Let $p$ be a prime number and $X\_0(p)/\mathbf{Q}$ be the classical modular curve for
$\Gamma\_0(p)$. Let $\tilde{X}\_0(p)/\mathbf{Z}$ be the projective arithmetic surface corresponding to the normalization of $\mathbb{P}\_{\mathbf{Z}[j]}^1$ inside $\mathbf{Q}(X\_0(p))$. Then when one reduces $\tilde{X}\_0(p)/\mathbf{Z}$ modulo $p$ one gets two copies of $\mathbb{P}\_{\mathbf{F}\_p}^1$ with normal crossings at points corresponding to supersingular elliptic curves over $\mathbf{F}\_{p^2}$. So let $\mathbf{Z}\_{p^2}$ be the unique quadratic unramified extension over $\mathbf{Z}\_p$. Having in mind Mumford's dictionary, this begs the following question:


Q: Is $\tilde{X}\_0(p)\otimes\mathbf{C}\_p$ uniformized by some Schottky group of $PGL\_2(\mathbf{Q}\_{p^2})$ ?



Is X_0(p) a Mumford curve over $Q_{p^2}$
Yes, unless I'm missing some subtlety.


Theorem 4.20 of [Mumford's paper](http://archive.numdam.org/ARCHIVE/CM/CM_1972__24_2/CM_1972__24_2_129_0/CM_1972__24_2_129_0.pdf) reads:



> 
> Every stable curve over $S$ with nonsingular generic fiber and
>  $k$-split degenerate closed fiber is isomorphic to $P\_{\Gamma}$ for a
>  unique$^\ast$ flat Schottky group $\Gamma \subset PGL(2,K)$.
> 
> 
> 


Here $S = \mathrm{Spec}\ A$ where $A$ is a complete integrally closed noetherian local ring, $K = \mathrm{Frac}\ A$, and $k$ is the residue field of $A$. Saying a stable curve $X$ over $k$ is $k$-split degenerate means (see shortly after Defn 3.2)


1. The components of the normalization $X$ are all $\mathbb{P}^1\_k$'s.
2. The nodes of $X$ are all $k$ points.
3. Any node locally looks like $k[x,y]/(xy)$. For example, working over $\mathbb{R}$, a node of the form $\mathbb{R}[x,y]/(x^2+y^2)$ is forbidden.


You say above (and I agree) that conditions (1) and (2) are true in your setting, and condition (3) follows from the fact that your nodes always join two different components, not one component and itself.


$^\*$ Mumford must mean "unique up to conjugacy". 


