## Distinguished affinoids

Fix a complete nonarchimedean field $K$ equipped with a fixed norm, with residue field $k$. Let $A$ be a $K$-affinoid algebra in the sense of classical rigid geometry. Here’s a funny definition I learned recently.

Definition. A surjection $\alpha : T_{n,K} \twoheadrightarrow A$ is distinguished if the associated residue norm $|\cdot|_\alpha$ equals the supremum seminorm $|\cdot|_{\mathrm{sup}}$. A $K$-affinoid algebra $A$ is distinguished if it admits a distinguished surjection from a Tate algebra.

Being distinguished imposes some obvious conditions on $A$: since the supremum seminorm is a norm iff $A$ is reduced, it certainly it implies
1) $A$ is reduced.
Since any residue norm takes values in $|K|$, it also implies
2) $|A|_{\mathrm{sup}} = |K|$.

If $K$ is stable (which holds if $K$ is discretely valued or algebraically closed), then the converse is true: 1) and 2) imply that $A$ is distinguished. Since 2) is automatic for $K$ algebraically closed, we see that any reduced $K$-affinoid is distinguished if $K$ is algebraically closed. It is also true that if $\alpha: T_{n,K} \to A$ is a distinguished surjection, then $\alpha^\circ: T_{n,K}^{\circ} \to A^\circ$ is surjective. Moreover, if $A$ satisfies 2), or $K$ is not discretely valued, then a surjection $\alpha: T_{n,K} \to A$ is distinguished iff $\alpha^\circ$ is surjective. Either way, if $A$ is distinguished then $A^\circ$ is a tft $K^\circ$-algebra.

All of this can be found in section 6.4.3 of BGR.

Question 1. If $A$ is reduced, is there a finite extension $L/K$ such that $A\otimes_K L$ is distinguished as an $L$-affinoid algebra?

This should be easy if it’s true. I didn’t think much about it.

Now suppose $A$ is distinguished, and let $\tilde{A} = A^{\circ} / A^{\circ \circ}$ be its reduction to a finite type $k$-algebra. As usual we have the specialization map $\mathrm{sp}: \mathrm{Sp}A \to \mathrm{Spec} \tilde{A}$. It is not hard to see that if $D(f) \subset \mathrm{Spec} \tilde{A}$ is a principal open, then $\mathrm{sp}^{-1}D(f)$ is a Laurent domain in $\mathrm{Sp}A$. Much less obvious is that for any open affine $U \subset \mathrm{Spec} \tilde{A}$, the preimage $\mathrm{sp}^{-1}U$ is an affinoid subdomain such that $A_U=\mathcal{O}(\mathrm{sp}^{-1}U)$ is distinguished and $\widetilde{A_U} = \mathcal{O}_{\mathrm{Spec} \tilde{A}}(U)$. This is buried in a paper of Bosch.

Loosely following Bosch, let us say an affinoid subdomain $V \subset \mathrm{Sp}A$ is formal if it can be realized as $\mathrm{sp}^{-1}U$ for some open affine $U \subset \mathrm{Spec} \tilde{A}$. Now let $X$ be a reduced quasicompact separated rigid space over $K$. Let us say a finite covering by open affinoids $U_1=\mathrm{Sp}A_1,\dots,U_n= \mathrm{Sp}A_n \subset X$ is a formal cover if
1) all $A_i$ are distinguished, and
2) for each $(i,j)$, the intersection $\mathrm{Sp}A_{ij}=U_{ij} := U_i \cap U_j$, which is automatically affinoid, is a formal affinoid subdomain in $U_i$ and in $U_j$.

This is a very clean kind of affinoid cover: we can immediately build a formal model for $X$ by gluing the tft formal affines $\mathrm{Spf}(A_i^\circ)$ along their common formal affine opens $\mathrm{Spf}(A_{ij}^\circ)$. Moreover, the special fiber of this formal model is just the gluing of the schemes $\mathrm{Spec}\widetilde{A_i}$ along the affine opens $\mathrm{Spec}\widetilde{A_{ij}}$.

