Choose an application

• Reference Manager
• EndNote
• RefWorks (Direct export to RefWorks)

This book is a sequel to Lectures on Complex Analytic Varieties: The Local Paranwtrization Theorem (Mathematical Notes 10, 1970). Its unifying theme is the study of local properties of finite analytic mappings between complex analytic varieties; these mappings are those in several dimensions that most closely resemble general complex analytic mappings in one complex dimension. The purpose of this volume is rather to clarify some algebraic aspects of the local study of complex analytic varieties than merely to examine finite analytic mappings for their own sake.Originally published in 1970.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.

Complex analysis --- Analytic spaces --- Mathematics --- Physical Sciences & Mathematics --- Calculus --- Spaces, Analytic --- Analytic functions --- Functions of several complex variables --- Algebra homomorphism. --- Algebraic curve. --- Algebraic extension. --- Algebraic surface. --- Algebraic variety. --- Analytic continuation. --- Analytic function. --- Associated prime. --- Atlas (topology). --- Automorphism. --- Bernhard Riemann. --- Big O notation. --- Branch point. --- Change of variables. --- Characterization (mathematics). --- Codimension. --- Coefficient. --- Cohomology. --- Complete intersection. --- Complex analysis. --- Complex conjugate. --- Complex dimension. --- Complex number. --- Connected component (graph theory). --- Corollary. --- Critical point (mathematics). --- Diagram (category theory). --- Dimension (vector space). --- Dimension. --- Disjoint union. --- Divisor. --- Equation. --- Equivalence class. --- Exact sequence. --- Existential quantification. --- Finitely generated module. --- Geometry. --- Hamiltonian mechanics. --- Holomorphic function. --- Homeomorphism. --- Homological dimension. --- Homomorphism. --- Hypersurface. --- Ideal (ring theory). --- Identity element. --- Induced homomorphism. --- Inequality (mathematics). --- Injective function. --- Integral domain. --- Invertible matrix. --- Irreducible component. --- Isolated singularity. --- Isomorphism class. --- Jacobian matrix and determinant. --- Linear map. --- Linear subspace. --- Local ring. --- Mathematical induction. --- Mathematics. --- Maximal element. --- Maximal ideal. --- Meromorphic function. --- Modular arithmetic. --- Module (mathematics). --- Module homomorphism. --- Monic polynomial. --- Monomial. --- Neighbourhood (mathematics). --- Noetherian. --- Open set. --- Parametric equation. --- Parametrization. --- Permutation. --- Polynomial ring. --- Polynomial. --- Power series. --- Quadratic form. --- Quotient module. --- Regular local ring. --- Removable singularity. --- Ring (mathematics). --- Ring homomorphism. --- Row and column vectors. --- Scalar multiplication. --- Scientific notation. --- Several complex variables. --- Sheaf (mathematics). --- Special case. --- Subalgebra. --- Submanifold. --- Subset. --- Summation. --- Surjective function. --- Taylor series. --- Theorem. --- Three-dimensional space (mathematics). --- Topological space. --- Vector space. --- Weierstrass preparation theorem. --- Zero divisor. --- Fonctions de plusieurs variables complexes --- Variétés complexes

Choose an application

• Reference Manager
• EndNote
• RefWorks (Direct export to RefWorks)

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.

Choose an application

• Reference Manager
• EndNote
• RefWorks (Direct export to RefWorks)

The purpose of this book is to provide a self-contained account, accessible to the non-specialist, of algebra necessary for the solution of the integrability problem for transitive pseudogroup structures.Originally published in 1981.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.

