Parametrizing Shimura subvarieties of $\mathrm{A}_1$ Shimura varieties and related geometric problems

TL;DR

This paper provides a complete parametrization of commensurability classes of totally geodesic subspaces in certain arithmetic quotients, specifically describing all $ ext{A}_1$ Shimura subvarieties and constructing examples with a fixed number of such classes.

Contribution

It introduces a comprehensive parametrization of these classes and constructs explicit examples with any finite number of commensurability classes, contrasting previous cases with infinitely many classes.

Findings

Complete parametrization of commensurability classes

Explicit examples with any finite number of classes

Contrasts with cases of hyperbolic 3-manifolds

Abstract

This paper gives a complete parametrization of the commensurability classes of totally geodesic subspaces of irreducible arithmetic quotients of $X_{a, b} = (\mathbf{H}^2)^a \times (\mathbf{H}^3)^b$. A special case describes all Shimura subvarieties of type $\mathrm{A}_1$ Shimura varieties. We produce, for any $n\geq 1$, 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.