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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 provides an accessible and self-contained introduction to the theory of algebraic curves over a finite field, a subject that has been of fundamental importance to mathematics for many years and that has essential applications in areas such as finite geometry, number theory, error-correcting codes, and cryptology. Unlike other books, this one emphasizes the algebraic geometry rather than the function field approach to algebraic curves. The authors begin by developing the general theory of curves over any field, highlighting peculiarities occurring for positive characteristic and requiring of the reader only basic knowledge of algebra and geometry. The special properties that a curve over a finite field can have are then discussed. The geometrical theory of linear series is used to find estimates for the number of rational points on a curve, following the theory of Stöhr and Voloch. The approach of Hasse and Weil via zeta functions is explained, and then attention turns to more advanced results: a state-of-the-art introduction to maximal curves over finite fields is provided; a comprehensive account is given of the automorphism group of a curve; and some applications to coding theory and finite geometry are described. The book includes many examples and exercises. It is an indispensable resource for researchers and the ideal textbook for graduate students.

Curves, Algebraic. --- Finite fields (Algebra) --- Modular fields (Algebra) --- Algebra, Abstract --- Algebraic fields --- Galois theory --- Modules (Algebra) --- Algebraic curves --- Algebraic varieties --- Abelian group. --- Abelian variety. --- Affine plane. --- Affine space. --- Affine variety. --- Algebraic closure. --- Algebraic curve. --- Algebraic equation. --- Algebraic extension. --- Algebraic function. --- Algebraic geometry. --- Algebraic integer. --- Algebraic number field. --- Algebraic number theory. --- Algebraic number. --- Algebraic variety. --- Algebraically closed field. --- Applied mathematics. --- Automorphism. --- Birational invariant. --- Characteristic exponent. --- Classification theorem. --- Clifford's theorem. --- Combinatorics. --- Complex number. --- Computation. --- Cyclic group. --- Cyclotomic polynomial. --- Degeneracy (mathematics). --- Degenerate conic. --- Divisor (algebraic geometry). --- Divisor. --- Dual curve. --- Dual space. --- Elliptic curve. --- Equation. --- Fermat curve. --- Finite field. --- Finite geometry. --- Finite group. --- Formal power series. --- Function (mathematics). --- Function field. --- Fundamental theorem. --- Galois extension. --- Galois theory. --- Gauss map. --- General position. --- Generic point. --- Geometry. --- Homogeneous polynomial. --- Hurwitz's theorem. --- Hyperelliptic curve. --- Hyperplane. --- Identity matrix. --- Inequality (mathematics). --- Intersection number (graph theory). --- Intersection number. --- J-invariant. --- Line at infinity. --- Linear algebra. --- Linear map. --- Mathematical induction. --- Mathematics. --- Menelaus' theorem. --- Modular curve. --- Natural number. --- Number theory. --- Parity (mathematics). --- Permutation group. --- Plane curve. --- Point at infinity. --- Polar curve. --- Polygon. --- Polynomial. --- Power series. --- Prime number. --- Projective plane. --- Projective space. --- Quadratic transformation. --- Quadric. --- Resolution of singularities. --- Riemann hypothesis. --- Scalar multiplication. --- Scientific notation. --- Separable extension. --- Separable polynomial. --- Sign (mathematics). --- Singular point of a curve. --- Special case. --- Subgroup. --- Sylow theorems. --- System of linear equations. --- Tangent. --- Theorem. --- Transcendence degree. --- Upper and lower bounds. --- Valuation ring. --- Variable (mathematics). --- Vector space.

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In this volume, the author covers profinite groups and their cohomology, Galois cohomology, and local class field theory, and concludes with a treatment of duality. His objective is to present effectively that body of material upon which all modern research in Diophantine geometry and higher arithmetic is based, and to do so in a manner that emphasizes the many interesting lines of inquiry leading from these foundations.

