Parametrizing Shimura subvarieties of A1A1 Shimura varieties and related geometric problems
This paper gives a complete parametrization of the commensurability classes of totally geodesic subspaces of irreducible arithmetic quotients of Xa,b=(H2)a×(H3)bXa,b=(H2)a×(H3)b . A special case describes all Shimura subvarieties of type A1A1 Shimura varieties. We produce, for any n≥1n≥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.
Linowitz, Benjamin, and Matthew Stover. 2016. “Parameterizing Shimura subvarieties of A1 Shimura varieties and related geometric problems.” Archiv der Mathematik 107(3): 213-226.
Archiv der Mathematik