##
**Topics in classical automorphic forms.**
*(English)*
Zbl 0905.11023

Graduate Studies in Mathematics. 17. Providence, RI: American Mathematical Society (AMS). xii, 259 p. (1997).

The theory of modular and automorphic forms constitutes an integral part of contemporary number theory (and of many other parts of mathematics and physics). Since its importance has grown tremendously in the recent past, it is quite natural that several monographs have appeared recently to cover the various aspects of the theory, and even more books are needed to make the enormous amount of new research accessible to prospective readers. The author of the book under review has chosen to deal with those parts of the theory which favour analytic methods and aim at number theoretic applications. We briefly indicate the contents of the various chapters.

Chapter 1 deals with introductory material on classical modular forms (elliptic functions, modular forms, Eisenstein series on \(SL_2 (\mathbb{Z})\), the linear space of modular forms). Chapter 2 contains some background material on automorphic forms in general (cofinite Fuchsian groups, multiplier systems, automorphic forms of real weight, and the eta function and the theta functions as examples). Eisenstein and Poincaré series for Fuchsian groups are introduced in Chapter 3, their Fourier expansion is derived, and the Petersson scalar product is used to prove the Petersson coefficient formulae relating the Fourier coefficients of an automorphic form \(f\) with the scalar product of \(f\) with a Poincaré series.

Chapters 4 and 5 belong to the field of research interest of the author and deal with various types of Kloosterman sums and Salié sums, including methods to compute or estimate these sums and with bounds for the Fourier coefficients of cusp forms. Here we find e.g. an elementary proof (not using Weil’s bound on the Kloosterman sums) for the bound \[ a(n)\ll n^{{k \over 2} -{1\over 4} + \varepsilon} \] for the Fourier coefficients of a modular cusp form of weight \(k\) on a Hecke congruence group \(\Gamma_0 (N)\), and even an improvement on this bound is derived which is used to establish the equidistribution of integral points on ellipsoids.

Chapter 6 is devoted to the theory of Hecke operators. To shorten the exposition, the author decides to replace the double coset construction by explicit choice of representatives. The theory is first explained in the case of the modular group. This is followed by the theory of Hecke operators on modular forms on \(\Gamma_0 (q)\) belonging to a character \(\chi \pmod q\). A brief overview of the theory of newforms is followed by a section giving proofs in the case of a primitive character \(\chi \bmod q\).

The theory of automorphic \(L\)-functions is started in Chapter 7, containing the basic results on Hecke \(L\)-functions associated with cusp forms, twisting of automorphic forms and \(L\)-functions and Weil’s converse theorem. The connection between elliptic curves and cusp forms is explained in Chapter 8, and the special case of the family of elliptic curves connected with the ancient problem of congruent numbers is discussed.

The following three chapters focus on the problem of representing integers by quadratic forms: Spherical functions on \(\mathbb{R}^n\) are introduced in Chapter 9 and an account of harmonic analysis on the sphere in \(\mathbb{R}^3\) is given. Theta functions associated with spherical functions are discussed in Chapter 10, and their automorphy behaviour is analyzed carefully. This is applied in Chapter 11 to representations of integers by positive definite integral quadratic forms (Siegel’s mass formula, representation of theta series by Eisenstein series and cusp forms, the circle method after Kloosterman, the singular series, equidistribution of integral points on ellipsoids).

Chapter 12 on automorphic forms associated with number fields aims at illustrating with numerous examples the unproven principle that the \(L\)-functions of algebraic number theory are derived from automorphic forms of various kinds. To accomplish this aim the author makes use of the Hecke-Weil converse theorems. A primary object here is to construct cusp forms from Hecke characters of a quadratic number field. The author also takes a direct approach to the problem and shows that the automorphic forms sought after can be expressed in terms of theta functions (at least in special cases). The chapter ends with a glimpse into the Langlands program in the context of two-dimensional Galois representations and modular forms of weight one (Theorem of P. Deligne and J.-P. Serre [Ann. Sci. Éc. Norm. Supér., IV. Sér. 7, 507-530 (1974; Zbl 0321.10026)]).

