Parametrizing Shimura subvarieties of A₁ Shimura varieties and related geometric problems

2015/10/13 by Linowitz, Benjamin, Stover, Matthew
#Algebraic Geometry (math.AG) #FOS: Mathematics #Geometric Topology (math.GT) #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1510.03728

Abstract

This paper gives a complete parametrization of the commensurability classes of totally geodesic subspaces of irreducible arithmetic quotients of X_{a, b} = (H²)^a × (H³)^b. A special case describes all Shimura subvarieties of type A₁ 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.

Related

• Geodesic curves on Shimura surfaces
• Subvarieties of Shimura varieties
• Shimura varieties
• Arithmetic of Shimura Varieties
• Algebraic flows on Shimura varieties