André Weil was one of the great mathematicians of the twentieth century. This 94-page book was originally published in 1975; it has now been reprinted in the Springer Classics in Mathematics series. His purpose, as stated in the Foreword, is to clean up some paths that have become overrun over time. Eisenstein and Kronecker were nineteenth century mathematicians, but their work on elliptic functions still has many applications in modern number theory.
Although explicating some nineteenth century mathematics, this text is not primarily a historical work; it was meant as a guide to mathematics useful in current research. The first half deals with Eisenstein’s contributions, while the second is devoted to Kronecker, who in 1891, just before he died, was to have given a lecture on Eisentein’s little-known work combining numnber theory and function theory, particularly elliptic functions.
To introduce Eisenstein’s results, Weil first presents the simpler case of trigonometric functions. He then goes on to the basic elliptic functions and their relationships, concluding the Eisenstein part with some variations on the major themes.
Kronecker and Eisenstein were both born in 1823, but Eisenstein died of tuberculosis at age 29. Kronecker wrote several papers on elliptic functions, only in the later stages realizing how his work was a generalization of Eisenstein’s. Weil details Kronecker’s work on elliptic functions, including a section showing how the modern theory of distributions allows a re-interpretation of some results. He concludes with some applications to number theory.
Weil has cleared a mathematical road rich in applications and connections with current research.
Art Gittleman (email@example.com) is Professor of Computer Science at California State University Long Beach.
Part 1: Eisenstein
The basic elliptic functions
Basic relations and infinite products
Part 2: Kronecker
Prelude to Kronecker
Kronecker's double series
Finale: Allegro con brio (Pell's equation and the Chowla-Selberg formula)
Index of Notations.