## 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?

## 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!

## Postdoc position at NUS

I’m looking to hire a postdoc here at NUS! This position is for two years, with the possibility of renewal for a third year, and carries no teaching duties. Ideally you will collaborate with me, but the job comes with near-total freedom to pursue your research. I also want to emphasize that Singapore is a beautiful country, with friendly people and amazing food, and it’s hard to imagine anyone regretting coming here for a few years.

Serious applicants, whose research interests are compatible with mine, are encouraged to apply via Mathjobs here. Although there is a quasi-official deadline of Jan. 31, in reality the position will stay open until I hire a suitable candidate, so late applications are welcome too.

I’m happy to answer further questions about the position via email.

## 2022 in review

Number of intercontinental moves: 1

Number of referee reports written: 6

Number of quick opinions written: Maybe 9 or 10?

Number of recommendation letters written: 6

Number of papers finished: 3

Percentage of those papers which use $\infty$-categories: 100%

Do I understand $\infty$-categories yet: of course not.

Number of papers accepted: 1

Number of papers rejected after a 20 month review process and a favorable referee report, on the basis of lazy and intellectually dishonest additional’ quick opinions: 1

Will I ever submit to that journal again: I doubt it!

Number of conferences attended: 4

Number of conferences attended which served cakes decorated in gold leaf: 1

Coolest things I read carefully this year:
-DC’s amazing new approach to Poincare duality: why struggle to define a trace map when you can define a cotrace map with elegance and panache? See here, and also here.
-DHKM’s beautiful and creative use of the Fargues-Scholze technology to settle a bunch of notorious foundational questions about modular representations of p-adic groups.
-LM’s thesis. Brutal and overwhelming, in the best possible way.
-EGH’s notes on p-adic Langlands.

Did I get COVID this year: yes.

Pitchfork-style rating of my COVID experience: 4.3/10

Do I honestly feel, in this moment, like I had any good ideas this year: meh.

Research goals for 2023:
-learn about motivic sheaves.
-figure out how to really’ picture the Drinfeld compactification of $\mathrm{Bun}_P$.
-formulate a precise categorical p-adic local Langlands conjecture.
-[10+ further items omitted due to vagueness]

## Questions on mod-p representations of p-adic groups

As the title says. There are more questions than answers in this subject, and for my own sake I decided to collect some of them here. I reserve the right to add to this list in the future, and I encourage readers to leave additional questions in the comments!

In this post, $F$ denotes a finite extension of $\mathbf{Q}_p$, $G$ denotes (the $F$-points of) a p-adic reductive group, $P=MU \subset G$ a parabolic with Levi decomposition. Let $\mathrm{Rep}(G)$ be the abelian category of smooth representations of $G$ on $\overline{\mathbf{F}_p}$-vector spaces, and let $D(G)$ be its derived category. On this category of representations, there are various finiteness conditions one can consider: admissible, locally admissible, finitely generated, finitely presented. These conditions will be abbreviated in obvious ways. Any admissible representation $\pi$ has a (Gelfand-Kirillov) dimension $\mathrm{dim}_G \pi$, which is an integer in the interval $[0,\dim G]$.

First, some basic structural questions.

Q1. (Emerton-Gee) Do finitely presented representations form an abelian subcategory of $\mathrm{Rep}(G)$? This reduces to: is the kernel of a map between finitely presented representations also finitely presented? Yes for $\mathrm{SL}_2(F)$ (Shotton). For higher rank groups there is some negative evidence (Timmins).

Q2. (Emerton) Is any finitely generated admissible representation necessarily of finite length? Yes for tori, and for $\mathrm{GL}_2(\mathbf{Q}_p)$ and related groups (Emerton).

Q3. If yes to Q2., then locally admissible representations form a locally finite abelian category, so they admit a block decomposition (Gabriel). Is this block decomposition interesting in general?

Q4. Is the evident functor $\alpha_G: D^+(\mathrm{Rep}(G)_{\mathrm{l.adm}}) \to D(G)$ fully faithful? Note that $\mathrm{Rep}(G)_{\mathrm{l.adm}}$ has enough injectives (Emerton). This reduces to: does $\mathrm{Ext}^i(A,B)$ vanish for all $i>0$, all locally admissible reps. $A$, and all locally admissible reps. $B$ which are injective in $\mathrm{Rep}(G)_{\mathrm{l.adm}}$?

Q5. Which integers can possibly occur as the dimension of an irreducible admissible representation?

