NMAK16007U Elliptic Curves

Volume 2021/2022

MSc Programme in Mathematics


The aim of this course is to discover the beautiful theory of elliptic curves. Elliptic curves are objects at the crossroads between geometry, analysis, algebra and number theory. They constitute one of the key ingredient in the proof of Fermat’s Last Theorem for instance, and famous open conjectures -for example the Birch and Swinnerton-Dyer conjecture- focus on these special curves. Studying compact Riemann surfaces, lattice theory and periodic functions, rational points and diophantine problems, projective and affine geometry of curves, schemes, higher Galois theory, modular forms and L functions, abelian varieties, local fields, global fields, finite fields, modern cryptography, each time these curves show up at a central place. 

As these objects really appear as a corner stone in the modern mathematical landscape, we offer a course presenting in details their various definitions and basic properties and focus on some modern applications.

Learning Outcome

Knowledge: The student should be familiar with the main results of the topics of the course. 

Skills: At the end of the course the student is expected to be able to follow and reproduce arguments at a high level corresponding to the contents of the course. 

Competences: The student should be able to apply the theory to solve problems of moderate difficulty within the topics of the course. 

Examples of course literature:

The Arithmetic of Elliptic Curves by Joseph Silverman.

Rational points on elliptic curves, UTM, Springer, by Joseph Silverman and John Tate.

Algebra 2 (Alg2) or similar.

Academic qualifications equivalent to a BSc degree is recommended.
6 hours of lectures and 2 hours of tutorials each week for 7 weeks.
Elliptic Curves definitely fits in the circle of ideas presented in these other courses: Algebra 3, Algebraic Number Theory, Analytic Number Theory and Algebraic Geometry.
  • Category
  • Hours
  • Lectures
  • 42
  • Preparation
  • 147
  • Exercises
  • 14
  • Exam
  • 3
  • Total
  • 206
Continuous feedback during the course of the semester
7,5 ECTS
Type of assessment
Continuous assessment
Written examination, 3 hours under invigilation
Two written assignments count each 20%. A final written exam counts the remaining 60% of the grade.
Only certain aids allowed

All aids allowed for the assignments. Only written aids allowed for the written exam.

Marking scale
7-point grading scale
Censorship form
No external censorship
One internal examiner.

30 minutes oral exam without preparation time, several internal examiners, all written aids allowed, counting for 100% of the grade.

Criteria for exam assesment

The student must in a satisfactory way demonstrate that he/she has mastered the learning outcome of the course.