Arithmetic purity of the Hardy-Littlewood property and geometric sieve for affine quadrics

2020/03/16 by Cao, Yang, Huang, Zhizhong · 1 citation
#11D45 #11N35 #14G12 #Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2003.07287

Abstract

We establish the Hardy-Littlewood property (à la Borovoi-Rudnick) for Zariski open subsets in affine quadrics of the form q(x₁,⋯,x_n)=m, where q is a non-degenerate integral quadratic form in n\geqslant 3 variables and m is a non-zero integer. This gives asymptotic formulas for the density of integral points taking coprime polynomial values, which is a quantitative version of the arithmetic purity of strong approximation property off infinity for affine quadrics.

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• Effective equidistribution, arithmetic purity of strong approximation, and geometric sieve for affine quadrics

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