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One of the most exciting new subjects in Algebraic Number Theory and Arithmetic Algebraic Geometry is the theory of Euler systems. Euler systems are special collections of cohomology classes attached to p-adic Galois representations. Introduced by Victor Kolyvagin in the late 1980's in order to bound Selmer groups attached to p-adic representations, Euler systems have since been used to solve several key problems. These include certain cases of the Birch and Swinnerton-Dyer Conjecture and the Main Conjecture of Iwasawa Theory. Because Selmer groups play a central role in Arithmetic Algebraic Geometry, Euler systems should be a powerful tool in the future development of the field. Here, in the first book to appear on the subject, Karl Rubin presents a self-contained development of the theory of Euler systems. Rubin first reviews and develops the necessary facts from Galois cohomology. He then introduces Euler systems, states the main theorems, and develops examples and applications. The remainder of the book is devoted to the proofs of the main theorems as well as some further speculations. The book assumes a solid background in algebraic Number Theory, and is suitable as an advanced graduate text. As a research monograph it will also prove useful to number theorists and researchers in Arithmetic Algebraic Geometry.

Algebraic number theory. --- p-adic numbers. --- Numbers, p-adic --- Number theory --- p-adic analysis --- Galois cohomology --- Cohomologie galoisienne. --- Algebraic number theory --- p-adic numbers --- Abelian extension. --- Abelian variety. --- Absolute Galois group. --- Algebraic closure. --- Barry Mazur. --- Big O notation. --- Birch and Swinnerton-Dyer conjecture. --- Cardinality. --- Class field theory. --- Coefficient. --- Cohomology. --- Complex multiplication. --- Conjecture. --- Corollary. --- Cyclotomic field. --- Dimension (vector space). --- Divisibility rule. --- Eigenvalues and eigenvectors. --- Elliptic curve. --- Error term. --- Euler product. --- Euler system. --- Exact sequence. --- Existential quantification. --- Field of fractions. --- Finite set. --- Functional equation. --- Galois cohomology. --- Galois group. --- Galois module. --- Gauss sum. --- Global field. --- Heegner point. --- Ideal class group. --- Integer. --- Inverse limit. --- Inverse system. --- Karl Rubin. --- Local field. --- Mathematical induction. --- Maximal ideal. --- Modular curve. --- Modular elliptic curve. --- Natural number. --- Orthogonality. --- P-adic number. --- Pairing. --- Principal ideal. --- R-factor (crystallography). --- Ralph Greenberg. --- Remainder. --- Residue field. --- Ring of integers. --- Scientific notation. --- Selmer group. --- Subgroup. --- Tate module. --- Taylor series. --- Tensor product. --- Theorem. --- Upper and lower bounds. --- Victor Kolyvagin. --- Courbes elliptiques --- Nombres, Théorie des

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Mumford-Tate groups are the fundamental symmetry groups of Hodge theory, a subject which rests at the center of contemporary complex algebraic geometry. This book is the first comprehensive exploration of Mumford-Tate groups and domains. Containing basic theory and a wealth of new views and results, it will become an essential resource for graduate students and researchers. Although Mumford-Tate groups can be defined for general structures, their theory and use to date has mainly been in the classical case of abelian varieties. While the book does examine this area, it focuses on the nonclassical case. The general theory turns out to be very rich, such as in the unexpected connections of finite dimensional and infinite dimensional representation theory of real, semisimple Lie groups. The authors give the complete classification of Hodge representations, a topic that should become a standard in the finite-dimensional representation theory of noncompact, real, semisimple Lie groups. They also indicate that in the future, a connection seems ready to be made between Lie groups that admit discrete series representations and the study of automorphic cohomology on "ients of Mumford-Tate domains by arithmetic groups. Bringing together complex geometry, representation theory, and arithmetic, this book opens up a fresh perspective on an important subject.

