This page contains some information that can be helpful before you register for the course. After registration, we will use ODTUclass. You should follow ODTUclass and check your emails regularly for important announcements during the semester.
Course Objectives
The objective of this course is to provide a deep and unified introduction to the theory of elliptic curves, covering their algebraic structure, arithmetic properties, and applications. Beginning with the group law and Weierstrass equations, the course explores topics such as isogenies, torsion points, and division polynomials. Students will study elliptic curves over finite fields, including point counting and Schoof’s algorithm, as well as applications to primality proving. The course further develops the theory over the rational and complex numbers, introducing the Mordell–Weil Theorem, complex multiplication, and the integrality of j-invariants, culminating in an overview of the role of elliptic curves in the proof of Fermat’s Last Theorem. The course aims to equip students with both theoretical foundations and computational tools, bridging modern algebra, number theory, and arithmetic geometry. This course provides a rigorous mathematical treatment of the subject while at the same time addressing issues of algorithmic implementations. This includes a thorough treatment of elliptic curve-related algorithms over finite fields and the theory of complex multiplication. These concepts are crucial to many theoretical and practical applications of elliptic curves..
Lecture Hours
The lecture hours will be decided in an organizational meeting. Follow the related announcement from this page and/or from the graduate students’ email group.
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Homework, Exams, and Grading
Homework will be assigned on a regular basis, and there will be 3-5 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 – around the 8th or 9th week.
- 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 in your own words. If you copy a solution, which is referred to as cheating, you will probably gain nothing and may encounter penalties.
Textbooks
- Washington, Elliptic Curves: Number Theory and Cryptography.
- Silverman, The Arithmetic of Elliptic Curves, 2nd edition.
- Silverman, Tate, Rational Points on Elliptic Curves.
Tentative Course Outline
The course content, together with a tentative course outline, can be found below. We will attempt to cover the related parts of the above textbooks each week.
- Week 1: (Sep 29 – Oct 3) Introduction.
- Week 2: (Oct 6 – Oct 10) The group law, Weierstrass equation.
- Week 3: (Oct 13 – Oct 17) Endomorphisms.
- Week 4: (Oct 20 – Oct 24) Torsion points, division polynomials.
- Week 5: (Oct 27 – Oct 31) Elliptic curves over finite fields.
- Week 6: (Nov 3 – Nov 7) Determining the group order. Schoof’s algorithm.
- Week 7: (Nov 10 – Nov 14) Supersingular curves.
- Week 8: (Nov 17 – Nov 21) DLP, factoring, and primality proving.
- Midterm (around the 8th or 9th week).
- Week 9: (Nov 21 – Nov 28) Elliptic curves over rational numbers.
- Week 10: (Dec 1 – Dec 5) Mordell-Weil theorem.
- Week 11: (Dec 8 – Dec 12) Elliptic curves over complex numbers.
- Week 12: (Dec 15 – Dec 19) Uniformization theorem.
- Week 13: (Dec 22 – Dec 26) Complex multiplication. Integrality of j-invariants.
- Week 14: (Dec 29 – Jan 2) Fermat’s last theorem.
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.