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We give a simplified and a direct proof of a special case of Ratner's theoremon closures and uniform distribution of individual orbits of unipotent flows;namely, the case of orbits of the diagonally embedded unipotent subgroup actingon $SL(2,K)/\Gamma_1\times ...\times SL(2,K)/\Gamma_n$, where $K$ is a locallycompact field of characteristic 0 and each $\Gamma_i$ is a cocompact discretesubgroup of $SL(2,K)$. This special case of Ratner's theorem plays a crucialrole in the proofs of uniform distribution of Heegner points by Vatsal, andMazur conjecture on Heegner points by C. Cornut; and their generalizations intheir joint work on CM-points and quaternion algebras. A purpose of the articleis to make the ergodic theoretic results accessible to a wide audience.

Unipotent flows on products of $SL(2,K)/\Gamma$'s