Small Solutions of generic ternary quadratic congruences

Abstract

We consider small solutions of quadratic congruences of the form $x_1^2+\hat{I}{\pm }_2{x}_2^2+\hat{I}{\pm }_3{x}_3^2\equiv 0\mathrm{mod}q$, where $q=p^m$ is an odd prime power. Here, $\hat{I}{\pm }_2$ is arbitrary but fixed and $\hat{I}{\pm }_3$ is variable, and we assume that $(\hat{I}{\pm }_2\hat{I}{\pm }_3,q)=1$. We show that for all $\hat{I}{\pm }_3$ modulo $q$ which are coprime to $q$ except for a small number of $\hat{I}{\pm }_3$'s, an asymptotic formula for the number of solutions $(x_1,x_2,x_3)$ to the congruence $x_1^2+\hat{I}{\pm }_2{x}_2^2+\hat{I}{\pm }_3{x}_3^2\equiv 0\mathrm{mod}q$ with $\max \{|x_1|,|x_2|,|x_3|\}\le N$ holds if $N\ge q^{11/24+\epsilon }$ 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 Milićević, replacing the exponent $11/24$ above by $11/25$. Under the Lindelöf hypothesis for Dirichlet $L$-functions, we are able to replace the exponent $11/24$ above by $1/3$.

10 pages