## When is it supercuspidal?

Today I want to talk about an amazing theorem of Moeglin, which I learned from WTG.

Let $F/\mathbf{Q}_p$ be a finite extension, $G=\mathrm{SO}_{2n+1}$ the split odd special orthogonal group over $F$, $G'$ its unique inner form. By work of Arthur and Moeglin, there is a natural bijection between discrete series representations of $G$ or $G'$, and pairs $(\phi,\chi)$ where $\phi: W_F \times \mathrm{SL}_2 \to \mathrm{Sp}_{2n}$ is a discrete L-parameter and $\chi$ is a character of the centralizer group $A_\phi$. In this setting, $\phi$ is discrete if it is the sum of $m$ pairwise-distinct irreducible representations $\phi_i = \sigma_i \boxtimes [d_i] : W_F \times \mathrm{SL}_2 \to \mathrm{Sp}_{2n_i}$ with $\sum_{1 \leq i \leq m} n_i = n$. Here $[d]:\mathrm{SL}_2 \to \mathrm{SL}_d$ is the usual d-1st symmetric power representation. The associated centralizer group $A_\phi$ is of the form $\{ \pm 1\}^m$, and has a canonical basis indexed by the irreducible summands $\phi_i$. Given $(\phi,\chi)$, let $\pi(\phi,\chi)$ be the associated discrete series representation. Note that $\pi(\phi,\chi)$ is a representation of $G$ if $\chi$ is trivial on the evident subgroup $\{ \pm 1 \} = Z(\mathrm{Sp}_{2n}) \subset A_\phi$, and is a representation of $G'$ otherwise. This splits the representations up evenly: for $\phi$ fixed, there are $2^m$ possible $\chi$‘s, and we get a Vogan L-packet $\Pi_\phi = \Pi_\phi(G) \cup \Pi_\phi(G')$ where $\Pi_\phi(G)$ and $\Pi_\phi(G')$ each contain $2^{m-1}$ elements.

Question. When is $\pi(\phi,\chi)$ a supercuspidal representation?

For $n=2$ I had previously memorized the (already complicated) answer to this question, so you can imagine my pleasure when I learned that Moeglin found a simple conceptual criterion which works in general! To state her theorem, we need a little vocabulary.

Definition. A discrete parameter $\phi=\oplus_i \phi_i$ is without gaps if for every $\sigma \boxtimes [d]$ occurring among the $\phi_i$‘s with $d \geq 3$, then also $\sigma \boxtimes [d-2]$ occurs among the $\phi_i$‘s.

Definition. Suppose $\phi$ is without gaps. A character $\chi$ of the component group is alternating if for every pair $\sigma \boxtimes [d]$ and $\sigma \boxtimes [d-2]$ (with $d \geq 3$) occurring among the $\phi_i$‘s, $\chi(\sigma \boxtimes [d]) = - \chi (\sigma \boxtimes [d-2])$. Moreover we require that on every summand of the form $\sigma \boxtimes [2]$, we have $\chi(\sigma \boxtimes [2])=-1.$

Theorem (Moeglin). The representation $\pi(\phi,\chi)$ is supercuspidal iff $\phi$ is without gaps and $\chi$ is alternating.

Example 0. By definition, $\phi$ is supercuspidal if $d_i =1$ for all summands. In this case, $\phi$ is (vacuously) without gaps and every $\chi$ is (vacuously) alernating, so $\Pi_\phi$ consists entirely of supercuspidal representations. The converse – if $\Pi_\phi$ consists only of supercuspidals then necessarily $\phi$ is supercuspidal – is also immediate!

Example 1. Let $\sigma_2, \sigma_2':W_F \to \mathrm{SL}_2$ be distinct supercuspidal parameters. Then $\phi = \sigma_2 \oplus \sigma_2' \oplus \sigma_2' \boxtimes [3]:W_F \times \mathrm{SL}_2 \to \mathrm{Sp}_{10}$ is a discrete parameter without gaps, with component group of size 8. It is easy to see that four of the possible $\chi$‘s are alternating, and two of these are trivial on the center of $\mathrm{Sp}_{10}$. Thus, the packets $\Pi_\phi(\mathrm{SO}_{11})$ and $\Pi_\phi(\mathrm{SO}_{11}')$ each contain four elements, two of which are supercuspidal and two of which are non-supercuspidal.

Example 2. Let $\sigma_2:W_F \to \mathrm{SL}_2$ and $\sigma_3:W_F \to \mathrm{O}_3$ be supercuspidal parameters. Then $\phi = \sigma_2 \oplus \sigma_3 \boxtimes [2] \oplus \sigma_3 \boxtimes [4]:W_F \times \mathrm{SL}_2 \to \mathrm{Sp}_{20}$ is a discrete parameter without gaps, with component group again of size 8. Now only two of the possible $\chi$‘s are alternating, and one of these is trivial on the center of $\mathrm{Sp}_{20}$. Thus, the packets $\Pi_\phi(\mathrm{SO}_{21})$ and $\Pi_\phi(\mathrm{SO}_{21}')$ each contain four elements, one of which is supercuspidal and three of which are non-supercuspidal.

Example 3. Let $\tau:W_F \to \{ \pm 1 \}$ be a nontrivial character. Then $\phi = 1 \boxtimes [2] \oplus \tau \boxtimes [2]:W_F \times \mathrm{SL}_2 \to \mathrm{Sp}_{4}$ is a discrete parameter (vacuously) without gaps, with component group of size 4. Now only one of the possible $\chi$‘s is alternating, and it is trivial on the center of $\mathrm{Sp}_{4}$. Thus, the packets $\Pi_\phi(\mathrm{SO}_{5})$ and $\Pi_\phi(\mathrm{SO}_{5}')$ each contain two elements, with $\Pi_\phi(\mathrm{SO}_{5})$ containing one supercuspidal and $\Pi_\phi(\mathrm{SO}_{5}')$ containing no supercuspidals.

