Synopsis
to Rational Points on Plane Curves.- 1 Elementary Properties of the Chord-Tangent Group Law on a Cubic Curve.- 2 Plane Algebraic Curves.- 3 Elliptic Curves and Their Isomorphisms.- 4 Families of Elliptic Curves and Geometric Properties of Torsion Points.- 5 Reduction mod p and Torsion Points.- 6 Proof of Mordell's Finite Generation Theorem.- 7 Galois Cohomology and Isomorphism Classification of Elliptic Curves over Arbitrary Fields.- 8 Descent and Galois Cohomology.- 9 Elliptic and Hypergeometric Functions.- 10 Theta Functions.- 11 Modular Functions.- 12 Endomorphisms of Elliptic Curves.- 13 Elliptic Curves over Finite Fields.- 14 Elliptic Curves over Local Fields.- 15 Elliptic Curves over Global Fields and ?-Adic Representations.- 16 L-Function of an Elliptic Curve and Its Analytic Continuation.- 17 Remarks on the Birch and Swinnerton-Dyer Conjecture.- Appendix Guide to the Exercises.
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