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Parametrizing Shimura subvarieties of A <inf>1</inf> Shimura varieties and related geometric problems
Linowitz, B Stover, M
Linowitz, B
Stover, M
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Journal Article
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2016-09-01
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10.1007/s00013-016-0944-9
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© 2016, Springer International Publishing. This paper gives a complete parametrization of the commensurability classes of totally geodesic subspaces of irreducible arithmetic quotients of Xa,b=(H2)a×(H3)b. A special case describes all Shimura subvarieties of type A 1 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.
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Springer Science and Business Media LLC
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Archiv der Mathematik
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