Equidistribution of CM points on a Shimura Curve modulo a ramified prime

TL;DR

This paper proves that Galois orbits of CM points on Shimura curves become uniformly distributed modulo ramified primes, using reduction techniques and Ratner's theorem to establish equidistribution.

Contribution

It introduces a new equidistribution result for CM points on Shimura curves at ramified primes, employing reduction analysis and Ratner's theorem.

Findings

Galois orbits of CM points are equidistributed modulo ramified primes.

Reduction of CM points can be effectively studied in the special fiber.

Ratner's theorem is applicable to the distribution of CM points on Shimura curves.

Abstract

We prove an equidistribution statement for the reduction of Galois orbits of CM points on the special fiber of a Shimura curve over a totally real field attached to some ramified primes. To do so we study the reduction of CM points in the special fiber and we use Ratner's theorem to obtain the desired equidistribution.