*(English)*Zbl 0955.11001

As *N. K. Takare* wrote in the review of the first edition of this very rich and stimulating book (1976; Zbl 0318.33004): “Professor André Weil tries to resurrect Eisenstein’s long-forgotten approach to the study of elliptic functions and, doing so, adds much value to this interesting part of special function theory. His opinion is that Eisenstein’s, together with Kronecker’s works, can be profitably applied to some current problems in the theory of elliptic functions and the modular group, and extended beyond, in particular to the arithmetical study of Eisenstein series for Hilbert modular group”.

The book is divided, roughly, into three parts. In the first one, the author presents the theory of elliptic functions as developed by Eisenstein as early as 1847 in Crelles Journal. This theory includes the definition, by means of “Eisenstein summation process”, of the function ${E}_{2}\left(z\right)$, which is $\wp \left(z\right)$ up to an additive constant, and the determination of its differential equation, fifteen years before Weierstrass. The connection beetween the function ${E}_{2}$ and the cotangent, as well as the fact that Eisenstein uses only elementary computations (avoiding Cauchy’s theory), turn out to be very illuminating. Eisenstein obtains, by his approach, most results of the theory of elliptic functions. André Weil completes the theory by proving the relations between theta series and infinite products, what Eisenstein refrained from doing by “lack of time” (Eisenstein died in 1852 aged 29).

The second part of the book is devoted to Kronecker’s memoirs on elliptic functions, which were published by the Berlin Academy in 1883, 1885, 1886, 1889 and 1990, mainly to those concerned with double series of the form

where $W$ is a lattice in the complex plane, and $\chi $ a character of the additive group $W$. One of the results presented by the author is Kronecker’s limit formula, which gives the expansion of $G(s,1)$ at $s=1$. Contrary to Eisenstein, Kronecker uses a “whole arsenal of powerful tools: Poisson summation, Cauchy’s theorem of residues, Dirichlet theory of Fourier series and, most important of all, Dirichlet transformation formula (essentially our Mellin transform)”. To these tools, the author adds the theory of distributions, which allows him a re-interpretation of Kronecker’s results. One of the most impressive applications of Kronecker’s work is the solution of Pell’s equation by means of elliptic functions.

The third part of the book (“Variations 1 and 2” and “Finale: allegro con brio”) connects Eisenstein’s and Kronecker’s works to advanced research in Analytic Number Theory, for example Damerell’s theorem [*R. M. Damerell*, Acta Arith. 17, 287-301 (1970; Zbl 0209.24603)] and Chowla-Selberg formula [*A. Selberg* and *S. Chowla*, J. Reine Angew. Math. 227, 86-110 (1967; Zbl 0166.05204)].