# The Langlands conjecture and the cohomology of Shimura varieties

Below are some really extended notes that I’ve written about work I’ve done recently alone (in my thesis) and with a collaborator (A. Bertoloni Meli).

While the explanation of my work was the original goal of the notes, they have since evolved into a motivation for the Langlands program in terms of the cohomology of Shimura varieties, as well as explaining some directions that the relationships between Shimura varieties and Langlands has taken in the last few decades (including my own work).

I hope that it’s useful to any reader out there. Part I was mostly written with me, four years ago, in mind. So, in a perfect world someone out there will be in the same headspace as I was, in which case it will (hopefully) be enlightening.

In case you’re wondering the intended level for the reader is probably: 1-3 year graduate student with interest in number theory and/or arithmetic geometry. In particular, for Part I there is an assumption that the reader has some basic knowledge about: Lie groups, algebraic geometry, number theory (e.g. be comfortable with what a Galois representation is), algebraic group theory, and etale cohomology (although this can be black-boxed in the standard way–e.g. all one needs to know is the contents of Section 3 of this set of notes). Part II is mostly written as an introduction to a research topic, and so requires more background.

Enjoy!

PS, feel encouraged to point out any mistakes/improvements that you think are worth mentioning.

The Langlands conjecture and the cohomology of Shimura varieties

# A fun (enough) talk

This is a rough transcription of a talk I gave to a class of algebraic number theory students at UC Berkeley with the goal of trying to understand how one might bring to bear modern techniques in number theory/geometry on some classical questions. I have essentially kept the format the same, while adding a bit of extra material (and adding in their responses to questions I asked).
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# 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.

# 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.

# Some examples of Geometric Galois representations

In this post we discuss the Galois representation associated to a projective scheme $X/K$, where $K$ is a number field. We also discuss how this representation can be computed in several simple cases.