Rational Points on Homogeneous Varieties and Equidistribution of Adelic Periods

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 an asymptotic for the number of rational points U(K) with height bounded by T as T -> a, 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.