## What does a general proper rigid space look like?

As the title says. Consider proper rigid spaces $X$ over some nonarchimedean field $K$. The “standard” examples of such things which don’t come from algebraic geometry are i) the Hopf surface $(\mathbf{A}^2 - 0)/p^\mathbf{Z}$, ii) non-algebraizable deformations of K3 surfaces over the residue field of $K$, and iii) generic abeloid varieties (which are analogous to generic compact complex tori).  But there must be gazillions of other examples, right? A “random” proper rigid space is hard to write down, sort of by definition. But there are certainly some natural questions one can ask:

-For every $n \geq 2$, does there exist a proper $n$-dimensional rigid space with no non-constant meromorphic functions, and admitting a formal model whose special fiber has components of general type? Can we find examples of such spaces with arbitrarily large dimension dimension which don’t come from lower-dimensional examples by simple operations (products, quotients by finite groups, etc.)? Same question but with “no non-constant meromorphic functions” replaced by the weaker requirement that $\mathrm{tr.deg}K(X)/K$ is small compared to $\dim X$.

-Do there exist non-algebraizable proper rigid spaces with “arbitrarily bad” singularities?

-Do there exist rigid analytic analogues of Kodaira’s class VII0 surfaces?

## Two questions and a story

$\bullet$ Let $f$ be some cuspidal Hecke eigenform, with associated Galois representation $\rho_{f}:G_{\mathbf{Q}}\to \mathrm{GL}_2(\overline{\mathbf{Q}_p})$. A notorious conjecture of Greenberg asserts that if $\rho_{f}|G_{\mathbf{Q}_p}$ is abelian (i.e. is a direct sum of characters), then $f$ is a CM form, or equivalently $\rho_f$ is induced from a character. At some point I was talking about this with Barry Mazur, and he suggested a possible generalization:

Let $\rho:G_{\mathbf{Q}} \to \mathrm{GL}_n(\overline{\mathbf{Q}_p})$ be an irreducible geometric Galois representation. Suppose that $\rho|G_{\mathbf{Q}_p}$ is a direct sum of characters. Is $\rho$ induced from a character?

Emerton found a nice argument which proves Greenberg’s conjecture conditionally on some kind of $p$-adic variational Hodge conjecture. Is there any similar evidence for this question in some higher-dimensional cases?

$\bullet$ Do separated etale maps of schemes satisfy effective descent with respect to fpqc covers? This is known if one restricts to quasi-compact separated etale maps. An analogous result is true for perfectoid spaces.

$\bullet$ Here’s a funny story I heard from Glenn Stevens a while back:

At some point in the early ’90s, before he announced his proof of Fermat, Wiles came to Boston and gave a seminar talk at BU. He spoke about what is now known as the Greenberg-Wiles duality formula. However, he didn’t mention his main motivations for this formula. The upshot is that Stevens came away from the talk with the sad feeling that Wiles had lost his touch.