A twisted version of the modular curve $X_1(N)$ The modular curve $Y_1(N)$ is known to parametrize pairs $(E,P)$ where $E$ is an elliptic curve and $P$ is a point in $E$ of exact order $N$ (at least if $N$ is invertible in the base; if not, one can use an approach suggested by Drinfeld, as seen Chapter 1 in the famous book by Katz and Mazur). It has a natural compatification $X_1(N)$.
I am interested in a "twisted version" of $Y_1(N)$, parametrizing pairs $(E,f)$, where $E$ is an elliptic curve and $f:\mu_N\to E$ is a monomorphism of groups schemes, with a map of such pairs $(E,f)$ must restrict to the identity on $\mu_N$ (if $N$ is invertible in the base). If the base contains the $N$-roots of unity, then we have an isomorphim between such curves (fixing an isomorphism between $\mu_N$ and $\mathbb{Z}/N\mathbb{Z}$).

Do you know a place where this curve has been studied? For example, it is known it's rational points over $\mathbb{Q}$?


What could be the definition of such curve if $N$ is not invertible in the base (using Drinfeld approach, for example, as in Katz-Mazur)?

 A: Aha, this is one of my favourites in the "subtle issues that people often overlook" categories :-)
The point is that this modular curve, let's call it $Y_\mu(N)$ (classifying pairs $(E, \mu_N \hookrightarrow E)$) is canonically isomorphic to $Y_1(N)$ (classifying embeddings of the constant group scheme): given $\alpha: \mu_N \hookrightarrow E$ you can consider the Cartier dual of $\alpha$ as a map $\alpha^\vee: (\mathbf{Z}/N) \hookrightarrow E'$, where $E' = E / \operatorname{image}(\mu)$. So $(E', \alpha^\vee)$ is a point of $Y_1(N)$. Similarly, there's a map going the other way. So there are canonical maps
$$Y_1(N) \to Y_\mu(N)$$
and
$$Y_\mu(N) \to Y_1(N)$$
whose composite is the identity; and this works over pretty much any base ring where you can make sense of the objects involved.
However, the above maps don't commute with the standard complex uniformisations of both $Y_1(N)$ and $Y_\mu(N)$ by the upper half-plane $\mathcal{H}$ modulo $\Gamma_1(N)$. (Actually, the map between them is precisely the Atkin--Lehner involution on the complex points, $z \mapsto -1/Nz$.) So these curves are "the same" as abstract curves; e.g. if $Y_1(N)$ has $\mathbf{Q}$-points, then so does $Y_\mu(N)$, but they aren't given by the images of the same points of $\mathcal{H}$.
On the other hand, if you work over a $\mathbf{Z}[1/N, \zeta_N]$-algebra (such as $\mathbf{C}$), you can identify $\mu_N$ with $(\mathbf{Z}/N)$ as group schemes; this gives you a second isomorphism between the two curves, which does commute with the complex uniformisation -- but is not defined over $\mathbf{Z}[1/N]$.
So these two objects are "nearly the same" in two different but incompatible ways: you can have an isomorphism between them that respects the Galois action, or one that respects the complex uniformisation, but not both. This causes no end of headaches (because a lot of authors are so sure they know which is the 'right' model of $Y_1(N)$ that they don't even bother to specify which one they're using).
