Mathematische Annalen

, Volume 356, Issue 2, pp 487–518 | Cite as

The basic stratum of some simple Shimura varieties

  • Arno KretEmail author


Under simplifying hypotheses we prove a relation between the \(\ell \)-adic cohomology of the basic stratum of a Shimura variety of PEL-type modulo a prime of good reduction of the reflex field and the cohomology of the complex Shimura variety. In particular we obtain explicit formulas for the number of points in the basic stratum over finite fields. We obtain our results using the trace formula and truncation of the formula of Kottwitz for the number of points on a Shimura variety over a finite fields.


Conjugacy Class Parabolic Subgroup Abelian Variety Division Algebra Automorphic Form 
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I thank my thesis advisor Laurent Clozel for all his help and providing me with the idea of truncating the formula of Kottwitz to the basic stratum, combined with his proposition on compact traces and then computing the result using the trace formula. I also thank my second thesis advisor Laurent Fargues for answering some of my geometric questions on Shimura varieties. Furthermore I thank Guy Henniart, Ioan Badulescu, Marko Tadic, Chunh-Hui Wang for their help with representation theory, Paul-James White for his help on the trace formula, Gerard Laumon for a discussion on compact traces, and finally Henniart, Clozel and the referee for their corrections and suggestions.


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© Springer-Verlag 2012

Authors and Affiliations

  1. 1.Université Paris-SudOrsay CedexFrance

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