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.


Remarks on Fargues-Scholze, part 2

Today I want to talk about section VII.3 in the manuscript. Here they define and study a functor f_{\natural} on solid sheaves which is left adjoint to the usual pullback functor. But why is this called relative homology?

To explain this name, you have to remember that in the usual formalism of etale cohomology, homology is realized as the compactly supported cohomology of the dualizing complex. Symbolically, if X is a variety with structure map f:X \to \mathrm{Spec} k, then the homology of X is given by Rf_! Rf^! \Lambda. Now, you might ask whether this works in families: if f:X \to Y is some map of varieties, maybe I can find some complex on Y whose stalk at y realizes the homology of X_y? If the constant sheaf is f-ULA, then formation of Rf^! \Lambda commutes with any base change, and exactly the same formula works, but in general there is no naive sheaf with this property.

The punchline now is that f_{\natural} \Lambda does have this property: when the constant sheaf is f-ULA (e.g. if Y is a point) it agrees with Rf_! Rf^! \Lambda by Proposition VII.5.2, and its formation commutes with arbitrary base change, so it really does give a complex on Y whose stalks realize the homology of the fibers of f. The only twist is that f_{\natural}\Lambda is a solid sheaf in general, not a classical etale sheaf.

Remarks on Fargues-Scholze

The Fargues-Scholze geometrization paper is available! In this post, and probably some future posts also, I’ll make some random comments on this paper. These won’t mean anything unless you’ve read (at the very least) the beautifully written introduction to the paper. To be clear, I have nothing of substance to say about the “big picture” – these will be purely technical remarks.

First of all, at the bottom of p. 324, one finds the slightly cryptic claim that although there are no general i_! functors in the D_{lis} setting (for i a locally closed immersion), one can define functors i^{b}_! in the D_{lis} setting, where i^b: \mathrm{Bun}_{G}^b \to \mathrm{Bun}_G is the inclusion of any Harder-Narasimhan stratum into \mathrm{Bun}_G. This is stated without proof. However, if you read carefully, you will notice that these lisse i^{b}_! functors are actually used in the paper, in the proof of Proposition VII.7.6. So maybe it’s worth saying something about how to construct them.

To build i^{b}_! in the D_{lis} setting, factor i^b as the composition \mathrm{Bun}_{G}^b \overset{i}{\to}  \mathrm{Bun}_{G}^{ \leq b} \overset{j}{\to} \mathrm{Bun}_G. Here \mathrm{Bun}_{G}^{ \leq b} is the open substack of bundles which are “more semistable” than \mathcal{E}_b. It will also be convenient to write \mathrm{Bun}_{G}^{ < b} = \mathrm{Bun}_{G}^{ \leq b} - \mathrm{Bun}_{G}^{ b}. Note that i is a closed immersion, and j is an open immersion, so j_! = j_{\natural} clearly preserves D_{lis}. The subtlety is in making sense of i_!, since then we can write i_{!}^{b} = j_! i_! as usual.

For i_!, we need the local chart \pi_{b}: \mathcal{M}_b \to \mathrm{Bun}_{G}^{\leq b} and its punctured version \pi_{b}^\circ : \mathcal{M}_{b}^\circ = \mathcal{M}_{b} \times_{\mathrm{Bun}_{G}^{\leq b}} \mathrm{Bun}_{G}^{< b} \to \mathrm{Bun}_{G}^{\leq b}. Recall that these charts also come with compatible maps q_b: \mathcal{M}_b \to [\ast / G_b(E)] and q_b^{\circ}: \mathcal{M}_{b}^{\circ} \to [\ast / G_b(E)]. Then for any A \in D_{lis}(\mathrm{Bun}_{G}^{b},\Lambda) \cong D_{lis}([\ast / G_b(E)],\Lambda), the correct definition turns out to be

i_! A = \mathrm{Cone}(\pi_{b \natural}^{\circ} q_{b}^{\circ \ast}A \to \pi_{b \natural} q_{b}^{\ast}A)\,\,\,\,(1).

The point here is that in the lisse world, the only pushforward functors which come for free are the functors f_{\natural} for cohomologically smooth maps f. Since \pi_{b} and \pi_{b}^{\circ} are cohomologically smooth – one of the hardest theorems in the paper! – the above construction preserves D_{lis}. Moreover, it’s easy to check that the formula above has the right properties. Indeed, the *-restriction of the RHS of (1) to \mathrm{Bun}_{G}^{b} is just A, by Proposition VII.7.2, while its complementary restriction to \mathrm{Bun}_{G}^{ < b} clearly vanishes.