Modular diagonal quotient surfaces
The modular diagonal quotient surfaces are obtained as quotients of \(X(N)^2\), where \(X(N)\) is the modular curve of full level structure \(N\). There are several reasons for interest in such surfaces: (i) as usual, the interplay between moduli and geometry provides rich and interesting structures; (ii) as arithmetic varieties they provide good test ground to arithmetic conjectures (such as the automorphy of their zeta functions); (iii) they are directly related to the problem of studying elliptic curves \(E_1, E_2\) whose Galois representations, derived from the \(N\)-torsion points, are isomorphic. Let \(G = \text{ SL}(2, {\mathbb{Z}}/N{\mathbb{Z}})/\{ \pm 1\}\), let \(Q_\varepsilon\) be a diagonal matrix in \(\text{ GL}(2, {\mathbb{Z}}/N{\mathbb{Z}})\) with diagonal elements \(\varepsilon, 1\), let \(\alpha_\varepsilon\) be the automorphism of \(G\) defined by conjugation by \(Q_\varepsilon\), and let \(\Delta_\varepsilon = \{(g, \alpha_\varepsilon(g)): g \in G\}\). The quotient surface \(Z_{N, \varepsilon} = \Delta_\varepsilon \backslash (X(N) \times X(N))\) is called a modular diagonal quotient surface. The authors show that if \(N\geq 5\), the surface \(Z_{N, \varepsilon}\) has precisely \(r_0+r_1+r_\infty\) singularities, where \(r_0 = h(-4N^2)\) and \(r_1 = h(-3N^2)\) are certain class numbers of binary quadratic forms, and \(r_\infty = \sum_{1< d|N} \varphi(d)\varphi(N/d)\) denotes the number of cusps of \(X_1(N)\). All are cyclic quotient singularities. The authors determine the geometric genus \(p_{g, N, \varepsilon}\) and the Chern numbers of an explicit desingularization \(\widetilde{Z}_{N, \varepsilon}\) of \(Z_{N, \varepsilon}\). They prove the following formula for the Kodaira dimension: \[ \kappa(\widetilde{Z}_{N, \varepsilon}) = \min(2, -1+ p_{g, N, \varepsilon}). \] They provide the classification of \(\widetilde{Z}_{N, \varepsilon}\) according to the classification of surfaces, providing complete lists of the cases when they are rational, \(K3\), elliptic and of general type. Some of the results of this paper were also obtained by \textit{C. F. Hermann} [Manuscr. Math. 72, No. 1, 95-110 (1991; Zbl 0749.14016)]. The authors often refer to an previous paper [\textit{E. Kani} and \textit{W. Schanz}, Manuscr. Math. 93, No. 1, 67-108 (1997; Zbl 0898.14017)].
scientific article; zbMATH DE number 1142982