##
**Modular forms. Transl. from the Japanese by Joshitaku Maeda.**
*(English)*
Zbl 0701.11014

Berlin etc.: Springer-Verlag. viii, 335 p. (1989).

The book under review is for the most part the English translation of the monograph by K. Doi and T. Miyake [Automorphic forms and number theory (Japanese) (1976; Zbl 0466.10012)]. The English version was prepared by T. Miyake who substantially revised the text. The contents of the book are roughly as follows.

Chapter I covers the basic facts on Fuchsian groups \(\Gamma\) acting on the upper half-plane \({\mathbb{H}}\). The careful discussion of \(\Gamma \setminus {\mathbb{H}}^*\) as a Riemann surface is noteworthy. Moreover, Siegel’s theorem is proved. Automorphic forms are introduced in Chapter II, and there is a discussion of the relation between automorphic forms of weight 2m on \({\mathbb{H}}\) and differentials of degree m on \(\Gamma \setminus {\mathbb{H}}^*\). This is used in order to compute the dimensions of various spaces of automorphic forms by means of the Riemann-Roch Theorem. Non-trivial automorphic forms are constructed using suitable Poincaré series. Special cases include Petersson’s “Poincaré series of parabolic type”, and the corresponding coefficient formulae and the completeness theorem are proved. The abstract Hecke algebra is introduced (following Shimura), and the action of the Hecke operators on spaces of automorphic forms is first studied for arbitrary cofinite Fuchsian groups.

In Chapter 3, some prerequisites on number theory, Dirichlet characters, Hecke characters and L-functions are briefly summarized.

Chapter 4 on modular groups and modular forms is the heart of the book. Here we find the basic material on \(SL_ 2({\mathbb{Z}})\) and the basic results on modular forms. The formulae for the dimensions of the spaces of integral forms and of cusp forms are deduced from the corresponding general results on cofinite Fuchsian groups. The section on congruence subgroups of \(SL_ 2({\mathbb{Z}})\) contains a substantial amount of information on relatively few pages. (The geometry of the corresponding fundamental domains is not investigated.) The strong number-theoretic flavour of the book becomes apparent from the subsequent sections and chapters.

Modular forms on \(\Gamma_ 0(N)\) and the associated Dirichlet series are discussed thoroughly, and the fundamental equivalence theorems of Hecke and of Weil are proved. The transformation formula of the Dedekind eta- function is proved by an elegant method of Weil which reduces the problem of the proof of the functional equation for \(\zeta (s)\zeta (s+1)\). This yields the product expansion for the discriminant function \(\Delta\).

An in depth study of Hecke algebras of modular groups and the relation between Fourier coefficients of modular forms and Hecke operators form the contents of the subsequent sections of Chapter 4. This includes a detailed discussion of primitive forms (following Atkin-Lehner, Miyake, Asai and Naganuma). Inverting a classical approach of Hecke, the author starts from Dirichlet L-series and constructs the associated modular forms (that is, the corresponding Eisenstein series) by an application of Weil’s theorem. Analogously, the cusp forms associated with L-functions of quadratic fields are also introduced via Weil’s theorem. Theta series with spherical functions serve as another important class of modular forms which are studied in detail.

Chapter 5 dealing with unit groups of quaternion algebras is quite unique in the present-day textbook literature on modular forms. The necessary background material on adèles and the theory of quaternion algebras is concisely and carefully explained, and the author proves the basic theorem that the unit group of norm 1 of an order R of a quaternion algebra B over \({\mathbb{Q}}\) is a cofinite Fuchsian group which is cocompact if and only if B is a division quaternion algebra. Then the Hecke algebras of unit groups of quaternion algebras are investigated.

Following Selberg, the author computes the traces of Hecke operators in Chapter 6. This includes a discussion of kernel functions, and the trace of Hecke operators is first computed in the case of a cofinite Fuchsian group. Then the general formula is applied to the case of a Fuchsian group which is obtained as the unit group of norm 1 of an order in an indefinite quaternion algebra over \({\mathbb{Q}}\). The final result, Theorem 6.8.4 covers almost two printed pages.

The study of Eisenstein series is resumed in Chapter 7. First, Eisenstein series of weight \(k\geq 3\) for modular groups are discussed and the relation with earlier investigations is established. For \(k\leq 2\) the author defines Eisenstein series with a complex parameter 1, computes their Fourier expansion and thus obtains their analytic continuation. This yields the definition and properties of Eisenstein series of weights \(k=1\) and \(k=2\). - As some applications of the trace formulae, the author includes some numerical tables on dimensions of spaces of cusp forms, eigenvalues and characteristic polynomials of Hecke operators and coefficients of primitive cusp forms of weight 2.

The book under review is a very welcome addition to the literature on modular forms. The newcomer gets a rapid introduction to this vast field which may lead him quickly to independent research. Summing up, this work can be warmly recommended to everybody with some interest in modular forms and their relations with number theory, and certainly any institutional library on pure mathematics should have this book.

