On octahedral extensions of \(\mathbb Q\) and quadratic \(\mathbb Q\)-curves.
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.
scientific article; zbMATH DE number 2058858