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Kottwitz, Robert E.

Points on some Shimura varieties over finite fields. (English) Zbl 0796.14014

J. Am. Math. Soc. 5, No. 2, 373-444 (1992).
The author proves the Langlands conjecture on zeta functions of Shimura varieties of PEL-type in cases \(A\) and \(C\). More precisely, he computes the number of fixed points of twisted Hecke correspondences \(\Phi^ j_ \wp \circ f\) inside a single isogeny class in terms of orbital and twisted orbital integrals.
An important rôle is played by a triple \((\gamma_ 0, \gamma, \delta)\) associated to points \((A, \lambda, i, \eta)\) on the Shimura variety in characteristic \(p\). It is shown that the Kottwitz invariant \(\alpha (\gamma_ 0, \gamma, \delta)\) of such a triple vanishes; conversely, if \(\alpha (\gamma_ 0, \gamma, \delta) = 1\) and two other obvious conditions are satisfied then \((\gamma_ 0, \gamma, \delta)\) comes from some \((A, \lambda,i)\). The proof of the vanishing of \(\alpha\) involves a consideration of the filtered Dieudonné module associated to a crystalline representation of the Galois group of a local field.
Apart from a lot of useful lemma’s, the article also includes Honda-Tate theory for abelian varieties with endomorphisms.
Reviewer: A.J.de Jong (Utrecht)

Cited in 14 Reviews
Cited in 157 Documents

MSC:

14G35 Modular and Shimura varieties
14G15 Finite ground fields in algebraic geometry
14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
11S40 Zeta functions and \(L\)-functions
14K05 Algebraic theory of abelian varieties
11G18 Arithmetic aspects of modular and Shimura varieties

Keywords:

Langlands conjecture on zeta functions of Shimura varieties; fixed points of twisted Hecke correspondences; Honda-Tate theory for abelian varieties with endomorphisms
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\textit{R. E. Kottwitz}, J. Am. Math. Soc. 5, No. 2, 373--444 (1992; Zbl 0796.14014)
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References:

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