H.-Kaletha-Weinstein study guide and FAQ

Tasho Kaletha, Jared Weinstein and I have just posted a joint paper, On the Kottwitz conjecture for local shtuka spaces. This is a heavily revised and rewritten version of a preprint by Tasho and Jared which has been available since 2017 (henceforth “KW”).

Is the main result here the same as in KW?
More or less, yes. The main theorem confirms the Kottwitz conjecture for the cohomology of moduli spaces of local shtukas, ignoring the Weil group action and allowing for an “error term” consisting of a non-elliptic virtual representation.

Is the rough idea of the proof still the same as in KW?
Yes: the idea is still to compute the cohomology by applying a suitable Lefschetz-Verdier trace formula, and explicitly computing the “local terms” at all of the elliptic fixed points.

OK, what’s new, then?
Well, basically all of the details are different. First and foremost, the discussion of the relevant Lefschetz-Verdier trace formula for v-stacks has been completely rewritten, using Lu-Zheng’s magical formalism of symmetric monoidal 2-categories of cohomological correspondences. This leads to huge conceptual simplifications, both in the statements and the proofs. Actually, we go beyond the formalism of Lu-Zheng, introducing certain 2-categories of “based cohomological correspondences”. This formalism play a crucial role in the proof of Theorem 4.5.3, which is a key ingredient in the proof of the main theorem. It’s also hard to imagine proving Theorem 4.6.1 without this formalism.

Secondly, a key calculation of local terms on the B_{\mathrm{dR}}-affine Grassmannian, Theorem 5.1.3, now has a completely different and correct (!) proof, via a degeneration to a similar calculation on the Witt vector affine Grassmannian. This degeneration argument relies on the observation that the local terms appearing in the Lefschetz-Verdier trace formula are compatible with any base change, Proposition 5.3.1; this seems to be a new observation even for schemes. (My first proof of Proposition 5.3.1 involved checking the commutativity of 500 diagrams, but eventually I realized that it follows immediately from the Lu-Zheng formalism!) Once we’ve degenerated to the Witt vector affine Grassmannian, we make an intriguing global-to-local argument using the trace formula and known properties of the weight functors in geometric Satake, and relying crucially on a recent result of Varshavsky.

There are some other notable improvements:

  • All assumptions in KW of the form “assume that some representation admits an invariant \overline{\mathbf{Z}_{\ell}}-lattice” have now been eliminated. This is far from formal, since we definitely don’t have a 6-functor formalism with \overline{\mathbf{Q}_{\ell}}-coefficients at hand. Instead, the idea is to reduce “by a continuity argument” to the case where things do admit lattices. The key ingredient here is Theorem 6.5.4, formerly a conjecture of Taylor, which should have lots of other applications.
  • The final part of the main theorem, giving practical criteria for the error term to vanish, is new.
  • Theorem 1.0.3 is new.

On a practical note, the Fargues-Scholze paper is now finished and available, so we can properly build on the powerful machinery and results in that paper.

Finally, the margins are no longer gigantic.

Any advice on how to read the paper?
Sure. The main Theorem 1.0.2/6.5.1 in the paper is essentially a combination of two separate results, Theorem 6.5.2 and Theorem 3.2.9. Theorem 3.2.9 is conditional on the refined local Langlands correspondence, and the material needed to formulate and prove this result is developed in Chapter 3 and Appendix A. The bulk of the paper (sections 4-6 and Appendices B-C) is devoted to building up enough material to prove Theorem 6.5.2.

Each of sections 3, 4, and 5 is fairly self-contained, although section 5 depends slightly on section 4. Section 6 builds heavily on sections 4 and 5, and also on [FS21]. Section 5 is actually devoted to proving a single result, Theorem 5.1.3 (and its reinterpretation Theorem 5.1.4), and can be treated as a black box. 

One thing you could do is read the introduction, then jump to Proposition 6.4.7 and work your way backwards as needed until all the notation makes sense. This proposition is the technical heart of the paper. The key inputs into the proof of Proposition 6.4.7 are Theorem 5.1.4, Theorem 4.5.3, and Corollary 4.3.8. The use of “the trace formula” in this whole argument is actually somewhat hidden: it is entirely encoded in the equality sign labelled “Cor. 4.3.8” at the bottom of p. 66.

