Possible Masters' Thesis Topics in Algebraic Geometry and Number Theory

Question

I am a graduate student at African Institute for Mathematical Sciences (AIMS) in a one-year structured masters program. I am currently looking for possible topics for my master's thesis in the area of Algebraic Geometry and (intersection with) Number Theory.

I have done some (not so rigorous) work on Elliptic Curve Cryptography in my Bachelor's Degree thesis, and I have done an Introductory Course on Algebraic and Analytic Number Theory. I am also currently taking an Introductory Course on Algebraic Geometry.

With my background and familiarity, I would really love to get a list of topics suitable for a master thesis in Algebraic Geometry and Number Theory.

P.S: The duration of the Master's Thesis is usually 6 weeks so this has got to be a topic I can work on within 6 weeks. Originality of work is not necessarily required but comprehension is paramount.

Answer 1

Here are some topics you could take a look at to get an idea of what you might want to do:

1. Galois theory and class field theory. This is a good way to learn more algebraic number theory
2. Properties of $L$-functions associated with Dirichlet characters, or $L$-functions associated with elliptic curves.
3. Basic theory of modular forms, and maybe the Eichler-Shimura congruence relation
4. The Chabauty-Coleman method, if you're comfortable with the $p$-adic numbers (if not, this could be a good way to learn about them!)

I think each of these topics are fairly flexible, and can be made more or less in-depth depending on your comfort with notions in algebraic geometry and commutative algebra, and depending on your time constraints.