Elliptic functions and modular forms.
(Elliptische Funktionen und Modulformen.)

*(German)*Zbl 0895.11001
Berlin: Springer. xi, 289 S. (1998).

The book under review provides another text on the classical theory of elliptic functions and modular forms of one variable. It grew out of a manuscript for lectures on these topics, which Max Koecher had prepared in 1989. After Professor Koecher’s death, the text of his manuscript has been revised and enlarged by A. Krieg, who perfectly succeeded in preserving Koecher’s spirit and style in mathematical writing. The outcome is the present textbook, which is mainly addressed to first-year graduate students who intend to amplify their background knowledge in one-dimensional complex function theory by studying two of the most fascinating classical topics in this area, especially so along the historical line of development of these two “pearls” in mathematics.

The text is subdivided into two essentially independent parts: elliptic functions (Chapter I; one third of the text) and modular forms (Chapters II–V; two thirds of the text). Chapter I explains the theory of elliptic functions in great detail, from their historical origin motivated by elliptic integrals up to their complete analytic description as doubly-periodic meromorphic functions of one complex variable. This discussion includes all the beautiful classical material like lattices in \(\mathbb{C}\) and their invariants, associated tori, periodic meromorphic functions, the field of elliptic functions, the (analytic and geometric) rôle of the Weierstrass \(\wp\)-function, elliptic curves and their moduli, product representations of elliptic functions, complex multiplication, Jacobi’s theta series, Jacobi forms (as an outlook), and further hints to applications of elliptic functions.

Chapter II lays the foundations for the subsequently developed theory of modular functions and modular forms. The authors describe the geometry of the Siegel upper-half plane modulo the Siegel modular group and its congruence subgroups, together with a brief discussion of general discontinuous subgroups of the modular group.

Modular functions and modular forms on the upper-half plane are then the central objects treated in Chapter III. After a lucid motivating introduction to the subject, and after a first compilation of the most elementary properties of modular functions and modular forms, some crucial examples (Eisenstein series, the discriminant and the absolute invariant) are illuminated. The weight formula and its far-reaching applications to holomorphic modular forms are then just as thoroughly discussed as the basic geometry of modular functions, including the conformal map defined by the absolute invariant \(j\) as well as the resulting “Picard’s Little Theorem”. This chapter is concluded by a well-presented account on Dedekind’s eta-function and modular forms for congruence subgroups, which incorporates the corresponding Fourier analysis and a brief treatment of cusp forms.

Chapter IV is devoted to the algebra of Hecke operators, the Petersson inner product, the allied theory of Eisenstein series, and the theory of Dirichlet series with prescribed functional equation. The concluding Chapter V then deals with theta series. This chapter contains quite a lot of material on these objects, ranging from theta series as holomorphic modular forms, the Leech lattice, a special case of Siegel’s Main Theorem, the action of Hecke operators on theta series, harmonic polynomials, quadratic forms of level \(N\), up to a concluding outlook to the Epstein zeta-function, Kronecker’s limit formula, and the Rankin convolution.

The extremely well-versed and beautiful presentation of the classical material on elliptic functions and modular forms is throughly interwoven with numerous historical remarks, short biographical comments on the brilliant pioneers in the field, hints to further-going and contemporary developments, and suggestions for further reading with regard to the various special topics touched upon in the course of the text. Each chapter comes with its own specific introduction, and each section (within a chapter) is followed by a list of related, carefully selected, further-leading, and continuously challenging exercises.

Altogether, this comprehensibly but rigorously written textbook is a didactic masterpiece which breathes the matchless style of Koecher’s mathematical teaching and writing. The book also teaches mathematical culture at its best, in an extremely enlightening manner, and it is very gratifying to study it for both aesthetically demanding students and pretentious teachers in the field. Needless to say, this textbook is also a perfect basis for subsequent studies in complex analysis, algebraic geometry (of curves and abelian varieties), and number theory. Wherever it is appropriate, the authors point out important links of the material to related topics in these disciplines. Thus, apart from being an excellent textbook on the classical theory of elliptic functions and modular forms, the book is a very appetizing invitation to higher mathematics in general.

