TY - BOOK ID - 80750372 TI - The Geometry and Cohomology of Some Simple Shimura Varieties. (AM-151). AU - Harris, Michael AU - Taylor, R L PY - 2001 SN - 1400837200 9781400837205 0691090920 9780691090924 PB - Princeton Princeton University Press DB - UniCat KW - Mathematics KW - Shimura varieties. KW - MATHEMATICS / Number Theory. KW - Varieties, Shimura KW - Arithmetical algebraic geometry KW - Math KW - Science KW - Abelian variety. KW - Absolute value. KW - Algebraic group. KW - Algebraically closed field. KW - Artinian. KW - Automorphic form. KW - Base change. KW - Bijection. KW - Canonical map. KW - Codimension. KW - Coefficient. KW - Cohomology. KW - Compactification (mathematics). KW - Conjecture. KW - Corollary. KW - Dimension (vector space). KW - Dimension. KW - Direct limit. KW - Division algebra. KW - Eigenvalues and eigenvectors. KW - Elliptic curve. KW - Embedding. KW - Equivalence class. KW - Equivalence of categories. KW - Existence theorem. KW - Field of fractions. KW - Finite field. KW - Function field. KW - Functor. KW - Galois cohomology. KW - Galois group. KW - Generic point. KW - Geometry. KW - Hasse invariant. KW - Infinitesimal character. KW - Integer. KW - Inverse system. KW - Isomorphism class. KW - Lie algebra. KW - Local class field theory. KW - Maximal torus. KW - Modular curve. KW - Moduli space. KW - Monic polynomial. KW - P-adic number. KW - Prime number. KW - Profinite group. KW - Residue field. KW - Ring of integers. KW - Separable extension. KW - Sheaf (mathematics). KW - Shimura variety. KW - Simple group. KW - Special case. KW - Spectral sequence. KW - Square root. KW - Subset. KW - Tate module. KW - Theorem. KW - Transcendence degree. KW - Unitary group. KW - Valuative criterion. KW - Variable (mathematics). KW - Vector space. KW - Weil group. KW - Weil pairing. KW - Zariski topology. UR - https://www.unicat.be/uniCat?func=search&query=sysid:80750372 AB - 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. ER -