Question 2. For $X$ a reduced qc separated rigid space over $K$, is there a finite extension $L/K$ such that $X_L$ admits a formal affinoid cover?

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## Report from Oberwolfach

In the first week of February, I attended an Oberwolfach workshop on Nonarchimedean geometry and applications. It was a great pleasure to attend a conference in person after such a long period of isolation. Thank you to the organizers for making this week so enjoyable! As usual, here are some scattered recollections from the workshop.

• Due to the hybrid nature of the workshop, the talks on several days didn’t start until 2 pm, presumably to accommodate participants in North America, with the final talk scheduled after dinner. This left a huge swath of unscheduled time, from 9 am until 2 pm, which actually turned out to be kind of great. I don’t know if it was the hunger for in-person interaction after 2 years of isolation, but people really seemed to take full advantage of this free time for vigorous discussion and collaboration. I actually liked this schedule better than the usual schedule.
• Best talks: Johannes Nicaise, Lucas Mann, Piotr Achinger, Yujie Xu, Ben Heuer.
• Worst talk: [redacted]
• Categorical Langlands for GL1 = Langlands for mice.
• Random Question 1 (via PS): Let $U \subset X$ be an open immersion of an affine scheme into a smooth projective variety. Is the complement $X \smallsetminus U$ an ample divisor on $X$?
• Random Question 1′ (via DC): Let $U \subset V$ be an open immersion of affine schemes. Is $U$ the nonvanishing locus of a section of an ample line bundle on $V$?
• One recurring theme throughout the week was the p-adic Simpson correspondence, with excellent talks from Ben Heuer and Matti Würthen. Here something quite amusing happened: in an informal conversation on Tuesday, Ben explained the complicated status of p-adic Simpon to me, and stressed that its most optimistic conjectural form isn’t actually written down, because no one wants to be the one to make a false conjecture. But then Matti Würthen did explicitly conjecture this exact statement in his Friday lecture! For the record, the hope is that for a smooth projective variety $X/ \mathbf{C}_p$, there is an equivalence of categories from $\mathbf{C}_p$-representations of $\pi_1^{\mathrm{et}}(X)$ towards semistable Higgs bundles on $X$ with vanishing Chern classes. As Ben stressed to me, this definitely fails if $\mathbf{C}_p$ is replaced with a larger algebraically closed nonarchimedean field. Hmm…
• One consequence of COVID measures is that seating for meals was not randomized as usual, but rather was fixed for the entire week. I was assigned to eat with Torsten Wedhorn, Bogdan Zavyalov, and François Loeser. This ended up being a really pleasant group to eat with! I already knew Torsten and Bogdan fairly well, but I’d never spoken with François before, and it turns out he’s a totally charming and delightful person. It was especially wonderful to hear him talk about his astonishing achievements in ultra long distance running. One memorable quote: “The first night without sleep is no problem. The second night is… interesting. And the third night… well… this I cannot recommend.”
• My relationship with Hochschild cohomology has gone from “???” to a vague understanding and a desire to learn more. Thanks to DC for some helpful explanations!
• There was much discussion among the younger participants about what Fargues’s categorical local Langlands conjecture should look like with mod-p coefficients (i.e. in the $\ell=p$ setting). Of course on the spectral side, one expects to see some category of quasicoherent or ind-coherent sheaves on the special fiber of the Emerton-Gee stack for $\hat{G}$. On the automorphic side, one should have some category of mod-p sheaves on $\mathrm{Bun}_G$, and the correct category should fall out of the general formalism developed by Mann in his thesis. One tantalizing fact, sketched out during some of these conversations, is that $\mathrm{Bun}_G$ is definitely $p$-cohomologically smooth (in a precise sense), not of dimension 0 as in the $\ell \neq p$ case, but of dimension equal to the dimension of $G(\mathbf{Q}_p)$ as a $p$-adic Lie group. It is surely no coincidence that this matches the expected dimension of the Emerton-Gee stack for $\hat{G}$.

