Small Solutions of generic ternary quadratic congruences

We consider small solutions of quadratic congruences of the form x₁²+α₂x₂²+α₃x₃²≡ 0 q, where q=pᵐ is an odd prime power. Here, α₂ is arbitrary but fixed and α₃ is variable, and we assume that (α₂α₃,p)=1. We show that for all α₃ modulo pᵐ which are coprime to p 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 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. If q is restricted to powers of a fixed prime p, we obtain a slight improvement of this result using the theory of p-adic exponent pairs, as developed by Mili\'cevi\'c, replacing the exponent $11/24$ above by $11/25$. Under the Lindel\"of hypothesis for Dirichlet L-functions, we are able to replace the exponent $11/24$ above by $1/3$.