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

2025/09/27 by Cao, Yang, Huang, Zhizhong, Zhang, Runlin
#11N35 #14G05 #37A17 #Dynamical Systems (math.DS) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2509.23320

Abstract

Let k be a number field. Let q(x₁,⋯,x_n) be a non-degenerate integral quadratic form in n≥ 3 variables with coefficients in k and m∈ k^×. Let X be the affine quadric defined by q=m in \mathbbAⁿ_k. Based on results on effective equidistribution of S-integral points in symmetric spaces, we establish the following: (i) The arithmetic purity of strong approximation off any single place of k for X; (ii) The geometric sieve for p₀-integral points on X when k=ℚ and p₀ is a prime number.

Citations

• The geometric sieve for quadrics
• Arithmetic purity of the Hardy-Littlewood property and geometric sieve for affine quadrics
• Arithmetic purity of strong approximation for semi-simple simply connected groups
• Arithmetic purity of strong approximation for homogeneous spaces
• Counting Integral Points in Certain Homogeneous Spaces
• Igusa integrals and volume asymptotics in analytic and adelic geometry
• Prime and almost prime integral points on principal homogeneous spaces
• Effective equidistribution of S-integral points on symmetric varieties
• Manin's and Peyre's conjectures on rational points and adelic mixing

Related

• Arithmetic purity of the Hardy-Littlewood property and geometric sieve for affine quadrics
• Quantitative strong approximation for ternary quadratic forms II
• Quantitative strong approximation for quaternary quadratic forms
• Strong approximation for the total space of certain quadric fibrations
• Arithmetic purity of strong approximation for semi-simple simply connected groups