Another tantalizing observation: the relationship between the Emerton-Gee stack and Wang-Erickson’s stack of Galois representations is perfectly analogous to the difference between the stacks $\mathrm{LocSys}_G$ and $\mathrm{LocSys}_{G}^{\mathrm{restr}}$ appearing in AGKRRV.

On the other hand, it also became clear that most of the analysis in Fargues-Scholze cannot carry over naively to the setting of p-adic coefficients, and that many of the crucial tools developed in their paper simply won’t help here. In particular, the magic charts $\pi_b: \mathcal{M}_b \to \mathrm{Bun}_G$ used by FS, which are $\ell$-cohomologically smooth for all primes $\ell \neq p$, are definitely NOT $p$-cohomologically smooth. This already fails for $G=\mathrm{GL}_2$. Likewise, their “strict Henselian” property should fail badly. New ideas are very much required!

## 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.

## 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.

## The dangers of naming things after people

I spent part of last weekend reading Alice Silverberg’s blog, which is simultaneously depressing and hilarious. Everyone should read it, but you probably won’t enjoy reading it unless you enjoy Coen brothers movies or the short stories of Kafka. Anyway, the following thoughts have been in my head for a few months, but I decided to record them here after reading this.

The cohomology of non-basic local Shimura varieties is described by the “Harris-Viehmann conjecture”, which is formally stated as Conjecture 8.4 in Rapoport-Viehmann’s paper. This story started with a daring and beautiful conjecture of Harris (conjecture 5.2 here), whose formulation however turned out to be slightly incorrect in general, cf. Example 8.3 in RV. The conjecture was then modified by Viehmann, and Rapoport named this modified conjecture the Harris-Viehmann conjecture (footnote 5 in RV).

Unfortunately, Conjecture 8.4 in RV is still not correct as stated: the Weil group action on the summands appearing on the right-hand side needs to be modified by certain half-integral Tate twists. As far as I know, Alexander Bertoloni-Meli is the only person who has publicly pointed out the need for this modification, and Conjecture 3.2.1 in his very cool paper is the only correct formulation of the Harris-Viehmann conjecture in print.

Since the need for these Tate twists was overlooked by a lot of very smart people, it only seems fair to me that Alexander should get credit for his contribution here. The obvious way to do this would be to refer to the Harris–Viehmann–Bertoloni-Meli conjecture, or the Bertoloni-Meli–Harris–Viehmann conjecture. You could pick the second option if you’re a stickler for alphabetical name orders in mathematics, or the first option if you feel (as I do) that Harris’s contribution here deserves priority.

But it gets even worse, because Harris also formulated another conjecture along similar lines (conjecture 5.4 in his article linked above), which has gotten somewhat less attention but which is nevertheless extremely interesting.* It turns out that one can formulate a unified conjecture which encompasses both Harris’s conjecture 5.4 and the Harris–Viehmann–Bertoloni-Meli conjecture. What should it be called? The Harris–Viehmann–Bertoloni-Meli—Harris conjecture? I guess not.

*Here’s a comment from MH: “I was (and am) much more attached to this conjecture than to the one that is called the Harris-Viehmann conjecture, because it required some work to find the right formalism (the parabolics that transfer between inner forms), whereas the other conjecture (independently of the incorrect formulation in my paper) was just the obvious extension of Boyer’s result.”

## 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.

## The six functors for Zariski-constructible sheaves in rigid geometry

In this post I want to talk about my recent paper with Bhargav Bhatt, which you can find here. This paper was a lot of fun to write, and I hope the toolkit we built will be useful for other researchers in this area. In this post I want to make some random remarks on this paper, which probably won’t mean anything if you don’t go read the real introduction to the paper first.

One funny point is that the proof of Theorem 1.6 leans on Theorem 1.7 fairly heavily, but in fact you can prove Theorem 1.6 without appealing to Theorem 1.7, at the price of much more intricate arguments. This was actually the state of the manuscript until mid-December, when we finally figured out how to prove Theorem 1.7.

