Arithmetic and Geometry Around Hypergeometric Functions
Arithmetic and Geometry Around Hypergeometric Functions

Author: Rolf-Peter Holzapfel

Publisher: Springer Science & Business Media

Published: 2007-06-28

Total Pages: 441

ISBN-13: 3764382848

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This volume comprises lecture notes, survey and research articles originating from the CIMPA Summer School Arithmetic and Geometry around Hypergeometric Functions held at Galatasaray University, Istanbul, June 13-25, 2005. It covers a wide range of topics related to hypergeometric functions, thus giving a broad perspective of the state of the art in the field.

Arithmetic and Geometry Around Hypergeometric Functions
Arithmetic and Geometry Around Hypergeometric Functions

Author: Rolf-Peter Holzapfel

Publisher: Birkhäuser

Published: 2009-09-03

Total Pages: 437

ISBN-13: 9783764391942

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This volume comprises lecture notes, survey and research articles originating from the CIMPA Summer School Arithmetic and Geometry around Hypergeometric Functions held at Galatasaray University, Istanbul, June 13-25, 2005. It covers a wide range of topics related to hypergeometric functions, thus giving a broad perspective of the state of the art in the field.

Arithmetic and Geometry Around Galois Theory
Arithmetic and Geometry Around Galois Theory

Author: Pierre Dèbes

Publisher: Springer Science & Business Media

Published: 2012-12-13

Total Pages: 411

ISBN-13: 3034804873

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This Lecture Notes volume is the fruit of two research-level summer schools jointly organized by the GTEM node at Lille University and the team of Galatasaray University (Istanbul): "Geometry and Arithmetic of Moduli Spaces of Coverings (2008)" and "Geometry and Arithmetic around Galois Theory (2009)". The volume focuses on geometric methods in Galois theory. The choice of the editors is to provide a complete and comprehensive account of modern points of view on Galois theory and related moduli problems, using stacks, gerbes and groupoids. It contains lecture notes on étale fundamental group and fundamental group scheme, and moduli stacks of curves and covers. Research articles complete the collection.​

Hypergeometric Functions in Arithmetic Geometry
Hypergeometric Functions in Arithmetic Geometry

Author: Adriana Julia Salerno

Publisher:

Published: 2009

Total Pages: 204

ISBN-13:

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Hypergeometric functions seem to be ubiquitous in mathematics. In this document, we present a couple of ways in which hypergeometric functions appear in arithmetic geometry. First, we show that the number of points over a finite field [mathematical symbol] on a certain family of hypersurfaces, [mathematical symbol] ([lamda]), is a linear combination of hypergeometric functions. We use results by Koblitz and Gross to find explicit relationships, which could be useful for computing Zeta functions in the future. We then study more geometric aspects of the same families. A construction of Dwork's gives a vector bundle of deRham cohomologies equipped with a connection. This connection gives rise to a differential equation which is known to be hypergeometric. We developed an algorithm which computes the parameters of the hypergeometric equations given the family of hypersurfaces.

Hessian Polyhedra, Invariant Theory And Appell Hypergeometric Functions
Hessian Polyhedra, Invariant Theory And Appell Hypergeometric Functions

Author: Lei Yang

Publisher: World Scientific

Published: 2018-03-13

Total Pages: 317

ISBN-13: 9813209496

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Our book gives the complex counterpart of Klein's classic book on the icosahedron. We show that the following four apparently disjoint theories: the symmetries of the Hessian polyhedra (geometry), the resolution of some system of algebraic equations (algebra), the system of partial differential equations of Appell hypergeometric functions (analysis) and the modular equation of Picard modular functions (arithmetic) are in fact dominated by the structure of a single object, the Hessian group $mathfrak{G}’_{216}$. It provides another beautiful example on the fundamental unity of mathematics.

Theory of Hypergeometric Functions
Theory of Hypergeometric Functions

Author: Kazuhiko Aomoto

Publisher: Springer Science & Business Media

Published: 2011-05-21

Total Pages: 327

ISBN-13: 4431539387

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This book presents a geometric theory of complex analytic integrals representing hypergeometric functions of several variables. Starting from an integrand which is a product of powers of polynomials, integrals are explained, in an open affine space, as a pair of twisted de Rham cohomology and its dual over the coefficients of local system. It is shown that hypergeometric integrals generally satisfy a holonomic system of linear differential equations with respect to the coefficients of polynomials and also satisfy a holonomic system of linear difference equations with respect to the exponents. These are deduced from Grothendieck-Deligne’s rational de Rham cohomology on the one hand, and by multidimensional extension of Birkhoff’s classical theory on analytic difference equations on the other.

Encyclopedia of Special Functions: The Askey-Bateman Project: Volume 2, Multivariable Special Functions
Encyclopedia of Special Functions: The Askey-Bateman Project: Volume 2, Multivariable Special Functions

Author: Tom H. Koornwinder

Publisher: Cambridge University Press

Published: 2020-10-15

Total Pages: 442

ISBN-13: 1108916554

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This is the second of three volumes that form the Encyclopedia of Special Functions, an extensive update of the Bateman Manuscript Project. Volume 2 covers multivariable special functions. When the Bateman project appeared, study of these was in an early stage, but revolutionary developments began to be made in the 1980s and have continued ever since. World-renowned experts survey these over the course of 12 chapters, each containing an extensive bibliography. The reader encounters different perspectives on a wide range of topics, from Dunkl theory, to Macdonald theory, to the various deep generalizations of classical hypergeometric functions to the several variables case, including the elliptic level. Particular attention is paid to the close relation of the subject with Lie theory, geometry, mathematical physics and combinatorics.

On Dwork's $p$-Adic Formal Congruences Theorem and Hypergeometric Mirror Maps
On Dwork's $p$-Adic Formal Congruences Theorem and Hypergeometric Mirror Maps

Author: E. Delaygue

Publisher: American Mathematical Soc.

Published: 2017-02-20

Total Pages: 106

ISBN-13: 1470423006

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Using Dwork's theory, the authors prove a broad generalization of his famous -adic formal congruences theorem. This enables them to prove certain -adic congruences for the generalized hypergeometric series with rational parameters; in particular, they hold for any prime number and not only for almost all primes. Furthermore, using Christol's functions, the authors provide an explicit formula for the “Eisenstein constant” of any hypergeometric series with rational parameters. As an application of these results, the authors obtain an arithmetic statement “on average” of a new type concerning the integrality of Taylor coefficients of the associated mirror maps. It contains all the similar univariate integrality results in the literature, with the exception of certain refinements that hold only in very particular cases.