Goren and Kassaei's paper "The Canonical Subgroup: a Subgroup-Free Approach" takes the position that the canonical subgroup of order $p$ for elliptic curves over $\mathbb{Z}_p$ with $\Gamma$-level structure that are "not too supersingular" can simply be viewed as a partially-defined section of the forgetful map $X(\Gamma_0(p) \cap \Gamma) \to X(\Gamma)$. (Passing to the congruence subgroup $\Gamma_0(p) \cap \Gamma$ just adds the extra information of an order $p$ subgroup, assuming $\Gamma$ has no level structure at $p$.)

My question is, what does this map look like? For simplicity, assume $\Gamma$ gives no level structure at $p$. The section as described above is a function, defined over $X(\Gamma)$ minus some disks, to $X(\Gamma_0(p) \cap \Gamma)$; it should be injective, so that it's like including this portion of $X(\Gamma)$ as a subspace of $X(\Gamma_0(p) \cap \Gamma)$. Can there be any of these missing disks where we could extend the section to the entirety of that area? Does it look like we just glue more $g$-holed tori to $X(\Gamma) \setminus \{\text{disks}\}$ along the edge of the deleted disks?

For a more explicit picture, assume that $X(\Gamma)$ is just a torus, and its supersingular locus is a single disk. Could $X(\Gamma_0(p) \cap \Gamma)$ also be a torus, so that the section given by the canonical subgroup could be extended to all of $X(\Gamma)$? Or would it have to be a higher genus surface to give some obstruction to extending it? Is there any restriction on the genus of $X(\Gamma_0(p) \cap \Gamma)$? It seems like just looking at the map to $X(\Gamma)$ means that it would have to be a genus $2$ surface. If the supersingular locus were two disks, would this restriction on the genus disappear?

EDIT: When we look at the canonical subgroup of order $p^n$ (which should be cyclic?), each canonical subgroup of order $p^{<n}$ lives inside of it, giving us a similar section $$X(\Gamma) \to X(\Gamma_0(p) \cap \Gamma) \to \dots \to X(\Gamma_0(p^n) \cap \Gamma).$$ Here the maps are again only defined outside of some (larger) disks in $X(\Gamma)$. I can ask similar questions about the geometry of each section $\to X(\Gamma_0(p^n)$. Are there any interesting pictures that can be drawn here?


1 Answer 1


Does it look like we just glue more g-holed tori to X(Γ)∖{disks} along the edge of the deleted disks?

In a nutshell, yes. The way to think about $X(\Gamma \cap \Gamma_0(p))$ is the following.

  • Take 2 copies of the (p-adic) modular curve $X(\Gamma)$.
  • Remove the supersingular residue discs from each copy.
  • Use a bit of rubber tubing to glue each supersingular "hole" in one copy to its partner in the other copy.

The result is $X(\Gamma \cap \Gamma_0(p))$.

In particular, this tells you that genus of the new curve is $g(\Gamma \cap \Gamma_0(p)) = 2 g(\Gamma) + n - 1$, where $n$ is the number of supersingular residue discs. So it's extremely rare for $X(\Gamma)$ and $X(\Gamma \cap \Gamma_0(p))$ to have the same genus -- this can only happen if $g(\Gamma) = 0$ and $p \in \{2, 3, 5, 7, 13\}$ (the primes such that there is only one supersingular j-invariant). Otherwise the curve "upstairs" has much larger genus than the one "downstairs".

What I've just described is a cartoon version of the main result of Deligne--Rapoport, which states that there is an integral model of $X(\Gamma \cap \Gamma_0(p))$ whose special fibre is the union of two copies of $X(\Gamma)$ intersecting transversely at the supersingular residue discs. The geometry is much more complicated for level $p^n$ with $n \ge 2$ (but the $n = 1$ case is already fun and interesting to think about).

I strongly recommend Matt Emerton's survey article "An introduction to the p-adic geometry of modular curves" if you want to get your head around these things.


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