The final Chapter 13 deals with convolution \(L\)-functions. The author’s main goal is to establish the analytic continuation of \(L(f \otimes g,s)\) and some kind of functional equation. This is achieved by the Rankin-Selberg method. The necessary prerequisites on Selberg’s theory of Eisenstein series are surveyed. The chapter ends with an outlook on symmetric power \(L\)-functions and their conjectural properties.

A highly attractive feature of the book under review is that it treats an enormous number of topics and that it tries to bridge the gap between an introductory course and contemporary research. This is achieved on roughly 250 pages, and clearly this is possible only since the author omits some proofs.

Unfortunately, in such instances the reader often will look in vain for a reference. In fact, the bibliography is a weak point of the book. Many standard references are missing, and there are quite a lot of instances where a result is hinted at and the bare name of the author is given without any further reference. The author points out in the preface that the book is based (largely verbatim) on a set of lecture notes from a graduate course.

The book is not free from obvious blunders. (E.g. it is stated on p. 4: “If \(f\) is continuous, the Fourier series \(\dots\) converges absolutely, hence uniformly.” Section 2.3 is marred by the error that a Fuchsian group of the first kind need not be finitely generated.) Historical remarks may not be reliable. (E.g. the first proof of the multiplicativity of the Ramanujan numbers \(\tau(n)\) is due to L. J. Mordell [Camb. Philos. Soc. Proc. 19, 117-124 (1917; JFM 46.0605.01)], not to E. Hecke, as stated on p. 91. The proofs of the normality of the Hecke operators and of the simultaneous diagonalization of the Hecke operators are due to H. Petersson, not to E. Hecke as claimed on p. 101, 104, 106; [see J. Elstrodt and F. Grunewald, The Petersson scalar product, Jahresber. Dtsch. Math.-Ver. 100, 253-283 (1998)]. C. G. J. Jacobi (1804-1851) obviously was not a “late eighteenth century” mathematician.)

Notwithstanding minor deficiencies of this kind, this is an excellent book, requiring hard work from the reader and giving rich reward for her or his effort.

Chapter 1 deals with introductory material on classical modular forms (elliptic functions, modular forms, Eisenstein series on \(SL_2 (\mathbb{Z})\), the linear space of modular forms). Chapter 2 contains some background material on automorphic forms in general (cofinite Fuchsian groups, multiplier systems, automorphic forms of real weight, and the eta function and the theta functions as examples). Eisenstein and Poincaré series for Fuchsian groups are introduced in Chapter 3, their Fourier expansion is derived, and the Petersson scalar product is used to prove the Petersson coefficient formulae relating the Fourier coefficients of an automorphic form \(f\) with the scalar product of \(f\) with a Poincaré series.

Chapters 4 and 5 belong to the field of research interest of the author and deal with various types of Kloosterman sums and Salié sums, including methods to compute or estimate these sums and with bounds for the Fourier coefficients of cusp forms. Here we find e.g. an elementary proof (not using Weil’s bound on the Kloosterman sums) for the bound \[ a(n)\ll n^{{k \over 2} -{1\over 4} + \varepsilon} \] for the Fourier coefficients of a modular cusp form of weight \(k\) on a Hecke congruence group \(\Gamma_0 (N)\), and even an improvement on this bound is derived which is used to establish the equidistribution of integral points on ellipsoids.

Chapter 6 is devoted to the theory of Hecke operators. To shorten the exposition, the author decides to replace the double coset construction by explicit choice of representatives. The theory is first explained in the case of the modular group. This is followed by the theory of Hecke operators on modular forms on \(\Gamma_0 (q)\) belonging to a character \(\chi \pmod q\). A brief overview of the theory of newforms is followed by a section giving proofs in the case of a primitive character \(\chi \bmod q\).

The theory of automorphic \(L\)-functions is started in Chapter 7, containing the basic results on Hecke \(L\)-functions associated with cusp forms, twisting of automorphic forms and \(L\)-functions and Weil’s converse theorem. The connection between elliptic curves and cusp forms is explained in Chapter 8, and the special case of the family of elliptic curves connected with the ancient problem of congruent numbers is discussed.

