Small solutions of generic ternary quadratic congruences to general moduli

We study small non-trivial solutions of quadratic congruences of the form x₁²+α₂x₂²+α₃x₃²≡ 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, α₂ is arbitrary but fixed and α₃ is variable, and we assume that (α₂α₃,q)=1. We show that for all α₃ modulo q which are coprime to q except for a small number of α₃'s, an asymptotic formula for the number of solutions (x₁,x₂,x₃) to the congruence x₁²+α₂x₂²+α₃x₃²≡ 0 q with max\|x₁|,|x₂|,|x₃|\≤ N and (x₃,q)=1 holds if N≥ q11/24+ε 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.