Parametrizing Shimura subvarieties of \({\mathrm{A}_1}\) Shimura varieties and related geometric problems. (English) Zbl 1410.11070
Summary: 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.
MSC:
| 11G18 | Arithmetic aspects of modular and Shimura varieties |
| 11F06 | Structure of modular groups and generalizations; arithmetic groups |
| 14G35 | Modular and Shimura varieties |
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