A note on equidistribution on a product of Shimura curves and André--Oort

Abstract

In this short note we show that Galois orbits of CM points equidistribute on a product of $r\ge 2$ non-isomorphic Shimura curves by applying the adelic toral-packet equidistribution theorem of Aka--Luethi--Michel--Wieser. As a consequence, we deduce André--Oort for the product of those curves, previously studied by Edixhoven and Yafaev, replacing GRH by a Linnik-type splitting condition at two auxiliary primes.