CONCENTRATION OF POINTS ON MODULAR QUADRATIC FORMS

Let Q( x, y) be a quadratic form with discriminant D not equal 0. We obtain non-trivial upper bound estimates for the number of solutions of the congruence Q( x, y) equivalent to lambda ( mod p), where p is a prime and x, y lie in certain intervals of length M, under the assumption that Q( x, y)- lambda is an absolutely irreducible polynomial modulo p. In particular, we prove that the number of solutions to this congruence is M degrees((1)) when M << p(1/4). These estimates generalize a previous result by Cilleruelo and Garaev on the particular congruence xy equivalent to lambda ( mod p).