Ebook: Elliptic Curves

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Content:
Front Matter....Pages I-XV
Introduction to Rational Points on Plane Curves....Pages 1-21
Elementary Properties of the Chord-Tangent Group Law on a Cubic Curve....Pages 22-42
Plane Algebraic Curves....Pages 43-61
Elliptic Curves and Their Isomorphisms....Pages 62-80
Families of Elliptic Curves and Geometric Properties of Torsion Points....Pages 81-98
Reduction mod p and Torsion Points....Pages 99-119
Proof of Mordell’s Finite Generation Theorem....Pages 120-137
Galois Cohomology and Isomorphism Classification of Elliptic Curves over Arbitrary Fields....Pages 138-151
Descent and Galois Cohomology....Pages 152-161
Elliptic and Hypergeometric Functions....Pages 162-182
Theta Functions....Pages 183-201
Modular Functions....Pages 202-221
Endomorphisms of Elliptic Curves....Pages 222-241
Elliptic Curves over Finite Fields....Pages 242-261
Elliptic Curves over Local Fields....Pages 262-271
Elliptic Curves over Global Fields and ?-Adic Representations....Pages 272-289
L-Function of an Elliptic Curve and Its Analytic Continuation....Pages 290-305
Remarks on the Birch and Swinnerton-Dyer Conjecture....Pages 306-314
Back Matter....Pages 315-350

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Content:
Front Matter....Pages I-XV
Introduction to Rational Points on Plane Curves....Pages 1-21
Elementary Properties of the Chord-Tangent Group Law on a Cubic Curve....Pages 22-42
Plane Algebraic Curves....Pages 43-61
Elliptic Curves and Their Isomorphisms....Pages 62-80
Families of Elliptic Curves and Geometric Properties of Torsion Points....Pages 81-98
Reduction mod p and Torsion Points....Pages 99-119
Proof of Mordell’s Finite Generation Theorem....Pages 120-137
Galois Cohomology and Isomorphism Classification of Elliptic Curves over Arbitrary Fields....Pages 138-151
Descent and Galois Cohomology....Pages 152-161
Elliptic and Hypergeometric Functions....Pages 162-182
Theta Functions....Pages 183-201
Modular Functions....Pages 202-221
Endomorphisms of Elliptic Curves....Pages 222-241
Elliptic Curves over Finite Fields....Pages 242-261
Elliptic Curves over Local Fields....Pages 262-271
Elliptic Curves over Global Fields and ?-Adic Representations....Pages 272-289
L-Function of an Elliptic Curve and Its Analytic Continuation....Pages 290-305
Remarks on the Birch and Swinnerton-Dyer Conjecture....Pages 306-314
Back Matter....Pages 315-350
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