Small solutions of generic ternary quadratic congruences to general moduli

We study small non-trivial solutions of quadratic congruences of the form x₁²+₂x₂²+₃x₃² 0 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, ₂ is arbitrary but fixed and ₃ is variable, and we assume that (₂₃, q) =1. We show that for all ₃ modulo q which are coprime to q except for a small number of ₃'s, an asymptotic formula for the number of solutions (x₁, x₂, x₃) to the congruence x₁²+₂x₂²+₃x₃² 0 q with \|x₁|, |x₂|, |x₃|\ N and (x₃, q) =1 holds if N q^11/24+ 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.