  We 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 $SL(2,K)/\Gamma_1\times ...\times SL(2,K)/\Gamma_n$, where $K$ is a locally
compact field of characteristic 0 and each $\Gamma_i$ 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.
