Parametrizing Shimura subvarieties of A <inf>1</inf> Shimura varieties and related geometric problems

Abstract

© 2016, Springer International Publishing. This paper gives a complete parametrization of the commensurability classes of totally geodesic subspaces of irreducible arithmetic quotients of Xa,b=(H2)a×(H3)b. A special case describes all Shimura subvarieties of type A 1 Shimura varieties. We produce, for any n≥ 1 , examples of manifolds/Shimura varieties with precisely n commensurability classes of totally geodesic submanifolds/Shimura subvarieties. This is in stark contrast with the previously studied cases of arithmetic hyperbolic 3-manifolds and quaternionic Shimura surfaces, where the presence of one commensurability class of geodesic submanifolds implies the existence of infinitely many classes.