NMAK16007U Elliptic Curves
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.
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.
Academic qualifications equivalent to a BSc degree is recommended.
- 7,5 ECTS
- Type of assessment
- Continuous assessmentWritten examination, 3 hours under invigilationTwo 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.