Introduction to the Classical Theory of Abelian Functions by A.I. Markushevich PDF

Algebraic Geometry

By A.I. Markushevich

The idea of Abelian capabilities, which was once on the heart of nineteenth-century arithmetic, is back attracting recognition. although, this present day it truly is often visible not only as a bankruptcy of the overall thought of features yet as a space of program of the information and strategies of commutative algebra.

This booklet provides an exposition of the basics of the idea of Abelian capabilities in line with the equipment of the classical thought of services. This idea comprises the idea of elliptic capabilities as a unique case. one of the themes coated are theta capabilities, Jacobians, and Picard kinds. the writer has aimed the publication basically at intermediate and complex graduate scholars, however it could even be available to the start graduate scholar or complex undergraduate who has a superb heritage in services of 1 advanced variable. This e-book will end up in particular valuable to those that aren't accustomed to the analytic roots of the topic. additionally, the specific historic creation cultivates a deep knowing of the topic. Thorough and self-contained, the publication will supply readers with a very good supplement to the standard algebraic approach.

Readership: top point undergraduates, graduate scholars, and examine mathematicians drawn to research.

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Extra resources for Introduction to the Classical Theory of Abelian Functions

Example text

Up , and moreover they will be representable in a neighborhood of any point as the ratio of two functions ... , u(°) P which are analytic at that point. See for example. (8) And this means that they are meromorphic functions of p complex variables. Let w, k = 1, ... , 2p be a system of 2p independent periods of the (8) E. Picard, Traite d'analyse, vol. , Gauthier-Villars, 1925, pp. 510-512. §9. MAIN DIRECTIONS OF DEVELOPMENT OF THE THEORY OF ABELIAN FUNCTIONS 41 integral f R( z,, w) d z (j = 1 , ...

In order to investigate the behavior of an algebraic function at the point at infinity, we make the substitution z = 1/z'. In the neighborhood of z' = 0 we get the equation ao(Z')w`t + ... + a`t (Z')= p where a1(z') are polynomials in z' (ao(z') whether we have: 0, an(z') 0). Depending on 30 I. HISTORICAL INTRODUCTION. THE JACOBI INVERSION PROBLEM (1) ao(0) 0 and the equation has no repeated roots, (2) ao(0) # 0 and the equation has repeated roots, or finally (3) a0(O)=O, we will have one of the cases outlined above for w = w(1/z').

Consequently, for example, we will not base our discussion on the theory of algebraic functions and Abelian integrals. Instead, we will need some facts from the theory of analytic functions of several complex variables. To be sure, the number of these facts is quite modest (see Chapter II ). An important role will be played by the Poincare-Cousin theorem on the existence of a representation of any meromorphic function as the quotient of entire functions which are relatively prime at each point (see Appendix B).

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