05878nam 22013815 450 991081489310332120230607230022.01-4008-3720-010.1515/9781400837205(CKB)2560000000080605(EBL)1744818(OCoLC)884646577(DE-B1597)446204(OCoLC)1013938914(OCoLC)979954342(DE-B1597)9781400837205(MiAaPQ)EBC1744818(EXLCZ)99256000000008060520190708d2001 fg engur|n|---|||||txtrdacontentcrdamediacrrdacarrierThe Geometry and Cohomology of Some Simple Shimura Varieties. (AM-151), Volume 151 /Richard Taylor, Michael HarrisPrinceton, NJ : Princeton University Press, [2001]©20021 online resource (288 p.)Annals of Mathematics Studies ;163Description based upon print version of record.0-691-09092-0 Frontmatter -- Contents -- Introduction -- Acknowledgements -- Chapter I. Preliminaries -- Chapter II. Barsotti-Tate groups -- Chapter III. Some simple Shimura varieties -- Chapter IV. Igusa varieties -- Chapter V. Counting Points -- Chapter VI. Automorphic forms -- Chapter VII. Applications -- Appendix. A result on vanishing cycles / Berkovich, V. G. -- Bibliography -- IndexThis book aims first to prove the local Langlands conjecture for GLn over a p-adic field and, second, to identify the action of the decomposition group at a prime of bad reduction on the l-adic cohomology of the "simple" Shimura varieties. These two problems go hand in hand. The results represent a major advance in algebraic number theory, finally proving the conjecture first proposed in Langlands's 1969 Washington lecture as a non-abelian generalization of local class field theory. The local Langlands conjecture for GLn(K), where K is a p-adic field, asserts the existence of a correspondence, with certain formal properties, relating n-dimensional representations of the Galois group of K with the representation theory of the locally compact group GLn(K). This book constructs a candidate for such a local Langlands correspondence on the vanishing cycles attached to the bad reduction over the integer ring of K of a certain family of Shimura varieties. And it proves that this is roughly compatible with the global Galois correspondence realized on the cohomology of the same Shimura varieties. The local Langlands conjecture is obtained as a corollary. Certain techniques developed in this book should extend to more general Shimura varieties, providing new instances of the local Langlands conjecture. Moreover, the geometry of the special fibers is strictly analogous to that of Shimura curves and can be expected to have applications to a variety of questions in number theory.Annals of Mathematics StudiesMathematicsGeometryGeneralMathematicsNumber TheoryShimura varietiesMATHEMATICS / Number TheorybisacshAbelian variety.Absolute value.Algebraic group.Algebraically closed field.Artinian.Automorphic form.Base change.Bijection.Canonical map.Codimension.Coefficient.Cohomology.Compactification (mathematics).Conjecture.Corollary.Dimension (vector space).Dimension.Direct limit.Division algebra.Eigenvalues and eigenvectors.Elliptic curve.Embedding.Equivalence class.Equivalence of categories.Existence theorem.Field of fractions.Finite field.Function field.Functor.Galois cohomology.Galois group.Generic point.Geometry.Hasse invariant.Infinitesimal character.Integer.Inverse system.Isomorphism class.Lie algebra.Local class field theory.Maximal torus.Modular curve.Moduli space.Monic polynomial.P-adic number.Prime number.Profinite group.Residue field.Ring of integers.Separable extension.Sheaf (mathematics).Shimura variety.Simple group.Special case.Spectral sequence.Square root.Subset.Tate module.Theorem.Transcendence degree.Unitary group.Valuative criterion.Variable (mathematics).Vector space.Weil group.Weil pairing.Zariski topology.MathematicsMathematicsShimura varieties.MATHEMATICS / Number Theory.516.3/5Harris Michael, 66777Taylor Richard, DE-B1597DE-B1597BOOK9910814893103321The Geometry and Cohomology of Some Simple Shimura Varieties. (AM-151), Volume 1513914879UNINA