Group theory --- Finite groups --- Algebraic number theory --- 512.73 --- 512.66 --- Homology theory --- Number theory --- Cohomology theory --- Contrahomology theory --- Algebraic topology --- Groups, Finite --- Modules (Algebra) --- Cohomology theory of algebraic varieties and schemes --- Homological algebra --- 512.66 Homological algebra --- 512.73 Cohomology theory of algebraic varieties and schemes --- Groupes, Théorie des. --- Group theory. --- Homology theory. --- Finite groups. --- Algebraic number theory. --- Abelian group. --- Alexander Grothendieck. --- Algebraic closure. --- Algebraic extension. --- Algebraic geometry. --- Algebraic number field. --- Brauer group. --- Category of abelian groups. --- Category of sets. --- Characterization (mathematics). --- Class field theory. --- Cohomological dimension. --- Cohomology. --- Cokernel. --- Commutative diagram. --- Composition series. --- Computation. --- Connected component (graph theory). --- Coset. --- Cup product. --- Dedekind domain. --- Degeneracy (mathematics). --- Diagram (category theory). --- Dimension (vector space). --- Diophantine geometry. --- Discrete group. --- Equivalence of categories. --- Exact sequence. --- Existential quantification. --- Explicit formula. --- Exponential function. --- Family of sets. --- Field extension. --- Finite group. --- Fundamental class. --- G-module. --- Galois cohomology. --- Galois extension. --- Galois group. --- Galois module. --- Galois theory. --- General topology. --- Geometry. --- Grothendieck topology. --- Group cohomology. --- Group extension. --- Group scheme. --- Hilbert symbol. --- Hopf algebra. --- Ideal (ring theory). --- Inequality (mathematics). --- Injective sheaf. --- Inner automorphism. --- Inverse limit. --- Kummer theory. --- Lie algebra. --- Linear independence. --- Local field. --- Mathematical induction. --- Mathematician. --- Mathematics. --- Module (mathematics). --- Morphism. --- Natural topology. --- Neighbourhood (mathematics). --- Normal extension. --- Normal subgroup. --- Number theory. --- P-adic number. --- P-group. --- Polynomial. --- Pontryagin duality. --- Power series. --- Prime number. --- Principal ideal. --- Profinite group. --- Quadratic reciprocity. --- Quotient group. --- Ring of integers. --- Sheaf (mathematics). --- Special case. --- Subcategory. --- Subgroup. --- Supernatural number. --- Sylow theorems. --- Tangent space. --- Theorem. --- Topological group. --- Topological property. --- Topological ring. --- Topological space. --- Topology. --- Torsion group. --- Torsion subgroup. --- Transcendence degree. --- Triviality (mathematics). --- Unique factorization domain. --- Variable (mathematics). --- Vector space. --- Groupes, Théorie des --- Nombres, Théorie des

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A general principle, discovered by Robert Langlands and named by him the "functoriality principle," predicts relations between automorphic forms on arithmetic subgroups of different reductive groups. Langlands functoriality relates the eigenvalues of Hecke operators acting on the automorphic forms on two groups (or the local factors of the "automorphic representations" generated by them). In the few instances where such relations have been probed, they have led to deep arithmetic consequences. This book studies one of the simplest general problems in the theory, that of relating automorphic forms on arithmetic subgroups of GL(n,E) and GL(n,F) when E/F is a cyclic extension of number fields. (This is known as the base change problem for GL(n).) The problem is attacked and solved by means of the trace formula. The book relies on deep and technical results obtained by several authors during the last twenty years. It could not serve as an introduction to them, but, by giving complete references to the published literature, the authors have made the work useful to a reader who does not know all the aspects of the theory of automorphic forms.

511.33 --- Analytical and multiplicative number theory. Asymptotics. Sieves etc. --- 511.33 Analytical and multiplicative number theory. Asymptotics. Sieves etc. --- Automorfe vormen --- Automorphic forms --- Formes automorphes --- Representation des groupes --- Representations of groups --- Trace formulas --- Vertegenwoordiging van groepen --- Formulas, Trace --- Discontinuous groups --- Group representation (Mathematics) --- Groups, Representation theory of --- Group theory --- Automorphic functions --- Forms (Mathematics) --- Analytical and multiplicative number theory. Asymptotics. Sieves etc --- Representations of groups. --- Trace formulas. --- Automorphic forms. --- 0E. --- Addition. --- Admissible representation. --- Algebraic group. --- Algebraic number field. --- Approximation. --- Archimedean property. --- Automorphic form. --- Automorphism. --- Base change. --- Big O notation. --- Binomial coefficient. --- Canonical map. --- Cartan subalgebra. --- Cartan subgroup. --- Central simple algebra. --- Characteristic polynomial. --- Closure (mathematics). --- Combination. --- Computation. --- Conjecture. --- Conjugacy class. --- Connected component (graph theory). --- Continuous function. --- Contradiction. --- Corollary. --- Counting. --- Coxeter element. --- Cusp form. --- Cyclic permutation. --- Dense set. --- Density theorem. --- Determinant. --- Diagram (category theory). --- Discrete series representation. --- Discrete spectrum. --- Division algebra. --- Eigenvalues and eigenvectors. --- Eisenstein series. --- Exact sequence. --- Existential quantification. --- Field extension. --- Finite group. --- Finite set. --- Fourier transform. --- Functor. --- Fundamental lemma (Langlands program). --- Galois extension. --- Galois group. --- Global field. --- Grothendieck group. --- Group representation. --- Haar measure. --- Harmonic analysis. --- Hecke algebra. --- Hilbert's Theorem 90. --- Identity component. --- Induced representation. --- Infinite product. --- Infinitesimal character. --- Invariant measure. --- Irreducibility (mathematics). --- Irreducible representation. --- L-function. --- Langlands classification. --- Laurent series. --- Lie algebra. --- Lie group. --- Linear algebraic group. --- Local field. --- Mathematical induction. --- Maximal compact subgroup. --- Multiplicative group. --- Nilpotent group. --- Orbital integral. --- P-adic number. --- Paley–Wiener theorem. --- Parameter. --- Parametrization. --- Permutation. --- Poisson summation formula. --- Real number. --- Reciprocal lattice. --- Reductive group. --- Root of unity. --- Scientific notation. --- Semidirect product. --- Special case. --- Spherical harmonics. --- Subgroup. --- Subset. --- Summation. --- Support (mathematics). --- Tensor product. --- Theorem. --- Trace formula. --- Unitary representation. --- Weil group. --- Weyl group. --- Zero of a function.