Q5′. Does every irreducible admissible representation have dimension $\leq \dim \gamma_G$? Here $\gamma_G$ is the dimension of the flag variety associated with the quasisplit form of $G$, so e.g. $\gamma_G = [F:\mathbf{Q}_p]n(n-1)/2$ for $G=\mathrm{GL}_n(F)$. This question reduces to the case of supersingular representations. Yes for $\mathrm{GL}_2(\mathbf{Q}_p)$ and related groups.

Next, recall the derived duality functor $\mathcal{S}_G : D(G) \to D(G)$, which is given by derived internal hom towards the trivial representation. This restricts to a self-equivalence on $D^b_{\mathrm{adm}}(G)$ satisfying biduality. If $\pi$ is an admissible rep in degree zero, then $\mathcal{S}^i_G(\pi) = H^i(\mathcal{S}_G(\pi))$ agrees with the i’th derived dual of $\pi$ defined by Kohlhaase (H.-Mann) – moreover, it is admissible of dimension $\leq i$, with equality in degree $\dim_G \pi$, and it vanishes for $i > \dim_G \pi$.

Q6. Does $\mathcal{S}^i_G(-)$ preserve the property of being admissible of finite length?

Q7. If $\pi$ is irreducible admissible and supersingular, can some $\mathcal{S}^i_G(\pi)$ admit a non-supersingular subquotient?

Recall that an admissible $\pi$ is Cohen-Macaulay if $\mathcal{S}^i_G(\pi)$ is nonzero for a single $i$.

Q8. Are there examples of irreducible admissible supersingular representations which are not Cohen-Macaulay?

Q9. Does the AHHV classification imply a clean classification of irreducible admissible Cohen-Macaulay representations in terms of supersingular Cohen-Macaulay representations of Levi subgroups?

Next, recall that parabolic induction induces a t-exact functor $\mathrm{Ind}_P^G : D(M) \to D(G)$ which preserves (local) admissibility. This functor has a right adjoint $\mathbf{R}_G^P$ (easy) and a left adjoint $\mathbf{L}_G^P$ (Heyer). Heyer showed that $\mathbf{L}_G^P$ preserves $D^b_{\mathrm{adm}}$. It is also true that $\mathbf{R}_G^P$ preserves $D^b_{\mathrm{adm}}$, and in fact that $\mathbf{R}_G^P \cong \chi_P \otimes \mathcal{S}_M \circ \mathbf{L}_G^P \circ \mathcal{S}_G [-\dim U]$ as functors on $D^b_{\mathrm{adm}}(G)$ (H.).

Q10. Is there some a priori interplay between the functors $\mathbf{L}_G^P$ and $\mathbf{R}_G^P$ and GK dimension? Note that $\mathrm{Ind}_P^G$ increases the dimension exactly by $\dim U$.

Q11. Writing $R\mathrm{Ord}_{\overline{P}}: D^+(\mathrm{Rep}(G)_{\mathrm{l.adm}}) \to D^+(\mathrm{Rep}(M)_{\mathrm{l.adm}})$ for Emerton’s derived ordinary parts functor, is it true that $\alpha_M \circ R\mathrm{Ord}_{\overline{P}} \cong \mathbf{R}_G^P \circ \alpha_G$?

Q12. Does second adjointness hold, in the sense that $\mathbf{L}_G^P \cong \mathbf{R}_G^{\overline{P}}[\dim U] \otimes \chi_P$ as functors on $D^b_{\mathrm{adm}}(G)$? This is equivalent to asking for an isomorphism $\mathcal{S}_M \circ \mathbf{L}_G^P \cong \mathbf{L}_G^{\overline{P}} \circ \mathcal{S}_G$ as functors on $D^b_{\mathrm{adm}}(G)$. Note that second adjointness cannot hold on all of $D(G)$, because $\mathbf{R}_G^{\overline{P}}$ does not preserve arbitrary direct sums (Abe-Henniart-Vigneras), hence cannot be a left adjoint.

Q13. Does some form of the geometric lemma hold, describing $\mathbf{L}_{G}^{P'} \circ \mathrm{Ind}_{P}^{G}$ as an iterated extension of functors of the form $\mathrm{Ind}_{?}^{M'} \circ \mathrm{Weyl\,twist} \circ \mathbf{L}_M^{?}[\mathrm{shift}]$?

## p-adic Langlands is weird

Parabolic induction is one of the most basic operations in the representation theory of p-adic groups. In the classical setting of complex coefficients, parabolic induction has a left adjoint given by the Jacquet module, and also a right adjoint given – miraculously – by the (twisted) Jacquet module for the opposite parabolic. Here all three functors are exact. One then defines supercuspidal representations as those which are killed by all Jacquet module functors, or equivalently as those which don’t occur in any parabolic induction.

