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The description for this book, Ramification Theoretic Methods in Algebraic Geometry (AM-43), Volume 43, will be forthcoming.

Algebraic fields. --- Geometry, Algebraic. --- Abelian group. --- Abstract algebra. --- Additive group. --- Affine variety. --- Algebraic closure. --- Algebraic curve. --- Algebraic equation. --- Algebraic function field. --- Algebraic function. --- Algebraic geometry. --- Algebraic number theory. --- Algebraic surface. --- Algebraic variety. --- Big O notation. --- Birational geometry. --- Branch point. --- Cardinal number. --- Cardinality. --- Complex number. --- Degrees of freedom (statistics). --- Dimension. --- Equation. --- Equivalence class. --- Existential quantification. --- Field extension. --- Field of fractions. --- Foundations of Algebraic Geometry. --- Function field. --- Galois group. --- Generic point. --- Ground field. --- Homomorphism. --- Ideal theory. --- Integer. --- Irrational number. --- Irreducible component. --- Linear algebra. --- Local ring. --- Mathematics. --- Max Noether. --- Maximal element. --- Maximal ideal. --- Natural number. --- Nilpotent. --- Noetherian ring. --- Null set. --- Order by. --- Order type. --- Parameter. --- Primary ideal. --- Prime ideal. --- Prime number. --- Projective variety. --- Quantity. --- Quotient ring. --- Ramification group. --- Rational function. --- Rational number. --- Real number. --- Resolution of singularities. --- Riemann surface. --- Ring (mathematics). --- Special case. --- Splitting field. --- Subgroup. --- Subset. --- Theorem. --- Theory of equations. --- Transcendence degree. --- Two-dimensional space. --- Uniformization. --- Valuation ring. --- Variable (mathematics). --- Vector space. --- Zero divisor. --- Zorn's lemma.

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In recent years, considerable progress has been made in studying algebraic cycles using infinitesimal methods. These methods have usually been applied to Hodge-theoretic constructions such as the cycle class and the Abel-Jacobi map. Substantial advances have also occurred in the infinitesimal theory for subvarieties of a given smooth variety, centered around the normal bundle and the obstructions coming from the normal bundle's first cohomology group. Here, Mark Green and Phillip Griffiths set forth the initial stages of an infinitesimal theory for algebraic cycles. The book aims in part to understand the geometric basis and the limitations of Spencer Bloch's beautiful formula for the tangent space to Chow groups. Bloch's formula is motivated by algebraic K-theory and involves differentials over Q. The theory developed here is characterized by the appearance of arithmetic considerations even in the local infinitesimal theory of algebraic cycles. The map from the tangent space to the Hilbert scheme to the tangent space to algebraic cycles passes through a variant of an interesting construction in commutative algebra due to Angéniol and Lejeune-Jalabert. The link between the theory given here and Bloch's formula arises from an interpretation of the Cousin flasque resolution of differentials over Q as the tangent sequence to the Gersten resolution in algebraic K-theory. The case of 0-cycles on a surface is used for illustrative purposes to avoid undue technical complications.

512.73 --- Cohomology theory of algebraic varieties and schemes --- 512.73 Cohomology theory of algebraic varieties and schemes --- Algebraic cycles. --- Hodge theory. --- Geometry, Algebraic. --- Algebraic geometry --- Geometry --- Complex manifolds --- Differentiable manifolds --- Geometry, Algebraic --- Homology theory --- Cycles, Algebraic --- Algebraic cycles --- Hodge theory --- Addition. --- Algebraic K-theory. --- Algebraic character. --- Algebraic curve. --- Algebraic cycle. --- Algebraic function. --- Algebraic geometry. --- Algebraic number. --- Algebraic surface. --- Algebraic variety. --- Analytic function. --- Approximation. --- Arithmetic. --- Chow group. --- Codimension. --- Coefficient. --- Coherent sheaf cohomology. --- Coherent sheaf. --- Cohomology. --- Cokernel. --- Combination. --- Compass-and-straightedge construction. --- Complex geometry. --- Complex number. --- Computable function. --- Conjecture. --- Coordinate system. --- Coprime integers. --- Corollary. --- Cotangent bundle. --- Diagram (category theory). --- Differential equation. --- Differential form. --- Differential geometry of surfaces. --- Dimension (vector space). --- Dimension. --- Divisor. --- Duality (mathematics). --- Elliptic function. --- Embedding. --- Equation. --- Equivalence class. --- Equivalence relation. --- Exact sequence. --- Existence theorem. --- Existential quantification. --- Fermat's theorem. --- Formal proof. --- Fourier. --- Free group. --- Functional equation. --- Generic point. --- Geometry. --- Group homomorphism. --- Hereditary property. --- Hilbert scheme. --- Homomorphism. --- Injective function. --- Integer. --- Integral curve. --- K-group. --- K-theory. --- Linear combination. --- Mathematics. --- Moduli (physics). --- Moduli space. --- Multivector. --- Natural number. --- Natural transformation. --- Neighbourhood (mathematics). --- Open problem. --- Parameter. --- Polynomial ring. --- Principal part. --- Projective variety. --- Quantity. --- Rational function. --- Rational mapping. --- Reciprocity law. --- Regular map (graph theory). --- Residue theorem. --- Root of unity. --- Scientific notation. --- Sheaf (mathematics). --- Smoothness. --- Statistical significance. --- Subgroup. --- Summation. --- Tangent space. --- Tangent vector. --- Tangent. --- Terminology. --- Tetrahedron. --- Theorem. --- Transcendental function. --- Transcendental number. --- Uniqueness theorem. --- Vector field. --- Vector space. --- 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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The description for this book, Meromorphic Functions and Analytic Curves. (AM-12), will be forthcoming.

