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Reciprocity laws of various kinds play a central role in number theory. In the easiest case, one obtains a transparent formulation by means of roots of unity, which are special values of exponential functions. A similar theory can be developed for special values of elliptic or elliptic modular functions, and is called complex multiplication of such functions. In 1900 Hilbert proposed the generalization of these as the twelfth of his famous problems. In this book, Goro Shimura provides the most comprehensive generalizations of this type by stating several reciprocity laws in terms of abelian varieties, theta functions, and modular functions of several variables, including Siegel modular functions. This subject is closely connected with the zeta function of an abelian variety, which is also covered as a main theme in the book. The third topic explored by Shimura is the various algebraic relations among the periods of abelian integrals. The investigation of such algebraicity is relatively new, but has attracted the interest of increasingly many researchers. Many of the topics discussed in this book have not been covered before. In particular, this is the first book in which the topics of various algebraic relations among the periods of abelian integrals, as well as the special values of theta and Siegel modular functions, are treated extensively.

Ordered algebraic structures --- 512.74 --- Abelian varieties --- Modular functions --- Functions, Modular --- Elliptic functions --- Group theory --- Number theory --- Varieties, Abelian --- Geometry, Algebraic --- Algebraic groups. Abelian varieties --- 512.74 Algebraic groups. Abelian varieties --- Abelian varieties. --- Modular functions. --- Abelian extension. --- Abelian group. --- Abelian variety. --- Absolute value. --- Adele ring. --- Affine space. --- Affine variety. --- Algebraic closure. --- Algebraic equation. --- Algebraic extension. --- Algebraic number field. --- Algebraic structure. --- Algebraic variety. --- Analytic manifold. --- Automorphic function. --- Automorphism. --- Big O notation. --- CM-field. --- Characteristic polynomial. --- Class field theory. --- Coefficient. --- Complete variety. --- Complex conjugate. --- Complex multiplication. --- Complex number. --- Complex torus. --- Corollary. --- Degenerate bilinear form. --- Differential form. --- Direct product. --- Direct proof. --- Discrete valuation ring. --- Divisor. --- Eigenvalues and eigenvectors. --- Embedding. --- Endomorphism. --- Existential quantification. --- Field of fractions. --- Finite field. --- Fractional ideal. --- Function (mathematics). --- Fundamental theorem. --- Galois extension. --- Galois group. --- Galois theory. --- Generic point. --- Ground field. --- Group theory. --- Groupoid. --- Hecke character. --- Homology (mathematics). --- Homomorphism. --- Identity element. --- Integer. --- Irreducibility (mathematics). --- Irreducible representation. --- Lie group. --- Linear combination. --- Linear subspace. --- Local ring. --- Modular form. --- Natural number. --- Number theory. --- Polynomial. --- Prime factor. --- Prime ideal. --- Projective space. --- Projective variety. --- Rational function. --- Rational mapping. --- Rational number. --- Real number. --- Residue field. --- Riemann hypothesis. --- Root of unity. --- Scientific notation. --- Semisimple algebra. --- Simple algebra. --- Singular value. --- Special case. --- Subgroup. --- Subring. --- Subset. --- Summation. --- Theorem. --- Vector space. --- Zero element.

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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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This book presents many of the main developments of the past two decades in the study of real submanifolds in complex space, providing crucial background material for researchers and advanced graduate students. The techniques in this area borrow from real and complex analysis and partial differential equations, as well as from differential, algebraic, and analytical geometry. In turn, these latter areas have been enriched over the years by the study of problems in several complex variables addressed here. The authors, M. Salah Baouendi, Peter Ebenfelt, and Linda Preiss Rothschild, include extensive preliminary material to make the book accessible to nonspecialists. One of the most important topics that the authors address here is the holomorphic extension of functions and mappings that satisfy the tangential Cauchy-Riemann equations on real submanifolds. They present the main results in this area with a novel and self-contained approach. The book also devotes considerable attention to the study of holomorphic mappings between real submanifolds, and proves finite determination of such mappings by their jets under some optimal assumptions. The authors also give a thorough comparison of the various nondegeneracy conditions for manifolds and mappings and present new geometric interpretations of these conditions. Throughout the book, Cauchy-Riemann vector fields and their orbits play a central role and are presented in a setting that is both general and elementary.

