Small solutions of generic ternary quadratic congruences
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 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 $1/3$.
