Algebraic Geometry III: Complex Algebraic Varieties Algebraic Curves and Their Jacobians / Edition 1

by A.N. Parshin, I. Rivin, V.S. Kulikov, I.R. Shafarevich, P.F. Kurchanov

Overview

Starting with the end of the seventeenth century, one of the most interesting directions in mathematics (attracting the attention as J. Bernoulli, Euler, Jacobi, Legendre, Abel, among others) has been the study of integrals of the form r dz l Aw(T) = -, TO W where w is an algebraic function of z. Such integrals are now called abelian. Let us examine the simplest instance of an abelian integral, one where w is defined by the polynomial equation (1) where the polynomial on the right hand side has no multiple roots. In this case the function Aw is called an elliptic integral. The value of Aw is determined up to mv + nv , where v and v are complex numbers, and m and n are 1 2 1 2 integers. The set of linear combinations mv+ nv forms a lattice H C C, and 1 2 so to each elliptic integral Aw we can associate the torus C/ H. 2 On the other hand, equation (1) defines a curve in the affine plane C = 2 2 {(z,w)}. Let us complete C2 to the projective plane lP' = lP' (C) by the addition of the "line at infinity", and let us also complete the curve defined 2 by equation (1). The result will be a nonsingular closed curve E C lP' (which can also be viewed as a Riemann surface). Such a curve is called an elliptic curve.

Product Details

ISBN-13:	9783642081187
Publisher:	Springer Berlin Heidelberg
Publication date:	12/07/2010
Series:	Encyclopaedia of Mathematical Sciences , #36
Edition description:	Softcover reprint of the original 1st ed. 1998
Pages:	270
Product dimensions:	6.10(w) x 9.25(h) x 0.24(d)

Table of Contents

I. Complex Algebraic Varieties: Periods of Integrals and Hodge Structures.- II. Algebraic Curves and Their Jacobians.