Rational points on certain homogeneous 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. Let $$\Delta (H)$$ be the image of H diagonally embedded in $$L^n$$ . 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.