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.

Joël Bellaïche, in memoriam

I was shocked and saddened to learn that Joël Bellaïche passed away last week. Joël 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 Joël.

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 Joël, Critical p-adic L-functions, then available as a preprint. This paper was a total revelation: Joël 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 Joël’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 Joël’s ideas generalize to the setting of Hilbert modular forms.

One of Joël’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, Joël 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 Joël’s was perhaps the first indication of the true richness of this question.

Finally, I also want to mention Joël’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-François 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 Würthen. Here something quite amusing happened: in an informal conversation on Tuesday, Ben explained the complicated status of p-adic Simpon to me, and stressed that its most optimistic conjectural form isn’t actually written down, because no one wants to be the one to make a false conjecture. But then Matti Würthen did explicitly conjecture this exact statement in his Friday lecture! For the record, the hope is that for a smooth projective variety X/ \mathbf{C}_p, there is an equivalence of categories from \mathbf{C}_p-representations of \pi_1^{\mathrm{et}}(X) towards semistable Higgs bundles on X with vanishing Chern classes. As Ben stressed to me, this definitely fails if \mathbf{C}_p is replaced with a larger algebraically closed nonarchimedean field. Hmm…
  • One consequence of COVID measures is that seating for meals was not randomized as usual, but rather was fixed for the entire week. I was assigned to eat with Torsten Wedhorn, Bogdan Zavyalov, and François Loeser. This ended up being a really pleasant group to eat with! I already knew Torsten and Bogdan fairly well, but I’d never spoken with François before, and it turns out he’s a totally charming and delightful person. It was especially wonderful to hear him talk about his astonishing achievements in ultra long distance running. One memorable quote: “The first night without sleep is no problem. The second night is… interesting. And the third night… well… this I cannot recommend.”
  • My relationship with Hochschild cohomology has gone from “???” to a vague understanding and a desire to learn more. Thanks to DC for some helpful explanations!
  • There was much discussion among the younger participants about what Fargues’s categorical local Langlands conjecture should look like with mod-p coefficients (i.e. in the \ell=p setting). Of course on the spectral side, one expects to see some category of quasicoherent or ind-coherent sheaves on the special fiber of the Emerton-Gee stack for \hat{G}. On the automorphic side, one should have some category of mod-p sheaves on \mathrm{Bun}_G, and the correct category should fall out of the general formalism developed by Mann in his thesis. One tantalizing fact, sketched out during some of these conversations, is that \mathrm{Bun}_G is definitely p-cohomologically smooth (in a precise sense), not of dimension 0 as in the \ell \neq p case, but of dimension equal to the dimension of G(\mathbf{Q}_p) as a p-adic Lie group. It is surely no coincidence that this matches the expected dimension of the Emerton-Gee stack for \hat{G}.

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

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

Question for homotopical readers

I’m trying to learn some \infty-categorical stuff more seriously, and I have a vague question, which maybe a generous reader can offer some insight on. In Lurie’s books, there are a great many different types of “fibration” conditions one can impose on a map of simplicial sets, as in Remark 2.0.0.5 of HTT. The actual definitions aside, how should one “really” think about these different conditions in practice? Basically, I am looking to get some intuition. The only one of these conditions I’ve managed to get a (slight) feeling for is that of a categorical fibration.

Several things part 4

-If you need a ten minute break from whatever important thing you’re doing, and you read German, you could do worse than looking at this short essay I wrote for the 2021 Jahrbuch of the Max-Planck-Gesellschaft. Big thanks to Christian Kaiser for doing the translation!

-In an earlier post, I mentioned that I didn’t appreciate monads. Thanks to some very lucid explanations from CJ, this is no longer the case. In retrospect, two things were confusing me:
i. If A\to B is a ring map, then describing B-modules in terms of A-modules with extra structure (namely as modules for the monad given by -\otimes_A B) is the canonical example of monadic descent, and it works for any ring map, while the more familiar faithfully flat descent is an example of comonadic descent.
ii. The perverted and misleading terminology of “algebras” for a monad, used e.g. on wikipedia, for the concept which should obviously be called “modules”. At least Lurie uses the word module in his books.

Fall roundup

Apologies for the lack of blogging. This has been an unusually busy fall.

