# What is the modular curve for level 1, 2?

An elliptic curve over a scheme $$S$$ is the data of a proper smooth morphism of schemes $$\pi: E\rightarrow S$$ whose geometric fibres are connected curves of genus $$1$$ and a section $$e: S \rightarrow E$$.

A level $$\Gamma(N)$$–structure on an elliptic curve $$E$$ over a scheme $$S$$ is an isomorphism $$E[N] \rightarrow (\mathbb{Z}/N\mathbb{Z})_S\times_S(\mathbb{Z}/N\mathbb{Z})_S$$ of group schemes over $$S$$.

Let $$R$$ be an integral domain. The category $$R-Ell$$ is the category whose objects are elliptic curves $$E\rightarrow S$$ with $$S$$ an $$R$$-scheme and whose morphisms are cartesian square diagrams of $$R$$-morphisms.

$$\Gamma(N)$$ is a functor $$R-Ell^{op}\rightarrow Set$$ sending an elliptic curve to the set of level $$\Gamma(N)$$-structures on it.

For $$N\geq 3$$ and $$R=\mathbb{Z}[1/N]$$, results of Katz and Mazur imply that $$\Gamma(N)$$ is representable by an elliptic curve whose base is a smooth affine $$\mathbb{Z}[1/n]$$-curve.

What I am asking is for $$N=1, 2$$ is the moduli problem representable by something geometric (algebraic space, DM stack etc.)? Is there a reference establishing its good properties? I find navigating the literature difficult.

• The map $\Gamma(p) \to \Gamma(1)$ is etale away from $p$, which makes it not very hard to check that $\Gamma(1)$ has an etale surjective cover by a scheme and is a DM stack. – Will Sawin Aug 2 '19 at 17:28
• @WillSawin thanks. – Srakotan Aug 2 '19 at 17:35
• As Katz and Mazur point out in the introduction to their book, their whole approach amounts to using stacks without explicitly admitting it. This question has some references which explain the material from a more explicitly "stacky" viewpoint: mathoverflow.net/questions/309265/… – David Loeffler Aug 3 '19 at 7:46