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

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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.

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  1. Department of Mathematics, University of Michigan, Ann Arbor, MI, 48109, USA

    Benjamin Linowitz

  2. Department of Mathematics, Temple University, 1805 N. Broad Street, Philadelphia, PA, 19122, USA

    Matthew Stover

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  1. Benjamin Linowitz
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  2. Matthew Stover
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Correspondence to Benjamin Linowitz.

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Linowitz, B., Stover, M. Parametrizing Shimura subvarieties of \({\mathrm{A}_1}\) Shimura varieties and related geometric problems. Arch. Math. 107, 213–226 (2016). https://doi.org/10.1007/s00013-016-0944-9

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

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