Choose an application

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

This is the first book to deal with invariant theory and the representations of finite groups. By restricting attention to finite groups Dr Benson is able to avoid recourse to the technical machinery of algebraic groups, and he develops the necessary results from commutative algebra as he proceeds. Thus the book should be accessible to graduate students. In detail, the book contains an account of invariant theory for the action of a finite group on the ring of polynomial functions on a linear representation, both in characteristic zero and characteristic p. Special attention is paid to the role played by pseudoreflections, which arise because they correspond to the divisors in the polynomial ring which ramify over the invariants. Also included is a new proof by Crawley-Boevey and the author of the Carlisle-Kropholler conjecture. This volume will appeal to all algebraists, but especially those working in representation theory, group theory, and commutative or homological algebra.

Invariants. --- Finite groups. --- Divisor theory. --- Categories (Mathematics) --- Ideals (Algebra) --- Modules (Algebra) --- Groups, Finite --- Group theory --- Groups (Mathematics)

Choose an application

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

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

Choose an application

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

The description for this book, Topics in Transcendental Algebraic Geometry. (AM-106), Volume 106, will be forthcoming.

Geometry, Algebraic. --- Hodge theory. --- Torelli theorem. --- Géométrie algébrique --- Théorie de Hodge --- Geometry, Algebraic --- Hodge theory --- Torelli theorem --- 512.7 --- Torelli's theorem --- Curves, Algebraic --- Jacobians --- Complex manifolds --- Differentiable manifolds --- Homology theory --- Algebraic geometry --- Geometry --- Algebraic geometry. Commutative rings and algebras --- 512.7 Algebraic geometry. Commutative rings and algebras --- Géométrie algébrique --- Théorie de Hodge --- Abelian integral. --- Algebraic curve. --- Algebraic cycle. --- Algebraic equation. --- Algebraic geometry. --- Algebraic integer. --- Algebraic structure. --- Algebraic surface. --- Arithmetic genus. --- Arithmetic group. --- Asymptotic analysis. --- Automorphism. --- Base change. --- Bilinear form. --- Bilinear map. --- Cohomology. --- Combinatorics. --- Commutative diagram. --- Compactification (mathematics). --- Complete intersection. --- Complex manifold. --- Complex number. --- Computation. --- Deformation theory. --- Degeneracy (mathematics). --- Differentiable manifold. --- Dimension (vector space). --- Divisor (algebraic geometry). --- Divisor. --- Elliptic curve. --- Elliptic surface. --- Equation. --- Exact sequence. --- Fiber bundle. --- Function (mathematics). --- Fundamental class. --- Geometric genus. --- Geometry. --- Hermitian symmetric space. --- Hodge structure. --- Homology (mathematics). --- Homomorphism. --- Homotopy. --- Hypersurface. --- Intersection form (4-manifold). --- Intersection number. --- Irreducibility (mathematics). --- Isomorphism class. --- Jacobian variety. --- K3 surface. --- Kodaira dimension. --- Kronecker's theorem. --- Kummer surface. --- Kähler manifold. --- Lie algebra bundle. --- Lie algebra. --- Linear algebra. --- Linear algebraic group. --- Line–line intersection. --- Mathematical induction. --- Mathematical proof. --- Mathematics. --- Modular arithmetic. --- Module (mathematics). --- Moduli space. --- Monodromy matrix. --- Monodromy theorem. --- Monodromy. --- Nilpotent orbit. --- Normal function. --- Open set. --- Period mapping. --- Permutation group. --- Phillip Griffiths. --- Point at infinity. --- Pole (complex analysis). --- Polynomial. --- Projective space. --- Pullback (category theory). --- Quadric. --- Regular singular point. --- Resolution of singularities. --- Riemann–Roch theorem for surfaces. --- Scientific notation. --- Set (mathematics). --- Special case. --- Spectral sequence. --- Subgroup. --- Submanifold. --- Surface of general type. --- Surjective function. --- Tangent bundle. --- Theorem. --- Topology. --- Transcendental number. --- Vector space. --- Zariski topology. --- Zariski's main theorem.

Choose an application

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

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.

Choose an application

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

"Modular forms are tremendously important in various areas of mathematics, from number theory and algebraic geometry to combinatorics and lattices. Their Fourier coefficients, with Ramanujan's tau-function as a typical example, have deep arithmetic significance. Prior to this book, the fastest known algorithms for computing these Fourier coefficients took exponential time, except in some special cases. The case of elliptic curves (Schoof's algorithm) was at the birth of elliptic curve cryptography around 1985. This book gives an algorithm for computing coefficients of modular forms of level one in polynomial time. For example, Ramanujan's tau of a prime number P can be computed in time bounded by a fixed power of the logarithm of P. Such fast computation of Fourier coefficients is itself based on the main result of the book: the computation, in polynomial time, of Galois representations over finite fields attached to modular forms by the Langlands program. Because these Galois representations typically have a nonsolvable image, this result is a major step forward from explicit class field theory, and it could be described as the start of the explicit Langlands program. The computation of the Galois representations uses their realization, following Shimura and Deligne, in the torsion subgroup of Jacobian varieties of modular curves. The main challenge is then to perform the necessary computations in time polynomial in the dimension of these highly nonlinear algebraic varieties. Exact computations involving systems of polynomial equations in many variables take exponential time. This is avoided by numerical approximations with a precision that suffices to derive exact results from them. Bounds for the required precision--in other words, bounds for the height of the rational numbers that describe the Galois representation to be computed--are obtained from Arakelov theory. Two types of approximations are treated: one using complex uniformization and another one using geometry over finite fields. The book begins with a concise and concrete introduction that makes its accessible to readers without an extensive background in arithmetic geometry. And the book includes a chapter that describes actual computations"-- "This book represents a major step forward from explicit class field theory, and it could be described as the start of the 'explicit Langlands program'"--

Galois modules (Algebra) --- Class field theory. --- Algebraic number theory --- Galois module structure (Algebra) --- Galois's modules (Algebra) --- Modules (Algebra) --- Arakelov invariants. --- Arakelov theory. --- Fourier coefficients. --- Galois representation. --- Galois representations. --- Green functions. --- Hecke operators. --- Jacobians. --- Langlands program. --- Las Vegas algorithm. --- Lehmer. --- Peter Bruin. --- Ramanujan's tau function. --- Ramanujan's tau-function. --- Ramanujan's tau. --- Riemann surfaces. --- Schoof's algorithm. --- Turing machines. --- algorithms. --- arithmetic geometry. --- arithmetic surfaces. --- bounding heights. --- bounds. --- coefficients. --- complex roots. --- computation. --- computing algorithms. --- computing coefficients. --- cusp forms. --- cuspidal divisor. --- eigenforms. --- finite fields. --- height functions. --- inequality. --- lattices. --- minimal polynomial. --- modular curves. --- modular forms. --- modular representation. --- modular representations. --- modular symbols. --- nonvanishing conjecture. --- p-adic methods. --- plane curves. --- polynomial time algorithm. --- polynomial time algoriths. --- polynomial time. --- polynomials. --- power series. --- probabilistic polynomial time. --- random divisors. --- residual representation. --- square root. --- square-free levels. --- tale cohomology. --- torsion divisors. --- torsion.