Quantitative strong approximation for ternary quadratic forms III

TL;DR

This paper develops an asymptotic counting method for primitive integral points on ternary quadratic forms, incorporating local conditions and relating results to the Brauer--Manin obstruction, using advanced circle method techniques.

Contribution

It introduces a novel application of the δ-variant of the Hardy--Littlewood circle method to count points on ternary quadratic forms with local conditions, connecting to Brauer--Manin obstruction.

Findings

Derived asymptotic formulas for counting points

Established connections to Brauer--Manin obstruction

Extended circle method techniques to new settings

Abstract

We prove asymptotic formulas for counting (primitive) integral points with local conditions on the (punctured) affine cone defined by a non-singular integral ternary quadratic form, and we relate our results to the Brauer--Manin obstruction. Our approach is based on the $\delta$-variant of the Hardy--Littlewood circle method developed by Heath-Brown.