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(q 5/8+e ), for any fixed e > 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.