Inhomogeneous cubic congruences and rational points on del Pezzo surfaces

TL;DR

This paper studies the distribution of solutions to a specific cubic congruence and applies these results to prove the Manin conjecture for a particular del Pezzo surface, also exploring elliptic curves with square-free discriminant.

Contribution

It establishes the Manin conjecture for a singular degree 2 del Pezzo surface and analyzes the density of solutions to inhomogeneous cubic congruences.

Findings

Proved the Manin conjecture for the specified del Pezzo surface.

Derived density estimates for solutions to the cubic congruence.

Explored the distribution of elliptic curves with square-free discriminant.

Abstract

For given non-zero integers a,b,q we investigate the density of integer solutions (x,y) to the binary cubic congruence ax^2+by^3=0 (mod q). We use this to establish the Manin conjecture for a singular del Pezzo surface of degree 2 defined over the rationals and to examine the distribution of elliptic curves with square-free discriminant.