Explicit form of quaternion modular embeddings
Let \(\mathbb{B}\) be an indefinite quaternion algebra over \(\mathbb{Q}\), and let \({\mathcal O}\) be an Eichler order of \(\mathbb{B}\). If \(Nr\) denotes the reduced norm of \(\mathbb{B}\), then \(\Gamma = \{\gamma \in {\mathcal O} \mid Nr (\gamma) = 1\}\) is a discrete subgroup of \(\mathbb{B}^{(1)}\cong SL_2 (\mathbb{R})\), and the quotient \(\Gamma \backslash {\mathcal H}\) of the upper half plane \({\mathcal H}\) by \(\Gamma\) is the \(\mathbb{C}\)-valued points of the Shimura curve \(S\) attached to \({\mathcal O}\). It is well-known that \(S(\mathbb{C})\) can be interpreted as the moduli space of principal polarized abelian surfaces having quaternion multiplication by \({\mathcal O}\), and the associated correspondence between the points of \(S(\mathbb{C})\) and the abelian surfaces can also be described by a quaternion modular embedding. More precisely, if \({\mathcal H}_2\) denotes the Siegel upper half space of degree two, then there is a holomorphic embedding \(\Phi : {\mathcal H} \to {\mathcal H}_2\), \(z \mapsto \Omega (z)\) that is compatible with the actions of \(\Gamma\) and \(Sp(4, \mathbb{Z})\) through an embedding \(\varphi : \Gamma \hookrightarrow Sp(4, \mathbb{Z})\). In this paper the author describes the maps \(\varphi\) and \(\Phi\) explicitly by constructing a concrete model of \({\mathcal O}\) and using its \(\mathbb{Z}\)-basis. He also determines some arithmetic properties of such embeddings.
scientific article; zbMATH DE number 871819