Modular diagonal quotient surfaces
type: Item
instance of: scholarly article
title: Modular diagonal quotient surfaces
zbMATH Open document ID: 0996.14012
DOI: 10.1007/PL00004379
author: E. Kani
author: Wolfgang Schanz
published in: Mathematische Zeitschrift
publication date: 1998-12-09T00:00:00Z
review text: 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\…
reviewed by: Eyal Z. Goren
Mathematics Subject Classification ID: 14G35
Mathematics Subject Classification ID: 14M17
Mathematics Subject Classification ID: 11G18
Mathematics Subject Classification ID: 14H25
Mathematics Subject Classification ID: 11F41
zbMATH DE Number: 1142982
zbMATH Keywords: modular diagonal quotient surfaces
zbMATH Keywords: modular curve
zbMATH Keywords: Kodaira dimension
zbMATH Keywords: Galois representation
MaRDI profile type: Publication
full work available at URL: pl00004379
OpenAlex ID: W2016584582
description: scientific article; zbMATH DE number 1142982