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I see everywhere the following:

Let $B$ be an indefinite quaternion algebra over $\mathbb{Q}$ of discriminant $D$, $\mathcal{O}_B$ be a maximal order, $N$ be an positive integer coprime to $D$.

Consider the moduli problem: for any scheme $S$, let $F(S)=\{(A,\iota,\alpha)\}$, where $A$ is an abelian surface over $S$, $\iota:\mathcal{O}\to End_S(A)$, and $\alpha$ is a level $N$ structure on $A$. Then $F$ is represented by a Deligne-Mumford stack.

However, I cannot find a reference which actually proves that $F$ satisfies those axioms that one uses to a stack.

Thanks a lot.

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    $\begingroup$ You mean to say "$F$ is a Deligne-Mumford stack" (over $\mathbf{Z}[1/DN]$). The data determines a canonical (principal) polarization on $A$, ensuring effective descent and allowing one to build the stack in terms of relatively representable Hom-functors over the DM-stack $\mathscr{A}_{g,d,N}$ (a stack which is better-documented in the literature!) for $g=2$ and $d=1$. Locating a published construction of this polarization seems to defeat Google, but such references are given in the first paragraph of page 3 of wwwf.imperial.ac.uk/~buzzard/maths/research/papers/shimura.pdf $\endgroup$
    – grghxy
    Commented Aug 13, 2015 at 15:16

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