With mod $p$ coefficients, parabolic induction is still exact, so it easily passes to a functor on derived categories. More precisely, fix $G$ a p-adic reductive group, and let $D(G)$ be the derived category of the category of smooth $\overline{\mathbf{F}_p}$ representations of $G$. Let $P=MU \subset G$ be any parabolic subgroup. Then the usual operation of parabolic induction upgrades to a t-exact functor $\mathrm{Ind}_P^G:D(M) \to D(G)$ which preserves $D^b_{\mathrm{adm}}$. By general nonsense, $\mathrm{Ind}_P^G$ commutes with all direct sums, and hence admits a right adjoint $\mathbf{R}_{G}^{P}: D(G) \to D(M)$. Much less obviously, a recent theorem of Heyer shows that $\mathrm{Ind}_P^G$ commutes with direct products, and hence admits a left adjoint $\mathbf{L}_{G}^{P}: D(G) \to D(M)$. Moreover, Heyer also shows that $\mathbf{L}_{G}^{P}$ preserves $D^{b}_{\mathrm{adm}}$, and computes its values in some examples.

Exercise. Show that $\mathbf{R}_{G}^{P}$ restricted to $D^{b}_{\mathrm{adm}}(G)$ satisfies the isomorphism $\mathbf{R}_{G}^{P} \cong (\mathcal{S}_M \circ \mathbf{L}_{G}^{P} \circ \mathcal{S}_G)[-\dim U] \otimes \chi_P$, where $\mathcal{S}_G$ is Kohlhaase’s derived duality funtor and $\chi_P:M \to \mathbf{F}_p^\times$ is the integral modulus character. Deduce that $\mathbf{R}_{G}^{P}$ preserves $D^{b}_{\mathrm{adm}}$.

Now, if you start with an irreducible representation $\pi$ in degree zero, it is formal that $\mathbf{R}_{G}^{P}(\pi)$ resp. $\mathbf{L}_{G}^{P}(\pi)$ will be concentrated in nonnegative resp. nonpositive degrees, and $H^0$ of it is something explicit: $H^0 \mathbf{L}_{G}^{P}(\pi)$ is basically the (naive) Jacquet module, and $H^0 \mathbf{R}_{G}^{P}(\pi)$ is Emerton’s functor $\mathrm{Ord}_{\overline{P}}(\pi)$ of ordinary parts. In particular, when $\pi$ is supersingular, both of these things vanish in degree zero. But of course, they might be nonzero in other degrees, since $\mathbf{L}_{G}^{P}(\pi)$ and $\mathbf{R}_{G}^{P}(\pi)$ are not t-exact.

In the special case where $G=\mathrm{GL}_2(\mathbf{Q}_p)$ and $P=B$ is the Borel, Heyer showed that $\mathbf{L}_{G}^{B}(\pi)$ vanishes identically for any irreducible supersingular representation $\pi$, and the above exercise then implies that also $\mathbf{R}_{G}^{B}(\pi)$ vanishes identically. However, if there’s one thing we’ve learned in recent years, it’s that p-adic Langlands is only simple for $\mathrm{GL}_2(\mathbf{Q}_p)$ – for every other group, the whole story is completely different.

Theorem (Yongquan Hu). If $G=\mathrm{GL}_2(\mathbf{Q}_{p^2})$, there are plenty of irreducible admissible supersingular representations $\pi$ such that $\mathbf{L}_{G}^{B}(\pi)$ and $\mathbf{R}_{G}^{B}(\pi)$ are both nonzero!

This is actually immediate from Corollary 1.2 here and basic adjunctions.

On further reflection, it is probably true that “most” supersingular representations of a given group have the property that some $\mathbf{L}_{G}^{P}(-)$ or $\mathbf{R}_{G}^{P}(-)$ is nonzero. If you believe in some version of the mod p Langlands correspondence, this is reflected in the fact that “most” mod p Galois representations are reducible (e.g., they are Zariski-dense in the Emerton-Gee stack).

Question. Is it true that “second adjointness” holds in this setting, in the sense that $\mathbf{L}_{G}^{P} \cong \mathbf{R}_{G}^{\overline{P}}[\dim U] \otimes \chi_P$ as functors on on $D^b_{\mathrm{adm}}(G)$, or even on all of $D(G)$?

