Counting integral points on some homogeneous varieties with large reductive stabilizers

TL;DR

This paper studies the asymptotic distribution of integral points on certain homogeneous varieties with large reductive stabilizers, combining equidistribution results from Lie groups and algebraic geometry.

Contribution

It introduces a novel approach combining two equidistribution results to analyze integral points on homogeneous varieties with large stabilizers.

Findings

Asymptotic count of integral points up to an implicit constant.

Extension of equidistribution techniques to varieties with intermediate subgroups.

Framework accommodating large stabilizers in counting problems.

Abstract

Let G be a semisimple group over rational numbers and H is a subgroup over rational numbers. Given a representation of G and an integral vector x whose stabilizer is equal to H. In this paper we investigate the asymptotic of integral points on Gx with bounded height. We find its asymptotic up to an implicit constant when H is large in G but we allow the presence of intermediate subgroups. This is achieved by a novel combination of two equidistribution results in two different settings: one is that of Eskin, Mozes and Shah on a Lie group modulo a lattice and the other one is a result of Chamber-Loir and Tschinkel on a smooth projective variety with a normal crossing divisor.