From Proposition 6.4.7, Theorem 6.4.9 (a weak form of Theorem 6.5.2) follows quickly, and then Theorem 6.5.2 follows from Theorem 6.4.9 by a continuity argument which relies crucially on Theorem 6.5.4.

This sounds cool, but you didn’t punt any key ingredients into a separate and currently nonexistent paper, did you?
I’m glad you asked. In working out the trace formula formalism in enough generality to give a conceptual proof of the main theorem, it became clear at some point that we needed the existence and expected properties of the functors Rf_! and Rf^! in etale cohomology for “smooth-locally nice” maps f between Artin v-stacks. For maps which are actually “nice”, and in particular non-stacky, these functors were constructed by Scholze, but extending them to the setting of stacky maps is not formal. Already in the analogous setting of schemes versus Artin stacks, these functors for Artin stacks were only constructed recently by Liu-Zheng, making heavy use of \infty-categories. In a forthcoming paper, Jared and I together with my postdoc Dan Gulotta will explain how to construct Rf_! and Rf^! for smooth-locally nice maps of Artin v-stacks. The arguments will very closely follow the arguments of Liu-Zheng, with the exception of one key non-formal piece of input (which took me three years of intermittent work).

Why do you only prove an explicit formula for the virtual character of \mathrm{Mant}_{b,\mu}(\rho) restricted to elliptic elements of G(F)? What’s so special about elliptic elements in this context?
The fixed points of elliptic elements lie in the “admissible locus” inside the relevant closed Schubert cell. This is discussed in detail in the introduction.

Fine, but maybe there’s still an obvious explicit formula for the virtual character of \mathrm{Mant}_{b,\mu}(\rho) at any strongly regular semisimple element.
Here’s an example to illustrate why I think this is a hard problem. I’ll be slightly heuristic about the details; if you want more precision, please leave a comment.

Let’s take G= \mathrm{GL}_2 and \mu=(1,0), so we’re in the Lubin-Tate/Drinfeld setting with G_b(F)=D^\times the units in the quaternion algebra over F. Let \rho be the trivial representation of D^\times. Then \mathrm{Mant}_{b,\mu}(\rho) = \mathrm{St}-\mathbf{1} as a virtual representation of G(F), by an old calculation of Schneider-Stuhler. Note that \mathrm{St}+\mathbf{1} is a principal series representation, hence non-elliptic, so the virtual character of \mathrm{St}-\mathbf{1} on elliptic elements of G(F) is the constant function -2. This matches perfectly with the fact that any elliptic g\in G(F) has two fixed points in \mathbf{P}^1, both contained in \Omega^{1}, and the “naive” local terms of the relevant sheaf j_!\mathbf{Z}_{\ell}[1] at both these points are -1. Here j:\Omega^1 \to \mathbf{P}^1 is the evident open immersion.

On the other hand, if g \in G(F) is regular semisimple and nonelliptic, then it’s conjugate to some t=\mathrm{diag}(t_1,t_2) with t_1 \neq t_2 \in F^\times. In this case there are still two fixed points, but they both lie in the “boundary” \mathbf{P}^1-\Omega^{1}. Since j_!\mathbf{Z}_{\ell}[1] restricted to the boundary is identically zero, its naive local terms at both fixed points are zero. Nevertheless, the trace formula still works, so the contribution of the true local terms at these two points must add up to the virtual character of \mathrm{St}-\mathbf{1} evaluated at t. This character value can be computed explicitly by van Dijk’s formula, and turns out to be -2+\tfrac{|t_1| + |t_2|}{|t_1-t_2|}. So this slightly strange expression needs to emerge from the sum of these two local terms. 

Putting these observations together, we see that in this example, the true local terms do not equal the naive local terms at the non-elliptic fixed points. Morally, the analysis in Section 3 of the paper can and should be interpreted as an unravelling of naive local terms (at elliptic fixed points). Hopefully this makes the difficulty clear.

Can you give an example where the error term in Theorem 1.0.2 is nonzero?
Sure, look at the two-dimensional Lubin-Tate local Shimura datum and take \rho trivial again, so \rho \in \Pi_{\phi}(G_b) with \phi the Steinberg parameter. Then \mathrm{Mant}_{b,\mu}(\rho) = \mathrm{St}-\mathbf{1} as discussed in the previous answer, while the sum on the right-hand side in Theorem 1.0.2 is just 2 \mathrm{St}. So in this case \mathrm{err}= -(\mathrm{St}+\mathbf{1}) is, up to sign, a reducible principal series representation.

