Classical modular forms, modular symbols, and congruences.

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This book gives a thorough introduction to several theories
that are fundamental to recent research on modular forms. Most of the
material, despite its importance, had previously been unavailable in textbook
form. Complete and readable proofs are given. Especially valuable and
clearly written are the treatments of the Eichler-Shimura isomorphism
on SL_2(Z), the Atkin-Lehner theory of new forms, and the relation between
congruences and Galois representations. The contents of the book are as follows: Part I gives the basic "classical theory", including Hecke operators and the Petersson scalar product. An appendix proves the Eichler-Selberg trace formula for the Hecke operators on SL_2(Z), which expresses this trace in terms of class numbers of imaginary quadratic fields. Part II introduces modular symbols, proves the Manin-Drinfelcprime d theorem that the cusps are of finite order in the divisor class group of the modular curve, derives the Manin relations for the action of the Hecke operators on the periods of cusp forms on SL_2(Z), and proves the Eichler-Shimura isomorphism. The latter proof is down-to-earth, and no knowledge of cohomology is presumed; the connection with Eichler cohomology is discussed briefly at the end. Part III is devoted to modular forms for congruence sub-groups, including a proof of the main theorem of Atkin-Lehner theory. An appendix discusses the Dedekind symbol arising from the logeta function. Part IV, on congruence properties and Galois representations, proves the structure theorem for modular forms modulo a prime p, Serre's filtration theorem for the theta-operator, and the q-expansion principle on p-divisibility of q-expansion coefficients of modular forms. Then the author proves the basic facts about finite subgroups of PGL_2 of a field of characteristic l (with the complete story concerning the exceptional subgroups gathered in an appendix to Part IV). He derives the connection between congruences mod},l satisfied by the trace of Frobenius, and the image of a Galois representation in GL_2 of the field F_l of l elements being contained in a small subgroup. Then, assuming the existence proved by Serre and Deligne of representations in GL_2(F_l) and GL_2(Z_l) associated to cusp forms, he shows how the results apply to modular forms, and proves that for a fixed cusp form the associated representations do not have small image (i.e., the congruences cannot occur) for l sufficiently large. Part V discusses p-adic distributions, the Iwasawa algebra, Bernoulli polynomials, p-adic and complex L-functions, and Hecke-Eisenstein and Klein forms. Mazur's approach to p-adic L-functions via p-adic Bernoulli distributions is used. Part IV leads right up to, but does not include, work by Serre and Katz on p-adic interpolation of modular forms. At the end of Part IV it would have taken only a few more pages to give Serre's derivation of the p-adic zeta-function as the constant term in the p-adic limit of Eisenstein series. Since this derivation would help motivate Part V (and explain its inclusion in a book on modular forms), the reader may want to pause between Parts IV and V to read §1 of Serre's article on p-adic modular forms [ Modular functions of one variable, III/ (Proc. Internat. Summer School, Univ. Antwerp, 1972), pp. 191--268, Lecture Notes in Math., Vol. 350, Springer, Berlin, 1973; MR 53 #7949a]. In conclusion, this book is a welcome addition to the literature for the growing number of students and mathematicians in other fields who want to understand the recent developments in the theory of modular forms. Reviewed by Neal Koblitz |