Uniformizations of modular curves
type: Item
instance of: scholarly article
title: Uniformizations of modular curves
zbMATH Open document ID: 0860.11022
author: Hershel M. Farkas
author: Yaacov Kopeliovich
author: Irwin Kra
published in: Communications in Analysis and Geometry
publication date: 1997-04-24T00:00:00Z
review text: This long and subtle paper stands within a series of notes devoted to the intimate relations between theta constants with rational characteristics, meromorphic modular forms for the principal congruence subgroup \(\Gamma(k)\) of the modular group \(\Gamma = \mathbb{P}\text{SL} (2,\mathbb{Z})\) and the geometry of the punctured Riemann surface \(\mathbb{H}^2/ \Gamma(k)\). These connections are established by associating the theta characteristics to the cusps of the group which leads in turn to new theta constant identities and explicit constructions of covering maps. In the present paper this approach is refined and extended to the larger group \(\Gamma_0(k)\). For this let \(V(k)\) be the finite-dimensional Hilbert space of holomorphic functions on the upper half plane \(\mathbb{H}^2\) spanned by the modified theta constants \(\varphi_\ell : \tau\mapsto \theta [\chi_\ell] (0,k. \tau)\) where \(\chi_\ell\) varies over the classes of rational characteristics. Then for \(k\) odd the full modular group operates on \(\mathbb{P} V(k) \simeq \mathbb{P} \mathbb{C}^{(k-3)/2}\) and induces a holomorphic map of the compactification \(\overline{\mathbb{H}^2/ \Gamma(k)}\) into \(\mathbb{P}\mathbb{C}^{(k-3)/2}\). Moreover for \(k\) prime this map reveals the automorphism group \(\Gamma/ \Gamma(k)\) of the surface as a group of projective transformations hence of isometries. It is this fact which yields a lot of geometric insight into the surface. Since the \({k-1 \over 2}\) ``distinguished'' punctures of the surface are permuted by the group \(\Gamma_0 (k)\) and support the divisor of \(\varphi_\ell/ \varphi_\ell\), two explicit generators of the function field \(K\overline {(\mathbb{H}^2/ \Gamma(k)})\) are constructed in terms of \(\varphi_1/ \varphi_0\) and \(\varphi_2/ \varphi_1\) for \(k\) prime \(> 7\). In general this field is a Galois extension of \(K\overline {(\mathbb{H}^2/ \Gamma)}\) generated by \(\varphi_\ell/ \varphi_{k-3 \over 2}\), \(\ell= 0,1, \dots, {k-5 \over 2}…
Mathematics Subject Classification ID: 11F11
Mathematics Subject Classification ID: 30F10
Mathematics Subject Classification ID: 11F27
Mathematics Subject Classification ID: 20H05
Mathematics Subject Classification ID: 14E20
zbMATH DE Number: 929683
zbMATH Keywords: uniformisation
zbMATH Keywords: modular curves
zbMATH Keywords: theta constants with rational characteristics
zbMATH Keywords: meromorphic modular forms
zbMATH Keywords: principal congruence subgroup
zbMATH Keywords: geometry of the punctured Riemann surface
zbMATH Keywords: explicit constructions of covering maps
MaRDI profile type: Publication
full work available at URL: cag.1996.v4.n2.a2
OpenAlex ID: W2736269269
DOI: 10.4310/CAG.1996.V4.N2.A2
description: scientific article; zbMATH DE number 929683