The current paintings treats p-adic houses of ideas of the hypergeometric differential equation d2 d ~ ( x(l - x) dx + (c(l - x) + (c - 1 - a - b)x) dx - ab)y = zero, 2 with a, b, c in four) n Zp, by way of developing the linked Frobenius constitution. For this building we draw upon the equipment of Alan Adolphson  in his 1976 paintings on Hecke polynomials. we're additionally indebted to him for the account (appearing as an appendix) of the relation among this differential equation and sure L-functions. we're indebted to G. Washnitzer for the strategy utilized in the development of our twin idea (Chapter 2). those notes characterize an increased kind of lectures given on the U. L. P. in Strasbourg throughout the fall time period of 1980. We take this chance to thank Professor R. Girard and IRMA for his or her hospitality. Our subject-p-adic analysis-was based through Marc Krasner. We enjoy dedicating this paintings to him. Contents 1 advent . . . . . . . . . . 1. the distance L (Algebraic idea) eight 2. twin thought (Algebraic) 14 three. Transcendental idea . . . . 33 four. Analytic twin idea. . . . . forty eight five. uncomplicated homes of", Operator. seventy three 6. Calculation Modulo p of the Matrix of ~ f,h ninety two 7. Hasse Invariants . . . . . . 108 eight. The a --+ a' Map . . . . . . . . . . . . a hundred and ten nine. Normalized answer Matrix. . . . . .. 113 10. Nilpotent Second-Order Linear Differential Equations with Fuchsian Singularities. . . . . . . . . . . . . 137 eleven. Second-Order Linear Differential Equations Modulo Powers ofp ..... .
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