Elliptic Curves
Math 778 - Spring 2022
Announcements
This page contains some information that can be helpful before you register for the course. After the registration, we will be using odtuclass. You should follow odtuclass and check your emails regularly for important announcements during the semester.
Lectures and Office Hours
Lectures will be face to face in M231 during the following times:
Wednesday 13:40-16:30, M231.
Office hours will be on thursday afternoons by appointment.
Homeworks, Exams and Grading
Homeworks will be assigned on a regular basis and there will be 4-6 homework sets by the end of the semester. There will be one midterm and a final. The time and the method of each exam will be announced later.
Midterm, 30 points - Thursday, April 28 at 11:00.
Final, 30 points - during the final exam period.
Homework, 40 points.
Homework Policy: You should write your solutions on your own. You are allowed to consult other people's solutions for homework problems, but you must express everything with your own words. If you copy a solution, which is referred to as cheating, you will probably gain nothing and may encounter penalties.
Textbooks and Tentative Course Outline
There are three textbooks. However we will mostly use the first one.
1) Lawrence C. Washington, Elliptic Curves: Number Theory and Cryptography,
2) Silverman, The Arithmetic of Elliptic Curves, and
3) Silverman & Tate, Rational Points on Elliptic Curves.
Find a tentative outline below for the whole semester. For each week, we will attempt to cover the indicated pages.
(Mar 07 - Mar 11) Introduction.
(Mar 14 - Mar 18) The group law, Weierstrass equation.
(Mar 21 - Mar 25) Endomorphisms.
(Mar 28 - Apr 01) Torsion points, division polynomials.
(Apr 04 - Apr 08) Elliptic curves over finite fields.
(Apr 11 - Apr 15) Determining the group order. Schoof's algorithm.
(Apr 18 - Apr 22) Supersingular curves.
(Apr 25 - Apr 29) DLP, factoring, and primality proving.
Midterm - Thursday, April 28 at 11:00
(May 02 - May 06) Spring Break.
(May 09 - May 13) Elliptic curves over rational numbers.
(May 16 - May 20) Mordell-Weil theorem.
(May 23 - May 27) Elliptic curves over complex numbers.
(May 30 - Jun 03) Uniformization theorem.
(Jun 06 - Jun 10) Complex multiplication. Integrality of j-invariants.
(Jun 13 - Jun 17) Fermat's last theorem.
Final - TBA
PARI / GP
The software PARI / GP is very simple to learn and
extremely strong to do computations with.
A note for undergraduate students
If you are an undergraduate student, your CGPA must be higher than 3.00 in order to take this course. If you are willing to take this course, please send me an email before the registration.