Lectures on p-adic Differential Equations

Bernard Dwork (auth.)0387907149, 9780387907147

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 ….. .

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Table of contents :
Front Matter….Pages i-viii
Introduction….Pages 1-7
The Space L (Algebraic Theory)….Pages 8-13
Dual Theory (Algebraic)….Pages 14-32
Transcendental Theory….Pages 33-47
Analytic Dual Theory….Pages 48-72
Basic Properties of ψ Operator….Pages 73-91
Calculation Modulo p of the Matrix of α f,h ….Pages 92-107
Hasse Invariants….Pages 108-109
The a → a ′ Map….Pages 110-112
Normalized Solution Matrix….Pages 113-136
Nilpotent Second-Order Linear Differential Equations with Fuchsian Singularities….Pages 137-144
Second-Order Linear Differential Equations Modulo Powers of p ….Pages 145-158
Dieudonné Theory….Pages 159-167
Canonical Liftings ( l ≥ 1)….Pages 168-174
Abelian Differentials….Pages 175-177
Canonical Lifting for l = 1….Pages 178-183
Supersingular Disks….Pages 184-194
The Function τ on Supersingular Disks ( l = 1)….Pages 195-201
The Defining Relation for the Canonical Lifting ( l = 1)….Pages 202-219
Semisimplicity….Pages 220-231
Analytic Factors of Power Series….Pages 232-241
p -adic Gamma Functions….Pages 242-249
p -adic Beta Functions….Pages 250-256
Beta Function as Residues….Pages 257-263
Singular Disks, Part I….Pages 264-271
Singular Disks, Part II. Nonlogarithmic Case….Pages 272-279
Singular Disks, Part III. Logarithmic Case….Pages 280-285
Back Matter….Pages 287-312