Algebraic Geometry III

Complex Algebraic Varieties Algebraic Curves and Their Jacobians

A.N. Parshin, I.R. Shafarevich (Herausgeber)

Buch | Softcover

VIII, 270 Seiten

2010 | 1. Softcover reprint of the original 1st ed. 1998
Springer Berlin (Verlag)
978-3-642-08118-7 (ISBN)

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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.

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

Erscheint lt. Verlag	1.12.2010
Reihe/Serie	Encyclopaedia of Mathematical Sciences
Co-Autor	V.S. Kulikov, P.F. Kurchanov, V.V. Shokurov
Übersetzer	I. Rivin
Zusatzinfo	VIII, 270 p.
Verlagsort	Berlin
Sprache	englisch
Maße	155 x 235 mm
Gewicht	426 g
Themenwelt	Mathematik / Informatik ► Mathematik ► Analysis
	Mathematik / Informatik ► Mathematik ► Arithmetik / Zahlentheorie
	Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
Schlagworte	algebraic curves • Algebraic Varieties • algebraische Kurven • Analysis • complex algebraic varieties • differential equation • Hodgesche Struktur • Hodge structures • Jacobian varieties • Jacobischen Varietät • komplexe algebraische Varietät • Mathematical Physics • Sätze von • Torelli theorems
ISBN-10	3-642-08118-5 / 3642081185
ISBN-13	978-3-642-08118-7 / 9783642081187
Zustand	Neuware