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Power series --- Root systems (Algebra) --- Laurent series

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Conformal mapping --- Analytic functions --- Complex variables(Functions of) --- Laurent series --- Power series

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Meromorphic functions --- Laurent series --- Power series --- Holomorphic functions --- Harmonic functions --- Functions of complex variables

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In this monograph, the authors develop a new theory of p-adic cohomology for varieties over Laurent series fields in positive characteristic, based on Berthelot's theory of rigid cohomology. Many major fundamental properties of these cohomology groups are proven, such as finite dimensionality and cohomological descent, as well as interpretations in terms of Monsky-Washnitzer cohomology and Le Stum's overconvergent site. Applications of this new theory to arithmetic questions, such as l-independence and the weight monodromy conjecture, are also discussed. The construction of these cohomology groups, analogous to the Galois representations associated to varieties over local fields in mixed characteristic, fills a major gap in the study of arithmetic cohomology theories over function fields. By extending the scope of existing methods, the results presented here also serve as a first step towards a more general theory of p-adic cohomology over non-perfect ground fields. Rigid Cohomology over Laurent Series Fields will provide a useful tool for anyone interested in the arithmetic of varieties over local fields of positive characteristic. Appendices on important background material such as rigid cohomology and adic spaces make it as self-contained as possible, and an ideal starting point for graduate students looking to explore aspects of the classical theory of rigid cohomology and with an eye towards future research in the subject.

Number theory. --- Geometry --- Mathematics --- Physical Sciences & Mathematics --- Cohomology operations. --- Laurent series. --- Series, Laurent --- Operations (Algebraic topology) --- Functions of complex variables --- Series, Taylor's --- Algebraic topology --- Geometry, algebraic. --- Algebraic Geometry. --- Number Theory. --- Number study --- Numbers, Theory of --- Algebra --- Algebraic geometry --- Algebraic geometry.

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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.