One can check by hand that this isomorphism is OK on irreps of $\mathrm{GL}_2(\mathbf{Q}_{p})$ using the calculations in Heyer’s paper and the exercise above. If this question has an affirmative answer, then $\mathbf{L}_{G}^{P}(\pi)$ is necessarily concentrated in degrees $[1-\dim U,-1]$ for any supersingular $\pi$, and similarly for $\mathbf{R}_{G}^{P}(\pi)$. In particular, in the setting of Hu’s example above, we would get that $\mathbf{L}_{G}^{B}(\pi)$ is concentrated in degree $-1$, and $\mathbf{R}_{G}^{B}(\pi)$ is concentrated in degree $1$.

## JoÃ«l BellaÃ¯che, in memoriam

I was shocked and saddened to learn that JoÃ«l BellaÃ¯che passed away last week. JoÃ«l was a wonderful mathematician, a fantastic expositor, and an ever-cheerful smiling figure in the number theory community. In this post I want to highlight some of the beautiful mathematics of JoÃ«l.

When I was writing my thesis, learning about “rigid analytic modular symbols” was rather difficult, due to a scarcity of written references. The main references were a gnomic preprint of Stevens, and a rather forbidding manuscript of Ash-Stevens. However, there was also a wonderful new paper of JoÃ«l, Critical p-adic L-functions, then available as a preprint. This paper was a total revelation: JoÃ«l simply and elegantly reconstructs the eigencurve using rigid analytic modular symbols, and gives a beautifully clear construction of p-adic L-functions varying over it, relating their behavior directly to the geometry of the eigencurve near classical points. This paper, together with JoÃ«l’s more leisurely exposition of this circle of ideas in his course notes, was a critical inspiration and reference for me for several projects. It’s only a slight exaggeration to say that my paper with JB is a student exercise, whose point is to check that JoÃ«l’s ideas generalize to the setting of Hilbert modular forms.

One of JoÃ«l’s main interests was the interplay between the local geometry of eigenvarieties, especially their (non)smoothness at classical points, p-adic L-functions, and Galois cohomology. One major effort in this direction was his pioneering and highly influential book with Chenevier, which was an instant classic and is still an invaluable reference for many different things. Also very beautiful is his paper with Chenevier, showing that the eigencurve is smooth at critical Eisenstein points. In a complementary direction, JoÃ«l showed that U(3) eigenvarieties can exhibit surprisingly unpleasant behavior near classical points, giving an example of such a point where the local ring is not even a UFD. It’s a fascinating question to understand the singularities of eigenvarieties, and I think this example of JoÃ«l’s was perhaps the first indication of the true richness of this question.

Finally, I also want to mention JoÃ«l’s wonderful notes on the Bloch-Kato conjecture. Despite the title, these notes are actually a stroll through a large portion of modern arithmetic geometry, and should be read by every graduate student.

Rest in peace.

## the March of progress

Several things to report on:

• The (hopefully) final version of HKW is done and dusted! The most significant change from the previous version is that GHW is finally done, so we can finally refer to precise results from that paper. For a quick summary of the main results from GHW, take a look at section 4.1 of HKW. The key new vocab is decent v-stacks and fine morphisms between them. Writing GHW was not good for my health (take a look at the acknowledgments…), and reading it might not be so good for yours. But if you really want to look at GHW, one thing you could do to get oriented is read Definitions 1.1 & 1.3, Theorem 1.4, and everything in section 4.1.

Bonus Question: Let $g:X \to Y$ and $f:Y \to Z$ be separated morphisms of locally spatial diamonds such that $g$ is surjective and universally open, and $f \circ g$ is compactifiable (in the sense of Definition 22.2 here). Is $f$ compactifiable? If the answer to this question is yes, then the nonsense about “strict” surjectivity in GHW could be eliminated (although it is harmless in applications, thank God). If you can settle this, or some slight weakening of it, please let me know!

Note that the numbering in HKW has changed slightly in comparison to the previous version, which I wrote about here, so the numbering in that post refers to v3 of the arxiv posting.
• In a previous post, I mentioned a bunch of roughly equivalent open problems about smooth representations of p-adic groups with coefficients in general $\mathbf{Z}[1/p]$-algebras, which I learned about from Jean-FranÃ§ois Dat. But now these problems have been more or less all solved, in a beautiful and shockingly short paper by Dat-Helm-Kurinczuk-Moss. It is quite curious that their results, which are statements in pure representation theory, depend in a crucial way on the Fargues-Scholze machinery. For more information, I can’t to any better than suggesting that you simply read their paper.

## 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 WÃ¼rthen. 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 WÃ¼rthen 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 FranÃ§ois 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 FranÃ§ois 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!