Grundlehren der Mathematischen Wissenschaften, 281. Berlin etc.: Springer-Verlag. XI, 189 p. DM 138.00 (1985).
Nowadays there seems to be a renewed interest in the theory of elliptic functions. At the end of the 19-th and the beginning of the 20-th century there were classical books on the subject. We recall only H. Burkhardt and G. Faber, Elliptische Funktionen (1920); R. Fricke, Die Elliptischen Funktionen und ihre Anwendungen. I,II (1916- 1922); J. Tannery and J. Molk, Eléments de la théorie des fonctions elliptiques. I,II,III,IV (1893-1902).
Lateron we studied elliptic functions as a special subject in classical books on analysis, more special complex function theory; such as E. T. Whittaker and G. N. Watson’s, A course of modern analysis (1952; Zbl 0108.269), K. Knopp’s Funktionentheory II (1981; Zbl 0452.30001) and the many cours d’Analyse of the French mathematicians. And nearly always elliptic functions were also used as an introduction to modular functions, the last being an essential tool for number theory. In the classical book there was the application to the spherical pendulum.
In recent times interest was asked for the history of elliptic functions by the reedition of a chapter out of C. Jordan’s, Fonctions elliptiques. Reprint of ”Cours d’Analyse de l’école Polytechnique” (1981; Zbl 0472.33002) and the historical study of A. Weil, Elliptic functions according to Eisenstein and Kronecker (1976; Zbl 0318.33004).
Apart from these several modern books concerning elliptic functions were published. The volume under review can be compared with S. Lang, Elliptic functions (1973; Zbl 0316.14001) and B. Schoeneberg, Elliptic modular functions (1974; Zbl 0285.10016). Lang’s book is quite modern, some people will call it high-low, it deals also with recent research, we mention his chapters on Tate-parametrization, on Ihara’s theory, on modern aspects of complex multiplication and of Kronecker’s limit formula. Schoeneberg’s book is written on the more classical topics in the tradition set by Erich Hecke.
The author’s book is largely inspired by many ideas and contributions to the field made by C. L. Siegel. The book is published in the series ”Grundlehren”, its title could have been (following the French tradition) ”Elements of the theory of elliptic functions”. It gives a nice elementary introduction in the subject, presuppositing only the fundamentals of complex analysis. It deals with the elliptic functions as well in the Weierstraß setting, starting with the -function and the thereof derived functions and , as with the -functions, as with the Jacobian functions sn, cn and dn and all the relations between these elliptic functions.
In the direction of modular functions the author deals with the modular invariant J( and with Dedekind’s -function. And, of course, having this material at hand, the author gives a number of applications in number theory, e.g. the law of quadratic reciprocity using Gaußian sums, the representation of integers by a positive-definite quadratic form, such as the sum of four squares. For the beginning student and for the specialist in the field very extensive notes at the end of every chapter are of great interest. There is a lot of explicit mentioning of the work of many classical authors and of new proofs of classical theorems given in our generation. As was said above many of the new, shorter and more elegant proofs are due to C. L. Siegel. The last sentence of the author’s book reads:
It is these papers of Siegel, in which the methods of Minkowski and of Hardy are combined and adapted, about which H. Weyl (in his Ges. Abhandlungen, No.136 (1944/45), Bd. IV, p. 232) wrote: ”In a series of publications which continue the great tradition of Gauß, Dirichlet, Hermite, Minkowski and which I am sure will be admired by generations of mathematicians to come Siegel has vastly added to our knowledge in the theory of definite and indefinite quadratic forms and of arithmetical reduction”.
One could add that the work of A. Weil continuing in the great tradition of Siegel has once more vastly added to our knowledge in the theory of definite and indefinite quadratic forms and to our insight of the rôle of this theory as a special (and simple) case of a more general, group theoretical, adelic and algebraic-geometrical approach of number theory.