Isogeny classes for points of Shimura varieties with values in a finite field. (Isogenieklassen von Punkten von Shimuramannigfaltigkeiten mit Werten in einem endlichen Körper.) (German) Zbl 0604.14029

The article considers the Shimura variety associated to the moduli problem of classifying polarised abelian varieties with a compatible action of a semisimple algebra with positive involution. A suitable definition of CM-type (i.e. with compatibility with the action of the algebra) is introduced. The main result says that each point on the Shimura variety over a finite field is isogenous to a CM-object which can be lifted to a CM-object definable over a number field. Using this result one can then, following an idea of Langlands, to each isogeny class of points over a finite field associate a conjugacy class of the group of points over the rationals of the group associated to the Shimura variety. These results were earlier announced by Langlands and Rapoport with an apparently faulty proof. After some preliminary discussions of general type the proof reduces to a computation with Dieudonné modules.
Reviewer: T.Ekedahl


14K10 Algebraic moduli of abelian varieties, classification
14G15 Finite ground fields in algebraic geometry
14G35 Modular and Shimura varieties
14K22 Complex multiplication and abelian varieties
14L05 Formal groups, \(p\)-divisible groups
Full Text: DOI


[1] , Algebraic Number Theory, Academic Press, London 1967.
[2] Travaux de Shimura, Sém. Bourbaki Exposé 389 (1971).
[3] Langlands, Proc. Symp. Pure Math. AMS 28 pp 401– (1976)
[4] and , 1974.
[5] Milne, Proc. Symp. Pure Math. AMS 33 pp 165– (1979)
[6] Abelian varieties, Oxford 1970. · Zbl 0223.14022
[7] Tate, Sém. Bourbaki Exposé 352 (1969)
[8] Zink, Comp. Math. 45 pp 15– (1981)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.