Elliptic Curves  Daniel Naylor  Contents  1 Fermat s Method of Infinite Descent  1 1 A variant for polynomials  2 Some Remarks on Plane Curves  2 1 The degree of a morphism  3 Weierstrass Equations  4 The Group Law  5 Isogenies  6 The Invariant Differential  7 Elliptic Curves over Finite Fields  7 1 Zeta functions  8 Formal Groups  9 Elliptic Curves over Local Fields  10 Elliptic Curves over Number Fields  The torsion subgroup  11 Kummer Theory  12 Elliptic Curves over Number Fields  The weak Mordell Weil Theorem  13 Heights  14 Dual isogenies and the Weil pairing  15 Galois Cohomology  16 Descent by Cyclic Isogeny  16 1 Descent by 2 isogeny  16 2 Birch Swinnerton Dyer Conjecture  Index  Index