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# Lectures on p-adic Differential Equations

• Bernard Dwork
Book

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

## Table of contents

1. Front Matter
Pages i-viii
2. Bernard Dwork
Pages 1-7
3. Bernard Dwork
Pages 8-13
4. Bernard Dwork
Pages 14-32
5. Bernard Dwork
Pages 33-47
6. Bernard Dwork
Pages 48-72
7. Bernard Dwork
Pages 73-91
8. Bernard Dwork
Pages 92-107
9. Bernard Dwork
Pages 108-109
10. Bernard Dwork
Pages 110-112
11. Bernard Dwork
Pages 113-136
12. Bernard Dwork
Pages 137-144
13. Bernard Dwork
Pages 145-158
14. Bernard Dwork
Pages 159-167
15. Bernard Dwork
Pages 168-174
16. Bernard Dwork
Pages 175-177
17. Bernard Dwork
Pages 178-183
18. Bernard Dwork
Pages 184-194
19. Bernard Dwork
Pages 195-201
20. Bernard Dwork
Pages 202-219
21. Bernard Dwork
Pages 220-231
22. Bernard Dwork
Pages 232-241
23. Bernard Dwork
Pages 242-249
24. Bernard Dwork
Pages 250-256
25. Bernard Dwork
Pages 257-263
26. Bernard Dwork
Pages 264-271
27. Bernard Dwork
Pages 272-279
28. Bernard Dwork
Pages 280-285
29. Back Matter
Pages 287-312

## About this book

### Introduction

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 ..... .

### Keywords

Equations Hypergeometrische Differentialgleichung differential equation logarithm p-adische Analysis

#### Authors and affiliations

• Bernard Dwork
• 1
1. 1.Department of MathematicsPrinceton UniversityPrincetonUSA

### Bibliographic information

• DOI https://doi.org/10.1007/978-1-4613-8193-8
• Copyright Information Springer-Verlag New York 1982
• Publisher Name Springer, New York, NY
• eBook Packages
• Print ISBN 978-1-4613-8195-2
• Online ISBN 978-1-4613-8193-8
• Series Print ISSN 0072-7830
• Buy this book on publisher's site