# Objects with no rational models

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 $\overline{\mathbb{Q}}$ which have no models over smaller subfields and explain how moduli theory can help clarify their discovery in certain situations.

# A new paper draft

I have not been able to post of late as I’ve been quite busy working on several projects.

I wanted to make a post though discussing a new draft with my collaborator A. Bertoloni Meli that I’m quite excited about. In it we discuss a method for characterizing the local Langlands conjecture for certain groups $G$ as in Scholze’s paper [Sch]. Namely we show that for certain classes of groups an equation like that in the Scholze–Shin conjecture (see [Conjecture 7.1, SS]) is enough to characterize the local Langlands conjecture (for supercuspidal parameters) at least if one is willing to assume that other expected properties of the local Langlands conjecture hold.

The main original idea of this paper is the realization that while the Langlands–Kottwitz–Scholze method only deals with Hecke operators at integral level (e.g. see the introduction to [Sch]) that one can circumvent the difficult questions this raises (e.g. see [Question 7.5,SS]) if one is willing to not only consider the local Langlands conjecture for $G$ in isolation, but also the local Langlands conjecture for certain groups closely related to $G$ (so-called elliptic hyperendoscopic groups). Another nice byproduct of this approach is that while the Scholze–Shin conjecture is stated as a set of equations for all endoscopic triples for $G$ our paper shows that one needs only consider the trivial endoscopic situation (for elliptic hyperendoscopic groups of $G$).

This paper is closely related to the paper mentioned in this previous post where me and A. Bertoloni Meli discuss the proof of the Scholze–Shin conjecture for unramified unitary groups in the trivial endoscopic triple setting.

References

[Sch] Scholze, Peter. The Local Langlands Correspondence for GL_n over $p$-adic fields, Invent. Math. 192 (2013), no. 3, 663–715.

[SS] Scholze, P. and Shin, S., 2013. On the cohomology of compact unitary group Shimura varieties at ramified split places. Journal of the American Mathematical Society26(1), pp.261-294.

# Punctured disks and punctured curves and cohomology, oh my! (Part I: the proper curve case)

This is the first in a series of posts whose goal is to compute the cohomology of the (several times) punctured closed $p$-adic disk $\mathbb{B}_{\mathbb{C}_p}-\{p_1,\ldots,p_m\}$ as well as the cohomology of smooth (and some non-smooth) algebraic curves.

# A smattering of representation theory

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 $p$) representations of $p$-adic groups. So, it discusses some of the classical theory of representations of $p$-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 $G(F)$ and $G(F')$ where $F$ and $F'$are soemthing like the tilts of $F$ and $F'$.

Enjoy!

Modular reps

# 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.

torsors

# Exercises in etale cohomology

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!

Nov. 7th version of exercises

# 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