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

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
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