Report from Oberwolfach

Recently returned from a workshop on “Arithmetic of Shimura varieties.”

  • The organizers did a great job choosing the speakers. For one thing, the overlap between speakers this time and speakers at the previous edition of this event was a singleton set, which I think is a reasonable choice. Moreover, the majority of the speakers were junior people, which is also totally reasonable. It was great to hear what everyone is doing.
  • Best talks: Ana Caraiani, Teruhisa Koshikawa, Keerthi Madapusi, Sug Woo Shin, Joao Lourenco
  • Best chaotic talk with amazingly strong theorems: Ian Gleason
  • SWS has advised a very disproportionate number of Alexanders.
  • The usual hike wasn’t possible, due to snow in the mountains. Ah well. Instead we hiked along a path parallel to the road. But there was still cake.
  • The food was slightly better than usual: they didn’t serve the notorious bread casserole, and one dinner (the polenta thing) was actually really good.
  • “I mean, you know Ben. He’s pretty unflappable. But, yeah… [redacted], uh… flaps him.”
  • During the workshop, LM and I hit upon a conceptual explanation for Bernstein-Zelevinsky duality, which works both for group representations and for sheaves on \mathrm{Bun}_G, even when \ell=p! More on this later.
  • “What was the motivation for this conjecture?” “The motivation was that it is true.”
  • Some young people have extremely weird expectations for how the postdoc job market should work.
  • The notion of “genericity” in various guises, and its relevance for controlling the cohomology of local and global Shimura varieties, was very much in the air. This came up in Caraiani and Koshikawa’s talks, and also in my (prepared but undelivered, see the first bullet above) talk. My handwritten notes are here, and may be of some interest. Conjectures 3 and 5, in particular, seem quite fun.
  • Had some interesting conversations with VL about nearby cycles and related topics. Here’s a concrete question: can the results in this paper be adapted to etale cohomology? There are definite obstructions in positive characteristic related to Artin-Schreier sheaves, but in characteristic zero it should be ok.
  • During the workshop, Ishimoto posted a beautiful paper completing Arthur’s results for inner forms of odd special orthogonal groups, at least for generic discrete parameters. I was vaguely sure for several years that this was the (only) missing ingredient in proving compatibility of the Fargues-Scholze LLC and the Arthur(-Ishimoto) LLC for \mathrm{SO}_{2n+1} and its unique inner form. After reading this paper, and with some key assists from SWS and WTG, I now see how to prove this compatibility (at least over unramified extensions F/\mathbf{Q}_p with p>2). It shouldn’t even take many pages to write down!
  • On a related note, shortly before the workshop, Li-Huerta posted his amazing results comparing Genestier-Lafforgue and Fargues-Scholze in all generality!

As always, Oberwolfach remains one of my favorite places to do mathematics. Thank you to the organizers for putting together a wonderful workshop!


Fall roundup

Apologies for the lack of blogging. This has been an unusually busy fall.

  • My student Linus Hamann has a website! Please go there and check out his beautiful preprints, especially his paper on comparing local Langlands correspondences for GSp4.
  • There have been a lot of great papers this year, but I was especially struck by these gorgeous ideas from Teruhisa Koshikawa. Readers might recall that the seminal Caraiani-Scholze papers contain a fun part (p-adic geometry of Shimura varieties and their Hodge-Tate fibers, semiperversity of Hodge-Tate pushforwards) and a not fun part (arguments with the twisted stable trace formula and Shin’s stable trace formula for Igusa varieties). Koshikawa completely eliminates the not fun part, replacing it with an extremely clever use of the Fargues-Scholze machinery. Even in the setting of the CS papers, Koshikawa’s main theorem is stronger; moreover, his technique opens the door to a wide generalization of the CS vanishing results beyond the specific unitary Shimura varieties they treated. (Note for ambitious readers: The problem of working out these generalizations has already been “taken” by specific people.)
  • Eagle-eyed readers of H.-Kaletha-Weinstein might’ve noticed that the entire paper depends crucially on a non-existent preprint cited as [GHW]. As discussed in a previous post, the point of GHW is to construct the functor Rf_! in etale cohomology for certain stacky maps of Artin v-stacks, by adapting some machinery of Liu-Zheng which they built to solve the analogous problem in the setting of Artin stacks. Since the above-mentioned papers of Hamann and Koshikawa both depend directly on HKW, and thus indirectly on GHW, I’ve felt some increased pressure recently* to actually produce this paper!
    However, I think this pressure helped push me past the final points of confusion in this project, and I’m pleased to report that after nearly 4 years of struggle, the details of GHW have finally come together. I’m cautiously optimistic that the paper will be publicly available within a few months. The arguments are an infernal mixture of delicate p-adic geometry and general \infty-categorical constructions. Actually, this is the most intense and frustrating project I’ve ever worked on. It will be good to finish it.
  • As always, David Roberts offers a voice of clarity against the nonsense burbling out from the IUT cultists.

    *Both from myself and from the referee for HKW.

