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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.

Mathematics --- Shimura varieties. --- MATHEMATICS / Number Theory. --- Varieties, Shimura --- Arithmetical algebraic geometry --- Math --- Science --- 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.

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Written for advanced undergraduate and first-year graduate students, this book aims to introduce students to a serious level of p-adic analysis with important implications for number theory. The main object is the study of G-series, that is, power series y=aij=0 Ajxj with coefficients in an algebraic number field K. These series satisfy a linear differential equation Ly=0 with LIK(x) [d/dx] and have non-zero radii of convergence for each imbedding of K into the complex numbers. They have the further property that the common denominators of the first s coefficients go to infinity geometrically with the index s. After presenting a review of valuation theory and elementary p-adic analysis together with an application to the congruence zeta function, this book offers a detailed study of the p-adic properties of formal power series solutions of linear differential equations. In particular, the p-adic radii of convergence and the p-adic growth of coefficients are studied. Recent work of Christol, Bombieri, André, and Dwork is treated and augmented. The book concludes with Chudnovsky's theorem: the analytic continuation of a G -series is again a G -series. This book will be indispensable for those wishing to study the work of Bombieri and André on global relations and for the study of the arithmetic properties of solutions of ordinary differential equations.

Analyse p-adique --- H-fonction --- H-functie --- H-function --- p-adic analyse --- p-adic analysis --- H-functions --- H-functions. --- p-adic analysis. --- Analysis, p-adic --- Algebra --- Calculus --- Geometry, Algebraic --- Fox's H-function --- G-functions, Generalized --- Generalized G-functions --- Generalized Mellin-Barnes functions --- Mellin-Barnes functions, Generalized --- Hypergeometric functions --- Adjoint. --- Algebraic Method. --- Algebraic closure. --- Algebraic number field. --- Algebraic number theory. --- Algebraic variety. --- Algebraically closed field. --- Analytic continuation. --- Analytic function. --- Argument principle. --- Arithmetic. --- Automorphism. --- Bearing (navigation). --- Binomial series. --- Calculation. --- Cardinality. --- Cartesian coordinate system. --- Cauchy sequence. --- Cauchy's theorem (geometry). --- Coefficient. --- Cohomology. --- Commutative ring. --- Complete intersection. --- Complex analysis. --- Conjecture. --- Density theorem. --- Differential equation. --- Dimension (vector space). --- Direct sum. --- Discrete valuation. --- Eigenvalues and eigenvectors. --- Elliptic curve. --- Equation. --- Equivalence class. --- Estimation. --- Existential quantification. --- Exponential function. --- Exterior algebra. --- Field of fractions. --- Finite field. --- Formal power series. --- Fuchs' theorem. --- G-module. --- Galois extension. --- Galois group. --- General linear group. --- Generic point. --- Geometry. --- Hypergeometric function. --- Identity matrix. --- Inequality (mathematics). --- Intercept method. --- Irreducible element. --- Irreducible polynomial. --- Laurent series. --- Limit of a sequence. --- Linear differential equation. --- Lowest common denominator. --- Mathematical induction. --- Meromorphic function. --- Modular arithmetic. --- Module (mathematics). --- Monodromy. --- Monotonic function. --- Multiplicative group. --- Natural number. --- Newton polygon. --- Number theory. --- P-adic number. --- Parameter. --- Permutation. --- Polygon. --- Polynomial. --- Projective line. --- Q.E.D. --- Quadratic residue. --- Radius of convergence. --- Rational function. --- Rational number. --- Residue field. --- Riemann hypothesis. --- Ring of integers. --- Root of unity. --- Separable polynomial. --- Sequence. --- Siegel's lemma. --- Special case. --- Square root. --- Subring. --- Subset. --- Summation. --- Theorem. --- Topology of uniform convergence. --- Transpose. --- Triangle inequality. --- Unipotent. --- Valuation ring. --- Weil conjecture. --- Wronskian. --- Y-intercept.

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This book presents a classification of all (complex)irreducible representations of a reductive group withconnected centre, over a finite field. To achieve this,the author uses etale intersection cohomology, anddetailed information on representations of Weylgroups.

