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 Astérisque 2004; 331 pp; softcover Number: 291 ISBN-10: 2-85629-150-3 ISBN-13: 978-2-85629-150-4 List Price: US$95 Individual Members: US$85.50 Order Code: AST/291 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. The book is suitable for graduate students and research mathematicians interested in number theory and algebraic geometry. A publication of the Société Mathématique de France, Marseilles (SMF), distributed by the AMS in the U.S., Canada, and Mexico. Orders from other countries should be sent to the SMF. Members of the SMF receive a 30% discount from list. Readership Graduate students and research mathematicians interested in number theory and algebraic geometry. Table of Contents L. Fargues. Cohomologie des espaces de modules de groupes $$p$$-divisibles et correspondances de Langlands locales Introduction Variétés de Shimura de type P.E.L. non ramifiées Espaces de Rapoport-Zink Uniformisation des variétés de Shimura de type P.E.L. Une suite spectrale de Hochschild-Serre pour l'uniformisation de Rapoport-Zink Formule de Lefschetz sur la fibre spéciale Formule de Lefschetz sur la fibre générique Contribution de la cohomologie de la strate basique Application à la cohomologie des espaces Rapoport-Zink de type E.L. et P.E.L. Appendices Références E. Mantovan. On Certain Unitary Shimura Varieties Introduction Preliminaries Igusa varieties A system of covers of the Newton polygon strata Group action on cohomology Formally lifting to characteristic zero Shimura varieties with level structure at $$p$$ The cohomology of Shimura varieties References