1) Prerequisites
Basic linear algebra (vector spaces, linear maps, characteristic polynomial); group theory including the structure theorem for finitely generated abelian groups; ring theory: rings, ideals, polynomial rings; basic field theory including finite fields. For example, chapters I-V of Lang's "Algebra" would be sufficient, or the following chapters of the Leiden undergraduate courses Algebra 1, 2, 3: 1–9, 11–14, 21, 22. The notes for the Leiden Algebra courses can be found here: http://websites.math.
We do not assume any prior knowledge of algebraic geometry.
For two weeks of the course, we will also need a small amount of complex analysis: meromorphic functions, Cauchy's theorem, residues.
For another week of the course, we will use a small amount of Galois theory, but a course in Galois theory is not essential.
2) Aim of the course
Along various historical paths, the origins of elliptic curves can be traced back to calculus, complex analysis, and algebraic geometry. Their arithmetic aspects have made elliptic curves into key objects in modern cryptography and in Wiles's proof of Fermat's last theorem.
This course is an introduction to both the theoretical and the computational aspects of elliptic curves. The topics treated include:
elliptic curves and their group law,
elliptic curves over the complex numbers, and lattices in the complex plane,
reduction of elliptic curves modulo primes, and its application to torsion points,
isogenies between elliptic curves,
heights of points on elliptic curves,
Mordell's theorem: the group of rational points on an elliptic curve over the rationals is finitely generated,
elliptic curves over finite fields, and their applications (e.g. in cryptography or integer factorisation),
computational aspects, including an introduction to the computer algebra system SageMath and its application to the study of elliptic curves.
3) Rules about Homework/Exam
The final grade will be based on a combination of a final exam (85%) and homework (15%). To pass the course, you need to achieve a minimum of 5.0 in the final exam and a 5.5 for the weighted average of homework and exam.
Exams will be standard 3-hour written closed book exams. The resit exam will give an opportunity to improve the exam score only; there is no second chance for the homework. Even after a retake the homework will count towards your final grade.
There will be exercises every week, and every other week some of those exercises must be handed in as homework in groups of at most 2. Only those exercises will be marked. This will total to approximately 7 regular homework assignments. The two lowest of these homework grades will not count. The average of the other grades will count for 12% of the final grade.
There will be a computer class, and the homework problems from that class will count for another 3% of the final grade.
Rules for homework
The only reason we have homework is the experience that students benefit from regularly working out exercises in detail and getting feedback. The goal of homework is to help you to work out the theory in depth.
You may submit each of the homework sets together with at most one other student, but you may not submit more than one homework set with the same student.
Teachers reserve the right to invite a student to further explain their submitted work. This can happen upon suspicion of cheating or plagiarism, but also as a random check. Suspicion of plagiarism is easy to avoid: make sure that, when writing up your exercises, you don't have any text in reach that is wholly or partially written by somebody else – so no book or web site, or the solution of a student that you are not submitting with.
4) Lecture notes/Literature
The main reference for the course is 1. below. For the first few lectures, we supplement this with 2. We occasionally use some of the other references listed below, but the ones we rely on will be freely available.
- Docent: Steffen Muller
- Docent: Haowen Zhang