Quadratic congruences on average and rational points on cubic surfaces

TL;DR

This paper studies the average solutions to quadratic congruences and applies the findings to prove Manin's conjecture for a specific singular cubic surface.

Contribution

It introduces new methods to analyze quadratic congruences on average and applies these results to a significant case of rational points on cubic surfaces.

Findings

Established Manin's conjecture for the A_5+A_1 singular cubic surface.

Provided new insights into the distribution of solutions to quadratic congruences.

Connected quadratic congruences to rational point counting on algebraic surfaces.

Abstract

We investigate the average number of solutions of certain quadratic congruences. As an application, we establish Manin's conjecture for a cubic surface whose singularity type is A_5+A_1.