Number Theory

Some notes on local/global Galois groups

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:

  1. 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‘.
  2. 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:

  1. 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.
  2. 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.

Shimura Varieties: motivation

EDIT: While these notes might be still useful to read, if one wants a more in-depth explanation of the ideas below see the notes from this post.

——

This will be the first in a series of posts discussing Shimura varieties. In particular, we will focus here on a sort of broad motivation for the subject—why Shimura varieties are a natural thing to study and, in particular, what they give us.

(more…)

Kummer theory and the weak Mordell-Weil theorem

In this post we discuss the notion of Kummer theory in its general form, and how this leads to a proof of the (weak) Mordell-Weil theorem.

(more…)

p-divisible groups, formal groups, and the Serre-Tate theorem

In this post we discuss the basic theory of p-divisible groups, their relationship to formal groups, and the Serre-Tate theorem.

(more…)