Abstract
We study the cusps of Shimura varieties arising from indefinite lattices splitting two hyperbolic planes. We determine the number of 0-dimensional cusps for a given variety and, when the lattice is maximal, we relate the genus of the lattice to the number of \(1\)-dimensional cusps and determine an explicit formula. As every lattice is contained as a sublattice of finite index in a maximal lattice, the results we obtain are useful in a general analysis.
Résumé
Nous étudions les pointes des variétés de Shimura apparaissant dans les treillis indéfinis séparant deux plans hyperboliques. Nous déterminons le nombre de pointes de dimension 0 pour une variété donnée et, lorsque le treillis est maximal, nous relions le genre du treillis et le nombre de pointes de dimension 1 via une formule explicite. Comme tout treillis est contenu, en tant que sous-treillis, dans un treillis maximal, les résultats obtenus sont utiles en général.
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Acknowledgments
The author would like to thank his supervisor Professor Eyal Goren for suggestions and help in editing. The author would particularly like to thank his anonymous referee who gave a very detailed review of this paper and offered up a number of insightful comments, including correcting an error in an earlier version of Example 6.1. The author would also like to thank Andrew Fiori for stimulating discussions, and Donald James for providing part of his student’s thesis [11].
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Attwell-Duval, D. On the number of cusps of orthogonal Shimura varieties. Ann. Math. Québec 38, 119–131 (2014). https://doi.org/10.1007/s40316-014-0026-y
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DOI: https://doi.org/10.1007/s40316-014-0026-y