The following three chapters focus on the problem of representing integers by quadratic forms: Spherical functions on \(\mathbb{R}^n\) are introduced in Chapter 9 and an account of harmonic analysis on the sphere in \(\mathbb{R}^3\) is given. Theta functions associated with spherical functions are discussed in Chapter 10, and their automorphy behaviour is analyzed carefully. This is applied in Chapter 11 to representations of integers by positive definite integral quadratic forms (Siegel’s mass formula, representation of theta series by Eisenstein series and cusp forms, the circle method after Kloosterman, the singular series, equidistribution of integral points on ellipsoids).

Chapter 12 on automorphic forms associated with number fields aims at illustrating with numerous examples the unproven principle that the \(L\)-functions of algebraic number theory are derived from automorphic forms of various kinds. To accomplish this aim the author makes use of the Hecke-Weil converse theorems. A primary object here is to construct cusp forms from Hecke characters of a quadratic number field. The author also takes a direct approach to the problem and shows that the automorphic forms sought after can be expressed in terms of theta functions (at least in special cases). The chapter ends with a glimpse into the Langlands program in the context of two-dimensional Galois representations and modular forms of weight one (Theorem of P. Deligne and J.-P. Serre [Ann. Sci. Éc. Norm. Supér., IV. Sér. 7, 507-530 (1974; Zbl 0321.10026)]).

The final Chapter 13 deals with convolution \(L\)-functions. The author’s main goal is to establish the analytic continuation of \(L(f \otimes g,s)\) and some kind of functional equation. This is achieved by the Rankin-Selberg method. The necessary prerequisites on Selberg’s theory of Eisenstein series are surveyed. The chapter ends with an outlook on symmetric power \(L\)-functions and their conjectural properties.

A highly attractive feature of the book under review is that it treats an enormous number of topics and that it tries to bridge the gap between an introductory course and contemporary research. This is achieved on roughly 250 pages, and clearly this is possible only since the author omits some proofs.

Unfortunately, in such instances the reader often will look in vain for a reference. In fact, the bibliography is a weak point of the book. Many standard references are missing, and there are quite a lot of instances where a result is hinted at and the bare name of the author is given without any further reference. The author points out in the preface that the book is based (largely verbatim) on a set of lecture notes from a graduate course.

The book is not free from obvious blunders. (E.g. it is stated on p. 4: “If \(f\) is continuous, the Fourier series \(\dots\) converges absolutely, hence uniformly.” Section 2.3 is marred by the error that a Fuchsian group of the first kind need not be finitely generated.) Historical remarks may not be reliable. (E.g. the first proof of the multiplicativity of the Ramanujan numbers \(\tau(n)\) is due to L. J. Mordell [Camb. Philos. Soc. Proc. 19, 117-124 (1917; JFM 46.0605.01)], not to E. Hecke, as stated on p. 91. The proofs of the normality of the Hecke operators and of the simultaneous diagonalization of the Hecke operators are due to H. Petersson, not to E. Hecke as claimed on p. 101, 104, 106; [see J. Elstrodt and F. Grunewald, The Petersson scalar product, Jahresber. Dtsch. Math.-Ver. 100, 253-283 (1998)]. C. G. J. Jacobi (1804-1851) obviously was not a “late eighteenth century” mathematician.)

Notwithstanding minor deficiencies of this kind, this is an excellent book, requiring hard work from the reader and giving rich reward for her or his effort.

Reviewer: J.Elstrodt (Münster)

### 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 |

33E05 | Elliptic functions and integrals |

11E25 | Sums of squares and representations by other particular quadratic forms |

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

11F03 | Modular and automorphic functions |

11F11 | Holomorphic modular forms of integral weight |

11F12 | Automorphic forms, one variable |

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

11F30 | Fourier coefficients of automorphic forms |

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

11F25 | Hecke-Petersson operators, differential operators (one variable) |

11F72 | Spectral theory; trace formulas (e.g., that of Selberg) |

11G05 | Elliptic curves over global fields |