512 --- Characters of groups --- Finite fields (Algebra) --- Finite groups --- Groups, Finite --- Group theory --- Modules (Algebra) --- Modular fields (Algebra) --- Algebra, Abstract --- Algebraic fields --- Galois theory --- Characters, Group --- Group characters --- Groups, Characters of --- Representations of groups --- Rings (Algebra) --- Algebra --- 512 Algebra --- Finite groups. --- Characters of groups. --- Addition. --- Algebra representation. --- Algebraic closure. --- Algebraic group. --- Algebraic variety. --- Algebraically closed field. --- Bijection. --- Borel subgroup. --- Cartan subalgebra. --- Character table. --- Character theory. --- Characteristic function (probability theory). --- Characteristic polynomial. --- Class function (algebra). --- Classical group. --- Coefficient. --- Cohomology with compact support. --- Cohomology. --- Combination. --- Complex number. --- Computation. --- Conjugacy class. --- Connected component (graph theory). --- Coxeter group. --- Cyclic group. --- Cyclotomic polynomial. --- David Kazhdan. --- Dense set. --- Derived category. --- Diagram (category theory). --- Dimension. --- Direct sum. --- Disjoint sets. --- Disjoint union. --- E6 (mathematics). --- Eigenvalues and eigenvectors. --- Endomorphism. --- Equivalence class. --- Equivalence relation. --- Existential quantification. --- Explicit formula. --- Explicit formulae (L-function). --- Fiber bundle. --- Finite field. --- Finite group. --- Fourier transform. --- Green's function. --- Group (mathematics). --- Group action. --- Group representation. --- Harish-Chandra. --- Hecke algebra. --- Identity element. --- Integer. --- Irreducible representation. --- Isomorphism class. --- Jordan decomposition. --- Line bundle. --- Linear combination. --- Local system. --- Mathematical induction. --- Maximal torus. --- Module (mathematics). --- Monodromy. --- Morphism. --- Orthonormal basis. --- P-adic number. --- Parametrization. --- Parity (mathematics). --- Partially ordered set. --- Perverse sheaf. --- Pointwise. --- Polynomial. --- Quantity. --- Rational point. --- Reductive group. --- Ree group. --- Schubert variety. --- Scientific notation. --- Semisimple Lie algebra. --- Sheaf (mathematics). --- Simple group. --- Simple module. --- Special case. --- Standard basis. --- Subset. --- Subtraction. --- Summation. --- Surjective function. --- Symmetric group. --- Tensor product. --- Theorem. --- Two-dimensional space. --- Unipotent representation. --- Vector bundle. --- Vector space. --- Verma module. --- Weil conjecture. --- Weyl group. --- Zariski topology.

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For hundreds of years, the study of elliptic curves has played a central role in mathematics. The past century in particular has seen huge progress in this study, from Mordell's theorem in 1922 to the work of Wiles and Taylor-Wiles in 1994. Nonetheless, there remain many fundamental questions where we do not even know what sort of answers to expect. This book explores two of them: What is the average rank of elliptic curves, and how does the rank vary in various kinds of families of elliptic curves? Nicholas Katz answers these questions for families of ''big'' twists of elliptic curves in the function field case (with a growing constant field). The monodromy-theoretic methods he develops turn out to apply, still in the function field case, equally well to families of big twists of objects of all sorts, not just to elliptic curves. The leisurely, lucid introduction gives the reader a clear picture of what is known and what is unknown at present, and situates the problems solved in this book within the broader context of the overall study of elliptic curves. The book's technical core makes use of, and explains, various advanced topics ranging from recent results in finite group theory to the machinery of l-adic cohomology and monodromy. Twisted L-Functions and Monodromy is essential reading for anyone interested in number theory and algebraic geometry.

