ELLIPTIC CURVES J.S. MILNE August 21, 1996; v1.01 Abstract. These are the notes for Math 679, University of Michigan, Winter 1996, exactly as they were handed out during the course except for some minor corrections. Please send comments and corrections to me at
[email protected] using “Math679” as the subject. Contents Introduction Fast factorization of integers Congruent numbers Fermat’s last theorem 1 1. Review of Plane Curves Affine plane curves Projective plane curves 2 2. Rational Points on Plane Curves. Hensel’s lemma A brief introduction to the p-adic numbers Some history 6 3. The Group Law on a Cubic Curve 12 4. Functions on Algebraic Curves and the Riemann-Roch Theorem Regular functions on affine curves Regular functions on projective curves The Riemann-Roch theorem The group law revisited Perfect base fields 14 5. Definition of an Elliptic Curve Plane projective cubic curves with a rational inflection point General plane projective curves Complete nonsingular curves of genus 1 19 Copyright 1996 J.S. Milne. You may make one copy of these notes for your own personal use. i ii J.S. MILNE The canonical form of the equation The group law for the canonical form 6. 7. Reduction of an Elliptic Curve Modulo p Algebraic groups of dimension 1 Singular cubic curves Reduction of an elliptic curve Semistable reduction Reduction modulo 2 and 3 Other fields Elliptic Curves over Qp 23 29 8. Torsion Points Formulas Solution to Exercise 4.8 32 9. N´ eron Models Weierstrass minimal models The work of Kodaira