Elliptic Curves

Dale Husemöller - Collection Graduate Texts in Mathematics

490 pages, parution le 04/02/2004 (2^eme édition)

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Résumé

This book is an introduction to the theory of elliptic curves, ranging from its most elementary aspects to current research. The first part, which grew out of Tate's Haverford lectures, covers the elementary arithmetic theory of elliptic curves over the rationals. The next two chapters recast the arguments used in the proof of the Mordell theorem into the context of Galois cohomology and descent theory. This is followed by three chapters on the analytic theory of elliptic curves, including such topics as elliptic functions, theta functions, and modular functions. Next, the theory of endomorphisms and elliptic curves over infinite and local fields are discussed. The book then continues by providing a survey of results in the arithmetic theory, especially those related to the conjecture of the Birch and Swinnerton-Dyer. This new edition contains three new chapters which explore recent directions and extensions of the theory of elliptic curves and the addition of two new appendices. The first appendix, written by Stefan Theisan, examines the role of Calabi-Yau manifolds in string theory, while the second, by Otto Forster, discusses the use of elliptic curves in computing theory and coding theory.
Dale Husemöller is a member of the faculty at the Max Planck Institute of Mathematics in Bonn.

Written for:
Graduate mathematics students, mathematicians

Sommaire

Introduction to Rational Points on Plane Curves
Elementary Properties of the Chord-Tangent Group Law on a Cubic Curve
Plane Algebraic Curves
Factorial Rings and Elimination Theory
Elliptic Curves and Their Isomorphism
Families of Elliptic Curves and Geometric Properties of Torsion Points
Reduction mod p and Torsion Points
Proof of Mordell's Finite Generation Theorem
Galois Cohomology and Isomorphism Classification of Elliptic Curves over Arbitrary Fields
Descent and Galois Cohomology
Elliptic and Hypergeometric Functions
Theta Functions
Modular Functions
Endomorphisms of Elliptic Curves
Elliptic Curves over Finite Fields
Elliptic Curves over Local Fields
Elliptic Curves over Global Fields and l-adic Representations
L-Functions of an Elliptic Curve and Its Analytic Continuation
Remarks on the Birch and Swinnerton-Dyer Conjecture
Remarks on the Modular Curves Conjecture and Fermat's Last Theorem
Higher Dimensional Analogs of Elliptic Curves: Calabi-Yau Varieties
Families of Elliptic Curves

Appendix I: Calabi-Yau Manifolds and String Theory
Appendix II: Elliptic Curves in Algorithmic Number Theory
Appendix III: Guide to the Exercises
Bibliography
Index

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	PAPIER
Éditeur(s)	Springer
Auteur(s)	Dale Husemöller
Collection	Graduate Texts in Mathematics
Parution	04/02/2004
Édition	2^eme édition
Nb. de pages	490
Format	16 x 24
Couverture	Relié
Poids	825g
Intérieur	Noir et Blanc
EAN13	9780387954905
ISBN13	978-0-387-95490-5