Lectures on p-adic Differential Equations
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
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Editors
M. Artin s. E. Heinz F. W. Magnus W. Schmidt
S. Chern J. L. Doob A. Grothendieck Hirzebruch L. Hormander S. Mac Lane C. C. Moore J. K. Moser M. Nagata D. S. Scott J. Tits B. L. van der Waerden
Managing Editors
M. Berger B. Eckmann S. R. S. Varadhan
Bernard Dwork
Lectures on
p-adic Differential Equations
Springer-Verlag New Yark Heidelberg Berlin
Bernard Dwork Department of Mathematics Princeton University Princeton, NJ 08540 U.S.A.
AMS Subject Classifications (1980): IOD30, 14G20, 33A30
Library of Congress Cataloging in Publication Data Dwork, Bernard M. Lectures on p-adic differential equations. (Grundlehren der mathematischen Wissenschaften; 253) Bibliography: p. I. Differential equations. 2. p-adic numbers. I. Title. II. Series. QA372.D86 515.3'5 82-5764 AACR2 With 5 Illustrations
© 1982 by Springer-Verlag New York Inc. Softcover reprint of the hardcover 1st edition 1982 All rights reserved. No part of this book may be translated or reproduced in any form without written permission from Springer-Verlag, 175 Fifth Avenue, New York, New York 10010, U.S.A. Typeset by Composition House Ltd., Salisbury, England. Printed by Malloy Lithographing, Inc., Ann Arbor, MI.
9 8 7 6 543 2 1
ISBN-13 :978-1-4613-8195-2 e-ISBN-13 :978-1-4613-8193-8 001: 10.1007/978-1-4613-8193-8
Preface
The present work treats p-adic properties of solutions of the hypergeometric differential equation d2 ( x(l - x) dx 2
+ (c(l -
x)
+ (c -
d ~ 1 - a - b)x) dx - ab)y = 0,
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
Introduction . . . . . . . . . . 1. The Space L (Algebraic Theory) 2. Dual Theory (Algebraic) 3. Transcendental Theory . . . . 4. Analytic Dual Theory. . . . . 5. Basic Properties of", Operator. 6. Calculation Modulo p of the Matrix of ~ f,h 7. Hasse Invariants . . . . . . 8. The a --+ a' Map . . . . . . . . . . . . 9. Normalized Solution Matrix. . . . . . . 10. Nilpotent Second-Order Linear Differential Equations with Fuchsian Singularities. . . . . . . . . . . . . 11. Second-Order Linear Differential Equations Modulo Powers ofp . . . . . . 12. Dieudonne Theory . . . . 13. Canonical Liftings (l ~ 1) . 14. Abelian Differentials . . . 15. Canonical Lifting for / = 1 16. Supersingular Disks . . . 17. The Function 'C on Supersingular Disks (/ = 1) 18. The Defining Relation for the Canonical Lifting (I = 1) 19.
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