More generally, if $\phi$ is without gaps and all $d_i$‘s are even, then only one $\chi$ is alternating, so the packet $\Pi_\phi$ contains a single supercuspidal representation (which may be a representation of $G$ or $G'$ – both possibilities occur) swimming in a sea of discrete series representations.

References:

Colette Moeglin, Classification des series discretes pour certains groupes classiques p-adiques, 2006

Bin Xu, On the cuspidal support of discrete series for p-adic quasisplit Sp(N) and SO(N), 2015

## Report from Oberwolfach

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

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

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

## Comparing local Langlands correspondences

At least six people have independently asked me some variant of the question:

What are the prospects for showing that the Fargues-Scholze construction of L-parameters is compatible with other constructions of the local Langlands correspondence?

In this post I’ll briefly lay out the answer as I see it.

For reductive groups $G$ over finite extensions $F/\mathbf{Q}_p$, the situation is complicated, since the status of LLC is complicated.

1. $\mathrm{GL}_n$ and $D_{1/n}^{\times}$. Compatibility for these groups is known and already proved in Fargues-Scholze, and follows from the realization of local Langlands and local Jacquet-Langlands in the cohomology of the Lubin-Tate tower.
2. Any inner form of $\mathrm{GL}_n$. Compatibility here is Theorem 1.0.3 in H.-Kaletha-Weinstein.
3. $\mathrm{SL}_n$ and inner forms. Compatibility should follow from the previous two points, but I guess it’s not completely trivial. Someone should write it down.
4. $\mathrm{GSp}_4$ and $\mathrm{Sp}_4$, and their unique inner forms. Compatibility for these groups has been proved by my student Linus Hamann. His preprint should be available very soon, and I’ll write a detailed blog post about it at that time. The arguments here rely on a number of special features of the group $\mathrm{GSp}_4$.
5. Split $\mathrm{SO}_{2n+1}$ and closely related groups. Partial results here are definitely possible by extending Hamann’s arguments, but it’s not clear to me whether complete results can be expected. I’ll say more about this when I write about Hamann’s paper.
6. Unitary groups. Partial results should be possible by combining some aspects of Hamann’s methods with recent works of Nguyen and Bertoloni-Meli–Nguyen.
7. $\mathrm{GSp}_{2n}$ and $\mathrm{Sp}_{2n}$ and their inner forms, $n>2$. This seems out of reach.
8. Even special orthogonal groups. I’m frankly confused about what’s going on here. Is there even an unambiguous LLC? In any case, this also seems hard.
9. Exceptional groups. There’s no “other” LLC here. Go home. (OK, for $G_2$ there’s a very cool recent paper of Harris-Khare-Thorne.)
10. General groups splitting over a tame extension, $p$ not too small. Here Kaletha has given a general construction which attaches a supercuspidal L-packet to any supercuspidal L-parameter. Compatibility of this construction with Fargues-Scholze might be approachable by purely local methods, but it seems to require substantial new ideas. An extremely weak partial result – constancy on Kaletha’s packets of the FS map from reps to L-parameters – is probably within reach, using the main results in H.-Kaletha-Weinstein.

The key point in many of the above situations is the following. Let’s say a group $G$ is accessible if it admits a geometric conjugacy class of minuscule cocharacters $\mu$ such that
1. The pair $(G,\mu)$ is totally Hodge-Newton reducible in the sense of Chen-Fargues-Shen.
2. Any L-parameter $\varphi: W_F \to \phantom{}^L G$ can be recovered up to isomorphism from the composition $r_{\mu} \circ \varphi$. (In practice one asks for slightly weaker versions of this.)
3. The local Shimura varieties attached to the local Shimura datum $(G,\mu,b)$ (with $b \in B(G,\mu)$ the unique basic element) uniformize the basic locus in a global Shimura variety of abelian type.

For groups satisfying this condition, there is hope. Very roughly, condition 2. implies that the FS construction is incarnated in the cohomology of a single local Shimura variety, whose cohomology can also be tightly related to the cohomology of a global Shimura variety using conditions 1. and 3. One then needs to know enough about the cohomology of these global Shimura varieties, namely that it realizes the “other” LLC you care about. Of course, this short outline veils substantial technical difficulties.

It turns out that $\mathrm{GL}_{n}$, $\mathrm{GU}_n$, $\mathrm{GSp}_4$, and $\mathrm{SO}_{2n+1}$ are all accessible, and this accounts for the definitive results in scenarios 1.-4. above and my optimism in scenarios 5.-6. On the other hand, $\mathrm{GSp}_{2n}$ is not accessible for $n>2$, and neither is $\mathrm{SO}_{2n}$ for $n>3$, and no exceptional groups are accessible. Hence my pessimism in scenarios 7.-9.

For reductive groups over finite extensions $F/\mathbf{F}_{p}((t))$, the situation is completely different. Here Genestier-Lafforgue have constructed a local Langlands correspondence for all groups, uniquely characterized by its compatibility with V. Lafforgue’s construction of global Langlands parameters. It is an extremely attractive problem to compare the Genestier-Lafforgue LLC with the Fargues-Scholze LLC. This should absolutely be within reach! After all, both constructions are realized in the cohomology of moduli spaces of shtukas, so the only “real” task should be to physically relate the moduli spaces of shtukas used by GL with those used by FS. This is probably not trivial: the spaces used by FS are local and totally canonical, while those used by GL seem to depend on a globalization and some auxiliary choices in a messy way. Nevertheless, I’d be surprised if this comparison is still an open problem two years from now.

## 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.