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