## Perfectoid uniformization, part one (ed.: and part two)

Let $C$ be a complete algebraically closed extension of $\mathbf{Q}_p$, and let $X$ be a nice rigid analytic space/locally Noetherian adic space over $C$.  What should it mean for $X$ to be covered by a perfectoid space?  Here’s one possible definition:

Definition. perfectoid covering space $Y$ of $X$ is a perfectoid space $Y \in X_{\mathrm{proet}}$ with the following property:

• Locally on some covering $X = \cup U_i$ of $X$ by connected affinoids $U_i$, the map $f: Y \to X$ can be factored as $Y_i := f^{-1}(U_i) \to Z_i \to U_i$, where $Z_i \to U_i$ is an etale covering map in de Jong’s sense and $Y_i \to Z_i$ realizes $Y_i$ as a perfectoid object in $(Z_i)_{\mathrm{profet}}$.

Note that $Y_i$ has an induced structure as a perfectoid space, so the final condition makes sense. Note also that each $Z_i$ is still rigid analytic/locally Noetherian/whatever, and that each connected component of $Z_i$ surjects onto $U_i$ (this is slightly nontrivial if $X$ isn’t normal). If you don’t know about etale covering maps, you should go read the paper where de Jong defines them, “Etale fundamental groups of nonarchimedean analytic spaces” – you won’t regret it.  I should also note that the connected components of $Z_i$ need not be affinoids!  This is one of those funky twists in the nonarchimedean universe: one might think that “connected and locally finite-etale-surjective with connected affinoid target” would imply affinoid, and yet it doesn’t.

OK, what should it mean for such a covering to be Galois?

Definition. A perfectoid covering space $Y/X$ is Galois with Galois group $G:= \mathrm{Aut}(Y/X)$ if the following two conditions are true:

• The fiber of the map $Y \to X$ over any geometric point of $X$ is a $G$-torsor.
• Make $G$ into a topological group by declaring that stabilizers of open affinoid perfectoid subsets of $Y$ give a basis of open subgroups. Then $Y \times_{X} Y$ is homeomorphic to $G \times Y$.

If $X$ is qcqs then I’d guess that $G$ is always locally profinite, but I haven’t tried to check this. In any case, in the examples below $G$ will always be locally profinite and it’ll be clear what to do. For a more careful treatment of some of these things, take a look at Peter’s recent paper on the Lubin-Tate tower.

Basic example. Take $X= \mathbf{A}^1$, the rigid analytic affine line over $C$, and take $Z = (\mathrm{Spf}(\mathcal{O}_C [[T]]))_{\eta}$ to be the open unit disk over $C$. The map $Z \to X$ sending $x$ to $\log (1+x)$ is the primordial example of an etale covering map: it is of infinite degree, with connected source and target, and Galois with Galois group $\mu_{p^{\infty}}$ (which acts by sending $T$ to $\zeta (1+T)-1$).
Now, living over $Z$ we have the perfectoid covering $Y$ given by adjoining all $p$-power roots of $1+T$, i.e. $Y \sim \lim_{\leftarrow \varphi} Z$ where $\varphi$ acts via $T \mapsto (1+T)^p -1$. (Exercise: Write $Y$ as the generic fiber of a suitable $\mathrm{Spf}$.) This is Galois over $Y$ with Galois group $\mathbf{Z}_p (1)$. Putting these maps together, we get:
The map $Y \to \mathbf{A}^1$ realizes $Y$ as a perfectoid covering space of $\mathbf{A}^1$, and this covering is connected and Galois with Galois group $\mathbf{Q}_p(1) \simeq \mathbf{Q}_p$.

Next time, we’re going to prove the following theorem, which (to me at least) is very surprising, and even somewhat disturbing:

Theorem. Let $X$ be the analytic space associated with a normal projective variety over $C$. Then one can find a perfectoid covering space $\tilde{X}/X$ with the following properties:

• The covering $\tilde{X}/X$ is connected and Galois with locally profinite Galois group $G$, and $X^{\lozenge} \cong \tilde{X}^{\lozenge} / G$ as diamonds over $\mathrm{Spd}\,C$. In particular, $X$ can be recovered from the data of $\tilde{X}$ with its action of $G$.
• The space $\tilde{X}$ is quasi-Stein in the sense that it has a covering by a rising union $V_1 \subset V_2 \subset \cdots \subset V_i \subset \cdots$ of affinoid perfectoid spaces.

So every normal projective variety can be uniformized by a “contractible” perfectoid space! This is wildly divergent from the situation over $\mathbf{C}$.

Edit (8/9/2015): It turns out I’m too lazy to carefully write up a proof of this theorem right now, so here’s a quick sketch of the idea.  One treats the case of $X= \mathbf{P}^n$ directly, taking $\tilde{X}$ to be a piece of the Lubin-Tate tower at infinite level, and the claimed properties of $\tilde{X}$ follow from Jared and Peter’s explicit description of the Lubin-Tate space at infinite level; in this case, one ends up with $G= \mathrm{SL}_{n+1}(\mathbf{Q}_p)$.  The general case follows from taking a projective embedding, and using the fact that closed immersions into perfectoid spaces are perfectoid.