Algebraic aspects of superspecial abelian surfaces with RM and CM

The Ibukiyama-Katsura-Oort correspondence establishes an algebraic framework for principally polarized superspecial abelian surfaces. This framework is powerful enough such that surfaces with extra structures can also be incorporated. We will discuss abelian varieties with real multiplication (an embedding of a totally real quadratic order into the endomorphism ring in a way that the image is symmetric with respect to the Rosati involution) and complex multiplication (an embedding of a quartic CM order which is stable under Rosati). Surfaces with RM admit a Deuring-like correspondence and we show how this can be used to solve algebraic pathfinding for surfaces with RM that have strict class number 1. Surfaces with CM give rise to the Shimura class group action that can be defined even in characteristic p and we provide some cryptographic and number theoretic applications of it.