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Modular forms and periods. (Formes modulaires et périodes.) (French) Zbl 1104.11017
Fischler, Stéphane (ed.) et al., Modular forms and transcendence. Colloquium of young researchers. Based on the conference on the links between modular forms and transcendence, Marseille, France, May 26–30, 2003. Paris: Société Mathématique de France (ISBN 2-85629-176-7/pbk). Séminaires et Congrès 12, 1-117 (2005).
This is a very readable introduction to the theory of modular forms, focussing mainly on \(\Gamma_0(N),\) leading up to quite recent results and connections with arithmetic geometry. It does not treat the Shimura-Taniyama conjecture, however, but deals with other aspects instead.
In the first part – 51 pages – the classical results on modular forms are being described; in particular, we find dimension formulae, the Petersson scalar product, Hecke operators, Atkin-Lehner theory and \(L\)-series. The discussion leads up to the conjectures of N. M. Katz and P. Sarnak [Random matrices, Frobenius eigenvalues, and monodromy. Colloquium Publications. American Mathematical Society (AMS). 45. Providence, RI: American Mathematical Society (AMS). (1999; Zbl 0958.11004)].
Part II – 26 pages – deals with rational structures and periods and in particular introduces the isomorphism of Eichler and Shimura.
Part III – 21 pages – introduces differential operators on modular forms and quasimodular forms. It culminates in the statement of a conjecture of Deligne-Beilinson-Scholl concerning special values of derivatives of modular \(L\)-series. Here, periods in the sense of M. Kontsevich and D. Zagier [Periods. Mathematics unlimited – 2001 and beyond. Berlin: Springer. 771-808 (2001; Zbl 1039.11002)] play a role.
In a last part, some appendices are sampled.
The paper is an invaluable introduction to the aspects of modular forms dealing with special values. It gives a fairly detailed account on the relevant literature and is spiced up with lots of nicely chosen concrete examples.
For the entire collection see [Zbl 1078.11001].

MSC:
11F11 Holomorphic modular forms of integral weight
11F25 Hecke-Petersson operators, differential operators (one variable)
11F37 Forms of half-integer weight; nonholomorphic modular forms
11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
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