Small Solutions of Quadratic Congruences, and Character Sums with Binary Quadratic Forms

Let $Q(x,y,z)$ be an integral quadratic form with determinant coprime to some modulus $q$. We show that $q\mid Q$ for some non-zero integer vector $(x,y,z)$ of length $O(q^{5\mathrm{/}8+\epsilon })$, for any fixed $\epsilon >0$. Without the coprimality condition on the determinant one could not achieve an exponent below $2\mathrm{/}3$. The proof uses a bound for short character sums involving binary quadratic forms, which extends a result of Chang.