Functions. --- Algebraic curve. --- Algebraic equation. --- Algebraic function. --- Algebraic surface. --- Analytic continuation. --- Analytic function. --- Arc (geometry). --- Argument principle. --- Basis (linear algebra). --- Bernhard Riemann. --- Betti number. --- Big O notation. --- Boundary value problem. --- C-function. --- C0. --- Characteristic function (probability theory). --- Circumference. --- Coefficient. --- Combination. --- Compact Riemann surface. --- Compact space. --- Complex analysis. --- Complex number. --- Computation. --- Concentric. --- Conformal map. --- Continuous function. --- Coordinate system. --- Degeneracy (mathematics). --- Derivative. --- Diameter. --- Differential form. --- Dimension. --- Disk (mathematics). --- Dual curve. --- Entire function. --- Equation. --- Equidistant. --- Euler characteristic. --- Existential quantification. --- Exponential function. --- Exterior (topology). --- Floor and ceiling functions. --- Fundamental theorem. --- Gauge factor. --- General position. --- Geometry. --- Harmonic function. --- Heine–Borel theorem. --- Hermann Weyl. --- Homogeneous coordinates. --- Improper integral. --- Integer. --- Interior (topology). --- Inverse function. --- Limit superior and limit inferior. --- Line integral. --- Linear differential equation. --- Linear map. --- Local parameter. --- Logarithm. --- Logarithmic derivative. --- Mathematics. --- Maximum principle. --- Meromorphic function. --- Modular form. --- Modular group. --- Moduli (physics). --- Monodromy theorem. --- Multiple integral. --- Natural number. --- Notation. --- Order by. --- Parallelepiped. --- Parameter. --- Polyad. --- Polynomial. --- Power series. --- Prime number. --- Probability. --- Projection (mathematics). --- Quantity. --- Rational function. --- Real variable. --- Rectangle. --- Residue theorem. --- Riemann integral. --- Riemann surface. --- Rotational symmetry. --- Second derivative. --- Simply connected space. --- Subset. --- Summation. --- Theorem. --- Theory. --- Topological space. --- Total order. --- Unit circle. --- Unit vector. --- Variable (mathematics).

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The description for this book, Contributions to the Theory of Riemann Surfaces. (AM-30), Volume 30, will be forthcoming.

Riemann surfaces. --- Abelian integral. --- Algebraic curve. --- Algebraic function. --- Algebraic geometry. --- Algebraic surface. --- Algebraic variety. --- Analytic continuation. --- Analytic function. --- Asymptotic formula. --- Automorphic function. --- Automorphism. --- Banach algebra. --- Bernhard Riemann. --- Boundary value problem. --- Bounded set (topological vector space). --- Coefficient. --- Compact Riemann surface. --- Compactification (mathematics). --- Complete metric space. --- Complex analysis. --- Complex manifold. --- Conformal map. --- Degeneracy (mathematics). --- Differential equation. --- Differential geometry. --- Differential of the first kind. --- Dimension (vector space). --- Dirichlet integral. --- Dirichlet problem. --- Dirichlet's principle. --- Divisor (algebraic geometry). --- Eigenvalues and eigenvectors. --- Elliptic function. --- Elliptic partial differential equation. --- Equation. --- Existence theorem. --- Existential quantification. --- Explicit formulae (L-function). --- Extremal length. --- Function (mathematics). --- Functional equation. --- Fundamental group. --- Fundamental theorem. --- Geometric function theory. --- Green's function. --- Harmonic conjugate. --- Harmonic function. --- Harmonic measure. --- Holomorphic function. --- Hyperbolic geometry. --- Hypergeometric function. --- Integral equation. --- Intersection (set theory). --- Interval (mathematics). --- Isometry. --- Isoperimetric inequality. --- Jordan curve theorem. --- Kähler manifold. --- Laplace's equation. --- Lebesgue integration. --- Linear differential equation. --- Linear map. --- Linear space (geometry). --- Mathematical physics. --- Mathematical theory. --- Mathematics. --- Meromorphic function. --- Metric space. --- Minkowski space. --- Operator (physics). --- Ordinary differential equation. --- Parametric equation. --- Parity (mathematics). --- Partial differential equation. --- Polynomial. --- Power series. --- Projection (linear algebra). --- Quadratic differential. --- Riemann mapping theorem. --- Riemann sphere. --- Riemann surface. --- Riemannian geometry. --- Riemannian manifold. --- Riemann–Roch theorem. --- Ring (mathematics). --- Scalar (physics). --- Sign (mathematics). --- Simultaneous equations. --- Special case. --- Surjective function. --- Tensor density. --- Theorem. --- Theory of equations. --- Theory. --- Topology. --- Uniformization theorem. --- Uniformization. --- Uniqueness theorem. --- Variable (mathematics). --- Weierstrass theorem.