Is my algebraic space a scheme?

Question

Consider $\mathcal{M}_{1,1}$ over $\bar{\mathbb{Q}}$.

I have an algebraic stack $\mathcal{M}$ finite etale over $\mathcal{M}_{1,1}$

I can prove that it is an algebraic space (essentially because all its "hidden fundamental groups" are trivial - ie, all geometric points of $\mathcal{M}$ have trivial 2-automorphism groups)

Must it be a scheme?

Answer 1

So the answer is yes, and follows from two facts:

1. For an abelian variety $A/S$, and $\sigma\in Aut_S(X)$, if there is a point $s\in S$ such that $\sigma|_{A_s} = id$, then $\sigma = id$. (This is called "rigidity", and can be found in Mumford's Geometric Invariant Theory)

2. For a stack affine over $\mathcal{M}_{1,1}$, it is representable if and only if every object has no nontrivial automorphisms. (This is Scholie's result, theorem 4.7.0 in Katz/Mazur's Arithmetic Moduli of Elliptic Curves)

Statement (2) tells us that we just need to check that our objects have no automorphisms, and (1) tells us that it suffices to check this for elliptic curves over fields, so we're done.