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|Q for some non-zero integer vector (x,y,z) of length O(q5/8+ε) , for any fixed ε>0 . Without the coprimality condition on the determinant one could not necessarily achieve an exponent below 2/3 . The proof uses a bound for short character sums involving binary quadratic forms, which extends a result of Chang