Counting Integral Points in Certain Homogeneous Spaces

TL;DR

This paper derives an asymptotic formula for counting integral points in certain homogeneous spaces, linking global counts to local solutions and addressing questions about the product of local solutions.

Contribution

It provides explicit asymptotic formulas for integral points on specific homogeneous spaces and clarifies the relationship between global counts and local solutions, extending previous results.

Findings

Asymptotic formulas relate global integral points to local solutions.

Explicit counts for integral points on norm equations.

Confirmed the product of local solutions equals the global count.

Abstract

The asymptotic formula of the number of integral points in non-compact symmetric homogeneous spaces of semi-simple simply connected algebraic groups is given by the average of the product of the number of local solutions twisted by the Brauer-Manin obstruction. The similar result is also true for homogeneous spaces of reductive groups with some restriction. As application, we will give the explicit asymptotic formulae of the number of integral points of certain norm equations and explain that the asymptotic formula of the number of integral points in Theorem 1.1 of \cite{EMS} is equal to the product of local integral solutions over all primes and answer a question raised by Borovoi related to the example 6.3 in \cite{BR95}.