##
**Quadratic forms and Hecke operators.**
*(English)*
Zbl 0613.10023

Grundlehren der Mathematischen Wissenschaften, 286. Berlin etc.: Springer-Verlag. XII, 374 p.; DM 184.00 (1987).

The book is an introduction to the theory of Siegel modular forms, with special consideration of the action of Hecke operators on theta series of quadratic forms. Starting from scratch (only elementary algebra and real and complex analysis is assumed to be known), the book leads up to recent research achievements, summing up and extending many of the author’s results in this area.

The contents in more detail: Chapter 1: Theta series of positive definite integral quadratic forms are defined and studied. Chapter 2: Modular forms in one and several variables are introduced and investigated, including singular forms. Chapter 3: Abstract Hecke rings, Hecke rings of the general linear group, the symplectic group and the triangular subgroup of the symplectic group are studied. Chapters 4 and 5: Hecke operators are investigated, especially their action on theta series, which is explicitly described. The book closes with three appendices on symmetric matrices, quadratic spaces and binary quadratic forms.

The book is clearly written, with emphasis on explicit description, which is sometimes not easy in the case of several variables. Connections with algebraic geometry or representation theory are not touched upon.

The contents in more detail: Chapter 1: Theta series of positive definite integral quadratic forms are defined and studied. Chapter 2: Modular forms in one and several variables are introduced and investigated, including singular forms. Chapter 3: Abstract Hecke rings, Hecke rings of the general linear group, the symplectic group and the triangular subgroup of the symplectic group are studied. Chapters 4 and 5: Hecke operators are investigated, especially their action on theta series, which is explicitly described. The book closes with three appendices on symmetric matrices, quadratic spaces and binary quadratic forms.

The book is clearly written, with emphasis on explicit description, which is sometimes not easy in the case of several variables. Connections with algebraic geometry or representation theory are not touched upon.

Reviewer: M.Peters

### MSC:

11F27 | Theta series; Weil representation; theta correspondences |

11-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to number theory |

11E45 | Analytic theory (Epstein zeta functions; relations with automorphic forms and functions) |

11-02 | Research exposition (monographs, survey articles) pertaining to number theory |