# The Fontaine-Winterberger theorem: going full tilt

This is the first in a series of posts whose goal is quite ambitious. Namely, we will attempt to give an intuitive explanation of why the recent push of several prominent mathematicians (Fargues, Scholze, etc.) to ‘geometrize’ the ‘arithmetic’ local Langlands program is intuitively feasible (at least, why it seems intuitive to me!) and, more to the point, to understand some of the major objects/ideas necessary to discuss it.

The goal of this post, in particular, is to try and understand why perfectoid fields (of which perfectoid spaces, their more corporeal counterparts) are natural objects to consider. This is far from a historical account of perfectoid fields and tilting, of which I am far from knowledgable. Instead, this is more in the style of Chow’s excellent You Could Have Invented Spectral Sequences explaining how one might have arrived at the definition of perfectoid fields by ‘elementary considerations’.

This post is somewhat out of order. In some magical world where I actually planned out my posts, this would have been situated less anteriorly but, as we’re constantly reminded, we do not live in a perfect world!

# Some motivation for p-adic Hodge theory

These are some notes that I wrote for a learning course at Berkeley–the goal being to understand the statement of the global Langlands conjecture.

The goal of the talk (that these notes were written for) was, specifically, to motivate $p$-adic Hodge theory with an eye, in particular, towards where it might be useful in understanding the statement of Langlands.

These are even less edited than usual, so I profusely apologize for any mistakes. As always, corrections/comments are very welcome!

Notes

# 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.

# Local class field theory: a discussion

In this post we discuss local class field theory (specifically looking at $p$-adic fields) with a focus on the broader picture, and the multiple approaches.