Small solutions of generic ternary quadratic congruences to general moduli

Abstract

We study small non-trivial solutions of quadratic congruences of the form $x_1^2+{\alpha }_2{x}_2^2+{\alpha }_3{x}_3^2\equiv 0\mathrm{mod}q$, with q being an odd natural number, in an average sense. This extends previous work of the authors in which they considered the case of prime power moduli q. Above, ${\alpha }_2$ 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 $x_1^2+{\alpha }_2{x}_2^2+{\alpha }_3{x}_3^2\equiv 0\mathrm{mod}q$ with $\max \left\{\right|x_1|,|x_2|,|x_3\left|\right\}\le N$ and $(x_3,q)=1$ holds if $N\ge q^{11/24+\epsilon }$ and q is large enough. It is of significance that we break the barrier 1/2 in the above exponent. Key tools in our work are Burgess's estimate for character sums over short intervals and Heath-Brown's estimate for character sums with binary quadratic forms over small regions whose proofs depend on the Riemann hypothesis for curves over finite fields. We also formulate a refined conjecture about the size of the smallest solution of a ternary quadratic congruence, using information about the Diophantine properties of its coefficients.