Small Solutions of Quadratic Congruences, and Character Sums with Binary Quadratic Forms

TL;DR

This paper establishes bounds on solutions to quadratic congruences with coprime determinants, utilizing character sum bounds to improve the understanding of solutions' size and distribution.

Contribution

It introduces new bounds for solutions of quadratic congruences with coprime determinants using extended character sum estimates, improving previous exponent limits.

Findings

Existence of solutions with size O(q^{5/8+ε}) for quadratic forms with coprime determinant.

Extension of Chang's character sum bounds to binary quadratic forms.

Improved exponent bounds compared to previous results when the determinant is coprime to q.

Abstract

Let $Q(x,y,z)$ be an integral quadratic form with determinant coprime to some modulus $q$. We show that $q\mid Q$ for some non-zero integer vector $(x,y,z)$ of length $O(q^{5/8+\varepsilon})$, for any fixed $\varepsilon>0$. Without the coprimality condition on the determinant one could not achieve an exponent below $2/3$. The proof uses a bound for short character sums involving binary quadratic forms, which extends a result of Chang.