Concentration of points on Modular Quadratic Forms

TL;DR

This paper establishes new upper bounds for the number of solutions to quadratic form congruences over finite fields, extending previous results and showing solutions are sparse when the interval size is small relative to the prime.

Contribution

It provides non-trivial upper bounds for solutions to quadratic form congruences in short intervals, generalizing earlier work on specific cases like the product form.

Findings

Number of solutions is $M^{o(1)}$ when $M \\ll p^{1/4}$

Bounds apply to quadratic forms with non-zero discriminant

Generalizes previous results on specific congruences

Abstract

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 $xy\equiv \lambda \pmod{p}$.