Arithmetic of p-adic Modular Forms
The central topic of this research monograph is the relation between p-adic modular forms and p-adic Galois representations, and in particular the theory of deformations of Galois representations recently introduced by Mazur. The classical theory of modul
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1304 Fernando Q. Gouvea
Arithmetic of p-adic Modular Forms
Spri nger-Verlag Berlin Heidelberg New York London Paris Tokyo
Author
Fernando Quadros Gouvea Instituto de Matematica e Estatfstica Universidade de Sao Paulo Caixa Postal 20570 (Ag. Iguatemi) 01498 Sao Paulo, SP, Brazil
Mathematics Subject Classification (1980, revised 1985): 11 F ISBN 3-540-18946-7 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-18946-7 Springer-Verlag New York Berlin Heidelberg
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Introduction
The theory of p-adic modular forms was initiated by Serre, Katz, and Dwork, who, in the early 1970's, attempted to define objects which would be recognizably modular forms but which would be truly p-adic, reflecting the p-adic topology in an essential way. Specifically, one wanted a theory where two modular forms with highly congruent q-expansion coefficients would be p-adically close, and where limits of modular forms (with respect to this topology) would exist. The initial motivation for this construction was the problem of p-adic interpolation of special values of L-functions. The first difficulty, of course, was to find the correct definitions. Serre's approach (see [Se73]) was the most elementary: modular forms were identified with their q-expansions and p-adic modular forms were considered as limits of such q-expansions. Serre showed that such limits have "weights" (in a suitable sense), and developed the theory sufficiently to be able to obtain p-adic L-functions by constructing suitable analytic families of p-adic modular forms. His theory of analytic families of modular forms will be briefly discussed in our third chapter. He was also the first to notice that modular forms of level N p (and appropriate nebentypus) are oflevel N when considered as p-adic modular forms, which is a crucial aspect of the p-adic theory. Dwork's approach was more analytic. In [Dw73], for example, he restricts himself to p-adic analytic functions on a modular curve (i.e., p-adic modular forms of weight zero); in the same article, he notes the importance of growth conditions and of the fact that the U operator is completely continuous (in the sense of Serre) in the case which he is considering. Katz's work brought these approaches together, showing how to define p-adic modular forms in modular terms, generalizing the results of Dwork on the U operator and o
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