On octahedral extensions of \(\mathbb Q\) and quadratic \(\mathbb Q\)-curves.
type: Item
instance of: scholarly article
title: On octahedral extensions of \(\mathbb Q\) and quadratic \(\mathbb Q\)-curves.
zbMATH Open document ID: 1049.11117
author: Julio Fernández
published in: Journal de Théorie des Nombres de Bordeaux
publication date: 2004-03-16T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/93874
review text: A \({\mathbb Q}\)-curve is an elliptic curve \(C\) defined over a number field \(k\) without complex multiplication and isogenous to all its Galois conjugates. In this article, a necessary condition is given for a surjective representation \({\rho} : \text{Gal}(\bar{\mathbb Q}/{\mathbb Q}) \to \text{PGL}_2(\mathbb F_3)\) to arise from the \(3\)-torsion of a \({\mathbb Q}\)-curve, that is, for \(\rho = \rho_C\) (up to conjugation in \(\text{PGL}_2(\mathbb F_3\)), where \(\rho_C\) comes from the Galois action on the \(3\)-torsion of the abelian varieties of \(\text{GL}_2\)-type having the curve \(C\) as a quotient. This necessary condition is given in terms of the discriminant and Witt invariant of the quadratic trace form associated to any quartic subextension of the fixed field of \(\rho\). If, moreover, \(\rho\) has a noncyclotomic determinant, then necessary and sufficient conditions are given for \(\rho\) to be elliptic over a quadratic field. These can again be described in terms of the determinant and the Witt invariant of the trace form of a quartic subextension.
Mathematics Subject Classification ID: 11R21
Mathematics Subject Classification ID: 11G05
Mathematics Subject Classification ID: 11E04
Mathematics Subject Classification ID: 11R32
Mathematics Subject Classification ID: 11R52
zbMATH DE Number: 2058858
zbMATH Keywords: elliptic curve
zbMATH Keywords: trace form
zbMATH Keywords: discriminant
zbMATH Keywords: Witt invariant
MaRDI profile type: Publication
OpenAlex ID: W2525104517
cites work: On the modularity of \(\mathbb Q\)-curves
cites work: Octahedral Galois Representations Arising From Q-Curves of Degree 2
cites work: Q4863564
cites work: Liftings of projective 2-dimensional Galois representations and embedding problems
cites work: \(\mathbb Q\)-curves and abelian varieties of \(\text{GL}_ 2\)-type.
cites work: Fields of definition of \(\mathbb{Q}\)-curves
cites work: Q4141905
cites work: The Witt invariant of the form \(\text{Tr}(x^ 2)\)
cites work: On stem extensions of \(S_ n\) as Galois group over number fields
DOI: 10.5802/JTNB.391
description: scientific article; zbMATH DE number 2058858