Quantitative strong approximation for quaternary quadratic forms

TL;DR

This paper establishes asymptotic formulas for the distribution of rational points on quadratic surfaces and investigates the growth of integral points on a related affine cone, advancing understanding of strong approximation in algebraic geometry.

Contribution

It provides the first quantitative strong approximation results for quaternary quadratic forms, including optimal error terms and growth estimates for integral points.

Findings

Asymptotic formulas with optimal error terms for rational points on quadratic surfaces.

Quantitative growth estimates for integral points on the punctured affine cone.

Extension of strong approximation results incorporating Brauer--Manin obstruction.

Abstract

The purpose of this article is twofold. On the one hand, we prove asymptotic formulas for the quantitative distribution of rational points on any smooth non-split projective quadratic surface. We obtain the optimal error term for the real place. On the other hand, we also study the growth of integral points on the three-dimensional punctured affine cone, as a quantitative version of strong approximation with Brauer--Manin obstruction for this quasi-affine variety.