Hyperelliptic modular curves
type: Item
instance of: scholarly article
title: Hyperelliptic modular curves
zbMATH Open document ID: 0771.14008
author: S. Singh
published in: Tsukuba Journal of Mathematics
publication date: 1992-09-27T00:00:00Z
review text: For a positive integer \(N\) and a subgroup \(\Delta\) of the multiplicative group \((\mathbb{Z}/N\mathbb{Z})^*\) containing \(-1\), let \(X_ \Delta(N)\) denote the modular curve, defined over \(\mathbb{Q}\), which corresponds to the modular group \[ \Gamma_ \Delta(N)=\left\{\left({a\;b\atop c\;d}\right)\in SL_ 2(\mathbb{Z});\;c\equiv 0\pmod N, (a\bmod N)\in\Delta\right\}. \] Specifically, for \(\Delta=\{\pm 1\}\) (resp. \(\Delta=(\mathbb{Z}/N\mathbb{Z})^*)\) one obtains the modular curves \(X_ 1(N)\) (resp. \(X_ 0(N))\) classifying pairs \((E,P)\) (resp. \((E,\langle P\rangle))\) of elliptic curves \(E\) with a point \(P\) (resp. cyclic group \(\langle P\rangle)\) of order \(N\) over the field of definition. The authors determine all hyperelliptic modular curves \(X_ \Delta(N)\) of genus \(g_ \Delta(N)\geq 2\). In fact they prove the following theorem: The hyperelliptic modular curves \(X_ \Delta(N)\) are the modular curves (of genus \(g_ 0(N)\geq 2)\) \(X_ 0(N)\) for \(N=22, 23, 26, 28, 29, 30, 31, 33, 35, 37, 39, 40, 41, 46, 47, 48, 50, 59, 71\) and the modular curves (of genus \(g_ 1(N)=2)\) \(X_ 1(N)\) for \(N=13, 16, 18\). -- The associated hyperelliptic involutions are also given. The proof is based on the fact that \(X_ \Delta(N)\) is a subcovering of the covering \(X_ 1(N)\to X_ 0(N)\), i.e. that we have a Galois covering \[ X_ 1(N)\to X_ \Delta(N)\to X_ 0(N): \quad (E,\pm P)\mapsto (E,\Delta P)\mapsto (E,\langle P\rangle). \] Use is made of work of \textit{A. P. Ogg} [Bull. Soc. Math. Fr. 102(1974), 449-462 (1974; Zbl 0314.10018)] who determined all hyperelliptic modular curves \(X_ 0(N)\). The automorphism groups of the hyperelliptic modular curves \(X_ \Delta(N)\) can also be given. Indeed, for square-free integers \(N\), the automorphism groups of the modular curves \(X_ 0(N)\) were determined by \textit{A. P. Ogg} [Math. Ann. 228, 279-292 (1977; Zbl 0336.14002)], and the automorphism groups of \(X_ 1(N)\) for square-free \(N\) were found by the second …
Mathematics Subject Classification ID: 14H52
Mathematics Subject Classification ID: 14G35
Mathematics Subject Classification ID: 14H10
Mathematics Subject Classification ID: 14H30
zbMATH DE Number: 61126
zbMATH Keywords: classification of modular curves
zbMATH Keywords: \(X_ 1(N)\)
zbMATH Keywords: \(X_ 0(N)\)
zbMATH Keywords: automorphism groups of the hyperelliptic modular curves
zbMATH Keywords: torsion points of elliptic curves over quadratic fields
reviewed by: Horst G. Zimmer
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.21099/tkbjm/1496161667
OpenAlex ID: W1516129493
DOI: 10.21099/TKBJM/1496161667
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description: scientific article; zbMATH DE number 61126