## 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!

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## Report from Oberwolfach

Just returned from a workshop on “Arithmetic of Shimura varieties” at Oberwolfach. Some scattered recollections:

• Gabber wasn’t there, but there were some Gabberesque moments anyway. In particular, during Xuhua He’s talk, Goertz observed that a point is an example of a Deligne-Lusztig variety, so any variety is a union of Deligne-Lusztig varieties. Gotta be careful…
• The food was about the same as usual. Worst Prize was tied between two dishes: a depressing vegetable soup which somehow managed to be flavorless and bitter simultaneously, and a dessert which looked like a lovely innocent custard but tasted like balsamic vinegar. The best dishes were all traditional German fare.
• Best Talks (in no particular order): Jean-Stefan Koskivirta, Miaofen Chen, Ben Howard, Timo Richarz.
• Apparently this paper can be boiled down to a page or two.
• There was (not surprisingly) some late-night discussion of the Stanford Mystery. [Redacted] proposed a theory so mind-bogglingly outrageous that it certainly won’t fit in this margin.
• “Fun was never really my goal.” – A representative UChicago alum.
• On Thursday it snowed, and a snowball fight broke out after dinner. This was a lot of fun, but I’m still glad we didn’t follow Pilloni’s suggestion of a match between Team Europe (Pilloni, Stroh, Morel, Anschutz, Richarz, Mihatsch, etc.) and Team USA (me).
• Here’s an innocent problem which turns out to be pretty tricky. Let $X$ be a (separated, smooth) rigid analytic space over $\mathbf{Q}_p$, and let $Y \to X$ be a map from a perfectoid space which is a $\underline{G}$-torsor for some profinite group $G$. In shorthand, you should think that $X =" Y/G$ with G acting freely (this is all literally true in the category of diamonds). It’s easy to cook up examples of this scenario: for instance, you can take $X=\mathrm{Spa}\mathbf{C}_p \left\langle T^{\pm 1} \right\rangle$ and $Y=\mathrm{Spa}\mathbf{C}_p \left\langle T^{\pm 1/p^\infty} \right\rangle$, so then $Y \to X$ is a torsor for the group $\mathbf{Z}_p$. However, there are also much more complicated examples which arise in nature. In particular, if $X$ is a Rapoport-Zink space or abelian-type Shimura variety at some finite level, and $Y$ is the associated infinite level perfectoid guy over it, then we’re in the situation above, with $G$ open in the $\mathbf{Q}_p$-points of some auxiliary reductive group.

Anyway, supposing we’re in the situation above, we can ask the following complementary questions:
Q1. Suppose that $Y$ is affinoid perfectoid. Does this imply that $X$ is an affinoid rigid space?
Q2. Suppose that $X$ is an affinoid rigid space. Does this imply that $Y$ is affinoid perfectoid?

It seems like both of these questions are actually really hard! For Q1, we can (by assumption) write $Y=\mathrm{Spa}(A,A^+)$ for some perfectoid Tate-Huber pair $(A,A^+)$, and then one might guess that $X$ coincides with $X'=\mathrm{Spa}(A^G,A^{+G})$. There is certainly a map $X \to X'$, but now one is faced with the problem of showing that $A^G$ is “big enough” for this map to be an isomorphism. This can be reduced to any one of a handful of auxiliary problems, but they all seem hard (at least to me). For instance, as a warmup one could try to prove either of the following implications:

W1. Under the hypothesis of Q1, $H^1(X,\mathcal{O}_X)$ vanishes.
W2. Under the hypothesis of Q1, $H^1(X,\mathcal{O}_{X}^+)$ is killed by a fixed power of $p.$

Both of these conclusions would certainly hold if we already knew that $X$ was affinoid: the first is just (a consequence of) Tate acyclicity, while the fact that $H^1(X,\mathcal{O}_{X}^+)$ is killed by some power of $p$ for smooth affinoids is a non-trivial theorem of Bartenwerfer. But I have totally failed to prove either W1 or W2.

In any case, the essential point with Q1 seems to be the following. If $H$ is some open subgroup, then $(A^+/p^n)^H$ will always have plenty of elements, and indeed taking the direct limit as $H$ shrinks recovers $A^+/p^n$. But the obstruction to lifting an element of $(A^+/p^n)^H$ to an element of $(A^+)^H$ is a torsion class in $H^1(H,A^+)$, and the latter group seems hard to control.

For Q2, there is maybe a slightly clearer path through the forest: it would follow from the following conjecture, which I explained during my talk in the workshop.

To set things up, let $(A,A^+)$ be any uniform Tate-Huber pair over $(\mathbf{Q}_p,\mathbf{Z}_p)$, and let $X=\mathrm{Spa}(A,A^+)$ be the associated pre-adic space. Let $X_v$ denote the site given by perfectoid spaces over $X$ with covers given by v-covers, and let $\mathcal{O}$ and $\mathcal{O}^+$ be the obvious structure sheaves on $X_v$. Set $\breve{A}^+ = H^0(X_v,\mathcal{O}^+)$ and $\breve{A} = \breve{A}^+ [1/p] = H^0(X_v,\mathcal{O})$, so the association $(A,A^+) \mapsto (\breve{A}, \breve{A}^+)$ is an endofunctor on the category of uniform Tate-Huber pairs over $\mathbf{Q}_p$. One can check that breve’ing twice is the same as breve’ing once, and that the natural map $(A,A^+) \to (\breve{A},\breve{A}^+)$ induces an isomorphism of diamonds. If $A$ is a smooth (or just seminormal) affinoid $K$-algebra for some $K/\mathbf{Q}_p$, or if $A$ is perfectoid, then breve’ing doesn’t change $A$.

Conjecture. Let $(A,A^+)$ be a uniform Tate-Huber pair over $\mathbf{Q}_p$ such that every completed residue field of $\mathrm{Spa}(A,A^+)$ is a perfectoid field. Then $\breve{A}$ is a perfectoid Tate ring.

Aside from disposing of Q2, this conjecture would settle another notorious problem: it would imply that if $A$ is a uniform sheafy Huber ring and $\mathrm{Spa}(A,A^+)$ is a perfectoid space, then $A$ is actually perfectoid.

It may be instructive to see an example of a non-perfectoid (uniform) Tate ring which satisfies the hypothesis of this conjecture. To make an example (with $p>2$), set $A=\mathbf{C}_p \left\langle T^{1/p^\infty} \right\rangle$, and let $B=A[\sqrt{T}]$ with the obvious topology. Set $C=\mathbf{C}_p \left\langle T^{1/2p^\infty} \right\rangle$, so there are natural maps $A \to B \to C$. Then $A$ and $C$ are perfectoid, but $B$ isn’t: the requisite $p$-power roots of $\sqrt{T}$ mod $p$ don’t exist. Nevertheless, every completed residue field of $B$ is perfectoid (exercise!), and the map $B \to C$ induces an isomorphism $\breve B \cong C$.

OK, this bullet point turned out pretty long, but these things have been in my head for the last couple months and it feels good to let them out. Besides, Yoichi Mieda asked me about Q1 during the workshop, so despite the technical nature of these questions, I might not be the only one who cares.

• Oberwolfach continues to be one of the best places in the world to do mathematics.

Thanks to the organizers for putting together such an excellent week!