## 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.

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 $\mathcal{E}_i$.

## 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!

## Vanishing for O+/p-cohomology of Stein spaces

OK I’m too lazy to convert this into a proper blog post, but Proposition 0.4 here is something I needed recently, which others might find useful.  Lemma 0.2 might also come in handy.  Please ask in the comments if you’d like more details!

Update (Sept. 29). James Newton pointed out that it’s not actually clear how to fill in the details in the “proof” of Lemma 0.1.i. Therefore, the results in Lemmas 0.1 and 0.2 of the document linked above are currently unproved in the generality stated there. Fortunately, I was able to recover some weaker versions of Lemmas 0.1 and 0.2 which still suffice for the intended application to cohomology of Stein spaces. The corrected writeup is available here.

## Zariski closed immersions

In p-adic geometry, what should it mean for a morphism to be a Zariski-closed immersion? For locally Noetherian adic spaces, the usual notion of a closed immersion of locally ringed spaces works just fine. For general analytic adic spaces, though, one quickly runs into annoying foundational issues. The issue is roughly as follows. Let $X=\mathrm{Spa}(A,A^+)$ be an (analytic) affinoid adic space. We can certainly define a reasonable notion of Zariski-closed subset, just by following our nose: a subset $Z \subset |X|$ should be Zariski-closed if there is an ideal $I \subset A$ such that $x \in |Z|$ iff $|f|_x = 0\,\forall f \in I$. These are exactly the subsets obtained by pulling back closed subsets of $\mathrm{Spec}(A)$ along the natural map $|\mathrm{Spa}(A,A^+)| \to |\mathrm{Spec}(A)|$. The problem, however, is that such a $Z$ need not come from an actual closed immersion of an affinoid adic space into $X$, because the quotient $A/I$ could just be some junky non-sheafy ring, and maybe there’s no canonical tweak (like replacing $I$ by its closure, or replacing $A/I$ by its uniform completion, or…) which will make it sheafy. And even if we can tweak $A/I$ to make it sheafy, how do we know that $A \to A/I$ is still surjective after going to some rational subset $U \subset X$? You get the picture.

Perhaps surprisingly, the situation for affinoid perfectoid spaces is a lot better. In particular, if $(A,A^+)$ is a perfectoid Tate-Huber pair, there are canonical bijections (satisfying some obvious compatibilities) between
1) closed subsets of $\mathrm{Spec}(A)$,
2) Zariski-closed subsets of $X=\mathrm{Spa}(A,A^+)$,
3) (isomorphism classes of) maps of Tate-Huber pairs $(A,A^+) \to (B,B^+)$ where $B$ is a perfectoid Tate ring, $A \to B$ is surjective, and $B^+$ is the integral closure of the image of $A^+$ in $B$.

We’ve already discussed the bijection 1) <–> 2). For 3) –> 1) or 2), just send $q:A \twoheadrightarrow B$ to the closed subset cut out by the ideal $\ker q$.  The miracle is the association 2) –> 3), which holds by an amazing theorem of Bhatt: if $I$ is a closed ideal in a perfectoid Tate ring $A$, then the uniform completion $B$ of $A/I$ is perfectoid and the natural map $A \to B$ is surjective, cf. Theorem 2.9.12 in Kedlaya’s notes here. Moreover, the map $A \to B$ remains surjective after rational localization on $A$. In particular, if $Z \subset X$ is a Zariski-closed subset, then 2) –> 3) gives an honest closed immersion $\mathrm{Spa}(B,B^+) \to \mathrm{Spa}(A,A^+)$ of locally ringed spaces, and $|\mathrm{Spa}(B,B^+)|$ maps homeomorphically onto $Z$.

The point of all this is that Zariski-closed immersions of affinoid perfectoid spaces behave as well as one could ever dream (with one caveat, which I’ll get to later). The following definition then suggests itself.

Definition. A map of small v-stacks $X \to Y$ is a Zariski-closed immersion if for any affinoid perfectoid space $W$ with a map $W \to Y$, the base change $X \times_{Y} W \to W$ is a Zariski-closed immersion of affinoid perfectoid spaces.

Now of course we’re free to make any definition we want in mathematics, but if it doesn’t capture some essential idea or experimentally observed phenomenon, then who cares? Let me now give some evidence that this definition passes this test.

Example 0. The property of being a Zariski-closed immersion is preserved under composition and base change. If $X \to Y$ is a Zariski-closed immersion and $Y$ is (a small v-sheaf, a diamond, a locally spatial diamond, qc or qs or separated or partially proper over a base $S$), then so is $X$.

Example 1. Let $f: X \to Y$ be a closed immersion of locally Noetherian adic spaces. If $Y$ is affinoid (so $X$ is too), then the map of diamonds $f^{\lozenge} : X^{\lozenge} \to Y^{\lozenge}$ is a Zariski-closed immersion. This is easy.

Example 2. Let $f: X \to Y$ be a closed immersion of locally Noetherian adic spaces again, but now assume that $f$ is the analytification of a closed immersion of quasiprojective varieties.  Then $f^{\lozenge}: X^{\lozenge} \to Y^{\lozenge}$ is a Zariski-closed immersion.  For this, we can use the assumption on $f$ to choose a vector bundle $\mathcal{E}$ on $Y$ together with a surjection $\mathcal{E} \twoheadrightarrow \mathcal{I}_{X} \subset \mathcal{O}_Y$. Then for any map $g: W \to Y$ from an affinoid perfectoid, the pullback $g^{\ast}\mathcal{E}$ (in the usual sense of ringed spaces) is a vector bundle on $W$, hence generated by finitely many global sections $e_1,\dots,e_n$ by Kedlaya-Liu. The images of $e_1,\dots,e_n$ along the natural map $(g^{\ast}\mathcal{E})(W) \to \mathcal{O}_{W}(W)$ generate an ideal, and the associated closed immersion of affinoid perfectoids $V \to W$ represents the fiber product $X^{\lozenge} \times_{Y^{\lozenge}} W$. (Hat tip to PS for suggesting this vector bundle trick.)

