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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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In this monograph p-adic period domains are associated to arbitrary reductive groups. Using the concept of rigid-analytic period maps the relation of p-adic period domains to moduli space of p-divisible groups is investigated. In addition, non-archimedean uniformization theorems for general Shimura varieties are established. The exposition includes background material on Grothendieck's "mysterious functor" (Fontaine theory), on moduli problems of p-divisible groups, on rigid analytic spaces, and on the theory of Shimura varieties, as well as an exposition of some aspects of Drinfelds' original construction. In addition, the material is illustrated throughout the book with numerous examples.

p-adic groups --- p-divisible groups --- Moduli theory --- 512.7 --- Theory of moduli --- Analytic spaces --- Functions of several complex variables --- Geometry, Algebraic --- Groups, p-divisible --- Group schemes (Mathematics) --- Groups, p-adic --- Group theory --- Algebraic geometry. Commutative rings and algebras --- 512.7 Algebraic geometry. Commutative rings and algebras --- p-divisible groups. --- Moduli theory. --- p-adic groups. --- Abelian variety. --- Addition. --- Alexander Grothendieck. --- Algebraic closure. --- Algebraic number field. --- Algebraic space. --- Algebraically closed field. --- Artinian ring. --- Automorphism. --- Base change. --- Basis (linear algebra). --- Big O notation. --- Bilinear form. --- Canonical map. --- Cohomology. --- Cokernel. --- Commutative algebra. --- Commutative ring. --- Complex multiplication. --- Conjecture. --- Covering space. --- Degenerate bilinear form. --- Diagram (category theory). --- Dimension (vector space). --- Dimension. --- Duality (mathematics). --- Elementary function. --- Epimorphism. --- Equation. --- Existential quantification. --- Fiber bundle. --- Field of fractions. --- Finite field. --- Formal scheme. --- Functor. --- Galois group. --- General linear group. --- Geometric invariant theory. --- Hensel's lemma. --- Homomorphism. --- Initial and terminal objects. --- Inner automorphism. --- Integral domain. --- Irreducible component. --- Isogeny. --- Isomorphism class. --- Linear algebra. --- Linear algebraic group. --- Local ring. --- Local system. --- Mathematical induction. --- Maximal ideal. --- Maximal torus. --- Module (mathematics). --- Moduli space. --- Monomorphism. --- Morita equivalence. --- Morphism. --- Multiplicative group. --- Noetherian ring. --- Open set. --- Orthogonal basis. --- Orthogonal complement. --- Orthonormal basis. --- P-adic number. --- Parity (mathematics). --- Period mapping. --- Prime element. --- Prime number. --- Projective line. --- Projective space. --- Quaternion algebra. --- Reductive group. --- Residue field. --- Rigid analytic space. --- Semisimple algebra. --- Sheaf (mathematics). --- Shimura variety. --- Special case. --- Subalgebra. --- Subgroup. --- Subset. --- Summation. --- Supersingular elliptic curve. --- Support (mathematics). --- Surjective function. --- Symmetric bilinear form. --- Symmetric space. --- Tate module. --- Tensor algebra. --- Tensor product. --- Theorem. --- Topological ring. --- Topology. --- Torsor (algebraic geometry). --- Uniformization theorem. --- Uniformization. --- Unitary group. --- Weil group. --- Zariski topology.

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Among the many differences between classical and p-adic objects, those related to differential equations occupy a special place. For example, a closed p-adic analytic one-form defined on a simply-connected domain does not necessarily have a primitive in the class of analytic functions. In the early 1980s, Robert Coleman discovered a way to construct primitives of analytic one-forms on certain smooth p-adic analytic curves in a bigger class of functions. Since then, there have been several attempts to generalize his ideas to smooth p-adic analytic spaces of higher dimension, but the spaces considered were invariably associated with algebraic varieties. This book aims to show that every smooth p-adic analytic space is provided with a sheaf of functions that includes all analytic ones and satisfies a uniqueness property. It also contains local primitives of all closed one-forms with coefficients in the sheaf that, in the case considered by Coleman, coincide with those he constructed. In consequence, one constructs a parallel transport of local solutions of a unipotent differential equation and an integral of a closed one-form along a path so that both depend nontrivially on the homotopy class of the path. Both the author's previous results on geometric properties of smooth p-adic analytic spaces and the theory of isocrystals are further developed in this book, which is aimed at graduate students and mathematicians working in the areas of non-Archimedean analytic geometry, number theory, and algebraic geometry.

