Orbits on Homogeneous Spaces of Arithmetic Origin and Approximations

Abstract

We prove an $S$-arithmetic version, in the context of algebraic groups defined over number fields, of Ratner’s theorem for closures of orbits of subgroups generated by unipotent elements. We apply this result in order to obtain a generalization of results of Margulis and of Borel–Prasad about values of irrational quadratic forms at integral points to the general setting of hermitian forms over division algebras with involutions of first or second kind. As a byproduct of our considerations we obtain another proof of the strong approximation theorem for algebraic groups defined over number fields.