Example 3. Let $X^{\ast}$ be a minimally compactified Hodge-type Shimura variety with infinite level at $p$. Then the boundary $Z \to X^{\ast}$ is a Zariski-closed immersion, and so is the diagonal $X^{\ast} \to X^{\ast} \times X^{\ast}$. (These both reduce to the previous example, using a small limit argument in the second case.) In particular, if $U,V \subset X^{\ast}$ are any open affinoid perfectoid subsets, then $U \cap V$ is also affinoid perfectoid. This small observation plays a non-negligible role in my forthcoming paper with Christian Johansson, where (among other things) we prove that any minimally compactified Shimura variety of pre-abelian type with infinite level at $p$ is perfectoid.

Example 4. Fix a perfectoid base field $K$ of characteristic zero. Then the inclusions $\mathrm{Fil}^n \mathrm{B}_{\mathrm{dR}}^{+} \subset \mathrm{B}_{\mathrm{dR}} \subset \mathrm{B}_{\mathrm{dR}}$ are Zariski-closed immersions of (ind-)diamonds over $\mathrm{Spd} K$. This can be proved by induction on $n$, and the base case reduces to the fact that the inclusion $\mathrm{Fil}^1 \mathrm{B}_{\mathrm{dR}}^{+} \subset \mathrm{B}_{\mathrm{dR}}^{+}$ is the pullback of $\{ 0 \} \to \mathbf{A}^{1}_{K}$ along $\theta$. (To make the induction work, you need to pick an element $\xi \in \mathrm{B_{dR}}^+(K)$ generating $\ker \theta$.)

Example 5. Fix a complete algebraically closed extension $C / \mathbf{Q}_p$. Fix a reductive group $G / \mathbf{Q}_p$ and a geometric conjugacy class of $G$-valued cocharacters $\mu$. Then $\mathrm{Gr}^{\mathrm{B_{dR}}}_{G, \leq \mu, C} \to \mathrm{Gr}^{\mathrm{B_{dR}}}_{G, C}$ is a Zariski-closed immersion. Also, if $\nu \leq \mu$, then $\mathrm{Gr}^{\mathrm{B_{dR}}}_{G, \leq \nu, C} \to \mathrm{Gr}^{\mathrm{B_{dR}}}_{G, \leq \mu, C}$ is a Zariski-closed immersion. These claims can be reduced to the case $G = \mathrm{GL}_n$, which in turn reduces to Example 4 by some trickery.

Example 6. Fix a complete algebraically closed nonarchimedean field $C$ of residue characteristic $p$, and let $\mathcal{E} \to \mathcal{F}$ be any injective map of coherent sheaves on the Fargues-Fontaine curve $X_C$. Then the associated map of Banach-Colmez spaces $\mathbb{V}(\mathcal{E}) \to \mathbb{V}(\mathcal{F})$ is a Zariski-closed immersion. This can also be reduced to Example 4.

Let me end with some caveats. First of all, I wasn’t able to prove that if $G \to H$ is a closed immersion of reductive groups, the induced map $\mathrm{Gr}^{\mathrm{B_{dR}}}_{G, C} \to \mathrm{Gr}^{\mathrm{B_{dR}}}_{H, C}$ is a Zariski-closed immersion, although it is surely true. The problem here is (roughly) that an $H$-torsor over some affinoid perfectoid $X$ can only be reduced to a $G$-torsor locally in the analytic topology on $X$, and we then run into the following open question:

Question. Is the property of being Zariski-closed local for the analytic topology? More precisely, if $X$ is affinoid perfectoid with a covering by rational subsets $U_i$, and $Z$ is a closed subset such that $Z \cap U_i$ is Zariski-closed in $U_i$ for all $i$, is $Z$ Zariski-closed?

There are also naturally occurring closed things which probably aren’t Zariski-closed immersions. For instance, I don’t think the map of Banach-Colmez spaces $0 \to H^1(\mathcal{O}(-1)) = \mathbf{A}^{1,\lozenge}_{C} / \underline{\mathbf{Q}_p}$ is a Zariski-closed immersion, because then pulling back would imply that $\underline{\mathbf{Q}_p} \to \mathbf{A}^{1,\lozenge}_{C}$ is a Zariski-closed immersion, which seems extremely unlikely to me. (But I didn’t manage to disprove it! Actually, can one give an explicit example of an affinoid perfectoid $X/C$ and a closed subset $S \subset X$ such that $C$ maps isomorphically to the completed residue field at every point in $S$ and such that $S$ is NOT Zariski-closed? Surely such examples exist.) I also don’t think (closures of) Newton strata in flag varieties are Zariski-closed immersions – they are just too weird.

I also wasn’t able to settle the following compatibility (but admittedly I didn’t try very hard).

Question. Let $f: X \to Y$ be a monomorphism of locally Noetherian adic spaces. If $f^{\lozenge}$ is a Zariski-closed immersion, is $f$ actually a closed immersion?

Happy new year!