath-Brown for such character sums to squarefree moduli. Our approach uses $p$-adic analysis. More precisely, we use tools from the $p$-adic theory of exponential sums, as initiated by Mili\'cevi\'c.