Another funny point is that the discussion of the “standard” / “constructible” t-structure on $D^{(b)}_{zc}(X,\mathbf{Z}_{\ell})$ turned out to be surprisingly subtle, cf. Theorem 3.39. Note that $D^{(b)}_{zc}(X,\mathbf{Z}_{\ell})$ is by definition a full subcategory of $D(X_v,\mathbf{Z}_{\ell})$, and the latter carries an obvious t-structure. Nevertheless, we weren’t able to settle the question of whether these t-structures are compatible:

Question. Do the cohomological functors $^c \mathcal{H}^n(-)$ on $D^{(b)}_{zc}(X,\mathbf{Z}_{\ell})$ produced by Theorem 3.39 agree with the usual cohomology sheaves on $D(X_v,\mathbf{Z}_{\ell})$?

I would be extremely interested to know the answer to this.

One thing missing from the paper is any discussion of ULA sheaves. (See Fargues-Scholze for the foundations of ULA sheaves in p-adic geometry. In what follows I take $\ell \neq p$, but the case $\ell = p$ should actually also be OK.) The first basic point to make is that for any rigid space $X/K$, any object $A \in D^{(b)}_{zc}(X,\mathbf{F}_{\ell})$ is ULA for the structure map $X \to \mathrm{Spa}K$. Sketch: The claim is local on $X$, so we can assume $X$ is quasicompact. By Proposition 3.6 and stability of ULA sheaves under proper pushforward, we reduce to the special case where $A = \mathbf{F}_{\ell}$ is constant. By an argument with resolution of singularities, we now reduce further to the case where $A$ is constant and $X$ is smooth, which is handled in Fargues-Scholze. Identical remarks apply with $\mathbf{Z}_{\ell}$-coefficients, or with general $\mathbf{Z}/n$ coefficients (but then only for objects of “finite tor-dimension”).

This is already enough to show that the Lefschetz trace formula works as expected for proper rigid spaces in characteristic zero. More precisely, suppose given a correspondence $c=(c_1,c_2): C \to X \times X$ of proper rigid spaces over an algebraically closed field, and a cohomological correspondence $u: c_1^{\ast}A \to Rc_2^{!}A$ on some $A \in D^b_{zc}(X,\mathbf{Z}_{\ell})$. Then the usual recipe to define local terms applies, and the expected equality $\mathrm{tr}(u|R\Gamma(X,A)) = \sum_{\beta \in \pi_0 \mathrm{Fix}(c)} \mathrm{loc}_{\beta}(u,A)$ holds true. (Note that $R\Gamma(X,A)$ is a perfect $\mathbf{Z}_{\ell}$-complex by Theorem 3.35.(3).)  This can be proved by imitating the unpleasant arguments with giant diagrams in SGA5, or by the amazing categorical magic of Lu-Zheng. Of course, the local terms $\mathrm{loc}_{\beta}(u,A)$ are just as mysterious as in the case of schemes.

It’s also natural to guess that an analogue of Deligne’s generic ULA theorem holds in this setting. Let me formally state this as a conjecture.

Conjecture. Let $f:X \to Y$ be a proper map of characteristic zero rigid spaces, and let $A \in D^{(b)}_{zc}(X,\mathbf{F}_{\ell})$ be any given object. Then there is a dense Zariski-open subset of $Y$ over which $A$ is $f$-ULA.

This should be within reach, but I didn’t think about it very much.

Finally, I want to highlight the open problems mentioned in Remark 4.10, Remark 4.14 and Section 4.5. Conjecture 4.16 is probably (for whatever reason) my favorite open problem right now. Actually, I don’t even know how to prove that $IH^{\ast}(X_C,\mathbf{Q}_p)$ is Hodge-Tate, or the even weaker statement that it has integral Hodge-Tate-Sen weights.

