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
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