In most sources, the field of definition of the modular curve $X(n)_\mathbb{C}$ (quotient of the upper half plane by the subgroup $\Gamma(n)$ of $SL_2(\mathbb{Z})$ congruent to $I\mod n$) is $\mathbb{Q}(\mu_n)$, where $\mu_n$ is the group of $n$th roots of unity.

However, in Deligne Rapoport V.4.4 (http://publications.ias.edu/sites/default/files/Number22.pdf), they define $\mathcal{G}[1/n]$ to be the stack over $\mathbb{Z}[1/n]$ classifying generalized elliptic curves $E/S$ whose singular geometric fibers are $n$-gons, equipped with an isomorphism of group schemes $$E[n]\cong\mu_n\times\mathbb{Z}/n$$ "of determinant 1'', by which they mean that the isomorphism is compatible with the weil pairing on $E[n]$ and the alternating pairing $\Lambda^2(\mu_n\times\mathbb{Z}/n)\rightarrow\mu_n$ defined by $\alpha\wedge k\mapsto \alpha^k$ if $\alpha\in\mu_n$ and $k\in\mathbb{Z}/n$.

It seems like by requiring that the level structures have ``determinant 1'', we've singled out a component of the usual moduli space for $\Gamma(n)$-structures, and our component based changed to $\mathbb{C}$ should be isomorphic to $X(n) := \Gamma(n)\backslash\overline{\mathcal{H}}$. However, this would seem to imply that $X(n)$ is defined over $\mathbb{Q}$ (or even over $\mathbb{Z}[1/n]$), which seems to contradict the usual wisdom that $X(n)$ is only defined over $\mathbb{Q}(\mu_n)$.

In fact, on the second page of Ogg's paper "Rational Points on Certain Elliptic Modular Curves", he mentions a ``canonical model of Shimura, as communicated to me by Casselman", which is a $\mathbb{Q}$-rational model of $X(n)$, such that the cusps are defined over $\mathbb{Q}(\mu_n)$.

Am I reading Deligne-Rapoport correctly? Is $X(n)$ actually definable over $\mathbb{Q}$? (I wonder if Ogg's 'canonical model of shimura' somehow coincides with DR's point of view)

Edit: Also does anyone know what it means for a morphism of stacks to be "localement représentable" (eg, right after definition IV.3.2)? I know the translation is locally representable, but that sounds like relatively representable (fiber products with a scheme is also a scheme).

defines$\zeta_n$ via pairing of $(1,0),(0,1)$! $\endgroup$ – user74230 Jan 5 '15 at 6:53