Simultaneous supersingular reductions of CM elliptic curves

We study the simultaneous reductions at several supersingular primes of elliptic curves with complex multiplication. We show - under additional congruence assumptions on the CM order - that the reductions are surjective (and even become equidistributed) on the product of supersingular loci when the discriminant of the order becomes large. This variant of the equidistribution theorems of Duke and Cornut-Vatsal is an(other) application of the recent work of Einsiedler and Lindenstrauss on the classification of joinings of higher-rank diagonalizable actions.