This is a continuation of this post.
This is the first in a series of posts whose goal is to compute the cohomology of the (several times) punctured closed -adic disk as well as the cohomology of smooth (and some non-smooth) algebraic curves.
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!
I have had the pleasure of helping to run a seminar on etale cohomology and, in the process, have been writing up questions for the participants to work on. In case it would be useful to any readers of my blog, I thought I’d include them here.
I will be continuing to edit this post with the most recent version of the exercises.
Please feel free to point out any errors and/or suggest any good problems!
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.
PS, feel encouraged to point out any mistakes/improvements that you think are worth mentioning.
In this post we compute the Galois representation where is the natural inclusion and is the inclusion of the origin.
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.
In this post we compute the group where is a number field.
In this post we compute the compactly supported cohomology of some simple varieties.