Submanifolds. --- Functions of several complex variables. --- Algebraic equation. --- Algebraic function. --- Algebraic manifold. --- Algebraic variety. --- Analytic function. --- Analytic geometry. --- Antiholomorphic function. --- Arbitrarily large. --- Automorphism. --- Banach space. --- Biholomorphism. --- Boundary value problem. --- CR manifold. --- Calculation. --- Canonical coordinates. --- Cauchy sequence. --- Cauchy–Riemann equations. --- Change of variables. --- Codimension. --- Commutative algebra. --- Commutator. --- Complex analysis. --- Complex dimension. --- Complex number. --- Complex plane. --- Complex space. --- Complexification (Lie group). --- Complexification. --- Connected space. --- Continuous function. --- Counterexample. --- Degenerate bilinear form. --- Diffeomorphism. --- Differentiable manifold. --- Differential operator. --- Dimension (vector space). --- Direct proof. --- Equation. --- Existential quantification. --- Exponential map (Lie theory). --- Field of fractions. --- First-order partial differential equation. --- Formal power series. --- Frobenius theorem (differential topology). --- Frobenius theorem (real division algebras). --- Function (mathematics). --- Geometry. --- Hermitian adjoint. --- Hilbert transform. --- Holomorphic function. --- Homogeneous coordinates. --- Hopf lemma. --- Hyperfunction. --- Hyperplane. --- Hypersurface. --- Implicit function theorem. --- Integrable system. --- Integral curve. --- Integral domain. --- Intersection (set theory). --- Interval (mathematics). --- Invertible matrix. --- Irreducible polynomial. --- Kobayashi metric. --- Lie algebra. --- Linear algebra. --- Linear subspace. --- Local diffeomorphism. --- Monodromy theorem. --- Neighbourhood (mathematics). --- Open set. --- Parametrization. --- Partial differential equation. --- Poisson kernel. --- Polynomial. --- Power series. --- Pseudoconvexity. --- Right inverse. --- Several complex variables. --- Special case. --- Stokes' theorem. --- Subbundle. --- Subharmonic function. --- Submanifold. --- Summation. --- Tangent bundle. --- Tangent space. --- Tangent vector. --- Taylor series. --- Theorem. --- Topological space. --- Topology. --- Transcendence degree. --- Transversal (geometry). --- Union (set theory). --- Unit vector. --- Variable (mathematics). --- Vector field. --- Vector space. --- Weierstrass preparation theorem.

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In this classic work, Anthony W. Knapp offers a survey of representation theory of semisimple Lie groups in a way that reflects the spirit of the subject and corresponds to the natural learning process. This book is a model of exposition and an invaluable resource for both graduate students and researchers. Although theorems are always stated precisely, many illustrative examples or classes of examples are given. To support this unique approach, the author includes for the reader a useful 300-item bibliography and an extensive section of notes.

Semisimple Lie groups. --- Representations of groups. --- Groupes de Lie semi-simples --- Représentations de groupes --- Semisimple Lie groups --- Representations of groups --- Semi-simple Lie groups --- Lie groups --- Group representation (Mathematics) --- Groups, Representation theory of --- Group theory --- Représentations de groupes --- 512.547 --- 512.547 Linear representations of abstract groups. Group characters --- Linear representations of abstract groups. Group characters --- Abelian group. --- Admissible representation. --- Algebra homomorphism. --- Analytic function. --- Analytic proof. --- Associative algebra. --- Asymptotic expansion. --- Automorphic form. --- Automorphism. --- Bounded operator. --- Bounded set (topological vector space). --- Cartan subalgebra. --- Cartan subgroup. --- Category theory. --- Characterization (mathematics). --- Classification theorem. --- Cohomology. --- Complex conjugate representation. --- Complexification (Lie group). --- Complexification. --- Conjugate transpose. --- Continuous function (set theory). --- Degenerate bilinear form. --- Diagram (category theory). --- Dimension (vector space). --- Dirac operator. --- Discrete series representation. --- Distribution (mathematics). --- Eigenfunction. --- Eigenvalues and eigenvectors. --- Existence theorem. --- Explicit formulae (L-function). --- Fourier inversion theorem. --- General linear group. --- Group homomorphism. --- Haar measure. --- Heine–Borel theorem. --- Hermitian matrix. --- Hilbert space. --- Holomorphic function. --- Hyperbolic function. --- Identity (mathematics). --- Induced representation. --- Infinitesimal character. --- Integration by parts. --- Invariant subspace. --- Invertible matrix. --- Irreducible representation. --- Jacobian matrix and determinant. --- K-finite. --- Levi decomposition. --- Lie algebra. --- Locally integrable function. --- Mathematical induction. --- Matrix coefficient. --- Matrix group. --- Maximal compact subgroup. --- Meromorphic function. --- Metric space. --- Nilpotent Lie algebra. --- Norm (mathematics). --- Parity (mathematics). --- Plancherel theorem. --- Projection (linear algebra). --- Quantifier (logic). --- Reductive group. --- Representation of a Lie group. --- Representation theory. --- Schwartz space. --- Semisimple Lie algebra. --- Set (mathematics). --- Sign (mathematics). --- Solvable Lie algebra. --- Special case. --- Special linear group. --- Special unitary group. --- Subgroup. --- Summation. --- Support (mathematics). --- Symmetric algebra. --- Symmetrization. --- Symplectic group. --- Tensor algebra. --- Tensor product. --- Theorem. --- Topological group. --- Topological space. --- Topological vector space. --- Unitary group. --- Unitary matrix. --- Unitary representation. --- Universal enveloping algebra. --- Variable (mathematics). --- Vector bundle. --- Weight (representation theory). --- Weyl character formula. --- Weyl group. --- Weyl's theorem. --- ZPP (complexity). --- Zorn's lemma.