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The Petersson scalar product. (English) Zbl 0934.11018
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.

##### 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