Elliptic Curves

Course Textbook: Silverman-Tate ("Rational Points of Elliptic Curves") --- Chapter 1-3 & a bit of Chapter 4, Appendix A-B.

Pre-requisite: Abstract algebra, Complex analysis & Elementary number theory

• Week 1 (Rational points on conics, "duplication" on cubics, with motivations)

1. Wed Lecture

2. Thurs Lecture

• Week 2 (Group law, Projective geometry I)

1. Wed Lecture

2. Thurs Lecture

• Week 3 (Projective Geometry II, Weierstrass' Form of Cubics)

1. Wed Lecture

2. Thurs Lecture

• Week 4 (Elliptic Functions: Liouville's Theory and Weierstrass' Theory)

1. Wed Lecture

2. Thurs Lecture

• Week 5 (The Uniformization Theorem)

1. Wed Lecture

2. Thurs Lecture

• Week 6 (Rational Torsion - Examples with Nagell-Lutz Theorem)

1. Wed Lecture

2. Thurs Lecture

• Week 7 (Proof of Nagell-Lutz via Localization)

1. Wed Lecture

2. Thurs Lecture

• Week 8 (Fermat's Infinite Descent & Neron-Tate's Height)

1. Wed Lecture

2. Thurs Lecture

• Week 9 (Mordell's Theorem - Sketch and Examples)

1. Wed Lecture

2. Thurs Lecture

• Week 10 (Further Remarks on Computations & Review)

1. Wed Lecture

2. Thurs Lecture

Exercise Sheets

• Sheet 1

• Sheet 2

• Sheet 3

• Sheet 4

Supplementary Materials

• Why Hasse-Minkowski is useful?

• Review of uniform convergence

• A worked-out example of the P-function