Special fiber of $X(p)$ in characteristic $p$

Question

Let $p \geq 5$ be a prime. Let $Y(p)$ be the fine moduli space representing elliptic curves + basis of the $p$-torsion over $\mathbb{Q}_p$ and let $Y_0(p)$ be the fine moduli space representing elliptic curves + point of $p$-torsion over $\mathbb{Q}_p$. We know that $Y_0(p)$ has a model over $\mathbb{Z}_p$ whose special fiber is a union of two copies of $Y_0(1) := \mathbb{P}^1$ meeting transversaly at supersingular points.

My question : what is the analogous description of the special fiber for $Y(p)$ (for some suitable model over $\mathbb{Z}_p$) ? What is the explicit description of the natural map $Y(p) \rightarrow Y_0(p)$ at the level of special fibers ?

I guess it's in Katz--Mazur, but the purpose of my question is to get an answer as self-contained as possible. I don't (necessarily) ask for the idea of the proof.

Answer 1

A bit of mastication of Katz-Mazur Theorem 13.7.6 and the surrounding text seems to yield the following description of the special fiber of $Y(p)$:

1. It is fundamentally $p+1$ copies of $\mathbb{P}^1$ (each with a nonempty finite set of punctures corresponding to cusps) all glued together at supersingular points.

2. The completed local ring at a $k$-rational supersingular point is isomorphic to $k[[x,y]]/(y\prod_{i=0}^{p-1}(x-iy))$, i.e., we have $p+1$ curves coming together at all possible $\mathbb{F}_p$-valued slopes.

3. The map to $Y_1(p)$ is $p$-to-1 on one component, and 1-to-1 on the other. This is because the map to $Y(1)$ has degree $p(p-1)$ on each component, while $Y_1(p)$ has a component of degree $p-1$ and a component of degree $p(p-1)$.