The Petersson scalar product. (English) Zbl 0934.11018

Elstrodt, J.; Grunewald, F.

Jahresber. Dtsch. Math.-Ver. 100, No. 4, 253-283 (1998).

The following topics are covered in this survey article: Petersson’s early work on automorphic forms. The Petersson scalar product and its applications to Poincaré series. Simultaneous diagonalization of Hecke operators. Hecke operators on \(\Gamma_0(N)\). Applications to the theory of real analytic automorphic forms. The Petersson scalar product in the theory of modular forms of several variables.
The concepts and results are carefully stated and explained, and precise references are given. The bibliography comprises 165 items. The article will be most valuable because it gives quick information on the results and their historical development, and because of its comments on the literature. It will be helpful as a guide for further reading.

Reviewer: G.Köhler (Würzburg)

MSC:

11F11	Holomorphic modular forms of integral weight
11-02	Research exposition (monographs, survey articles) pertaining to number theory
11F25	Hecke-Petersson operators, differential operators (one variable)
11F60	Hecke-Petersson operators, differential operators (several variables)
11F46	Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms

Keywords:

survey; Petersson scalar product; Poincaré series; diagonalization of Hecke operators; real analytic automorphic forms; modular forms of several variables