What objects do the cusps of Modular curve classify?

Question

We know that a modular curve has a moduli interpretation, i.e, it classify elliptic curves with additional structure. But after we add cusps on it, is the cusps also has a moduli interpretation? What objects do the cusps classify?

Answer 1

If you just want a classical picture over the complex numbers, the objects lying over the cusp points are Néron polygons equipped with some extra structure. To make a Néron $n$-gon, you take $\mathbb{Z}/n\mathbb{Z} \times \mathbb{P}^1$ ($n$-copies of the 2-sphere), and glue the components into a circular chain, with 0 in one sphere glued transversely to $\infty$ in the next. The free action of $\mathbb{Z}/n\mathbb{Z}$ is part of the structure of the $n$-gon.

The nature of the extra structure depends on the moduli problem describing the modular curve. Here are some examples:

• A Néron 1-gon is a nodal cubic curve, so the object lying over the cusp of $X(1)$ is a nodal projective line equipped with a point in the smooth locus. Silverman gives a description of an analytic neighborhood of the cusp in his Advanced Topics book, in the section on the Tate curve. There is a more formal description in chapter 8 of Katz-Mazur.

• For $X_0(N)$, the smooth locus parametrizes elliptic curves equipped with a cyclic subgroup of order $N$, and for each $m|N$ one has cusps that parametrize $m$-gons equipped with a distinguished point (the identity element) and a cyclic subgroup of order $N$ in the smooth locus.

• For $X_1(N)$, the smooth locus parametrizes elliptic curves equipped with a point of order $N$, and for each $m|N$ one has cusps that parametrize $m$-gons equipped with a distinguished point (the identity element) and point of order $N$ in the smooth locus.

• For $X(N)$, the smooth locus parametrizes elliptic curves equipped with an isomorphism from $(\mathbb{Z}/N\mathbb{Z})^2$ to the $N$-torsion, and for each $m|N$ one has cusps that parametrize $mN$-gons equipped with an isomorphism from $(\mathbb{Z}/N\mathbb{Z})^2$ to the $N$-torsion.

For each of these moduli problems, there are explicit formulas for the precise number of cusps for each $m$. For $X_0(N)$, you can find it in Shimura, section 1.6. For the others I think it isn't too hard to derive.

As others mentioned, Deligne-Rapoport's compactified moduli problems are described using the notion of generalized elliptic curve, which is a flat proper family of connected curves of arithmetic genus 1 and at most nodal singularities, equipped with a distinguished section and a "group law". Unlike the smooth case, the group law is a necessary extra datum here because we wouldn't have a uniquely defined $\mathbb{Z}/n\mathbb{Z}$-action on the $n$-gon over a cusp without it.

Answer 2

Yes, the moduli problem extends to the cusps by way of generalized elliptic curves, i.e., certain semistable curves of arithmetic genus one. For instance, with no level structure there is one point added in compactifying the $j$-line and this corresponds to a nodal cubic curve.

The basic ideas are pretty easy; the technical details less so. For a thorough modern treatment, I recommend this paper of Brian Conrad.