# Geometry of the section $X_0(N) \to X_0(pN)$ given by the canonical subgroup

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 $$p^{ lives inside of it, giving us a similar section $$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?

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.