Lie algebras. --- Ideals (Algebra) --- Pseudogroups. --- Global analysis (Mathematics) --- Lie groups --- Algebraic ideals --- Algebraic fields --- Rings (Algebra) --- Algebras, Lie --- Algebra, Abstract --- Algebras, Linear --- Lie algebras --- Pseudogroups --- 512.81 --- 512.81 Lie groups --- Ideals (Algebra). --- Lie, Algèbres de. --- Idéaux (algèbre) --- Pseudogroupes (mathématiques) --- Ordered algebraic structures --- Analytical spaces --- Addition. --- Adjoint representation. --- Algebra homomorphism. --- Algebra over a field. --- Algebraic extension. --- Algebraic structure. --- Analytic function. --- Associative algebra. --- Automorphism. --- Bilinear form. --- Bilinear map. --- Cartesian product. --- Closed graph theorem. --- Codimension. --- Coefficient. --- Cohomology. --- Commutative ring. --- Commutator. --- Compact space. --- Complex conjugate. --- Complexification (Lie group). --- Complexification. --- Conjecture. --- Constant term. --- Continuous function. --- Contradiction. --- Corollary. --- Counterexample. --- Diagram (category theory). --- Differentiable manifold. --- Differential form. --- Differential operator. --- Dimension (vector space). --- Dimension. --- Direct sum. --- Discrete space. --- Donald C. Spencer. --- Dual basis. --- Embedding. --- Epimorphism. --- Existential quantification. --- Exterior (topology). --- Exterior algebra. --- Exterior derivative. --- Faithful representation. --- Formal power series. --- Graded Lie algebra. --- Ground field. --- Homeomorphism. --- Homomorphism. --- Hyperplane. --- I0. --- Indeterminate (variable). --- Infinitesimal transformation. --- Injective function. --- Integer. --- Integral domain. --- Invariant subspace. --- Invariant theory. --- Isotropy. --- Jacobi identity. --- Levi decomposition. --- Lie algebra. --- Linear algebra. --- Linear map. --- Linear subspace. --- Local diffeomorphism. --- Mathematical induction. --- Maximal ideal. --- Module (mathematics). --- Monomorphism. --- Morphism. --- Natural transformation. --- Non-abelian. --- Partial differential equation. --- Pseudogroup. --- Pullback (category theory). --- Simple Lie group. --- Space form. --- Special case. --- Subalgebra. --- Submanifold. --- Subring. --- Summation. --- Symmetric algebra. --- Symplectic vector space. --- Telescoping series. --- Theorem. --- Topological algebra. --- Topological space. --- Topological vector space. --- Topology. --- Transitive relation. --- Triviality (mathematics). --- Unit vector. --- Universal enveloping algebra. --- Vector bundle. --- Vector field. --- Vector space. --- Weak topology. --- Lie, Algèbres de. --- Idéaux (algèbre) --- Pseudogroupes (mathématiques)

Choose an application

• Reference Manager
• EndNote
• RefWorks (Direct export to RefWorks)

An elliptic curve is a particular kind of cubic equation in two variables whose projective solutions form a group. Modular forms are analytic functions in the upper half plane with certain transformation laws and growth properties. The two subjects--elliptic curves and modular forms--come together in Eichler-Shimura theory, which constructs elliptic curves out of modular forms of a special kind. The converse, that all rational elliptic curves arise this way, is called the Taniyama-Weil Conjecture and is known to imply Fermat's Last Theorem. Elliptic curves and the modeular forms in the Eichler- Shimura theory both have associated L functions, and it is a consequence of the theory that the two kinds of L functions match. The theory covered by Anthony Knapp in this book is, therefore, a window into a broad expanse of mathematics--including class field theory, arithmetic algebraic geometry, and group representations--in which the concidence of L functions relates analysis and algebra in the most fundamental ways. Developing, with many examples, the elementary theory of elliptic curves, the book goes on to the subject of modular forms and the first connections with elliptic curves. The last two chapters concern Eichler-Shimura theory, which establishes a much deeper relationship between the two subjects. No other book in print treats the basic theory of elliptic curves with only undergraduate mathematics, and no other explains Eichler-Shimura theory in such an accessible manner.

Curves, Elliptic. --- Affine plane (incidence geometry). --- Affine space. --- Affine variety. --- Algebra homomorphism. --- Algebraic extension. --- Algebraic geometry. --- Algebraic integer. --- Algebraic number theory. --- Algebraic number. --- Analytic continuation. --- Analytic function. --- Associative algebra. --- Automorphism. --- Big O notation. --- Binary quadratic form. --- Birch and Swinnerton-Dyer conjecture. --- Bounded set (topological vector space). --- Change of variables. --- Characteristic polynomial. --- Coefficient. --- Compactification (mathematics). --- Complex conjugate. --- Complex manifold. --- Complex number. --- Conjecture. --- Coprime integers. --- Cusp form. --- Cyclic group. --- Degeneracy (mathematics). --- Dimension (vector space). --- Dirichlet character. --- Dirichlet series. --- Division algebra. --- Divisor. --- Eigenform. --- Eigenvalues and eigenvectors. --- Elementary symmetric polynomial. --- Elliptic curve. --- Elliptic function. --- Elliptic integral. --- Equation. --- Euler product. --- Finitely generated abelian group. --- Fourier analysis. --- Function (mathematics). --- Functional equation. --- General linear group. --- Group homomorphism. --- Group isomorphism. --- Hecke operator. --- Holomorphic function. --- Homomorphism. --- Ideal (ring theory). --- Integer matrix. --- Integer. --- Integral domain. --- Intersection (set theory). --- Inverse function theorem. --- Invertible matrix. --- Irreducible polynomial. --- Isogeny. --- J-invariant. --- Linear fractional transformation. --- Linear map. --- Liouville's theorem (complex analysis). --- Mathematical induction. --- Meromorphic function. --- Minimal polynomial (field theory). --- Modular form. --- Monic polynomial. --- Möbius transformation. --- Number theory. --- P-adic number. --- Polynomial ring. --- Power series. --- Prime factor. --- Prime number theorem. --- Prime number. --- Principal axis theorem. --- Principal ideal domain. --- Principal ideal. --- Projective line. --- Projective variety. --- Quadratic equation. --- Quadratic function. --- Quadratic reciprocity. --- Riemann surface. --- Riemann zeta function. --- Simultaneous equations. --- Special case. --- Summation. --- Taylor series. --- Theorem. --- Torsion subgroup. --- Transcendence degree. --- Uniformization theorem. --- Unique factorization domain. --- Variable (mathematics). --- Weierstrass's elliptic functions. --- Weil conjecture.

