# 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 automorphic representations

I was asked to give a series of talks in an `automorphic representations’ learning seminar at Berkeley. Specifically, I was asked to talk about the local representation theory of $\text{GL}_2(\mathbb{Q}_p)$ and aspects of the global representation theory of $\text{GL}_2(\mathbb{A}_\mathbb{Q})$.

These are ongoing but I thought I’d post my notes for the first two talks. The first is on the representation theory of $\text{GL}_2(\mathbb{Q}_p)$ and the second is on the representation theory of $\text{GL}_2(\mathbb{A}_\mathbb{Q}^\infty)$ with a focus on the relationship to modular forms.

These notes are a bit on the ‘basic’ side, leaving most technical details/proofs to the more comprehensive texts. This could be useful for some people who just want to get an overview of the ideas involved.

Anyways, feedback welcome as always!

Represenations of GL_2(Q_p)

Representations of GL_2(A_Q^\infty)

# An update of sorts

I get an email every week or so which says something to the effect of “When are you going to make another blog post?” I am continually amazed by this–it’s absolutely shocking to me to see so many people interested in the things I write, and I find the emails and comments that I get heartening to no end.

So, for those who have been asking, I have not been posting that many things for a combination of two reasons. First, I’ve just been super busy. I’ve been running many seminars here at Berkeley and, of course more importantly, been working on my thesis. But, perhaps even more of the issue, I’ve started multiple, multiple very long posts that are approaching completion, but (for semi-perfectionist reasons) I’d rather not post them quite yet.

For those curious, a sampler of these in-preparation posts are:

• A post on why the tilting functor of Scholze et al. is a reasonable thing to do. Sort of in the same mindset of You could have invented spectral sequences.
• A post on stacks, with a focus on understanding the statement “$H^1(S,\text{Aut}(\mathcal{F}))$ (i.e. $\text{Aut}(\mathcal{F})$-torsors) classify objects on $S$ locally isomorphic to $\mathcal{F}$” or, equivalently, with a focus on the “theory of twists”.
• A high-level discussion of the Eichler-Shimura construction and how it fits into the larger picture of the Langlands program. I find that using more advanced ideas (such as etale cohomology) not only makes the whole construction incredibly more natural, but leads onequite readily into the general idea of why Shimura varieties are important–why they should realize something like the global Langlands correspondence.
• A rambling discussion of motives, the Weight-Monodromy conjecture, and Galois representations.
• A semi-thorough discussion of modular curves, modular forms, and their relationship from an algebro-geometric standpoint (something like a ‘what I’ve needed to know from Katz-Mazur’).

Essentially all of these are ‘mostly done’, and I hope to post (some) of them soon–probably in the order they were listed above.

Anyways, thanks again for your continued support! A special thanks to Dr. Woit whose undeservedly kind words brought quite a bit of attention to my small blog.