Parametrizing Shimura subvarieties of A1A1 Shimura varieties and related geometric problems

Abstract

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.

Publisher

Springer Verlag

Publication Date

9-1-2016

Publication Title

Archiv der Mathematik

Department

Mathematics

Document Type

Article

DOI

https://dx.doi.org/10.1007/s00013-016-0944-9

Language

English

Format

text

Link to Full Text

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