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Dan Petersen
Dan Petersen
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The stack $\overline{\mathcal M}_{1,1}^r$ is usually denoted by $X_1(r)$. It is a modular curve. In fact one can write down by hand an isomorphism between the moduli functors for the Hurwitz stack interpretation of $\overline{\mathcal M}_{1,1}^r$ that you mention, and the modular interpretation of $X_1(r)$ described in Deligne-Rapoport: this isomorphism is described in Section 3 of http://www.ams.org/mathscinet-getitem?mr=2968638 . I should also say that there is a minor issue here -- there are two common definitions of $\overline{\mathcal M}_{1,1}^r$, and one is a $\mu_r$-gerbe over the other. For the isomorphism $\overline{\mathcal M}_{1,1}^r \cong X_1(r)$ you should use the "rigidified" version.

Anyway, the Picard group of a smooth twisted curve $\mathscr C$ is an extension $$ 0 \to \mathrm{Pic}(C)\to \mathrm{Pic}(\mathscr C) \to \prod_{i=1}^k \mu_{r_i} \to 0,$$$$ 0 \to \mathrm{Pic}(C)\to \mathrm{Pic}(\mathscr C) \to \prod_{i=1}^k \mu_{r_i} \to 1,$$ if there are $k$ stacky points with stabilizers of order $r_1, \ldots, r_k$, and $C$ is the coarse space. Most textbooks on modular forms will describe the genus of $X_1(r)$, its number of cusps and the respective stabilizers. For instance this is described very explicitly in Diamond-Shurman.

The stack $\overline{\mathcal M}_{1,1}^r$ is usually denoted by $X_1(r)$. It is a modular curve. In fact one can write down by hand an isomorphism between the moduli functors for the Hurwitz stack interpretation of $\overline{\mathcal M}_{1,1}^r$ that you mention, and the modular interpretation of $X_1(r)$ described in Deligne-Rapoport: this isomorphism is described in Section 3 of http://www.ams.org/mathscinet-getitem?mr=2968638 . I should also say that there is a minor issue here -- there are two common definitions of $\overline{\mathcal M}_{1,1}^r$, and one is a $\mu_r$-gerbe over the other. For the isomorphism $\overline{\mathcal M}_{1,1}^r \cong X_1(r)$ you should use the "rigidified" version.

Anyway, the Picard group of a smooth twisted curve $\mathscr C$ is an extension $$ 0 \to \mathrm{Pic}(C)\to \mathrm{Pic}(\mathscr C) \to \prod_{i=1}^k \mu_{r_i} \to 0,$$ if there are $k$ stacky points with stabilizers of order $r_1, \ldots, r_k$, and $C$ is the coarse space. Most textbooks on modular forms will describe the genus of $X_1(r)$, its number of cusps and the respective stabilizers. For instance this is described very explicitly in Diamond-Shurman.

The stack $\overline{\mathcal M}_{1,1}^r$ is usually denoted by $X_1(r)$. It is a modular curve. In fact one can write down by hand an isomorphism between the moduli functors for the Hurwitz stack interpretation of $\overline{\mathcal M}_{1,1}^r$ that you mention, and the modular interpretation of $X_1(r)$ described in Deligne-Rapoport: this isomorphism is described in Section 3 of http://www.ams.org/mathscinet-getitem?mr=2968638 . I should also say that there is a minor issue here -- there are two common definitions of $\overline{\mathcal M}_{1,1}^r$, and one is a $\mu_r$-gerbe over the other. For the isomorphism $\overline{\mathcal M}_{1,1}^r \cong X_1(r)$ you should use the "rigidified" version.

Anyway, the Picard group of a smooth twisted curve $\mathscr C$ is an extension $$ 0 \to \mathrm{Pic}(C)\to \mathrm{Pic}(\mathscr C) \to \prod_{i=1}^k \mu_{r_i} \to 1,$$ if there are $k$ stacky points with stabilizers of order $r_1, \ldots, r_k$, and $C$ is the coarse space. Most textbooks on modular forms will describe the genus of $X_1(r)$, its number of cusps and the respective stabilizers. For instance this is described very explicitly in Diamond-Shurman.

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Dan Petersen
Dan Petersen
  • 40.2k
  • 2
  • 114
  • 201

The stack $\overline{\mathcal M}_{1,1}^r$ is usually denoted by $X_1(r)$. It is a modular curve. In fact one can write down by hand an isomorphism between the moduli functors for the Hurwitz stack interpretation of $\overline{\mathcal M}_{1,1}^r$ that you mention, and the modular interpretation of $X_1(r)$ described in Deligne-Rapoport: this isomorphism is described in Section 3 of http://www.ams.org/mathscinet-getitem?mr=2968638 . I should also say that there is a minor issue here -- there are two common definitions of $\overline{\mathcal M}_{1,1}^r$, and one is a $\mu_r$-gerbe over the other. For the isomorphism $\overline{\mathcal M}_{1,1}^r \cong X_1(r)$ you should use the "rigidified" version.

Anyway, the Picard group of a smooth twisted curve $\mathscr C$ is an extension $$ 0 \to \mathrm{Pic}(C)\to \mathrm{Pic}(\mathscr C) \to \prod_{i=1}^k \mu_{r_i} \to 0,$$ if there are $k$ stacky points with stabilizers of order $r_1, \ldots, r_k$, and $C$ is the coarse space. Most textbooks on modular forms will describe the genus of $X_1(r)$, its number of cusps and the respective stabilizers. For instance this is described very explicitly in Diamond-Shurman.