  • My student Linus Hamann has a website! Please go there and check out his beautiful preprints, especially his paper on comparing local Langlands correspondences for GSp4.
  • There have been a lot of great papers this year, but I was especially struck by these gorgeous ideas from Teruhisa Koshikawa. Readers might recall that the seminal Caraiani-Scholze papers contain a fun part (p-adic geometry of Shimura varieties and their Hodge-Tate fibers, semiperversity of Hodge-Tate pushforwards) and a not fun part (arguments with the twisted stable trace formula and Shin’s stable trace formula for Igusa varieties). Koshikawa completely eliminates the not fun part, replacing it with an extremely clever use of the Fargues-Scholze machinery. Even in the setting of the CS papers, Koshikawa’s main theorem is stronger; moreover, his technique opens the door to a wide generalization of the CS vanishing results beyond the specific unitary Shimura varieties they treated. (Note for ambitious readers: The problem of working out these generalizations has already been “taken” by specific people.)
  • Eagle-eyed readers of H.-Kaletha-Weinstein might’ve noticed that the entire paper depends crucially on a non-existent preprint cited as [GHW]. As discussed in a previous post, the point of GHW is to construct the functor Rf_! in etale cohomology for certain stacky maps of Artin v-stacks, by adapting some machinery of Liu-Zheng which they built to solve the analogous problem in the setting of Artin stacks. Since the above-mentioned papers of Hamann and Koshikawa both depend directly on HKW, and thus indirectly on GHW, I’ve felt some increased pressure recently* to actually produce this paper!
    However, I think this pressure helped push me past the final points of confusion in this project, and I’m pleased to report that after nearly 4 years of struggle, the details of GHW have finally come together. I’m cautiously optimistic that the paper will be publicly available within a few months. The arguments are an infernal mixture of delicate p-adic geometry and general \infty-categorical constructions. Actually, this is the most intense and frustrating project I’ve ever worked on. It will be good to finish it.
  • As always, David Roberts offers a voice of clarity against the nonsense burbling out from the IUT cultists.

    *Both from myself and from the referee for HKW.

An appreciation

Like many people, I’ve had rather mixed experiences with referees. On one occasion, I had a paper rejected by a referee who explicitly admitted in their report that they weren’t qualified to evaluate it.  More recently, though, my experiences have been very positive:

  • The referee for my paper with Kiran was extremely careful, and provided us with many detailed comments which (hopefully) have lead to a huge increase in the correctness and readability of the paper. Time from submission to receipt of referee report: 8 months
  • The referee for my paper with Bhargav found many small typos, as well as one or two minor inaccuracies. Time from submission to receipt of referee report: 5 1/2 months
  • The referee for my paper with Tasho and Jared found many small typos and inaccuracies, together with one more significant conceptual blunder. Time from submission to receipt of referee report: 6 weeks (!)

New versions of these papers should all be available soon. In the mean time, if any of the referees mentioned above are reading this, please accept my gratitude for your hard work!

Several things part 3

  • Here’s a suggestive hallucination, useful to me for keeping some things straight, but maybe not useful to anyone else:
    Let G be a p-adic reductive group, C an algebraically closed field of characteristic zero (or maybe just of characteristic \neq p). The set X_G of isom. classes of smooth irreducible C-representations of G(\mathbf{Q}_p) really wants to be an algebraic variety, but it’s not. However, X_G has a canonical best approximation by an (ind-)algebraic variety, the Bernstein variety Z_G. The canonical map X_G \to Z_G is “quasifinite and birational”.  The algebraic functions on X_G are given by trace forms, i.e. by functions of the form \pi \mapsto \mathrm{tr}(f| \pi) for some arbitrary f \in C_c(G,C). There is also a canonical second-best approximation of X_G by an algebraic variety, the spectral Bernstein variety Z_{G}^{\mathrm{spec}}, i.e. the coarse quotient of the stack parametrizing (G-relevant) L-parameters W_{\mathbf{Q}_p} \to \phantom{}^L G. It is second-best in the sense that there is a quasifinite map Z_G \to Z_{G}^{\mathrm{spec}}. The composite map X_G \to Z_{G}^{\mathrm{spec}} sends \pi to its semisimple L-parameter.

    Should I post more hallucinations like this?

  • I have to admit that I struggle psychologically with things related to foundations, especially subtleties arising from “big” constructions and the usual prophylactics involving universes or cutoff cardinals or whatever. For one thing, I don’t really care. But more significantly, the idea that ZFC (or something like it) should be accepted as the “standard foundations” of mathematics is absolutely revolting and nonsensical to me. The fact that everything in ZFC is a set makes it a complete non-starter for me as a reflection of how mathematics really operates. In some sense, I don’t really believe in “naked” sets.
    \phantom{}
    Anyway, I was never able to articulate my thoughts about this stuff very precisely. It was thus something of a revelation when I read this article at the Xena project, and realized that type theory is what I’ve been craving all along. I also strongly recommend this article by Todd Trimble which articulates my problems with ZFC much more eloquently than I can. (I don’t really understand ETCS yet, but it also seems like it would satisfy me.)
  • Is the twitter account @GeoMoChi08 a parody? I would dearly love to know what’s going on with this account (and with @math_jin).

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.