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Let $X_H$ be the modular curve over $\mathbb{Q}$ associated to an open subgroup $H \le \operatorname{GL}_2(\widehat{\mathbb{Z}})$. Let $\pi_0((X_H)_{\overline{\Q}})$ be the set of geometric components. Since $X_H$ is integral, the action of $\Gal_{\Q}$ on $\pi_0((X_H)_{\overline{\Q}})$ is transitive. Let $C$ be one of the geometric components. The canonical field of definition of $C$ is the subfield $K$ of $\overline{\mathbb{Q}}$ fixed by the stabilizer $\operatorname{Stab}_{\Gal_{\Q}}(C)$.

Let $N$ be the level of $H$, which is the smallest $N \ge 1$ such that $H$ is the inverse image of a subgroup $H_N \le \operatorname{GL}_2(\mathbb{Z}/N\mathbb{Z})$. Then $K$ is a number field contained in $\mathbb{Q}(\zeta_N)$.

Since $K$ is abelian, it is independent of the choice of $C \in S$, and every geometric component of $X_H$ has a canonical model over $K$.

Equivalently, $K$ can be defined as the ring of global regular functions of the $\mathbb{Q}$-scheme $X_H$.

Also, $K$ is the subfield of $\mathbb{Q}^{\textup{ab}}$ fixed by $\det(H) \le \widehat{\mathbb{Z}}^\times$, or equivalently the subfield of $\mathbb{Q}(\zeta_N)$ fixed by $\det(H_N) \subset (\mathbb{Z}/N\mathbb{Z})^\times$.

It can happen that a geometric component of $X_H$ is definable over a field smaller than its canonical field of definition. For example, the canonical field of definition of a geometric component of the big modular curve $X(N)^{\text{big}}_{\mathbb{Q}}$ is $\mathbb{Q}(\zeta_N)$, but any such geometric component is the base extension of a curve over $\mathbb{Q}$, namely the arithmetic modular curve $X(N)^{\text{arith}}_{\mathbb{Q}}$.

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• Review status: beta
• Last edited by Bjorn Poonen on 2022-03-24 15:46:20
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