Concentration of points on Modular Quadratic Forms

Details Concentration of points on Modular Quadratic Forms Concentration of points on Modular Quadratic Forms

2010-12-16

arxiv.org/abs/1012.3569v2

Mathematics

Number Theory

Accepted for publication in the International Journal of Number Theory, 4 pages

Scientific paper

Let $Q(x,y)$ be a quadratic form with discriminant $D\neq 0$. We obtain non trivial upper bound estimates for the number of solutions of the congruence $Q(x,y)\equiv\lambda \pmod{p}$, 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 $M^{o(1)}$ when $M\ll p^{1/4}$. These estimates generalize a previous result by Cilleruelo and Garaev on the particular congruence $xy\equiv \lambda \pmod{p}$.

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