p-adic analysis. --- Analysis, p-adic --- Algebra --- Calculus --- Geometry, Algebraic --- Abelian category. --- Acting in. --- Addition. --- Aisle. --- Algebraic closure. --- Algebraic curve. --- Algebraic structure. --- Algebraic variety. --- Allegory (category theory). --- Analytic function. --- Analytic geometry. --- Analytic space. --- Archimedean property. --- Arithmetic. --- Banach algebra. --- Bertolt Brecht. --- Buttress. --- Centrality. --- Clerestory. --- Commutative diagram. --- Commutative property. --- Complex analysis. --- Contradiction. --- Corollary. --- Cosmetics. --- De Rham cohomology. --- Determinant. --- Diameter. --- Differential form. --- Dimension (vector space). --- Divisor. --- Elaboration. --- Embellishment. --- Equanimity. --- Equivalence class (music). --- Existential quantification. --- Facet (geometry). --- Femininity. --- Finite morphism. --- Formal scheme. --- Fred Astaire. --- Functor. --- Gavel. --- Generic point. --- Geometry. --- Gothic architecture. --- Homomorphism. --- Hypothesis. --- Imagery. --- Injective function. --- Irreducible component. --- Iterated integral. --- Linear combination. --- Logarithm. --- Marni Nixon. --- Masculinity. --- Mathematical induction. --- Mathematics. --- Mestizo. --- Metaphor. --- Morphism. --- Natural number. --- Neighbourhood (mathematics). --- Neuroticism. --- Noetherian. --- Notation. --- One-form. --- Open set. --- P-adic Hodge theory. --- P-adic number. --- Parallel transport. --- Patrick Swayze. --- Phrenology. --- Politics. --- Polynomial. --- Prediction. --- Proportion (architecture). --- Pullback. --- Purely inseparable extension. --- Reims. --- Requirement. --- Residue field. --- Rhomboid. --- Roland Barthes. --- Satire. --- Self-sufficiency. --- Separable extension. --- Sheaf (mathematics). --- Shuffle algebra. --- Subgroup. --- Suggestion. --- Technology. --- Tensor product. --- Theorem. --- Transept. --- Triforium. --- Tubular neighborhood. --- Underpinning. --- Writing. --- Zariski topology.

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Modular Forms and Special Cycles on Shimura Curves is a thorough study of the generating functions constructed from special cycles, both divisors and zero-cycles, on the arithmetic surface "M" attached to a Shimura curve "M" over the field of rational numbers. These generating functions are shown to be the q-expansions of modular forms and Siegel modular forms of genus two respectively, valued in the Gillet-Soulé arithmetic Chow groups of "M". The two types of generating functions are related via an arithmetic inner product formula. In addition, an analogue of the classical Siegel-Weil formula identifies the generating function for zero-cycles as the central derivative of a Siegel Eisenstein series. As an application, an arithmetic analogue of the Shimura-Waldspurger correspondence is constructed, carrying holomorphic cusp forms of weight 3/2 to classes in the Mordell-Weil group of "M". In certain cases, the nonvanishing of this correspondence is related to the central derivative of the standard L-function for a modular form of weight 2. These results depend on a novel mixture of modular forms and arithmetic geometry and should provide a paradigm for further investigations. The proofs involve a wide range of techniques, including arithmetic intersection theory, the arithmetic adjunction formula, representation densities of quadratic forms, deformation theory of p-divisible groups, p-adic uniformization, the Weil representation, the local and global theta correspondence, and the doubling integral representation of L-functions.

Arithmetical algebraic geometry. --- Shimura varieties. --- Varieties, Shimura --- Algebraic geometry, Arithmetical --- Arithmetic algebraic geometry --- Diophantine geometry --- Geometry, Arithmetical algebraic --- Geometry, Diophantine --- Arithmetical algebraic geometry --- Number theory --- Abelian group. --- Addition. --- Adjunction formula. --- Algebraic number theory. --- Arakelov theory. --- Arithmetic. --- Automorphism. --- Bijection. --- Borel subgroup. --- Calculation. --- Chow group. --- Coefficient. --- Cohomology. --- Combinatorics. --- Compact Riemann surface. --- Complex multiplication. --- Complex number. --- Cup product. --- Deformation theory. --- Derivative. --- Dimension. --- Disjoint union. --- Divisor. --- Dual pair. --- Eigenfunction. --- Eigenvalues and eigenvectors. --- Eisenstein series. --- Elliptic curve. --- Endomorphism. --- Equation. --- Explicit formulae (L-function). --- Fields Institute. --- Formal group. --- Fourier series. --- Fundamental matrix (linear differential equation). --- Galois group. --- Generating function. --- Green's function. --- Group action. --- Induced representation. --- Intersection (set theory). --- Intersection number. --- Irreducible component. --- Isomorphism class. --- L-function. --- Laurent series. --- Level structure. --- Line bundle. --- Local ring. --- Mathematical sciences. --- Mathematics. --- Metaplectic group. --- Modular curve. --- Modular form. --- Modularity (networks). --- Moduli space. --- Multiple integral. --- Number theory. --- Numerical integration. --- Orbifold. --- Orthogonal complement. --- P-adic number. --- Pairing. --- Prime factor. --- Prime number. --- Pullback (category theory). --- Pullback (differential geometry). --- Pullback. --- Quadratic form. --- Quadratic residue. --- Quantity. --- Quaternion algebra. --- Quaternion. --- Quotient stack. --- Rational number. --- Real number. --- Residue field. --- Riemann zeta function. --- Ring of integers. --- SL2(R). --- Scientific notation. --- Shimura variety. --- Siegel Eisenstein series. --- Siegel modular form. --- Special case. --- Standard L-function. --- Subgroup. --- Subset. --- Summation. --- Tensor product. --- Test vector. --- Theorem. --- Three-dimensional space (mathematics). --- Topology. --- Trace (linear algebra). --- Triangular matrix. --- Two-dimensional space. --- Uniformization. --- Valuative criterion. --- Whittaker function.