Arithmetic of Elliptic Curves

ISBN-10: 0387094938
ISBN-13: 9780387094939
Edition: 2nd 2009
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Description: The theory of elliptic curves is distinguished by its long history and by the diversity of the methods that have been used in its study. This book treats the arithmetic theory of elliptic curves in its modern formulation, through the use of basic  More...

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Book details

Edition: 2nd
Copyright year: 2009
Publisher: Springer
Publication date: 5/29/2009
Binding: Hardcover
Pages: 514
Size: 6.25" wide x 9.50" long x 1.00" tall
Weight: 1.936
Language: English

The theory of elliptic curves is distinguished by its long history and by the diversity of the methods that have been used in its study. This book treats the arithmetic theory of elliptic curves in its modern formulation, through the use of basic algebraic number theory and algebraic geometry. The book begins with a brief discussion of the necessary algebro-geometric results, and proceeds with an exposition of the geometry of elliptic curves, the formal group of an elliptic curve, and elliptic curves over finite fields, the complex numbers, local fields, and global fields. Included are proofs of the Mordell–Weil theorem giving finite generation of the group of rational points and Siegel's theorem on finiteness of integral points.For this second edition of The Arithmetic of Elliptic Curves, there is a new chapter entitled Algorithmic Aspects of Elliptic Curves, with an emphasis on algorithms over finite fields which have cryptographic applications. These include Lenstra's factorization algorithm, Schoof's point counting algorithm, Miller's algorithm to compute the Tate and Weil pairings, and a description of aspects of elliptic curve cryptography. There is also a new section on Szpiro's conjecture and ABC, as well as expanded and updated accounts of recent developments and numerous new exercises.The book contains three appendices: Elliptic Curves in Characteristics 2 and 3, Group Cohomology, and a third appendix giving an overview of more advanced topics.

Preface to the Second Edition
Preface to the First Edition
Introduction
Algebraic Varieties
Affine Varieties
Projective Varieties
Maps Between Varieties
Exercises
Algebraic Curves
Curves
Maps Between Curves
Divisors
Differentials
The Riemann-Roch Theorem
Exercises
The Geometry of Elliptic Curves
Weierstrass Equations
The Group Law
Elliptic Curves
Isogenies
The Invariant Differential
The Dual Isogeny
The Tate Module
The Weil Pairing
The Endomorphism Ring
The Automorphism Group
Exercises
The Formal Group of an Elliptic Curve
Expansion Around O
Formal Groups
Groups Associated to Formal Groups
The Invariant Differential
The Formal Logarithm
Formal Groups over Discrete Valuation Rings
Formal Groups in Characteristic p
Exercises
Elliptic Curves over Finite Fields
Number of Rational Points
The Weil Conjectures
The Endomorphism Ring
Calculating the Hasse Invariant
Exercises
Elliptic Curves Over C
Elliptic Integrals
Elliptic Functions
Construction of Elliptic Functions
Maps Analytic and Maps Algebraic
Uniformization
The Lefschetz Principle
Exercises
Elliptic Curves over Local Fields
Minimal Weierstrass Equations
Reduction Modulo �
Points of Finite Order
The Action of Inertia
Good and Bad Reduction
The Group E/E0
The Criterion of N�ron-Ogg-Shafarevich
Exercises
Elliptic Curves over Global Fields
The Weak Mordell-Weil Theorem
The Kummer Pairing via Cohomology
The Descent Procedure
The Mordell-Weil Theorem over Q
Heights on Projective Space
Heights on Elliptic Curves
Torsion Points
The Minimal Discriminant
The Canonical Height
The Rank of an Elliptic Curve
Szpiro's Conjecture and ABC
Exercises
Integral Points on Elliptic Curves
Diophantine Approximation
Distance Functions
Siegel's Theorem
The S-Unit Equation
Effective Methods
Shafarevich's Theorem
The Curve Y2 = X3+ D
Roth's Theorem-An Overview
Exercises
Computing the Mordell-Weil Group
An Example
Twisting-General Theory
Homogeneous Spaces
The Selmer and Shafarevich-Tate Groups
Twisting-Elliptic Curves
The Curve Y2 = X3 + DX
Exercises
Algorithmic Aspects of Elliptic Curves
Double-and-Add Algorithms
Lenstra's Elliptic Curve Factorization Algorithm
Counting the Number of Points in E(Fq)
Elliptic Curve Cryptography
Solving the ECDLP: The General Case
Solving the ECDLP: Special Cases
Pairing-Based Cryptography
Computing the Weil Pairing
The Tate-Lichtenbaum Pairing
Exercises
Elliptic Curves in Characteristics 2 and 3
Exercises
Group Cohomology (H0 and H1)
Cohomology of Finite Groups
Galois Cohomology
Nonabelian Cohomology
Exercises
Further Topics: An Overview
Complex Multiplication
Modular Functions
Modular Curves
Tate Curves
N�ron Models and Tate's Algorithm
L-Series
Duality Theory
Local Height Functions
The Image of Galois
Function Fields and Specialization Theorems
Variation of ap and the Sato-Tate Conjecture
Notes on Exercises
List of Notation
References
Index

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