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p-adic Differential Equations

Author: Kiran S. Kedlaya

Publisher: Cambridge University Press

Published: 2010-06-10

Total Pages: 399

ISBN-13: 1139489208

Over the last 50 years the theory of p-adic differential equations has grown into an active area of research in its own right, and has important applications to number theory and to computer science. This book, the first comprehensive and unified introduction to the subject, improves and simplifies existing results as well as including original material. Based on a course given by the author at MIT, this modern treatment is accessible to graduate students and researchers. Exercises are included at the end of each chapter to help the reader review the material, and the author also provides detailed references to the literature to aid further study.

Lectures on p-adic Differential Equations

Author: Bernard Dwork

Publisher: Springer Science & Business Media

Published: 2012-12-06

Total Pages: 318

ISBN-13: 1461381932

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The present work treats p-adic properties of solutions of the hypergeometric differential equation d2 d ~ ( x(l - x) dx + (c(l - x) + (c - 1 - a - b)x) dx - ab)y = 0, 2 with a, b, c in 4) n Zp, by constructing the associated Frobenius structure. For this construction we draw upon the methods of Alan Adolphson [1] in his 1976 work on Hecke polynomials. We are also indebted to him for the account (appearing as an appendix) of the relation between this differential equation and certain L-functions. We are indebted to G. Washnitzer for the method used in the construction of our dual theory (Chapter 2). These notes represent an expanded form of lectures given at the U. L. P. in Strasbourg during the fall term of 1980. We take this opportunity to thank Professor R. Girard and IRMA for their hospitality. Our subject-p-adic analysis-was founded by Marc Krasner. We take pleasure in dedicating this work to him. Contents 1 Introduction . . . . . . . . . . 1. The Space L (Algebraic Theory) 8 2. Dual Theory (Algebraic) 14 3. Transcendental Theory . . . . 33 4. Analytic Dual Theory. . . . . 48 5. Basic Properties of", Operator. 73 6. Calculation Modulo p of the Matrix of ~ f,h 92 7. Hasse Invariants . . . . . . 108 8. The a --+ a' Map . . . . . . . . . . . . 110 9. Normalized Solution Matrix. . . . . .. 113 10. Nilpotent Second-Order Linear Differential Equations with Fuchsian Singularities. . . . . . . . . . . . . 137 11. Second-Order Linear Differential Equations Modulo Powers ofp ..... .

P-adic Differential Equations

Author: Kiran Sridhara Kedlaya

Publisher:

Published: 2014-05-14

Total Pages: 400

ISBN-13: 9780511750182

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The first comprehensive, unified development of the theory of p-adic differential equations.

A Course in p-adic Analysis

Author: Alain M. Robert

Publisher: Springer Science & Business Media

Published: 2013-04-17

Total Pages: 451

ISBN-13: 1475732546

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Discovered at the turn of the 20th century, p-adic numbers are frequently used by mathematicians and physicists. This text is a self-contained presentation of basic p-adic analysis with a focus on analytic topics. It offers many features rarely treated in introductory p-adic texts such as topological models of p-adic spaces inside Euclidian space, a special case of Hazewinkel’s functional equation lemma, and a treatment of analytic elements.

P-adic Analysis and Mathematical Physics

Author: Vasili? Sergeevich Vladimirov

Publisher: World Scientific

Published: 1994

Total Pages: 350

ISBN-13: 9789810208806

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p-adic numbers play a very important role in modern number theory, algebraic geometry and representation theory. Lately p-adic numbers have attracted a great deal of attention in modern theoretical physics as a promising new approach for describing the non-Archimedean geometry of space-time at small distances.This is the first book to deal with applications of p-adic numbers in theoretical and mathematical physics. It gives an elementary and thoroughly written introduction to p-adic numbers and p-adic analysis with great numbers of examples as well as applications of p-adic numbers in classical mechanics, dynamical systems, quantum mechanics, statistical physics, quantum field theory and string theory.

P-Adic Functional Analysis

Author: A.K. Katsaras

Publisher: CRC Press

Published: 2001-07-03

Total Pages: 337

ISBN-13: 0203908147

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This volume collects together lectures presented at the Sixth International Conference held at the University of Ioannina, Greece, on p-adic functional analysis with applications in the fields of physics, differential equations, number theory, probability theory, dynamical systems, and algebraic number fields. It discusses the commutation relation AB-BA=I and its central role in quantum mechanics.

p-adic Differential Equations

Author: Kiran Kedlaya

Publisher: Cambridge University Press

Published: 2022-06-09

Total Pages: 517

ISBN-13: 1009123343

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A detailed and unified treatment of $P$-adic differential equations, from the basic principles to the current frontiers of research.

An Introduction to G-Functions. (AM-133), Volume 133

Author: Bernard Dwork

Publisher: Princeton University Press

Published: 2016-03-02

Total Pages: 349

ISBN-13: 1400882540

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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.

An Introduction to G-functions

Author: Bernard Dwork

Publisher: Princeton University Press

Published: 1994-05-22

Total Pages: 348

ISBN-13: 0691036810

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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.

Differential Equations and the Stokes Phenomenon

Author: B L J Braaksma

Publisher: World Scientific

Published: 2002-12-10

Total Pages: 344

ISBN-13: 9814487430

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This volume is the record of a workshop on differential equations and the Stokes phenomenon, held in May 2001 at the University of Groningen. It contains expanded versions of most of the lectures given at the workshop. To a large extent, both the workshop and the book may be regarded as a sequel to a conference held in Groningen in 1995 which resulted in the book The Stokes Phenomenon and Hilbert's 16th Problem (B L J Braaksma, G K Immink and M van der Put, editors), also published by World Scientific (1996). Both books offer a snapshot concerning the state of the art in the areas of differential, difference and q-difference equations. Apart from the asymptotics of solutions, Painlevé properties and the algebraic theory, new topics addressed in the second book include arithmetic theory of linear equations, and Galois theory and Lie symmetries of nonlinear differential equations. Contents:Toward p-Adic Stokes Phenomena? Singularities of p-Adic Differential Equations (Y André)Moduli Spaces for Linear Differential Equations (M Berkenbosch)Factorization of Differential Operators and Application to Differential Galois Theory (M Bouffet)Movable Singularities of Solutions of Nonlinear Differential and Difference Equations and the Painlevé Property (O Costin & M Kruskal)On a Conjecture of Sophus Lie (J Draisma)A Tale of Three Structures: the Arithmetic of Multizetas, the Analysis of Singularities, the Lie Algebra ARI (J Ecalle)A Differential Intermediate Value Theorem (J van der Hoeven)Integrable Systems and Number Theory (P H van der Kamp et al.)Towards the Galois Groupoid of Nonlinear O D E (F Loray)Galois Theory of q-Difference Equations: The “Analytical” Approach (J Sauloy)“Exact WKB Integration” of the Polynomial 1D Schrödinger (or Sturm-Liouville) Problem (A Voros)Une Sommation Discrète Pour des Équations Aux q-Différence Linéaires et à Coefficients Analytiques: Théorie Générale et Exemples (C-G Zhang) Readership: Graduate students, academics and researchers in analysis & differential equations, approximation theory and mathematical physics. Keywords:Ordinary Differential Equations (linear and non-linear), p-Adic Equations;Moduli;PainlevÃ¨ Property;Lie Algebras;Multizetas;Integrable Systems;q-Difference Equations;WKB Integration