# An example of a coarse moduli space of elliptic curves

Let $\mathcal{S}$ be the moduli stack (over $\text{Spec }\mathbb{Z}$) of elliptic curves endowed with a trivialisation of its Hodge bundle.

Classically, we know that $\mathcal{S}\otimes \mathbb{Z}[1/6]$ is represented by the affine scheme $$U=\text{Spec } \mathbb{Z}\left[\frac{1}{6},g_2,g_3,\frac{1}{g_2^3 - 27g_3^2}\right]$$ One can show that it is not representable over $\mathbb{Z}$, so I was wondering what is the coarse moduli space $S$, over $\text{Spec }\mathbb{Z}$, of $\mathcal{S}$?

First of all, can we say something by a priori general reasoning? For instance, do we know a priori that $S$ is a scheme? Affine? etc.

In fact, I have a candidate for $S$: $$S = \text{Spec }\frac{\mathbb{Z}[c_4,c_6,\Delta^{\pm 1}]}{(c_4^3-c_6^2 - 1728\Delta)}$$ It is a localisation of the ring of modular forms over $\mathbb{Z}$. It is easy to define a natural map $\mathcal{S}\to S$ and I think it is ok to show that $\mathcal{S}(k) \to S(k)$ is a bijection whenever $k$ is an algebraically closed field. But I get confused when trying to prove the universal property.

• In Chapter 7 of "Geometric Invariant Theory", Mumford constructs the coarse moduli space of principally polarized Abelian varieties as a scheme over $\text{Spec}(\mathbb{Z})$. He proves that, Zariski locally over $\text{Spec}(\mathbb{Z})$, this scheme is quasi-projective. It is true (and he notes this), that the scheme is globally quasi-projective. In the $1$-dimensional case, it is straightforward to prove that the scheme is affine by working Zariski locally over $\text{Spec}(\mathbb{Z})$ and using Hartshorne, Exercise II.2.17. – Jason Starr Feb 17 '16 at 9:59
• I'm not sure if I understood your point. So, this proves that the coarse moduli space $M$ associated to the moduli stack $\mathcal{M}$ of elliptic curves is affine. It is easy to show that $\mathcal{S} \to \mathcal{M}$ is representable and affine. Does this already imply that $S$ is an affine scheme? – Docteur Cottard Feb 17 '16 at 10:49
• I suggest you look at Chapter 7 of Mumford's book. – Jason Starr Feb 17 '16 at 11:33
• Ok, thanks, I think I get it now. By the same methods in Mumford's book one should be able to prove that the coarse moduli exists and is an affine scheme. In this case, I think it is easy to show that it is necessarily isomorphic to $\text{Spec }\Gamma(\mathcal{S},\mathcal{O}_{\mathcal{S}})$ and this should give the explict description I guessed above. – Docteur Cottard Feb 20 '16 at 22:30
• The coarse moduli look exactly like you say. A proof is analogous to the proof for the coarse moduli of the moduli of elliptic curves itself, whch was discussed in mathoverflow.net/questions/199008/… – Lennart Meier Feb 21 '16 at 2:16