It would be very interesting to formulate an extension of the Kottwitz conjecture which covered all discrete L-parameters. Since the Kottwitz conjecture is best regarded as a reflection of the Hecke eigensheaf property in Fargues’s conjecture, this is closely related to understanding Fargues’s conjecture for general discrete parameters.

It seems like the Kottwitz conjecture should also make sense for local shtukas over Laurent series fields. Why doesn’t the paper cover this case too? 
Good question! The short answer is that all of the p-adic geometry should work out uniformly over Laurent series fields and over finite extensions of \mathbf{Q}_p, and indeed Fargues-Scholze handles these two situations uniformly. However, the literature on local Langlands and harmonic analysis for reductive groups over positive characteristic local fields case is pretty thin. This is the main reason we stuck to the mixed characteristic case. For more on the state of the literature, take a look at section 1.6 in Kal16a.

Do the methods of this paper give any information about \mathrm{Mant}_{b,\mu} for non-basic b?
Yes: If b is non-basic, or b is basic and \rho is parabolically induced, our methods can be applied to prove that \mathrm{Mant}_{b,\mu}(\rho) is always a virtual combination of representations induced from proper parabolic subgroups of G. This is a soft form of the Harris-Viehmann conjecture. The details of this argument will be given elsewhere (in the sense of Deligne-Mumford).

Any surprising subtleties?
Lemma 6.5.5 was a bracing experience. My first “proof” was absolute nonsense, due to some serious misconceptions I had about harmonic analysis on p-adic groups – big thanks to JT for setting me straight. My second proof didn’t work because I cited a result in a famous article (>100 mathscinet citations) whose proof turns out to be fallacious – big thanks to JFD for setting me straight. Tasho and I devised the current proof, which in hindsight is actually a very natural argument. (I was hoping, apparently unreasonably, for a shortcut.) 

Closing thoughts:
-At least for me, the proof of Theorem 6.5.4 is a very striking illustration of the raw Wagnerian power of the machinery developed in [FS21]. This result was totally out of reach a few years ago, but with the machinery of [FS21] in hand, the proof is less than a page! Note that this argument crucially uses the D_{\mathrm{lis}} and Hecke operator formalism of [FS21] with general coefficient rings.
-Let me again emphasize how heavily the paper depends on the recent preprints [LZ20], [Var20], and (of course) [FS21].
-The paper depends, more significantly than most, on a choice of isomorphism \mathbf{C} \simeq \overline{\mathbf{Q}_{\ell}}. It’s naturally of interest to wonder how everything depends on this choice. Imai has formulated a conjectural answer to this question. Of course, “independence of \ell” results in the etale cohomology of diamonds are probably very hard.

 

Rampage!

I’m excited to announce a new weekly online-only research seminar on p-adic geometry and related topics, organized by Arthur-César Le Bras, Jared Weinstein, and myself. We will “meet” on Zoom on Thursdays at 16:00 UTC (that’s 9 am in California, noon in Boston, 5 pm in London, 6 pm in Bonn…). Bhargav Bhatt will give the first talk on June 18.

Please follow the instructions at the seminar website here to get the Zoom link. We will also keep an up-to-date schedule on researchseminars.org here.

All credit to Jared for the name!

Report from Tucson

Just back from the 2017 Arizona Winter School on perfectoid spaces.  First of all, I should say that everything was impressively well-organized, and that the lecturers did a fantastic job, especially considering the technical weight of this material. (Watch the videos if you don’t believe me.)  Jared Weinstein, in particular, has an almost supernatural ability to make a lecture on some technical thing feel comforting.

Now to the jokes.

  • In his opening lecture, Scholze called perfectoid spaces a “failed theory”, on account of his inability to completely settle weight-monodromy. “You see, I’m Prussian, and when a Prussian says he wants to do something, he really feels responsible for doing it.”
  • Audience member: “Why are they called diamonds?”
    Scholze: “[oral explanation of the picture on p. 63 of the Berkeley notes]”
    Weinstein: “Also, diamonds are hard.”
  • Anon.: “When you’re organizing a conference, the important thing is not to give in and be the first one who actually does stuff.  Because then you’ll end up doing everything!  Don’t do that!  Don’t be the dumb one!”
    Me: “Didn’t you organize [redacted] a couple of years ago?”
    Anon.: “Yeah… It turned out that Guido Kings was the dumb one.”
  • Mazur: “It just feels like the foundations of this area aren’t yet… hmm…”
    Me: “Definitive?”
    Mazur: “Yes, exactly.  I mean, if Grothendieck were here, he would be screaming.”
  • “Do you ever need more than two legs?”
  • During the hike, someone sat on a cactus.
  • Finally, here is a late night cartoon of what a universal cohomology theory over \mathbb{Z} might look like (no prizes for guessing who drew this):
    cartoon

What does an inadmissible locus look like?

