$X_0^1(\mathfrak{N})_{\mathfrak{b}}$ is the Hilbert modular surface $X(\Gamma)$ for $\Gamma = \Gamma_0^1(\mathfrak{N})_{\mathfrak{b}} \le \SL(\Z_F \oplus \frak{b})$ the inverse image of $\begin{pmatrix} \ast & \ast \\ 0 & \ast \end{pmatrix} \subset \SL((\Z_F/\frak{N}) \oplus (\frak{b} / \frak{N} \frak{b}))$. As a moduli space it parameterizes quadruples $(A, \iota, , \lambda, \phi)$, where $A$ is an abelian surface over $\Q$, $\lambda$ is a polarization whose polarization module is isomorphic to $(\frak{d}_F \frak{b})^{-1}$ where $\frak{d}_F$ is the different of $F$, $\iota : \Z_F \to \End(A_{\overline{\Q}})$ is an embedding and $\phi : A \to A'$ is a rational isogeny with kernel isomorphic to $\Z_F / \frak{N}$, compatible with $\iota$ and $\lambda$.
- Review status: beta
- Last edited by Eran Assaf on 2023-07-12 16:03:16
- 2023-07-12 16:03:16 by Eran Assaf
- 2023-07-12 15:59:48 by Eran Assaf
- 2023-07-12 15:58:55 by Eran Assaf
- 2023-07-12 14:44:01 by Eran Assaf
- 2023-07-12 14:42:41 by Eran Assaf
- 2023-07-12 14:38:51 by Eran Assaf