TITLE: A reference for $\mathbb{A}^1_R$ being a coarse moduli space of the stack of elliptic curves
QUESTION [13 upvotes]: Let $R$ be a ring and let $\mathcal{M}_{1, R}$ be the algebraic $R$-stack of elliptic curves (over $R$-schemes as bases). One knows that the coarse moduli space of $\mathcal{M}_{1, R}$ is supposed to be $\mathcal{M}_{1, R} \rightarrow \mathbb{A}^1_R$, with the map being given by the $j$-invariant. Is there a reference for this fact for an arbitrary $R$? I know that Deligne and Rapoport treat the case $R = \mathbb{Z}$ in Theorem VI.1.1 of their article, but they note that even this case is slightly painful to prove directly. The general case does not seem to follow formally because the formation of a coarse moduli space does not commute with non-flat base change.
For completeness, I recall the definition of the coarse moduli space $M$ of an algebraic $R$-stack $\mathcal{M}$ as is given in Definition I.8.1 of Deligne-Rapoport: $M$ is an $R$-algebraic space equipped with a morphism $\mathcal{M} \rightarrow M$ that induces a bijection on (isomorphism classes of objects of) $\overline{k}$-points for every algebraically closed $R$-field $\overline{k}$ and that is initial among morphisms from $\mathcal{M}$ to $R$-algebraic spaces.

REPLY [3 votes]: Here are some modern references: [1, Theorem 13.1.15] and [2, Lemma 2.1]
[1] Olsson, "Algebraic Spaces and Stacks", Colloquium Publications 62, AMS (2016)
[2] Fulton, Olsson, "The Picard group of $\mathscr{M}_{1,1}$", Algebra & Number Theory, vol. 4, no. 1 (2010) link