Small Solutions of generic ternary quadratic congruences

TL;DR

This paper establishes asymptotic formulas for small solutions to certain quadratic congruences modulo prime powers, breaking previous barriers and improving results under specific hypotheses.

Contribution

It provides new bounds for the size of solutions to quadratic congruences, surpassing the 1/2 barrier, and refines these bounds using advanced number theory techniques.

Findings

Breaks the 1/2 barrier for solution size exponent.

Improves bounds using p-adic exponent pairs.

Under Lindelöf hypothesis, achieves an exponent of 1/3.

Abstract

We consider small solutions of quadratic congruences of the form $x_1^2+\alpha_2x_2^2+\alpha_3x_3^2\equiv 0 \bmod{q}$, where $q=p^m$ is an odd prime power. Here, $\alpha_2$ is arbitrary but fixed and $\alpha_3$ is variable, and we assume that $(\alpha_2\alpha_3,q)=1$. We show that for all $\alpha_3$ modulo $q$ which are coprime to $q$ except for a small number of $\alpha_3$'s, an asymptotic formula for the number of solutions $(x_1,x_2,x_3)$ to the congruence $x_1^2+\alpha_2x_2^2+\alpha_3x_3^2\equiv 0 \bmod{q}$ with $\max\{|x_1|,|x_2|,|x_3|\}\le N$ holds if $N\ge q^{11/24+\varepsilon}$ as $q$ tends to infinity over the set of all odd prime powers. It is of significance that we break the barrier 1/2 in the above exponent. If $q$ is restricted to powers $p^m$ of a {\it fixed} prime $p$ and $m$ tends to infinity, we obtain a slight improvement of this result using the theory of $p$-adic exponent pairs, as developed by Mili\'cevi\'c, replacing the exponent $11/24$ above by $11/25$. Under the Lindel\"of hypothesis for Dirichlet $L$-functions, we are able to replace the exponent $11/24$ above by $1/3$.