Choose an application

• Reference Manager
• EndNote
• RefWorks (Direct export to RefWorks)

One of the most important mathematical achievements of the past several decades has been A. Grothendieck's work on algebraic geometry. In the early 1960s, he and M. Artin introduced étale cohomology in order to extend the methods of sheaf-theoretic cohomology from complex varieties to more general schemes. This work found many applications, not only in algebraic geometry, but also in several different branches of number theory and in the representation theory of finite and p-adic groups. Yet until now, the work has been available only in the original massive and difficult papers. In order to provide an accessible introduction to étale cohomology, J. S. Milne offers this more elementary account covering the essential features of the theory. The author begins with a review of the basic properties of flat and étale morphisms and of the algebraic fundamental group. The next two chapters concern the basic theory of étale sheaves and elementary étale cohomology, and are followed by an application of the cohomology to the study of the Brauer group. After a detailed analysis of the cohomology of curves and surfaces, Professor Milne proves the fundamental theorems in étale cohomology -- those of base change, purity, Poincaré duality, and the Lefschetz trace formula. He then applies these theorems to show the rationality of some very general L-series.Originally published in 1980.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.

Ordered algebraic structures --- 512.73 --- 512.66 --- Geometry, Algebraic --- Homology theory --- Sheaf theory --- Cohomology, Sheaf --- Sheaf cohomology --- Sheaves, Theory of --- Sheaves (Algebraic topology) --- Algebraic topology --- Cohomology theory --- Contrahomology theory --- Algebraic geometry --- Geometry --- Cohomology theory of algebraic varieties and schemes --- Homological algebra --- Geometry, Algebraic. --- Homology theory. --- Sheaf theory. --- 512.66 Homological algebra --- 512.73 Cohomology theory of algebraic varieties and schemes --- Abelian category. --- Abelian group. --- Adjoint functors. --- Affine variety. --- Alexander Grothendieck. --- Algebraic closure. --- Algebraic cycle. --- Algebraic equation. --- Algebraic space. --- Algebraically closed field. --- Artinian. --- Automorphism. --- Base change. --- Brauer group. --- CW complex. --- Cardinal number. --- Category of sets. --- Central simple algebra. --- Chow's lemma. --- Closed immersion. --- Codimension. --- Cohomology ring. --- Cohomology. --- Cokernel. --- Commutative diagram. --- Complex number. --- Dedekind domain. --- Derived category. --- Diagram (category theory). --- Direct limit. --- Discrete valuation ring. --- Divisor. --- Epimorphism. --- Equivalence class. --- Existential quantification. --- Fibration. --- Field of fractions. --- Fine topology (potential theory). --- Finite field. --- Finite morphism. --- Flat morphism. --- Functor. --- Fundamental class. --- Fundamental group. --- G-module. --- Galois cohomology. --- Galois extension. --- Galois group. --- Generic point. --- Group scheme. --- Gysin sequence. --- Henselian ring. --- Identity element. --- Inclusion map. --- Integral domain. --- Intersection (set theory). --- Inverse limit. --- Invertible sheaf. --- Isomorphism class. --- Lefschetz pencil. --- Local ring. --- Maximal ideal. --- Module (mathematics). --- Morphism of schemes. --- Morphism. --- Noetherian. --- Open set. --- Power series. --- Presheaf (category theory). --- Prime ideal. --- Prime number. --- Principal homogeneous space. --- Profinite group. --- Projection (mathematics). --- Projective variety. --- Quasi-compact morphism. --- Residue field. --- Riemann surface. --- Sheaf (mathematics). --- Sheaf of modules. --- Special case. --- Spectral sequence. --- Stein factorization. --- Subalgebra. --- Subcategory. --- Subgroup. --- Subring. --- Subset. --- Surjective function. --- Tangent space. --- Theorem. --- Topological space. --- Topology. --- Torsion sheaf. --- Torsor (algebraic geometry). --- Vector bundle. --- Weil conjecture. --- Yoneda lemma. --- Zariski topology. --- Zariski's main theorem. --- Geometrie algebrique --- Cohomologie