## Observations from Indiana

Just back from the p-adic Langlands party. Some random moments:

• Language options on the German American Banking ATM: English, Spanish.
• “We’ve encoded Kevin Buzzard’s brain into a two-variable power series!”
• Heated discussion: do you write $\overline{\mathbf{Q}}_p$ or $\overline{\mathbf{Q}_p}$?  Clearly the latter is right.
• “I’m not using adic spaces just to be hip.”
• Sean Howe asked me a great question.  Consider a pair $(G,\mu)$ with $G/\mathbf{Q}_p$ reductive and $\mu \in X_{\ast}(G_{\overline{\mathbf{Q}_p}})$ a minuscule cocharacter (defined over the reflex field $E$ of its conjugacy class), so we have an associated flag variety $\mathcal{F}\ell_{G,\mu}$ defined over $E$ as usual.  Caraiani-Scholze defined a Newton stratification of this space, with the strata $\mathcal{F}\ell^{b}_{G,\mu}$ indexed by the Kottwitz set $B(G,\mu)$.  Sean’s question is whether or not the $\mu$-ordinary stratum always coincides with the set $\mathcal{F}\ell_{G,\mu}(E)$.  I have a proof when $G$ is quasisplit (and maybe it works more generally, I haven’t checked), but it uses fancy pants things like shtukas, diamonds, etc.  Surely there is a direct argument.

Caraiani-Scholze showed that the $\mu$-ordinary stratum is always closed and generalizing; if one also knew it was zero-dimensional (which they prove in the PEL case), these things together would imply that it consists only of classical rigid analytic points.  For this last deduction, one can use the following cute fact: If $X$ is a locally tft adic space over some nonarchimedean field $K$, and $x \in X$ is any rank one point with associated residue field $K_x$, then the supremum of the ranks of all specializations of $x$ coincides with $1+\mathrm{tr.deg}(K_x /K)$. In particular, $x$ is a classical rigid point iff it has no specializations.

• Salted Butter Tea tastes exactly like, well, salted butter tea.
• Judith Ludwig gave a beautiful talk, exposing the following phenomenon:  Let $G=\mathrm{SL}_2/\mathbf{Q}$ and consider a suitable eigenvariety $X=X_{G,K^p}$ with its sheaf of p-adic automorphic forms $\mathcal{M}$. Then there exists a classical point $x \in X$ with the following two properties:
i. In some small neighborhood $U$ of $x$, classical points are dense and accumulating, and for all classical points $y \in U\smallsetminus {x}$, the fiber $\mathcal{M}_y$ contains only classical automorphic forms.
ii. The fiber $\mathcal{M}_x$ contains both classical and non-classical forms.

This cannot happen for the (cuspidal) $\mathrm{GL}_2$ eigenvariety, where $\mathcal{M}_x$ consists only of classical forms at any classical point $x$ (although this isn’t entirely obvious).

• “I have visual aids!”
• Why do people still use the Faltings site?  In hindsight, and with all due respect to Faltings, it’s obviously a failed attempt at defining the pro-etale site.  The fact that perfectoid Shimura varieties live in the pro-etale site, but not the Faltings site, of any finite-level Shimura variety under them should be reason enough for anyone working on p-adic automorphic forms to make the switch.
• The spectral halo and the ghost conjecture hovered over the proceedings in a manner entirely consistent with their names.

Thanks to Matthias Strauch and Keenan Kidwell for organizing such a great conference!