MA5114 - Elliptic Curves
Lecturer: Dr. Roberto Laface
Tutor: M.Sc. Kai Behrens
Monday, July 23rd, 2018 at 10:00AM in MI 03.08.011
Time and venue
- Lectures: Tuesdays and Thursdays, 8:30-10:00 in MI 02.10.11
- Exercise sessions: Fridays, 8:30-10:00 in MI 02.08.020
- Office hours: by appointment
This course offers an introduction to the (elementary) theory of elliptic curves. We will be discussing:
- algebraic varieties
- Zariski topology
- algebraic curves
- group law on cubic curves
- geometry of elliptic curves I
- theory of divisors on curves
- geometry of elliptic curves II
- elliptic curves over finite fields
- elliptic curves over C and complex tori
Here is a detailed syllabus.
The student is expected to have some knowledge in Algebra. Commutative algebra or algebraic geometry are helpful but not required. Preliminaries to the course will be given at the beginning of the course, together with some references.
- Exercise Sheet 1
- Exercise Sheet 2
- Exercise Sheet 3
- Exercise Sheet 4
- Exercise Sheet 5
- Exercise Sheet 6
- Exercise Sheet 7
- Exercise Sheet 8
- Exercise Sheet 9
- Exercise Sheet 10
- Exercise Sheet 11
Elliptic curves and their generalizations provide naturally a vast source of interesting problems both geometric and algebraic in nature. If you are looking for a topic for your bachelor/master thesis, you are welcome to come talk to me after class or make an appointment.)
References (in order of difficulty)
For commutative algebra:
- Reid, M.: "Undergraduate Commutative Algebra", Cambridge University Press (1995).
- Eisenbud, D.: "Commutative Algebra with a view towards Algebraic Geometry", Springer-Verlag New York (2004).
For algebraic geometry:
- Hulek, K.: "Elementary Algebraic Geometry", Student Mathematical Library (2003).
- Hartshorne, R.: "Algebraic Geometry", Springer-Verlag New York (1977).
For elliptic curves:
- Silverman, J.H. and Tate, J.: "Rational points on elliptic curves", Springer International Publishing (2015).
- Silverman, J.H.: "The arithmetic of elliptic curves", Springer-Verlag New York (2009).
- Husemöller, D.: "Elliptic curves", Springer-Verlag New York (2004).