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

TL;DR

This paper provides a simplified proof of a special case of Ratner's theorem concerning unipotent flows on products of $SL(2,K)/\Gamma$ spaces, with applications to number theory and distribution of special points.

Contribution

It offers a direct and simplified proof of a key special case of Ratner's theorem, facilitating broader accessibility and application in number theory.

Findings

Simplified proof of a special case of Ratner's theorem.

Application to distribution of Heegner points.

Connections to conjectures on CM-points and quaternion algebras.

Abstract

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.