Mumford-Tate groups. --- Geometry, Algebraic. --- Algebraic geometry --- Geometry --- Hodge theory --- Mumford-Tate groups --- Deligne torus integer. --- Galois group. --- Grothendieck conjecture. --- Hodge decomposition. --- Hodge domain. --- Hodge filtration. --- Hodge orientation. --- Hodge representation. --- Hodge structure. --- Hodge tensor. --- Hodge theory. --- Kubota rank. --- Lie algebra representation. --- Lie group. --- Mumford-Tate domain. --- Mumford-Tate group. --- Mumford-Tate subdomain. --- Noether-Lefschetz locus. --- Vogan diagram method. --- Weyl group. --- abelian variety. --- absolute Hodge class. --- algebraic geometry. --- arithmetic group. --- automorphic cohomology. --- classical group. --- compact dual. --- complex manifold. --- complex multiplication Hodge structure. --- complex multiplication. --- endomorphism algebra. --- exceptional group. --- holomorphic mapping. --- homogeneous complex manifold. --- homomorphism. --- mixed Hodge structure. --- moduli space. --- monodromy group. --- natural symmetry group. --- oriented imaginary number fields. --- period domain. --- period map. --- polarization. --- polarized Hodge structure. --- pure Hodge structure. --- reflex field. --- semisimple Lie algebra. --- semisimple Lie group. --- Γ-equivalence classes.

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The study of exponential sums over finite fields, begun by Gauss nearly two centuries ago, has been completely transformed in recent years by advances in algebraic geometry, culminating in Deligne's work on the Weil Conjectures. It now appears as a very attractive mixture of algebraic geometry, representation theory, and the sheaf-theoretic incarnations of such standard constructions of classical analysis as convolution and Fourier transform. The book is simultaneously an account of some of these ideas, techniques, and results, and an account of their application to concrete equidistribution questions concerning Kloosterman sums and Gauss sums.

Group theory --- Algebraic geometry --- Number theory --- 511.33 --- Analytical and multiplicative number theory. Asymptotics. Sieves etc. --- 511.33 Analytical and multiplicative number theory. Asymptotics. Sieves etc. --- Gaussian sums --- Homology theory --- Kloosterman sums --- Monodromy groups --- Kloostermann sums --- Sums, Kloosterman --- Sums, Kloostermann --- Exponential sums --- Cohomology theory --- Contrahomology theory --- Algebraic topology --- Gauss sums --- Sums, Gaussian --- Analytical and multiplicative number theory. Asymptotics. Sieves etc --- Gaussian sums. --- Kloosterman sums. --- Homology theory. --- Monodromy groups. --- Number theory. --- Nombres, Théorie des. --- Exponential sums. --- Sommes exponentielles. --- Arithmetic --- Arithmétique --- Geometry, Algebraic. --- Géométrie algébrique --- Abelian category. --- Absolute Galois group. --- Absolute value. --- Additive group. --- Adjoint representation. --- Affine variety. --- Algebraic group. --- Automorphic form. --- Automorphism. --- Big O notation. --- Cartan subalgebra. --- Characteristic polynomial. --- Classification theorem. --- Coefficient. --- Cohomology. --- Cokernel. --- Combination. --- Commutator. --- Compactification (mathematics). --- Complex Lie group. --- Complex number. --- Conjugacy class. --- Continuous function. --- Convolution theorem. --- Convolution. --- Determinant. --- Diagonal matrix. --- Dimension (vector space). --- Direct sum. --- Dual basis. --- Eigenvalues and eigenvectors. --- Empty set. --- Endomorphism. --- Equidistribution theorem. --- Estimation. --- Exactness. --- Existential quantification. --- Exponential sum. --- Exterior algebra. --- Faithful representation. --- Finite field. --- Finite group. --- Four-dimensional space. --- Frobenius endomorphism. --- Fundamental group. --- Fundamental representation. --- Galois group. --- Gauss sum. --- Homomorphism. --- Integer. --- Irreducibility (mathematics). --- Isomorphism class. --- Kloosterman sum. --- L-function. --- Leray spectral sequence. --- Lie algebra. --- Lie theory. --- Maximal compact subgroup. --- Method of moments (statistics). --- Monodromy theorem. --- Monodromy. --- Morphism. --- Multiplicative group. --- Natural number. --- Nilpotent. --- Open problem. --- P-group. --- Pairing. --- Parameter space. --- Parameter. --- Partially ordered set. --- Perfect field. --- Point at infinity. --- Polynomial ring. --- Prime number. --- Quotient group. --- Representation ring. --- Representation theory. --- Residue field. --- Riemann hypothesis. --- Root of unity. --- Sheaf (mathematics). --- Simple Lie group. --- Skew-symmetric matrix. --- Smooth morphism. --- Special case. --- Spin representation. --- Subgroup. --- Support (mathematics). --- Symmetric matrix. --- Symplectic group. --- Symplectic vector space. --- Tensor product. --- Theorem. --- Trace (linear algebra). --- Trivial representation. --- Variable (mathematics). --- Weil conjectures. --- Weyl character formula. --- 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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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.