# A ‘brief’ discussion about torsors

Attached below are notes written for two mentees I had in an independent study concerning the etale fundamental group. The goal was to motivate cohomology (in particular etale cohomology) via torsors and motivate torsors using the theory of twists. I think that the notes are well-intentioned and do genuniely have interesting didcatic value buried deep inside them. Unfortunately, they are long-winded, meandering, and overly self-indulgent. One day I intend to go back and tighten them up.

Please feel free to leave any constructive comments!

torsors

# 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

# 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!

# A computation a day: a pullback pushforward

In this post we compute the Galois representation $i^\ast R^m j_\ast\mathbb{Q}_\ell$ where $j:\mathbb{G}_{m,\overline{k}}\hookrightarrow\mathbb{A}^1_k$ is the natural inclusion and $i:\text{Spec}(k)\hookrightarrow \mathbb{A}^1_k$ is the inclusion of the origin.

# 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.

# A class field theoretic phenomenon

In this post we discuss one example of what’s called a ‘class field theoretic phenomenon’. In particular, we focus on the application of trying to understand the property of when $X^3-2$ has three distinct roots modulo $p$, for various primes $p$.

# An invitation to p-adic Hodge theory, or: How i learned to stop worrying and love fontaine

This is the rough outline of a talk I recently gave at the Berkeley Student Algebraic Geometry Seminar on the progression of ideas that might lead one to define the Hodge-Tate decomposition.