The Newton stratification is true

Let G be a connected reductive group over \mathbf{Q}_p, and let \mu be a G-valued (geometric) conjugacy class of minuscule cocharacters, with reflex field E. In their Annals paper, Caraiani and Scholze defined a very interesting stratification of the flag variety \mathcal{F}\ell_{G,\mu} (regarded as an adic space over E) into strata \mathcal{F}\ell_{G,\mu}^{b}, where b runs over the Kottwitz set B(G,\mu^{-1}). Let me roughly recall how this goes: any (geometric) point x \to \mathcal{F}\ell_{G,\mu} determines a canonical modification \mathcal{E}_x \to \mathcal{E}_{triv} of the trivial G-bundle on the Fargues-Fontaine curve, meromorphic at \infty and with “mermorphy \mu” in the usual sense. On the other hand, Fargues proved that G-bundles on the curve are classified up to isomorphism by B(G), and then Caraiani-Scholze and Rapoport proved that \mu-meromorphic modifications of the trivial bundle are exactly classified by the subset B(G,\mu^{-1}) (CS proved that only these elements can occur; R proved that all of these elements occur). The Newton stratification just records which element of this set parametrizes the bundle \mathcal{E}_x.

The individual strata are pretty weird. For example, if G=GL_n and \mu=(1,0,\dots,0), then \mathcal{F}\ell_{G,\mu} \simeq \mathbf{P}^{n-1} and the open stratum is just the usual Drinfeld space \Omega^{n-1}, but the other strata are of the form \Omega^{n-i-1} \times^{P_{n-i,i}(\mathbf{Q}_p)} GL_n(\mathbf{Q}_p), where P_{n-i,i} is the evident parabolic in GL_n and the action on \Omega^{n-i-1} is via the natural map P_{n-i,i}(\mathbf{Q}_p) \twoheadrightarrow GL_{n-i}(\mathbf{Q}_p). Qualitatively, this says that they’re unions of profinitely many copies of lower-dimensional Drinfeld spaces. In particular, the non-open strata are not rigid analytic spaces. There are also examples of strata which don’t have any classical rigid analytic points. However, the \mathcal{F}\ell_{G,\mu}^{b}‘s are always perfectly well-defined from the topological or diamond point of view.

Anyway, I’m getting to the following thing, which settles a question left open by Caraiani-Scholze.

Theorem. Topologically, the Newton stratification of \mathcal{F}\ell_{G,\mu} is a true stratification: the closure of any stratum is a union of strata.

The idea is as follows. After base-changing from E to the completed maximal unramified extension E' (which is a harmless move), there is a canonical map \zeta: \mathcal{F}\ell_{G,\mu,E'} \to \mathrm{Bun}_{G} sending x to the isomorphism class of \mathcal{E}_x. Here \mathrm{Bun}_{G} denotes the stack of G-bundles on the Fargues-Fontaine curve, regarded as a stack on the category of perfectoid spaces over \overline{\mathbf{F}_p}. This stack is stratified by locally closed substacks \mathrm{Bun}_{G}^{b} defined in the obvious way, and by construction the Newton stratification is just the pullback of this stratification along \zeta. Now, by Fargues’s theorem we get an identification |\mathrm{Bun}_{G}| = B(G), so it is completely trivial to see that the stratification of \mathrm{Bun}_{G} is a true stratification (at the level of topological spaces). We then conclude by the following observation:

Proposition. The map \zeta is universally open.

The idea is to observe that \zeta factors as a composition of two maps \mathcal{F}\ell_{G,\mu,E'} \to [\mathcal{F}\ell_{G,\mu,E'}/\underline{G(\mathbf{Q}_p)}] \to \mathrm{Bun}_{G}. Here the first map is a \underline{G(\mathbf{Q}_p)}-torsor by construction, so it’s universally open by e.g. Lemma 10.13 here. More subtly, the second map is also universally open. Why? Because it is cohomologically smooth in the sense of Definition 23.8 here; universal openness then follows by Proposition 23.11 in the same document.

For the cohomological smoothness claim, take any affinoid perfectoid space with a map T \to \mathrm{Bun}_{G}, corresponding to some bundle \mathcal{F} / \mathcal{X}_T. After some thought, one works out the fiber product X = T \times_{\mathrm{Bun}_{G}} [\mathcal{F}\ell_{G,\mu,E'}/\underline{G(\mathbf{Q}_p)}] “explicitly”: it parametrizes untilts of T over E' together with isomorphism classes of \mu^{-1}-meromorphic modifications \mathcal{E}\to \mathcal{F} supported along the section T^{\sharp} \to \mathcal{X}_T induced by our preferred untilt, with the property that \mathcal{E} is trivial at every geometric point of T. Without the final condition, we get a larger functor X' which etale-locally on T is isomorphic to T \times_{\mathrm{Spd}(\overline{\mathbf{F}_p})} \mathcal{F}\ell_{G,\mu^{-1},E'}^{\lozenge}. (To get the latter description, note that etale-locally on T we can trivialize \mathcal{F} on the formal completion of the curve along T^{\sharp}, and then use Beauville-Laszlo to interpret the remaining data as a suitably restricted modification of the trivial G-torsor on \mathrm{Spec} \mathbf{B}_{dR}^{+}(\mathcal{O}(T^{\sharp})). This is a Schubert cell in a Grassmannian. Then use Caraiani-Scholze’s results on the Bialynicki-Birula map.) Anyway anyway, after a little more fiddling around the point is basically that the projection X' \to T is cohomologically smooth because it’s the base change of a smooth map of rigid spaces. By Kedlaya-Liu plus epsilon, the natural map X \to X' is an open immersion, so X \to T is cohomologically smooth. Since T was arbitrary, this is enough.





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.