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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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A survey, thorough and timely, of the singularities of two-dimensional normal complex analytic varieties, the volume summarizes the results obtained since Hirzebruch's thesis (1953) and presents new contributions. First, the singularity is resolved and shown to be classified by its resolution; then, resolutions are classed by the use of spaces with nilpotents; finally, the spaces with nilpotents are determined by means of the local ring structure of the singularity.

Algebraic geometry --- Analytic spaces --- SINGULARITIES (Mathematics) --- 512.76 --- Singularities (Mathematics) --- Geometry, Algebraic --- Spaces, Analytic --- Analytic functions --- Functions of several complex variables --- Birational geometry. Mappings etc. --- Analytic spaces. --- Singularities (Mathematics). --- 512.76 Birational geometry. Mappings etc. --- Birational geometry. Mappings etc --- Analytic function. --- Analytic set. --- Analytic space. --- Automorphism. --- Bernhard Riemann. --- Big O notation. --- Calculation. --- Chern class. --- Codimension. --- Coefficient. --- Cohomology. --- Compact Riemann surface. --- Complex manifold. --- Computation. --- Connected component (graph theory). --- Continuous function. --- Contradiction. --- Coordinate system. --- Corollary. --- Covering space. --- Dimension. --- Disjoint union. --- Divisor. --- Dual graph. --- Elliptic curve. --- Elliptic function. --- Embedding. --- Existential quantification. --- Factorization. --- Fiber bundle. --- Finite set. --- Formal power series. --- Hausdorff space. --- Holomorphic function. --- Homeomorphism. --- Homology (mathematics). --- Intersection (set theory). --- Intersection number (graph theory). --- Inverse limit. --- Irreducible component. --- Isolated singularity. --- Iteration. --- Lattice (group). --- Line bundle. --- Linear combination. --- Line–line intersection. --- Local coordinates. --- Local ring. --- Mathematical induction. --- Maximal ideal. --- Meromorphic function. --- Monic polynomial. --- Nilpotent. --- Normal bundle. --- Open set. --- Parameter. --- Plane curve. --- Pole (complex analysis). --- Power series. --- Presheaf (category theory). --- Projective line. --- Quadratic transformation. --- Quantity. --- Riemann surface. --- Riemann–Roch theorem. --- Several complex variables. --- Submanifold. --- Subset. --- Tangent bundle. --- Tangent space. --- Tensor algebra. --- Theorem. --- Topological space. --- Transition function. --- Two-dimensional space. --- Variable (mathematics). --- Zero divisor. --- Zero of a function. --- Zero set. --- Variétés complexes --- Espaces analytiques

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The theory of infinite loop spaces has been the center of much recent activity in algebraic topology. Frank Adams surveys this extensive work for researchers and students. Among the major topics covered are generalized cohomology theories and spectra; infinite-loop space machines in the sense of Boadman-Vogt, May, and Segal; localization and group completion; the transfer; the Adams conjecture and several proofs of it; and the recent theories of Adams and Priddy and of Madsen, Snaith, and Tornehave.

Algebraic topology --- Loop spaces --- Espaces de lacets --- Infinite loop spaces. --- Abelian group. --- Adams spectral sequence. --- Adjoint functors. --- Algebraic K-theory. --- Algebraic topology. --- Automorphism. --- Axiom. --- Bott periodicity theorem. --- CW complex. --- Calculation. --- Cartesian product. --- Cobordism. --- Coefficient. --- Cofibration. --- Cohomology operation. --- Cohomology ring. --- Cohomology. --- Commutative diagram. --- Continuous function. --- Counterexample. --- De Rham cohomology. --- Diagram (category theory). --- Differentiable manifold. --- Dimension. --- Discrete space. --- Disjoint union. --- Double coset. --- Eilenberg. --- Eilenberg–Steenrod axioms. --- Endomorphism. --- Epimorphism. --- Equivalence class. --- Euler class. --- Existential quantification. --- Explicit formulae (L-function). --- Exterior algebra. --- F-space. --- Fiber bundle. --- Fibration. --- Finite group. --- Function composition. --- Function space. --- Functor. --- Fundamental class. --- Fundamental group. --- Geometry. --- H-space. --- Homology (mathematics). --- Homomorphism. --- Homotopy category. --- Homotopy group. --- Homotopy. --- Hurewicz theorem. --- Inverse limit. --- J-homomorphism. --- K-theory. --- Limit (mathematics). --- Loop space. --- Mathematical induction. --- Maximal torus. --- Module (mathematics). --- Monoid. --- Monoidal category. --- Moore space. --- Morphism. --- Multiplication. --- Natural transformation. --- P-adic number. --- P-complete. --- Parameter space. --- Permutation. --- Prime number. --- Principal bundle. --- Principal ideal domain. --- Pullback (category theory). --- Quotient space (topology). --- Reduced homology. --- Riemannian manifold. --- Ring spectrum. --- Serre spectral sequence. --- Simplicial set. --- Simplicial space. --- Special case. --- Spectral sequence. --- Stable homotopy theory. --- Steenrod algebra. --- Subalgebra. --- Subring. --- Subset. --- Surjective function. --- Theorem. --- Theory. --- Topological K-theory. --- Topological ring. --- Topological space. --- Topology. --- Universal bundle. --- Universal coefficient theorem. --- Vector bundle. --- Weak equivalence (homotopy theory). --- Topologie algébrique

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Written by Arthur Ogus on the basis of notes from Pierre Berthelot's seminar on crystalline cohomology at Princeton University in the spring of 1974, this book constitutes an informal introduction to a significant branch of algebraic geometry. Specifically, it provides the basic tools used in the study of crystalline cohomology of algebraic varieties in positive characteristic.Originally published in 1978.The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

