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 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 in isolation, but also the local Langlands conjecture for certain groups closely related to (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 our paper shows that one needs only consider the trivial endoscopic situation (for elliptic hyperendoscopic groups of ).
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.
[Sch] Scholze, Peter. The Local Langlands Correspondence for GL_n over -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 Society, 26(1), pp.261-294.
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.
EDIT: While these notes might be still useful to read, if one wants a more in-depth explanation of the ideas below see the notes from this post.
This will be the first in a series of posts discussing Shimura varieties. In particular, we will focus here on a sort of broad motivation for the subject—why Shimura varieties are a natural thing to study and, in particular, what they give us.