Let H/ \overline{\mathbf{F}_p} be some p-divisible group of dimension d and height h, and let \mathcal{M} be the rigid generic fiber (over \mathrm{Spa}\,\breve{\mathbf{Q}}_p) of the associated Rapoport-Zink space. This comes with its Grothendieck-Messing period map \pi: \mathcal{M} \to \mathrm{Gr}(d,h), where \mathrm{Gr}(d,h) is the rigid analytic Grassmannian paramatrizing rank d quotients of the (covariant) rational Dieudonne module M(H) /\breve{\mathbf{Q}}_p. Note that \mathrm{Gr}(d,h) is a very nice space: it’s a smooth connected homogeneous rigid analytic variety, of dimension d(h-d).

The morphism \pi is etale and partially proper (i.e. without boundary in Berkovich’s sense), and so the image of \pi is an open and partially proper subspace* of the Grassmannian, which is usually known as the admissible locus. Let’s denote this locus by \mathrm{Gr}(d,h)^a. The structure of the admissible locus is understood in very few cases, and getting a handle on it more generally is a famous and difficult problem first raised by Grothendieck (cf. the Remarques on p. 435 of his 1970 ICM article). About all we know so far is the following:

  • When d=1 (so \mathrm{Gr}(d,h) = \mathbf{P}^{h-1}) and H is connected, we’re in the much-studied Lubin-Tate situation. Here, Gross and Hopkins famously proved that \pi is surjective, not just on classical rigid points but on all adic points, so \mathrm{Gr}(d,h)^a = \mathrm{Gr}(d,h) is the whole space. This case (along with the “dual” case where h>2,d=h-1) turns out to be the only case where \mathrm{Gr}(d,h)^a = \mathrm{Gr}(d,h), cf. Rapoport’s appendix to Scholze’s paper on the Lubin-Tate tower.
  • When H \simeq \mathbf{G}_m^{d} \oplus (\mathbf{Q}_p/\mathbf{Z}_p)^{h-d}, i.e. when H has no bi-infinitesimal component, it turns out that \mathrm{Gr}(d,h)^a = \mathbf{A}^{d(h-d)} is isomorphic to rigid analytic affine space of the appropriate dimension, and can be identified with the open Bruhat cell inside \mathrm{Gr}(d,h). This goes back to Dwork, who proved it when d=1,h=2. (I don’t know a citation for the general result, but presumably for arbitrary d,h this is morally due to Serre-Tate/Katz?)
  • In general there’s also the so-called weakly admissible locus \mathrm{Gr}(d,h)^{wa} \subset \mathrm{Gr}(d,h), which contains the admissible locus and is defined in some fairly explicit way. It’s also characterized as the maximal admissible open subset of the Grassmannian with the same classical points as the admissible locus. In the classical rigid language, the map \mathrm{Gr}(d,h)^a \to \mathrm{Gr}(d,h)^{wa} is etale and bijective; this is the terminology used e.g. in Rapoport-Zink’s book.
  • In general, the admissible and weakly admissible loci are very different.  For example, when H is isoclinic and (d,h)=1 (i.e. when M(H) is irreducible as a \varphi-module), \mathrm{Gr}(d,h)^a contains every classical point, and \mathrm{Gr}(d,h)^{wa} = \mathrm{Gr}(d,h), so the weakly admissible locus tells you zilch about the admissible locus in this situation (and they really are different for any 1 < d < h-1).

That’s about it for general results.

