Rational points on certain homogeneous varieties

TL;DR

This paper establishes an asymptotic formula for counting rational points of bounded height on certain homogeneous varieties constructed from algebraic groups over number fields, under specific cohomological conditions.

Contribution

It provides the first asymptotic count for rational points on these particular equivariant compactifications, extending previous results in the area.

Findings

Asymptotic formula for rational points count

Conditions under which the formula holds

Application to specific algebraic group varieties

Abstract

Let $L$ be a simply-connected simple connected algebraic group over a number field $F$, and $H$ be a semisimple absolutely maximal connected $F$-subgroup of $L$. Under a cohomological condition, we prove an asymptotic formula for the number of rational points of bounded height on projective equivariant compactifications of $\Delta(H)\backslash L^n$ with respect to a balanced line bundle, where $\Delta(H)$ is the image of $H$ diagonally embedded in $L^n$.