wild ramification | Hard Arithmetic

While I’m posting things.

Here are some notes that I wrote for a Galois representations learning seminar. I was tasked with giving the first talk about local fields, global fields, their Galois groups, and their connection.

Since most participants were seasoned veterans (at least insofar as basic definitions/results go) I tried to sail towards slightly more interesting waters. Thus, these notes, while containing (basically) the bare-bones technical information, have a slightly different goal then a standard introductory talk on the subject. Namely, they focus on two things:

Trying to establish, via multiple analogies, a ‘geometric understanding’ of what $G_{\mathbb{Q}_p}$ is measuring with regards to $G_\mathbb{Q}$ –how it is studying the ‘local arithmetico-geometric data of a (punctured) $\text{Spec}(\mathbb{Z}$ ) at $p$ ‘.
Trying to emphasize the ‘credo’ that the hard part of something like $G_{\mathbb{Q}_p}$ is the wild ramification group $P_{\mathbb{Q}_p}$ . This is done by explaining how $G_{\mathbb{Q}_p}/P_{\mathbb{Q}_p}$ is ‘simple’ and explaining how one can understand geometrically (by thinking about the geometry of curves over finite fields) why wild ramification is hard.This, for people that know some Galois representations, should not be a shocking focus since the oomph of big results like Grothendieck’s $\ell$ -adic monodromy theorem is that $P_{\mathbb{Q}_p}$ is ‘almost killed’ when discussing $\ell$ -adic representations and, combining this with our credo, explains why $\ell$ -adic representations are ‘simpler’ than $p$ -adic ones.

There should be two warnings though:

I proof read these even less than I usually do for posts. So, please take the contents with an extra large grain of salt. Please let me know if any mistakes are present and I will (attempt to) correct them.
Apparently there is a phrase ‘simple’ in group theory, which is kind of a big deal. I kind of, perhaps, maybe forgot this while writing these notes. So the phrase ‘simple group’ should be translated to ‘not very complicated group’ in these notes.

Here are the notes: rachel-seminar-talk-3.