the March of progress

Several things to report on:

  • The (hopefully) final version of HKW is done and dusted! The most significant change from the previous version is that GHW is finally done, so we can finally refer to precise results from that paper. For a quick summary of the main results from GHW, take a look at section 4.1 of HKW. The key new vocab is decent v-stacks and fine morphisms between them. Writing GHW was not good for my health (take a look at the acknowledgments…), and reading it might not be so good for yours. But if you really want to look at GHW, one thing you could do to get oriented is read Definitions 1.1 & 1.3, Theorem 1.4, and everything in section 4.1.

    Bonus Question: Let g:X \to Y and f:Y \to Z be separated morphisms of locally spatial diamonds such that g is surjective and universally open, and f \circ g is compactifiable (in the sense of Definition 22.2 here). Is f compactifiable? If the answer to this question is yes, then the nonsense about “strict” surjectivity in GHW could be eliminated (although it is harmless in applications, thank God). If you can settle this, or some slight weakening of it, please let me know!

    Note that the numbering in HKW has changed slightly in comparison to the previous version, which I wrote about here, so the numbering in that post refers to v3 of the arxiv posting.
  • In a previous post, I mentioned a bunch of roughly equivalent open problems about smooth representations of p-adic groups with coefficients in general \mathbf{Z}[1/p]-algebras, which I learned about from Jean-François Dat. But now these problems have been more or less all solved, in a beautiful and shockingly short paper by Dat-Helm-Kurinczuk-Moss. It is quite curious that their results, which are statements in pure representation theory, depend in a crucial way on the Fargues-Scholze machinery. For more information, I can’t to any better than suggesting that you simply read their paper.

Elliptic curves over Q(i) are potentially automorphic

This spectacular theorem was announced by Richard Taylor on Thursday, in a lecture at the joint meetings.  Taylor credited this result and others to Allen-Calegari-Caraiani-Gee-Helm-Le Hung-Newton-Scholze-Taylor-Thorne (!), as an outcome of the (not so) secret mini-conference which took place at the IAS this fall.  The key new input here is work in progress of Caraiani-Scholze on the cohomology of non-compact unitary Shimura varieties, which can be leveraged to check (at least in some cases) the most difficult hypothesis in the Calegari-Geraghty method: local-global compatibility at l=p for torsion classes.

The slides from my talk can be found here. Naturally I managed to say “diamond” a bunch of times.