This post is short, and hopefully sweet.
For the past couple of months, I’ve been meaning to clean and upload these lecture notes from a talk I gave early this semester on p-adic Lie groups! The talk was given as part of a weekly seminar attended by faculty and grad students at GT and Emory to learn about Robert Coleman and Claude Chabauty’s remarkable attempts to provide an effective proof of the Mordell Conjecture (a.k.a. Falting’s Theorem) using methods from p-adic analysis. The seminar was organized by Matt Baker (whose own wonderful blog can be found here).
You’ll find very few proofs: the goal was to give the statements and intuition behind the main structure theorem. The target audience is someone with good general knowledge of (real) differential topology and of the p-adic numbers, but possibly little or no knowledge of (real) Lie group theory. (i.e., the audience is precisely myself prior to preparing these notes.) There are bits at the beginning and end which are intended to relate these ideas to Chabauty’s method and to Bill McCallum’s paper on Chabauty for Fermat curves. In the talk, they were intended as motivation, but for anyone seeking to learn about Lie theory, they’re basically beside the point.
Anyways, enough delay – the file can be downloaded by clicking here. I only ask that anyone using these notes let me know of any errors you may find!
P.S. Last semester (in the Yellow Pigs’ collective final semester at Michigan) I had a class with Bill Fulton on schemes and cohomology in which he presented a beautifully geometric proof of Serre duality for curves, purportedly due to George Kempf. At some point, I’d like to collect this proof from several badly organized notebooks and put it here as well, and in the meantime I’ll probably be posting a series of notes related to schemes, sheaves, and cohomology building towards that goal, so stay tuned!