The text is subdivided into two essentially independent parts: elliptic functions (Chapter I; one third of the text) and modular forms (Chapters II–V; two thirds of the text). Chapter I explains the theory of elliptic functions in great detail, from their historical origin motivated by elliptic integrals up to their complete analytic description as doubly-periodic meromorphic functions of one complex variable. This discussion includes all the beautiful classical material like lattices in \(\mathbb{C}\) and their invariants, associated tori, periodic meromorphic functions, the field of elliptic functions, the (analytic and geometric) rôle of the Weierstrass \(\wp\)-function, elliptic curves and their moduli, product representations of elliptic functions, complex multiplication, Jacobi’s theta series, Jacobi forms (as an outlook), and further hints to applications of elliptic functions.

Chapter II lays the foundations for the subsequently developed theory of modular functions and modular forms. The authors describe the geometry of the Siegel upper-half plane modulo the Siegel modular group and its congruence subgroups, together with a brief discussion of general discontinuous subgroups of the modular group.

Modular functions and modular forms on the upper-half plane are then the central objects treated in Chapter III. After a lucid motivating introduction to the subject, and after a first compilation of the most elementary properties of modular functions and modular forms, some crucial examples (Eisenstein series, the discriminant and the absolute invariant) are illuminated. The weight formula and its far-reaching applications to holomorphic modular forms are then just as thoroughly discussed as the basic geometry of modular functions, including the conformal map defined by the absolute invariant \(j\) as well as the resulting “Picard’s Little Theorem”. This chapter is concluded by a well-presented account on Dedekind’s eta-function and modular forms for congruence subgroups, which incorporates the corresponding Fourier analysis and a brief treatment of cusp forms.

Chapter IV is devoted to the algebra of Hecke operators, the Petersson inner product, the allied theory of Eisenstein series, and the theory of Dirichlet series with prescribed functional equation. The concluding Chapter V then deals with theta series. This chapter contains quite a lot of material on these objects, ranging from theta series as holomorphic modular forms, the Leech lattice, a special case of Siegel’s Main Theorem, the action of Hecke operators on theta series, harmonic polynomials, quadratic forms of level \(N\), up to a concluding outlook to the Epstein zeta-function, Kronecker’s limit formula, and the Rankin convolution.

The extremely well-versed and beautiful presentation of the classical material on elliptic functions and modular forms is throughly interwoven with numerous historical remarks, short biographical comments on the brilliant pioneers in the field, hints to further-going and contemporary developments, and suggestions for further reading with regard to the various special topics touched upon in the course of the text. Each chapter comes with its own specific introduction, and each section (within a chapter) is followed by a list of related, carefully selected, further-leading, and continuously challenging exercises.

Altogether, this comprehensibly but rigorously written textbook is a didactic masterpiece which breathes the matchless style of Koecher’s mathematical teaching and writing. The book also teaches mathematical culture at its best, in an extremely enlightening manner, and it is very gratifying to study it for both aesthetically demanding students and pretentious teachers in the field. Needless to say, this textbook is also a perfect basis for subsequent studies in complex analysis, algebraic geometry (of curves and abelian varieties), and number theory. Wherever it is appropriate, the authors point out important links of the material to related topics in these disciplines. Thus, apart from being an excellent textbook on the classical theory of elliptic functions and modular forms, the book is a very appetizing invitation to higher mathematics in general.

Reviewer: W.Kleinert (Berlin)

##### MSC:

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

11F11 | Holomorphic modular forms of integral weight |

33E05 | Elliptic functions and integrals |

14H52 | Elliptic curves |

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

11G15 | Complex multiplication and moduli of abelian varieties |

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

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

11G05 | Elliptic curves over global fields |