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.
In this post we classify one-dimensional connected group varieties of dimension .
Attached below are notes that I wrote for a seminar at Berkeley.
The goal of the notes was to understand some of the representation theory surrounding Scholze’s paper on the cohomology of the Lubin–Tate tower. In particular I, Koji Shimizu, DongGyu Lim, and Sander Mack-Crane were/are interested in understanding whether there is a function field analogue of this paper.
In particular, it has an eye towards modular (i.e. mod ) representations of -adic groups. So, it discusses some of the classical theory of representations of -adic groups from a categorical perspective which serves one better in the modular representation setting. It also discusses the fascinating theorem of Kazhdan relating Hecke algebras for and where and are soemthing like the tilts of and .
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.
The goal of this post is to introduce, in a very informal way, the notion of a reductive group, and discuss some examples.
In this post we discuss the basic theory of p-divisible groups, their relationship to formal groups, and the Serre-Tate theorem.