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The basic stratum of some simple Shimura varieties

Abstract

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.

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Notes

  1. See Definition 4 for the precise statement.

  2. In the definition of the Harish-Chandra map there are different sign conventions possible. For example [2] and [16] use the convention \(q^{ \langle \chi , H_M(m) \rangle } = |\chi (m)|_p\) instead. Our sign follows that of [39]. In the article [9] there is no definition of the Harish-Chandra map but we have checked that Clozel uses our normalization.

  3. This follows from the results in [33], combined with the method in [18], see the discussion on the end of page 172 and beginning of page 173 in the introduction to [33].

  4. The reason for this sign is that the formula conjectured by Kottwitz in the article [25] turned out to be slightly mistaken. When Kottwitz proved his conjecture in [26] he found that a different sign should be used. However he did not change the sign in the conclusion of his theorem, rather he introduced it at the beginning by replacing \(h\) by \(h^{-1}\). We follow the conventions of Kottwitz because we refer to both articles constantly.

  5. The only cases where we know it is smooth is when it is a finite variety.

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Acknowledgments

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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Kret, A. The basic stratum of some simple Shimura varieties. Math. Ann. 356, 487–518 (2013). https://doi.org/10.1007/s00208-012-0854-5

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Keywords

  • Conjugacy Class
  • Parabolic Subgroup
  • Abelian Variety
  • Division Algebra
  • Automorphic Form