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Shimuravarietäten und Gerben. (Shimura varieties and gerbs). (German) Zbl 0615.14014

The problem considered in this article is the description of the points in the reduction modulo \(p\) of a Shimura variety. The description, which is conjectural, is based on the concept of an admissible homomorphism \(\phi: {\mathcal P}\to {\mathcal G}_ G\) of an explicitly defined gerb \({\mathcal P}\) into the neutral gerb associated to the reductive group G over \({\mathbb{Q}}\) which defines the Shimura variety. To each such homomorphism \(\phi\) there is associated a double coset space of the form \(X_{\phi}(K)=I_{\phi}\setminus [X^ p/K^ p\times X_ p]\), together with an operator \(\Phi_{\phi}\) such that the set of points in the reduction \(modulo\quad p\) of \(Sh(G,h)_ K\) together with the action of the Frobenius, is the disjoint union indexed by the equivalence classes of admissible \(\phi\) of \(X_{\phi}(K)\). Here \(K\subset G({\mathbb{A}}_ f)\) is assumed to be of the form \(K=K^ p\cdot K_ p\) and on the rational prime p there are severe restrictions which however include the case where \(K_ p\) is hyperspecial, which conjecturally implies that the Shimura variety has good reduction at the primes above p.
This conjecture generalizes and makes more precise the original conjecture in terms of Frobenius pairs of R. P. Langlands [Math. Dev. Hilbert Probl., Proc. Symp. Pure Math. 28, De Kalb 1974, 401-418 (1976; Zbl 0345.14006)]. In fact, there is a bijective correspondence between equivalence classes of Frobenius pairs and local equivalence classes of admissible homomorphisms. Using concepts and results of R. E. Kottwitz, some of them still unpublished [cf. Duke Math. J. 51, 611-650 (1984; Zbl 0576.22020), Math. Ann. 269, 287-300 (1984; Zbl 0547.14013), Compos. Math. 56, 201-220 (1985; Zbl 0597.20038)] it is shown that the conjecture yields in the case where \(K_ p\) is hyperspecial an explicit numerical formula for the number of points with values in a finite field of characteristic p of the Shimura variety.
The origin of the description in terms of gerbs is A. Grothendieck’s conjectural theory of motives and of the description of their category in terms of gerbs [N. Saavedra Rivano, ”Categories Tannakiennes”, Lect. Notes Math. 265 (1972; Zbl 0241.14008)]. It is shown that assuming the standard conjectures on algebraic cycles, the Tate conjecture and the Hodge conjecture for abelian varieties of CM-type, the conjecture is valid for the Shimura varieties which are moduli spaces of abelian varieties with additional structure. This uses results of T. Zink [Math. Nachr. 112, 103-124 (1983; Zbl 0604.14029)].

MSC:

14G20 Local ground fields in algebraic geometry
14A20 Generalizations (algebraic spaces, stacks)
14C99 Cycles and subschemes
18G50 Nonabelian homological algebra (category-theoretic aspects)
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