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.


1 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 $p(p-1)$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.