Small Solutions of generic ternary quadratic congruences

We consider small solutions of quadratic congruences of the form $x_1^2+{\alpha }_2{x}_2^2+{\alpha }_3{x}_3^2\equiv 0q$, 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 $x_1^2+{\alpha }_2{x}_2^2+{\alpha }_3{x}_3^2\equiv 0q$ with $\max \{\mathrm{\mid }{x}_1\mathrm{\mid },\mathrm{\mid }{x}_2\mathrm{\mid },\mathrm{\mid }{x}_3\mathrm{\mid }\}\le N$ holds if $N\ge q^{11\mathrm{/}24+\epsilon }$ 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 $11\mathrm{/}24$ above by $11\mathrm{/}25$. Under the Lindelöf hypothesis for Dirichlet $L$-functions, we are able to replace the exponent $11\mathrm{/}24$ above by $1\mathrm{/}3$.