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 ..... .

Inhalt

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 of p.- 12. Dieudonné Theory.- 13. Canonical Liftings (l ? 1).- 14. Abelian Differentials.- 15. Canonical Lifting for l = 1.- 16. Supersingular Disks.- 17. The Function ? on Supersingular Disks (l = 1).- 18. The Defining Relation for the Canonical Lifting (l = 1).- 19. Semisimplicity.- 20. Analytic Factors of Power Series.- 21. p-adic Gamma Functions.- 22. p-adic Beta Functions.- 23. Beta Functions as Residues.- 24. Singular Disks, Part I.- 25. Singular Disks, Part II. Nonlogarithmic Case.- 26. Singular Disks, Part III. Logarithmic Case.- Index of Symbols.