Unipotent Flows on Products of Sl(2,k)/γ’s

We will give a simplified and a direct proof of a special case of Ratner’s theorem on closures and uniform distribution of individual orbits of unipotent flows; namely, the case of orbits of the diagonally embedded unipotent subgroup acting on SL(2, K)/Γ1 × · · ·×SL(2, K)/Γn, where K is a locally compact field of characteristic 0 and each Γi is a cocompact discrete subgroup of SL(2, K). This special case of Ratner’s theorem plays a crucial role in the proofs of uniform distribution of Heegner points by Vatsal, and Mazur conjecture on Heegner points by C. Cornut; and their generalizations in their joint work on CM-points and quaternion algebras. A purpose of the article is to make the ergodic theoretic results accessible to a wide audience.