Elliptic Curves - M1 - 8EC

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. 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.leidenuniv.nl/algebra/.

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 numbers,

  • 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,

  • elliptic curves over finite fields, and their applications (e.g. in cryptography or integer factorisation),

  • computational aspects.

3) Rules about Homework/Exam

The final grade will be based on a combination of a final exam (75%) and homework (25%). To pass the course, you need to achieve a minimum of 5.0 in the final exam and a 5.5 for a weighted average of homework and exam.

Exams will be standard 3-hour written 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. 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 20% of the final grade.

The homework problems from the computer class will count for another 5% of the final grade.

 

4) Lecture notes/Literature

Our main reference is “The Arithmetic of Elliptic Curves” (2nd Edition) by Joseph H. Silverman. We will occasionally use other notes or references, which are freely available.