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Abelian functions. Abel’s theorem and the allied theory of theta functions. Foreword by Igor Krichever (xvii-xxx). Repr. (English) Zbl 0848.14012
Cambridge Mathematical Library. Cambridge: Cambridge Univ. Press. xxxv, 684 p. (1995).
The theory of abelian functions is certainly amongst the greatest creations of mathematics that have been brought forth during the past two centuries. Historically emerged around 1829 from Jacobi’s famous statement of the inversion problem (“Umkehrproblem”) for integrals on algebraic curves, the development of the theory of abelian functions actually characterizes, to a large extent, the progress in mathematics achieved over the entire 19th century. The pioneering contributions by Jacobi himself, Abel, Riemann, Weierstrass, Picard, Poincaré, Humbert, Appell, Klein, Schottky, Krazer, Wirtinger, Baker, and others have made the theory of abelian functions into a dominating topic in mathematics, which provided, at that time, both one of the essential inspirations and a powerful toolkit for the development of algebraic geometry, complex analysis, and topology. Also, the theory of partial differential equations has been strongly influenced by the method of abelian functions, especially of theta functions, whereat the names of Painlevé, Poincaré, Kovalevskaya, and Neumann stand for some of the most important representatives. In particular, it is fair to say that, at the end of the 19th century, algebraic geometry – in its analytical aspects – widely coincided with the theory of abelian functions.
In the 1890s, the theory of abelian functions had reached a high degree of maturity and completeness. The report by A. Brill and M. Noether [in Jahresbericht Deutsch. Math. Ver. 3, 107-566 (1894); cf.: Fortschr. Math. 25, 70-75 (1894)] and the treatises of E. Picard [“Traité d’analyse”, Vol. II (1894)], C. Jordan [“Cours d’analyse”, Tome I, II (1894)], P. Appell and E. Goursat [“Théorie des fonctions algébriques et de leurs intégrales” (1895)], W. Wirtinger [“Untersuchungen über Thetafunktionen” (1895)], and H. Stahl [“Theorie der Abel’schen Functionen” (1896)] provided some accounts on this subject.
Then, in 1897, came the author’s encyclopedic book “Abel’s theorem and the allied theory including the theory of theta functions”. His admirably ambitious aim was to present the actual state of knowledge of algebraic geometry, of abelian functions and, in particular, of theta functions in a comprehensive, detailed and systematic textbook. In twenty-two chapters, and two appendices, the author managed to cover nearly all of the according material elaborated by that time. Using the framework of transcendental functions, and in particular the already far-developed theory of theta functions as the basic conceptual and methodical tool, he provided an encyclopedia of transcendental algebraic geometry, as it existed at the end of the last century, together with its allied analytic theories. Despite its importance, the author’s encyclopedic treatise has never seen any new edition since then. In the 1970s, when theta functions, and in particular his contributions to their applications, gained new interest in algebraic geometry, in the theory of nonlinear evolution equations, and in the mathematical physics of particles and fields, the author’s extremely valuable book was rediscovered as an indispensable source and reference. His treatise contains a wealth of information, including many special (sometimes almost forgotten) results and examples which are again of great interest to geometers, analysts, and physicists.
The second edition at hand, which is a reprint of the original edition from 1897, may therefore be regarded as a highly welcome addition to the contemporary literature on the subject. The undiminished significance of the author’s old encyclopedia is emphasized by a new foreword written by I. Krichever, in which the link between algebraic curves, theta functions, and soliton equations is briefly outlined. This beautifully written foreword throws a bridge between the author’s text and the frontiers of current research, and it explains why the book under review is still surprisingly up-to-date.

14H05 Algebraic functions and function fields in algebraic geometry
14H42 Theta functions and curves; Schottky problem
14-02 Research exposition (monographs, survey articles) pertaining to algebraic geometry
14K25 Theta functions and abelian varieties