To go further, let’s switch our perspective a little. Since \mathrm{Gr}(d,h)^a is an open and partially proper subspace of \mathrm{Gr}(d,h), the subset |\mathrm{Gr}(d,h)^a| \subseteq |\mathrm{Gr}(d,h)| is open and specializing, so its complement is closed and generalizing.  Now, according to a very general theorem of Scholze, namely Theorem 2.42 here (for future readers, in case the numbering there changes: it’s the main theorem in the section entitled “The miracle theorems”), if \mathcal{D} is any diamond and E \subset |\mathcal{D}| is any locally closed generalizing subset, there is a functorially associated subdiamond \mathcal{E} \subset \mathcal{D} with |\mathcal{E}| = E inside |\mathcal{D}|. More colloquially, one can “diamondize” any locally closed generalizing subset of |\mathcal{D}|, just as any locally closed subspace of |X| for a scheme X comes from a unique (reduced) subscheme of X.

Definition. The inadmissible/nonadmissible locus \mathrm{Gr}(d,h)^{na} is the subdiamond of \mathrm{Gr}(d,h)^{\lozenge} obtained by diamondizing the topological complement of the admissible locus, i.e. by diamondizing the closed generalizing subset |\mathrm{Gr}(d,h)^a|^c \subset |\mathrm{Gr}(d,h)| \cong |\mathrm{Gr}(d,h)^{\lozenge}|.

It turns out that one can actually get a handle on \mathrm{Gr}(d,h)^{na} in a bunch of cases!  This grew out of some conversations with Jared Weinstein – back in April, Jared raised the question of understanding the inadmissible locus in a certain particular period domain for \mathrm{GL}_2 with non-minuscule Hodge numbers, and we managed to describe it completely in that case (see link below). Last night, though, I realized we hadn’t worked out any interesting examples in the minuscule (i.e. p-divisible group) setting! Here I want to record two such examples, hot off my blackboard, one simple and one delightfully bizarre.

Example 1. Take h=4, d=2 and H isoclinic. Then |\mathrm{Gr}(d,h)^a|^c is a single classical point, corresponding to the unique filtration on M(H) with Hodge numbers 0,0,1,1 which is not weakly admissible. So \mathrm{Gr}(d,h)^a = \mathrm{Gr}(d,h)^{wa} in this case.

Example 2. Take h=5, d=2 and H isoclinic$.  Now things are much stranger.  Are you ready?
Theorem. In this case, the locus \mathrm{Gr}^{na} is naturally isomorphic to the diamond (X \smallsetminus 0)^{\lozenge} / \underline{D^\times}, where X is an open perfectoid unit disk in one variable over \breve{\mathbf{Q}}_p and D=D_{1/3} is the division algebra over \mathbf{Q}_p with invariant 1/3, acting freely on X \smallsetminus 0 in a certain natural way. Precisely, the disk X arises as the universal cover of the connected p-divisible group of dimension 1 and height 15, and its natural D-action comes from the natural D_{1/15}-action on X via the map D_{1/3} \to D_{1/3} \otimes D_{-2/5} \simeq D_{-1/15} \simeq D_{1/15}^{op}.

This explicit description is actually equivariant for the D_{2/5}-actions on X and Gr. As far as diamonds go, (X \smallsetminus 0)^{\lozenge}/\underline{D^{\times}} is pretty high-carat: it’s spatial (roughly, its qcqs with lots of qcqs open subdiamonds), and its structure morphism to \mathrm{Spd}\,\breve{\mathbf{Q}}_p is separated, smooth, quasicompact, and partially proper in the appropriate senses. Smoothness, in particular, is meant in the sense of Definition 6.1 here (cf. also the discussion in Section 4.3 here). So even though this beast doesn’t have any points over any finite extension of \breve{\mathbf{Q}}_p, it’s still morally a diamondly version of a smooth projective curve!

The example Jared and I had originally worked out is recorded in section 5.5 here. The reader may wish to try adapting our argument from that situation to the cases mentioned above – this is a great exercise in actually using the classification of vector bundles on the Fargues-Fontaine curve in a hands-on calculation.

Anyway, here’s a picture of (X \smallsetminus 0)^{\lozenge} / \underline{D^{\times}}, with some other inadmissible loci in the background:

diamond

 

 

*All rigid spaces here and throughout the post are viewed as adic spaces: in the classical language, \mathrm{Gr}(d,h)^a does not generally correspond to an admissible open subset of \mathrm{Gr}(d,h), so one would be forced to say that there exists a rigid space \mathrm{Gr}(d,h)^a together with an etale monomorphism \mathrm{Gr}(d,h)^a \to \mathrm{Gr}(d,h). But in the adic world it really is a subspace.