# 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!