Algebraic geometry --- Geometry, Algebraic. --- Homology theory. --- Functions, Zeta. --- Zeta functions --- Cohomology theory --- Contrahomology theory --- Algebraic topology --- Geometry --- Abelian category. --- Additive map. --- Adjoint functors. --- Adjunction (field theory). --- Adjunction formula. --- Alexander Grothendieck. --- Algebra homomorphism. --- Artinian. --- Automorphism. --- Axiom. --- Banach space. --- Base change map. --- Base change. --- Betti number. --- Calculation. --- Cartesian product. --- Category of abelian groups. --- Characteristic polynomial. --- Characterization (mathematics). --- Closed immersion. --- Codimension. --- Coefficient. --- Cohomology. --- Cokernel. --- Commutative diagram. --- Commutative property. --- Commutative ring. --- Compact space. --- Corollary. --- Crystalline cohomology. --- De Rham cohomology. --- Degeneracy (mathematics). --- Derived category. --- Diagram (category theory). --- Differential operator. --- Discrete valuation ring. --- Divisibility rule. --- Dual basis. --- Eigenvalues and eigenvectors. --- Endomorphism. --- Epimorphism. --- Equation. --- Equivalence of categories. --- Exact sequence. --- Existential quantification. --- Explicit formula. --- Explicit formulae (L-function). --- Exponential type. --- Exterior algebra. --- Exterior derivative. --- Formal power series. --- Formal scheme. --- Frobenius endomorphism. --- Functor. --- Fundamental theorem. --- Hasse invariant. --- Hodge theory. --- Homotopy. --- Ideal (ring theory). --- Initial and terminal objects. --- Inverse image functor. --- Inverse limit. --- Inverse system. --- K-theory. --- Leray spectral sequence. --- Linear map. --- Linearization. --- Locally constant function. --- Mapping cone (homological algebra). --- Mathematical induction. --- Maximal ideal. --- Module (mathematics). --- Monomial. --- Monotonic function. --- Morphism. --- Natural transformation. --- Newton polygon. --- Noetherian ring. --- Noetherian. --- P-adic number. --- Polynomial. --- Power series. --- Presheaf (category theory). --- Projective module. --- Scientific notation. --- Series (mathematics). --- Sheaf (mathematics). --- Sheaf of modules. --- Special case. --- Spectral sequence. --- Subring. --- Subset. --- Symmetric algebra. --- Theorem. --- Topological space. --- Topology. --- Topos. --- Transitive relation. --- Universal property. --- Zariski topology. --- Geometrie algebrique --- Topologie algebrique --- Varietes algebriques --- Cohomologie

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Over the field of real numbers, analytic geometry has long been in deep interaction with algebraic geometry, bringing the latter subject many of its topological insights. In recent decades, model theory has joined this work through the theory of o-minimality, providing finiteness and uniformity statements and new structural tools. For non-archimedean fields, such as the p-adics, the Berkovich analytification provides a connected topology with many thoroughgoing analogies to the real topology on the set of complex points, and it has become an important tool in algebraic dynamics and many other areas of geometry. This book lays down model-theoretic foundations for non-archimedean geometry. The methods combine o-minimality and stability theory. Definable types play a central role, serving first to define the notion of a point and then properties such as definable compactness. Beyond the foundations, the main theorem constructs a deformation retraction from the full non-archimedean space of an algebraic variety to a rational polytope. This generalizes previous results of V. Berkovich, who used resolution of singularities methods. No previous knowledge of non-archimedean geometry is assumed. Model-theoretic prerequisites are reviewed in the first sections.

Tame algebras. --- Algebras, Tame --- Associative algebras --- Abhyankar property. --- Berkovich space. --- Galois orbit. --- Riemann-Roch. --- Zariski dense open set. --- Zariski open subset. --- Zariski topology. --- algebraic geometry. --- algebraic variety. --- algebraically closed valued field. --- analytic geometry. --- birational invariant. --- canonical extension. --- connectedness. --- continuity criteria. --- continuous definable map. --- continuous map. --- curve fibration. --- definable compactness. --- definable function. --- definable homotopy type. --- definable set. --- definable space. --- definable subset. --- definable topological space. --- definable topology. --- definable type. --- definably compact set. --- deformation retraction. --- finite simplicial complex. --- finite-dimensional vector space. --- forward-branching point. --- fundamental space. --- g-continuity. --- g-continuous. --- g-open set. --- germ. --- good metric. --- homotopy equivalence. --- homotopy. --- imaginary base set. --- ind-definable set. --- ind-definable subset. --- inflation homotopy. --- inflation. --- inverse limit. --- iso-definability. --- iso-definable set. --- iso-definable subset. --- iterated place. --- linear topology. --- main theorem. --- model theory. --- morphism. --- natural functor. --- non-archimedean geometry. --- non-archimedean tame topology. --- o-minimal formulation. --- o-minimality. --- orthogonality. --- path. --- pro-definable bijection. --- pro-definable map. --- pro-definable set. --- pro-definable subset. --- pseudo-Galois covering. --- real numbers. --- relatively compact set. --- residue field extension. --- retraction. --- schematic distance. --- semi-lattice. --- sequence. --- smooth case. --- smoothness. --- stability theory. --- stable completion. --- stable domination. --- stably dominated point. --- stably dominated type. --- stably dominated. --- strong stability. --- substructure. --- topological embedding. --- topological space. --- topological structure. --- topology. --- transcendence degree. --- v-continuity. --- valued field. --- Γ-internal set. --- Γ-internal space. --- Γ-internal subset.