Affine congruences and rational points on a certain cubic surface

TL;DR

This paper proves Manin's conjecture for a specific cubic surface by establishing estimates for affine congruences, advancing understanding of rational points on algebraic surfaces.

Contribution

It introduces new estimates for affine congruences and applies them to confirm Manin's conjecture for a D_4 singular cubic surface, improving prior results.

Findings

Confirmed Manin's conjecture for the specified cubic surface

Developed new estimates for solutions of affine congruences

Answered a previously posed problem by Tschinkel

Abstract

We establish estimates for the number of solutions of certain affine congruences. These estimates are then used to prove Manin's conjecture for a cubic surface split over Q and whose singularity type is D_4. This improves on a result of Browning and answers a problem posed by Tschinkel.