Supplementary Lecture Notes on Elliptic Curves

1. What is an elliptic curve? 2 2. Mordell-Weil Groups 5 2.1. The Group Law on a Smooth, Plane Cubic Curve 5 2.2. Reminders on Commutative Groups 8 2.3. Some Elementary Results on Mordell-Weil Groups 9 2.4. The Mordell-Weil Theorem 11 2.5. K-Analytic Lie Groups 13 3. Background on Algebraic Varieties 15 3.1. Affine Varieties 15 3.2. Projective Varieties 18 3.3. Homogeneous Nullstellensätze 20 3.4. Bézout’s Theorem 22 3.5. Bézout’s Theorem and Singular Points 24 3.6. Projective Equivalence 26 4. The Riemann-Roch Express 29 4.1. Riemann-Roch Spaces 31 4.2. The Riemann-Roch Theorem 32 4.3. Descent for Riemann-Roch Spaces 33 4.4. Some Applications of Riemann-Roch 33 4.5. The Genus of a Smooth Plane Curve 33 4.6. Mappings into Projective Space 34 5. Weierstrass Cubics 38 5.1. Singular Weierstrass Cubics 39 5.2. Covariants of Weierstrass Cubics 40 6. The l-adic Tate module 44 7. Elliptic Curves Over Finite Fields 45 7.1. Hasse’s Theorem 45 7.2. The Endomorphism Ring 47 7.3. The Characteristic Polynomial of Frobenius 49 8. The Mordell-Weil Theorem I: Overview 52 8.1. Statement 52 8.2. Outline of the Proof 52 9. The Mordell-Weil Theorem II: Weak Mordell-Weil 53 9.1. Statement and Outline of the Proof 53 9.2. Outline of the proof 53 9.3. The preliminary reduction 54 9.4. The Kummer Pairing 55 9.5. Interlude: Weak Mordell-Weil Over Big Fields 56