This is the first in a series of 4 posts whose goal is to briefly introduce rigid geometry with a focus on examples. The ultimate goal is to continue this post (and its sequels) with the reader hopefully having a broad grasp of the rigid geometry of the objects involved.
The goal of this post, which is just a bit of fun between more serious posts/projects, is to discuss some examples of algebro-geometric objects over which have no models over smaller subfields and explain how moduli theory can help clarify their discovery in certain situations.
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 discuss a weird example of a finite map of varieties which doesn’t preserve projectivity.
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 .
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!