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Introduction to the classical theory of Abelian functions. Translated from the Russian by G. Bluher. Translation edited by Ralph P. Boas. (English) Zbl 0743.14033
Translations of Mathematical Monographs. 96. Providence, RI: American Mathematical Society (AMS). viii, 175 p. (1992).
The present book is a translation of the Russian original monograph (1979; Zbl 0493.14023). The theory of abelian functions played a central role in the mathematical research of the 19th century, and it is precisely that classical framework which the author presents here. That means, the book provides an introduction to the theory of abelian functions which is based on the classical theory of complex functions, and in this regard it is methodically closer to the famous earlier expositions by F. Conforto [“Abelsche Funktionen und algebraische Geometrie”, Grundlehren der Math. Wiss. 84 (1956; Zbl 0074.366)] and C. L. Siegel [“Topics in complex function theory”, Vol. III (New York 1973; Zbl 0257.32002)] than to the various recent books on the algebro-geometric theory of abelian varieties and theta functions. The main feature of the book under review is, now as before, that it is very well suited for teaching and learning at the upper undergraduate level. It just requires the standard knowledge of complex function theory in several variables, all the additional tools are given in three appendices.
MSC:
14K20Analytic theory; abelian integrals and differentials
14H05Algebraic functions; function fields
33E05Elliptic functions and integrals
14-01Textbooks (algebraic geometry)
32N05General theory of automorphic functions of several complex variables