Chapter I covers the basic facts on Fuchsian groups \(\Gamma\) acting on the upper half-plane \({\mathbb{H}}\). The careful discussion of \(\Gamma \setminus {\mathbb{H}}^*\) as a Riemann surface is noteworthy. Moreover, Siegel’s theorem is proved. Automorphic forms are introduced in Chapter II, and there is a discussion of the relation between automorphic forms of weight 2m on \({\mathbb{H}}\) and differentials of degree m on \(\Gamma \setminus {\mathbb{H}}^*\). This is used in order to compute the dimensions of various spaces of automorphic forms by means of the Riemann-Roch Theorem. Non-trivial automorphic forms are constructed using suitable Poincaré series. Special cases include Petersson’s “Poincaré series of parabolic type”, and the corresponding coefficient formulae and the completeness theorem are proved. The abstract Hecke algebra is introduced (following Shimura), and the action of the Hecke operators on spaces of automorphic forms is first studied for arbitrary cofinite Fuchsian groups.

In Chapter 3, some prerequisites on number theory, Dirichlet characters, Hecke characters and L-functions are briefly summarized.

Chapter 4 on modular groups and modular forms is the heart of the book. Here we find the basic material on \(SL_ 2({\mathbb{Z}})\) and the basic results on modular forms. The formulae for the dimensions of the spaces of integral forms and of cusp forms are deduced from the corresponding general results on cofinite Fuchsian groups. The section on congruence subgroups of \(SL_ 2({\mathbb{Z}})\) contains a substantial amount of information on relatively few pages. (The geometry of the corresponding fundamental domains is not investigated.) The strong number-theoretic flavour of the book becomes apparent from the subsequent sections and chapters.

Modular forms on \(\Gamma_ 0(N)\) and the associated Dirichlet series are discussed thoroughly, and the fundamental equivalence theorems of Hecke and of Weil are proved. The transformation formula of the Dedekind eta- function is proved by an elegant method of Weil which reduces the problem of the proof of the functional equation for \(\zeta (s)\zeta (s+1)\). This yields the product expansion for the discriminant function \(\Delta\).

An in depth study of Hecke algebras of modular groups and the relation between Fourier coefficients of modular forms and Hecke operators form the contents of the subsequent sections of Chapter 4. This includes a detailed discussion of primitive forms (following Atkin-Lehner, Miyake, Asai and Naganuma). Inverting a classical approach of Hecke, the author starts from Dirichlet L-series and constructs the associated modular forms (that is, the corresponding Eisenstein series) by an application of Weil’s theorem. Analogously, the cusp forms associated with L-functions of quadratic fields are also introduced via Weil’s theorem. Theta series with spherical functions serve as another important class of modular forms which are studied in detail.

Chapter 5 dealing with unit groups of quaternion algebras is quite unique in the present-day textbook literature on modular forms. The necessary background material on adèles and the theory of quaternion algebras is concisely and carefully explained, and the author proves the basic theorem that the unit group of norm 1 of an order R of a quaternion algebra B over \({\mathbb{Q}}\) is a cofinite Fuchsian group which is cocompact if and only if B is a division quaternion algebra. Then the Hecke algebras of unit groups of quaternion algebras are investigated.

Following Selberg, the author computes the traces of Hecke operators in Chapter 6. This includes a discussion of kernel functions, and the trace of Hecke operators is first computed in the case of a cofinite Fuchsian group. Then the general formula is applied to the case of a Fuchsian group which is obtained as the unit group of norm 1 of an order in an indefinite quaternion algebra over \({\mathbb{Q}}\). The final result, Theorem 6.8.4 covers almost two printed pages.

The study of Eisenstein series is resumed in Chapter 7. First, Eisenstein series of weight \(k\geq 3\) for modular groups are discussed and the relation with earlier investigations is established. For \(k\leq 2\) the author defines Eisenstein series with a complex parameter 1, computes their Fourier expansion and thus obtains their analytic continuation. This yields the definition and properties of Eisenstein series of weights \(k=1\) and \(k=2\). - As some applications of the trace formulae, the author includes some numerical tables on dimensions of spaces of cusp forms, eigenvalues and characteristic polynomials of Hecke operators and coefficients of primitive cusp forms of weight 2.

The book under review is a very welcome addition to the literature on modular forms. The newcomer gets a rapid introduction to this vast field which may lead him quickly to independent research. Summing up, this work can be warmly recommended to everybody with some interest in modular forms and their relations with number theory, and certainly any institutional library on pure mathematics should have this book.

Reviewer: J.Elstrodt

### MSC:

11Fxx | Discontinuous groups and automorphic forms |

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

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

11F11 | Holomorphic modular forms of integral weight |

11F12 | Automorphic forms, one variable |

11F20 | Dedekind eta function, Dedekind sums |

11F06 | Structure of modular groups and generalizations; arithmetic groups |

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

11F30 | Fourier coefficients of automorphic forms |

11F66 | Langlands \(L\)-functions; one variable Dirichlet series and functional equations |