Let $X(N)$ and $Y(N)$ be respectively the compactified and uncompactified modular curves parametrising elliptic curves with full level $N$ structure. In other words, a point on $Y(N)$ is (essentially) a pair $(E,\phi)$ where E is an elliptic curve and $\phi$ is an isomorphism $E[N]\rightarrow (\mathbb{Z}/N\mathbb{Z})^2$.
There is an action of $GL_2(\mathbb{Z}/N\mathbb{Z})$ on $X(N)$ and $Y(N)$, given by $M \cdot (E,\phi)=(E,M\phi)$.
In this paper http://www-personal.umich.edu/~bubaran/classlast.pdf, the author lets $H$ be a subgroup of $GL_2(\mathbb{Z}/N\mathbb{Z})$, and defines a modular curve $X_H(N)$ to be $X(N)$ quotiented out by the action of $H$. She then says that, provided $det(H)=(\mathbb{Z}/N\mathbb{Z})^\times$, $X_H$ is
(a) Projective (b) Nonsingular (c) Defined over $\mathbb{Q}$
My question is why is this true? I am especially interested in the 'defined over $\mathbb{Q}$' part. Does anybody know of a reference which explains conditions under which these hold?
Also, is $Y_H(N)$ a nonsingular projective curve defined over $\mathbb{Q}$? My gut feeling is that it would be projective and defined over $\mathbb{Q}$, but not nonsingular.
The "defined over $\mathbb{Q}$" part is nicely explained in Section 1.2, page 44, of Chapter III (by David Rohrlich) in the compilation of articles "Modular Forms and Fermat's Last Theorem" (Editors: Cornell, Silverman, and Stevens).