## What does a general proper rigid space look like?

As the title says. Consider proper rigid spaces $X$ over some nonarchimedean field $K$. The “standard” examples of such things which don’t come from algebraic geometry are i) the Hopf surface $(\mathbf{A}^2 - 0)/p^\mathbf{Z}$, ii) non-algebraizable deformations of K3 surfaces over the residue field of $K$, and iii) generic abeloid varieties (which are analogous to generic compact complex tori).  But there must be gazillions of other examples, right? A “random” proper rigid space is hard to write down, sort of by definition. But there are certainly some natural questions one can ask:

-For every $n \geq 2$, does there exist a proper $n$-dimensional rigid space with no non-constant meromorphic functions, and admitting a formal model whose special fiber has components of general type? Can we find examples of such spaces with arbitrarily large dimension dimension which don’t come from lower-dimensional examples by simple operations (products, quotients by finite groups, etc.)? Same question but with “no non-constant meromorphic functions” replaced by the weaker requirement that $\mathrm{tr.deg}K(X)/K$ is small compared to $\dim X$.

-Do there exist non-algebraizable proper rigid spaces with “arbitrarily bad” singularities?

-Do there exist rigid analytic analogues of Kodaira’s class VII0 surfaces?

## Geometry of the B_dR affine Grassmannian

As many readers of this blog already know, one key result in modern p-adic geometry is Scholze’s theorem that the $B_{\mathrm{dR}}$-affine Grassmannian is an ind-spatial diamond. The proof of this given in the Berkeley notes is a bit tricky and technical: it uses covering by infinite-dimensional objects in a crucial way, as well as an abstract Artin-type representability criterion.  So I’m very pleased to report that Bence Hevesi has given a beautiful new proof of this theorem in his Bonn master’s thesis. Bence’s proof avoids representability criteria or coverings by huge objects. Instead, his idea is to reduce to $\mathrm{GL}_n$ and then construct explicit charts for closed Schubert cells, using moduli of local shtukas at infinite level. You can read Bence’s outstanding thesis here.

## Brain teaser: mysterious moduli and local Langlands

Fix an integer $n>1$. Let $X$ denote the moduli space of triples $(\mathcal{E}_1, \mathcal{E}_2,f)$ where $\mathcal{E}_i$ is a vector bundle of rank $n$ on the Fargues-Fontaine curve which is trivial at all geometric points, and $f: \mathcal{E}_1 \oplus \mathcal{E}_2 \to \mathcal{O}(1/2n)$ is an injection which is an isomorphism outside the closed Cartier divisor at infinity.

Brain teaser a. Prove that $X$ is a locally spatial diamond over $\breve{\mathbf{Q}}_p$ with a Weil descent datum to $\mathbf{Q}_p$.

Now, let $D$ be the division algebra over $\mathbf{Q}_p$ of invariant $1/2n$, and let $\tau$ be an irreducible representation of $D^\times$ whose local (inverse) Jacquet-Langlands correspondent is supercuspidal. Note that $D^\times$ acts on $X$ by its natural identification with $\mathrm{Aut}(\mathcal{O}(1/2n))$.

Brain teaser b. Prove that the geometric etale cohomology of $X$ satisfies the following:

$R\Gamma_c(X_{\mathbf{C}_p},\overline{\mathbf{Q}_\ell})\otimes_{D^\times} \tau \cong \varphi_{\tau}[1-2n](\tfrac{1-2n}{2})$ if $\tau$ is orthogonal, and $R\Gamma_c(X_{\mathbf{C}_p},\overline{\mathbf{Q}_\ell})\otimes_{D^\times} \tau \cong 0$ if $\tau$ is not orthogonal.

Here $\varphi_\tau$ denotes the Langlands parameter of $\tau$.

It is probably not fair to call these brain teasers. Anyway, here is one big hint: the infinite-level Lubin-Tate space for $\mathrm{GL}_{2n}$ is naturally a $\mathrm{GL}_n(\mathbf{Q}_p)^2$-torsor over $X$, by trivializing the bundles $\mathcal{E}_i$.