Rational points on homogeneous varieties and Equidistribution of Adelic periods

TL;DR

This paper studies the distribution of rational points on certain homogeneous varieties over number fields, proving new cases of Manin's conjecture by leveraging equidistribution of adelic periods and unipotent flow theory.

Contribution

It introduces new asymptotic formulas for rational points on homogeneous varieties and advances the understanding of Manin's conjecture for wonderful varieties.

Findings

Established asymptotics for rational points with bounded height.

Proved new cases of Manin's conjecture for wonderful varieties.

Demonstrated equidistribution of semisimple adelic periods using unipotent flows.

Abstract

Let U:=L\G be a homogeneous variety defined over a number field K, where G is a connected semisimple K-group and L is a connected maximal semisimple K-subgroup of G with finite index in its normalizer. Assuming that G(K_v) acts transitively on U(K_v) for almost all places v of K, we obtain the asymptotic of the number of rational points in U(K) with height bounded by T, and settle new cases of Manin's conjecture for many wonderful varieties. The main ingredient of our approach is the equidistribution of semisimple adelic periods, which is established using the theory of unipotent flows.