$X_{\text{sp}}(N)$ (awaiting review)

$X_{\text{sp}}(N)$ is the modular curve for the subgroup $H\le \GL_2(\widehat\Z)$ given by the inverse image of a Cartan subgroup $\begin{pmatrix} * & 0\\ 0& * \end{pmatrix}$ that is split at every prime dividing $N$. As a moduli space it parameterizes triples $(E,C,D)$ where $E$ is an elliptic curve over $k$, and $C$ and $D$ are $\Gal_k$-stable cyclic subgroups such that $E[N](\overline{k})\simeq C \oplus D$.