Let $\mathscr{M}\_{1,1,\mathbb{Z}}$ denote the moduli stack of elliptic curves.



> 
> Does there exist a scheme $X$ and a finite group $G$ acting on $X$ such that $\mathscr{M}\_{1,1,\mathbb{Z}}$ is isomorphic to the quotient stack $[X/G]$?
> 
> 
> 


Remarks/thoughts: For any scheme $S$, set $\mathscr{M}\_{1,1,S} := \mathscr{M}\_{1,1,\mathbb{Z}} \times\_{\operatorname{Spec}\mathbb{Z}} S$.


1. We have $\mathscr{M}\_{1,1,\mathbb{Z}} \simeq [W/H]$ where $W = \operatorname{Spec} \mathbb{Z}[a\_{1},a\_{2},a\_{3},a\_{4},a\_{6},\Delta^{-1}]$ where $\Delta \in \mathbb{Z}[a\_{1},a\_{2},a\_{3},a\_{4},a\_{6}]$ is the discriminant and $H$ is a subgroup scheme of $\mathrm{GL}\_{3,\mathbb{Z}}$ of relative dimension 4 over $\mathbb{Z}$, see [1, [Tag 072S](http://stacks.math.columbia.edu/tag/072S)] and [6, Section 3].
2. By [3, 4.7.2], for $N \ge 3$, the restrictions $\mathscr{M}\_{1,1,\mathbb{Z}[\frac{1}{N}]}$ are isomorphic to $[Y(N)/\mathrm{GL}\_{2}(\mathbb{Z}/N)]$ where $Y(N) \to \mathbb{Z}[\frac{1}{N}]$ is a smooth affine morphism of relative dimension 1. (Thus there is an affine open covering of $\mathbb{Z}$ on which the restriction of $\mathscr{M}\_{1,1,\mathbb{Z}}$ is a quotient stack by a finite group.) For $N=2$, see Remark 2.8 and the following paragraph of [4] and also Section 4 of [5].
3. Such scheme $X$ would have to be affine and smooth over $\mathbb{Z}$ of relative dimension 1. (Reason why $X$ is affine: Fix $N \ge 3$ and set $T\_{N}' := Y(N) \times\_{\mathscr{M}\_{1,1,\mathbb{Z}[\frac{1}{N}]}} X[\frac{1}{N}]$; then $T\_{N}' \to Y(N)$ is a $G$-torsor, hence $T\_{N}'$ is affine; moreover $T\_{N}' \to X[\frac{1}{N}]$ is a $\mathrm{GL}\_{2}(\mathbb{Z}/N)$-torsor so $X[\frac{1}{N}]$ is affine; thus $X \to \mathbb{Z}$ is Zariski-locally on the target an affine morphism; thus $X$ is affine.)
4. Such scheme $X$ cannot have a $\mathbb{Z}$-point (otherwise $\mathscr{M}\_{1,1,\mathbb{Z}}$ itself has a $\mathbb{Z}$-point, but there are no elliptic curves over $\mathbb{Z}$).


References:


[1] Stacks Project


[2] Olsson, "Algebraic Spaces and Stacks", Colloquium Publications 62, AMS (2016)


[3] Katz, Mazur, "Arithmetic Moduli of Elliptic Curves", volume 108 of Annals of Mathematics Studies, Princeton University Press, Princeton, NJ, 1985


[4] Conrad, "Isogenies and level structures", Notes for Stanford Math 248B, [link](http://math.stanford.edu/~conrad/248BPage/handouts/level.pdf)


[5] Antieau, Meier, "The Brauer group of the moduli stack of elliptic curves", [arxiv](https://arxiv.org/abs/1608.00851)


[6] Fulton, Olsson, "The Picard group of $\mathscr{M}\_{1,1}$", Algebra & Number Theory, vol. 4, no. 1 (2010) [link](https://math.berkeley.edu/~molsson/Picard1.pdf)



Is $\mathscr{M}_{1,1,\mathbb{Z}}$ isomorphic to a quotient stack by a finite group?
I guess I'll post my comments as an answer.


The definition of the quotient stack makes $p : X\rightarrow [X/G]$ into a $G$-torsor (in whatever topology $\mathcal{T}$ one chooses). Since here we're working with a finite abstract group, $X\rightarrow[X/G] = \mathcal{M}\_{1,1,\mathbb{Z}}$ is $\mathcal{T}$-locally isomorphic to a disjoint union of $|G|$ copies of $\mathcal{M}\_{1,1,\mathbb{Z}}$, and hence since being finite etale is local on the target for pretty much any choice of topology $\mathcal{T}$, we find that $p$ is finite etale.


However, since [the fundamental group of $\mathcal{M}\_{1,1,\mathbb{Z}}$ is trivial](https://mathoverflow.net/questions/105047/fundamental-group-of-the-moduli-stack-of-elliptic-curves?rq=1), $X$ is itself a disjoint union of copies of $\mathcal{M}\_{1,1,\mathbb{Z}}$, and hence cannot be a scheme.


EDIT: Actually, it seems all this also follows from [Chapter 6 of LMB's book](http://stacks.math.columbia.edu/tag/04UZ) (again using the triviality of $\pi\_1(\mathcal{M}\_{1,1,\mathbb{Z}})$).


