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Let $N \in \mathbb{N}$, and let $G \subseteq GL_2(\Z / N \Z)$. The modular curve $X(G)$ is an algebraic curve defined over $\Q(\zeta_N)$, whose points, with the exception of particular points called cusps, correspond to equivalence classes of pairs $(E,\varphi)$, where $E$ is an elliptic curve and $\varphi : E[N] \rightarrow \Z / N \Z \times \Z / N \Z$ is an isomorphism of abelian groups. Two such pairs $(E, \varphi)$ and $(E', \varphi')$ are equivalent if there exist an isomorphism $\iota : E \rightarrow E'$ and an element $g \in G$ such that $\varphi' \circ \iota = g \circ \varphi$. The cusps correspond to degenerate such pairs.

If $\det(G) = (\Z / N \Z)^{\times}$, then $X(G)$ is defined over $\Q$ and its rational points are the elliptic curves $E$, whose associated Galois representation $\rho_{E, N} : \Gal_{\Q} \rightarrow GL_2(\Z / N \Z)$ has image contained in $G$.

The set of its complex points $X(G)(\C)$ is naturally isomorphic to the quotient of the completed upper half plane $\Gamma_G \backslash \mathcal{H}^*$ as a Riemann surface, where $\Gamma_G \subseteq SL_2(\Z)$ is the preimage of $G \cap SL_2(\Z / N \Z)$.

Assume $G \trianglelefteq G' \subseteq GL_2(\Z / N \Z)$. If $c \in G'$, and $(E, \varphi)$ is a pair as above, then so is $(E, c \varphi)$. This yields an action of the group $G' / G$ on $X(G)$. The quotient of $X(G)$ by $G' / G$ is the modular curve $X(G')$.

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  • Last edited by Eran Assaf on 2020-10-29 12:25:52
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