Rational and elliptic parametrizations of \(\mathbb Q\)-curves
type: Item
instance of: scholarly article
title: Rational and elliptic parametrizations of \(\mathbb Q\)-curves
zbMATH Open document ID: 0932.11037
author: Josep González
author: Joan-C. Lario
published in: Journal of Number Theory
publication date: 2000-03-19T00:00:00Z
review text: An elliptic curve \(E\) defined over \(\overline{\mathbb Q}\) is called \(\mathbb Q\)-curve if \(E\) and its Galois conjugates \(E^\sigma\) are isogenous over \(\mathbb Q\) for any \(\sigma\in \text{Gal} (\overline{\mathbb Q}/\mathbb Q)\). Let \(N\) be a square free positive integer and \(B(N)\) be the group generated by the Atkin-Lehner involutions of the modular curve \(X_0(N)\). Every non-cusp rational point of the curve \(X^*(N)= X_0(N)/B(N)\) lifts to \(X_0(N)\) giving \(\mathbb Q\)-curves defined over abelian extensions of type \((2,\dots, 2)\). In this paper the authors first determine all square-free values of \(N\) such that \(X_0(N)\) has genus \(g=0\) or 1. In case where \(g=0\), they describe the construction of a Hauptmodul on \(X^*(N)\) and obtain families of \(\mathbb Q\)-curves over quadratic, biquadratic and triquadratic extensions. As application of these results some parametric families of \(\mathbb Q\)-curves are given, obtained from the rational curves \(X^*(n)\), where \(n= 6,11,23,30\). In case where \(g=1\), they obtain explicit modular parametrizations of a reduced Neron model of \(X^*(N)\) and obtain families of \(\mathbb Q\)-curves parametrized by its Mordell-Weil group defined over quadratic, biquadratic, triquadratic and tetraquadratic extensions. Furthermore, they illustrate this method giving some examples from the curves \(X^*(n)\), where \(n= 37, 74, 82\).
Mathematics Subject Classification ID: 11G05
Mathematics Subject Classification ID: 11G18
zbMATH DE Number: 1229760
zbMATH Keywords: elliptic curve
zbMATH Keywords: \(\mathbb Q\)-curve
zbMATH Keywords: Galois conjugates
zbMATH Keywords: Atkin-Lehner involutions
zbMATH Keywords: modular curve
zbMATH Keywords: Hauptmodul
zbMATH Keywords: reduced Neron model
reviewed by: Dimitros Poulakis
describes a project that uses: ecdata
MaRDI profile type: Publication
full work available at URL: jnth.1998.2259
OpenAlex ID: W2046204299
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DOI: 10.1006/JNTH.1998.2259
description: scientific article; zbMATH DE number 1229760