It’s the start of a new year, so I’m going to start up something new here. My research interests have recently tended towards arithmetic geometry, so my plan for 2012 is to write some basics of algebraic number theory and arithmetic geometry that I use a lot. I’ll try to avoid redoing some of the stuff that has already been done at Climbing Mount Bourbaki.
I’d like to explain what modular forms are and what some of their basic properties are. I may detour a little into Galois representations at some point. I definitely want to talk about L-functions of varieties and what it means for a variety to be modular. This may lead to some discussion about Fermat’s Last Theorem and the Taniyama-Shimura conjecture. Scattered throughout I’ll probably have to cover some more classical algebraic number theory.
If you’re interested in related topics just post a comment and maybe I’ll get to it. Maybe in a few weeks I’ll scratch this whole idea and do something else. Who knows?