## Shimura varieties, Rapoport-Zink spaces and local Langlands correspondences. (Variétés de Shimura, espaces de Rapoport-Zink et correspondances de Langlands locales.)(French, English)Zbl 1050.11002

Astérisque 291. Paris: Société Mathématique de France (ISBN 2-85629-150-3). xii, 331 p. (2004).
The two articles contained in this volume will be reviewed individually.
Summary: This volume contains two articles. Both deal with generalizations of Michael Harris’ and Richard Taylor’s work on the cohomology of P.E.L. type Shimura varieties of signature $$(1,n-1)$$ and on the cohomology of Lubin-Tate spaces. They are based on the work of Robert Kottwitz on those varieties in the general signature case, and on the work of Michael Rapoport and Thomas Zink on moduli spaces of $$p$$-divisible groups generalizing the one of Lubin-Tate and Drinfeld. In the first article it is proved that the $$\ell$$-adique étale cohomology of some of those “supersingular” moduli spaces of $$p$$-divisible groups realizes some cases of local Langlands correspondences. For this the author establishes a formula linking the cohomology of those spaces to the one of the “supersingular” locus of a Shimura variety. Then he proves that the supercuspidal part of the cohomology of those varieties is completely contained in the one of the “supersingular” locus.
The second article links the cohomology of a Newton stratum of the Shimura variety, for example the “supersingular” stratum, to the cohomology of the attached local moduli space of $$p$$-divisible groups and to the cohomology of some global varieties in positive characteristic named Igusa varieties that generalize the classical Igusa curves attached to modular curves.

### MSC:

 11-06 Proceedings, conferences, collections, etc. pertaining to number theory 00B15 Collections of articles of miscellaneous specific interest 14-06 Proceedings, conferences, collections, etc. pertaining to algebraic geometry 11G18 Arithmetic aspects of modular and Shimura varieties 14G35 Modular and Shimura varieties 11S37 Langlands-Weil conjectures, nonabelian class field theory