L-functions. --- Monodromy groups. --- Functions, L --- -L-functions. --- Group theory --- -Number theory --- L-functions --- Monodromy groups --- Abelian variety. --- Absolute continuity. --- Addition. --- Affine space. --- Algebraically closed field. --- Ambient space. --- Average. --- Betti number. --- Birch and Swinnerton-Dyer conjecture. --- Blowing up. --- Codimension. --- Coefficient. --- Computation. --- Conjecture. --- Conjugacy class. --- Convolution. --- Critical value. --- Differential geometry of surfaces. --- Dimension (vector space). --- Dimension. --- Direct sum. --- Divisor (algebraic geometry). --- Divisor. --- Eigenvalues and eigenvectors. --- Elliptic curve. --- Equation. --- Equidistribution theorem. --- Existential quantification. --- Factorization. --- Finite field. --- Finite group. --- Finite set. --- Flat map. --- Fourier transform. --- Function field. --- Functional equation. --- Goursat's lemma. --- Ground field. --- Group representation. --- Hyperplane. --- Hypersurface. --- Integer matrix. --- Integer. --- Irreducible component. --- Irreducible polynomial. --- Irreducible representation. --- J-invariant. --- K3 surface. --- L-function. --- Lebesgue measure. --- Lefschetz pencil. --- Level of measurement. --- Lie algebra. --- Limit superior and limit inferior. --- Minimal polynomial (field theory). --- Modular form. --- Monodromy. --- Morphism. --- Numerical analysis. --- Orthogonal group. --- Percentage. --- Polynomial. --- Prime number. --- Probability measure. --- Quadratic function. --- Quantity. --- Quotient space (topology). --- Representation theory. --- Residue field. --- Riemann hypothesis. --- Root of unity. --- Scalar (physics). --- Set (mathematics). --- Sheaf (mathematics). --- Subgroup. --- Summation. --- Symmetric group. --- System of imprimitivity. --- Theorem. --- Trivial representation. --- Zariski topology.

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This book contains accounts of talks held at a symposium in honorof John C. Moore in October 1983 at Princeton University, The workincludes papers in classical homotopy theory, homological algebra,rational homotopy theory, algebraic K-theory of spaces, and othersubjects.

Algebraic topology --- K-theory --- 512.73 --- 515.14 --- 512.73 Cohomology theory of algebraic varieties and schemes --- Cohomology theory of algebraic varieties and schemes --- 515.14 Algebraic topology --- Moore, John C. --- Abelian group. --- Adams spectral sequence. --- Adjoint functors. --- Adjunction (field theory). --- Algebraic K-theory. --- Algebraic closure. --- Algebraic geometry. --- Algebraic group. --- Algebraic number field. --- Algebraic space. --- Algebraic topology. --- Algebraically closed field. --- Associative algebra. --- Boundary (topology). --- CW complex. --- Classification theorem. --- Closure (mathematics). --- Coalgebra. --- Cofibration. --- Cohomology. --- Commutative diagram. --- Commutative property. --- Coproduct. --- Deformation theory. --- Degenerate bilinear form. --- Diagram (category theory). --- Differentiable manifold. --- Dimension (vector space). --- Division algebra. --- Eilenberg–Moore spectral sequence. --- Epimorphism. --- Exterior (topology). --- Formal power series. --- Free Lie algebra. --- Free algebra. --- Freudenthal suspension theorem. --- Function (mathematics). --- Function space. --- Functor. --- G-module. --- Galois extension. --- Global dimension. --- Group cohomology. --- Group homomorphism. --- H-space. --- Hilbert's Theorem 90. --- Homology (mathematics). --- Homomorphism. --- Homotopy category. --- Homotopy group. --- Homotopy. --- Hopf algebra. --- Hopf invariant. --- Hurewicz theorem. --- Inclusion map. --- Inequality (mathematics). --- Integral domain. --- Isometry. --- Isomorphism class. --- K-theory. --- Lie algebra. --- Lie group. --- Limit (category theory). --- Loop space. --- Mathematician. --- Mathematics. --- Noetherian ring. --- Order topology. --- P-adic number. --- Polynomial ring. --- Polynomial. --- Prime number. --- Principal bundle. --- Principal ideal domain. --- Projective module. --- Projective plane. --- Pullback (category theory). --- Pushout (category theory). --- Ring of integers. --- Series (mathematics). --- Sheaf (mathematics). --- Simplicial category. --- Simplicial complex. --- Simplicial set. --- Special case. --- Spectral sequence. --- Square (algebra). --- Stable homotopy theory. --- Steenrod algebra. --- Superalgebra. --- Theorem. --- Topological K-theory. --- Topological space. --- Topology. --- Triviality (mathematics). --- Uniqueness theorem. --- Universal enveloping algebra. --- Vector bundle. --- Weak equivalence (homotopy theory). --- William Browder (mathematician). --- Géométrie algébrique --- K-théorie