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Introductory lectures on Siegel modular forms. (English) Zbl 0693.10023

Cambridge Studies in Advanced Mathematics, 20. Cambridge: Cambridge University Press. x, 162 p. £22.50; $ 39.50 (1990).
The book under review provides an excellent introduction to the theory of Siegel modular forms. It contains Siegel’s classical theory for the full modular group. If one compares this monograph with the books of E. Freitag [Siegelsche Modulfunktionen (Grundlehren Math. Wiss. 254) Berlin etc.: Springer (1983; Zbl 0498.10016)] or A. N. Andrianov [Quadratic forms and Hecke operators (Grundlehren Math. Wiss. 286) Berlin etc.: Springer (1987; Zbl 0613.10023)], this book is written in a more elementary style. The character of lecture notes is emphasized by the fact that it is accessible to graduate students after a one-complex-variable course.
Part I refers to the modular group. Minkowski’s reduction theory is included and a fundamental domain for the Siegel modular group is constructed. In part II it is shown that the space of Siegel modular forms has finite dimension. Klingen-Eisenstein series are introduced and the metrization theory is pointed out. Part III is devoted to large weights. All the different types of Poincaré series are investigated. Part IV refers to small weights. The theory of singular modular forms is described, where special attention is devoted to theta series. Following E. Freitag [Nachr. Akad. Wiss. Göttingen 1965, 151–157 (1965; Zbl 0133.33202)] the graded ring of Siegel modular forms of degree two is determined. Siegel modular functions are dealt with in part V. They are shown to be quotients of Siegel modular forms and the transcendence degree is determined. The final part VI contains a description of the analytic properties of a class of Dirichlet series attached to Siegel modular forms, which was introduced by Maaß and Koecher.

MSC:

11F46 Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms
11-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to number theory
11F66 Langlands \(L\)-functions; one variable Dirichlet series and functional equations
11F27 Theta series; Weil representation; theta correspondences
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