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The Geometry and Cohomology of Some Simple Shimura Varieties. (AM-151), Volume 151 / / Richard Taylor, Michael Harris



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Autore: Harris Michael Visualizza persona
Titolo: The Geometry and Cohomology of Some Simple Shimura Varieties. (AM-151), Volume 151 / / Richard Taylor, Michael Harris Visualizza cluster
Pubblicazione: Princeton, NJ : , : Princeton University Press, , [2001]
©2002
Descrizione fisica: 1 online resource (288 p.)
Disciplina: 516.3/5
Soggetto topico: Mathematics
Shimura varieties
MATHEMATICS / Number Theory
Soggetto non controllato: Abelian 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
Persona (resp. second.): TaylorRichard
Note generali: Description based upon print version of record.
Nota di contenuto: 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 -- Index
Sommario/riassunto: This 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.
Titolo autorizzato: The Geometry and Cohomology of Some Simple Shimura Varieties. (AM-151), Volume 151  Visualizza cluster
ISBN: 1-4008-3720-0
Formato: Materiale a stampa
Livello bibliografico Monografia
Lingua di pubblicazione: Inglese
Record Nr.: 9910814893103321
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Serie: Annals of Mathematics Studies