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Lectures on p-adic Differential Equations

  • Book
  • © 1982

Overview

Authors:
  1. Bernard Dwork

    1. Department of Mathematics, Princeton University, Princeton, USA

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Part of the book series: Grundlehren der mathematischen Wissenschaften (GL, volume 253)

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Table of contents (27 chapters)

  1. Front Matter

    Pages i-viii
    Download chapter PDF

  2. Introduction

    • Bernard Dwork
    Pages 1-7

  3. The Space L (Algebraic Theory)

    • Bernard Dwork
    Pages 8-13

  4. Dual Theory (Algebraic)

    • Bernard Dwork
    Pages 14-32

  5. Transcendental Theory

    • Bernard Dwork
    Pages 33-47

  6. Analytic Dual Theory

    • Bernard Dwork
    Pages 48-72

  7. Basic Properties of ψ Operator

    • Bernard Dwork
    Pages 73-91

  8. Calculation Modulo p of the Matrix of α f,h

    • Bernard Dwork
    Pages 92-107

  9. Hasse Invariants

    • Bernard Dwork
    Pages 108-109

  10. The a → a′ Map

    • Bernard Dwork
    Pages 110-112

  11. Normalized Solution Matrix

    • Bernard Dwork
    Pages 113-136

  12. Nilpotent Second-Order Linear Differential Equations with Fuchsian Singularities

    • Bernard Dwork
    Pages 137-144

  13. Second-Order Linear Differential Equations Modulo Powers of p

    • Bernard Dwork
    Pages 145-158

  14. Dieudonné Theory

    • Bernard Dwork
    Pages 159-167

  15. Canonical Liftings (l ≥ 1)

    • Bernard Dwork
    Pages 168-174

  16. Abelian Differentials

    • Bernard Dwork
    Pages 175-177

  17. Canonical Lifting for l = 1

    • Bernard Dwork
    Pages 178-183

  18. Supersingular Disks

    • Bernard Dwork
    Pages 184-194

  19. The Function Ï„ on Supersingular Disks (l = 1)

    • Bernard Dwork
    Pages 195-201

  20. The Defining Relation for the Canonical Lifting (l = 1)

    • Bernard Dwork
    Pages 202-219
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Keywords

About this book

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 ..... .

Authors and Affiliations

  • Department of Mathematics, Princeton University, Princeton, USA

    Bernard Dwork

Bibliographic Information

  • Book Title: Lectures on p-adic Differential Equations

  • Authors: Bernard Dwork

  • Series Title: Grundlehren der mathematischen Wissenschaften

  • DOI: https://doi.org/10.1007/978-1-4613-8193-8

  • Publisher: Springer New York, NY

  • eBook Packages: Springer Book Archive

  • Copyright Information: Springer-Verlag New York Inc. 1982

  • Softcover ISBN: 978-1-4613-8195-2Published: 06 November 2011

  • eBook ISBN: 978-1-4613-8193-8Published: 06 December 2012

  • Series ISSN: 0072-7830

  • Series E-ISSN: 2196-9701

  • Edition Number: 1

  • Number of Pages: 310

  • Topics: Analysis

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