Modular cover (awaiting review)

Each inclusion of open subgroups $H\leq G$ of $\GL_2(\widehat\Z)$ induces a morphism of modular curves $X_H\to X_G$: every elliptic curve with level $H$ structure gives rise to an elliptic curve with level $G$ structure. We call such a morphism of modular curves a modular cover.

The degree of the morphism $X_H\to X_G$ is equal to the ratio of the $\PSL_2$-index of $H$ by that of $G$. Note that when $-1$ is in $G$ but not $H$, we might have $H < G$ strictly contained but the map $X_H \to X_G$ may have degree $1$ (hence an isomorphism).

A modular cover is minimal if $H < G$ is a maximal (proper) subgroup.

The morphism $X_H\to X_G$ induces a surjective homomorphism $\phi \colon \operatorname{Jac}(X_H)\to\operatorname{Jac}(X_G)$. The kernel of the modular cover is the connected component of $\ker \phi$. This is an abelian variety, possibly of dimension zero.