On the geometry and arithmetic of some Siegel modular threefolds (Q1896585)

The goal of this article is a detailed investigation of certain Siegel modular threefolds. Let \(\Gamma (2n, 4n)\) be the subgroup of matrices \(\left (\begin{smallmatrix} A \\ C \end{smallmatrix} \begin{smallmatrix} B \\ D \end{smallmatrix} \right) \in \Gamma (2n) \subseteq Sp_2 (\mathbb{Z}) \subseteq SL_4 (\mathbb{Z})\) for which \(A,B\) have diagonal \(\equiv 0 \text{mod} 4n\) and denote by \(\Gamma (2,4,8)\) the unique normal subgroup of \(Sp_2 (\mathbb{Z})\) of index 2 in \(\Gamma (4,8)\) and containing \(\Gamma (8)\). Then the associated Siegel modular threefold is embedded using theta constants into \(\mathbb{P}^{13}\). Omitting a 6-tuple of distinct even theta characteristics from the embedding defines a map from the Siegel upper half space \({\mathfrak S}_2\) into \(\mathbb{P}^7\) and a subgroup \(\Gamma' \subseteq \Gamma (2,4)\) for which the associated Siegel modular threefold embeds into the closure of the image in \(\mathbb{P}^7\) of \({\mathfrak S }_2\). The product of the 6 theta constants gives then a modular form of weight 3 for \(\Gamma'\). The varieties corresponding in this way to the three \(Sp_2 (\mathbb{Z})\)-orbits of such 6-tuples of theta constants are then studied. In particular their Betti and Hodge numbers are computed and in the two cases where the associated modular forms of weight 3 are cuspidal the associated Galois representations in the \(l\)-adic \(H^3\) of the variety are determined. It turns out that in one case its \(L\)-function is that of the elliptic modular form \(g = \eta (2z)^4 \eta (8z)^4\) and in the other case it is related to the \(L\)-function of a Hecke character of \(\mathbb{Q} (i)\). In an appendix it is shown that the cusp form of weight 3 arising in the first case is the Saito-Kurokawa lifting of \(g\).