It is known that only finitely many elliptic curves can be presented
as quotients of Fermat curves. I have recently given a very simple
proof of this fact based on Gauss sums and zeta functions. The main
ideas will be explained in the lectures and then the project will be
to determine explicitly just which elliptic curves do appear as
quotients of Fermat curves.
Lecture topics:
1. Finite fields and character sums
2. Counting points on Fermat curves
3. Zeta functions
4. Project overview: maps from Fermat curves to elliptic curves