Octahedral Galois representations arising from Q-curves of degree 2
Abstract
Generically, one can attach to a Q-curve C octahedral representations Gal(Qbar/Q) --> GL(2,Fbar_3) coming from the Galois action on the 3-torsion of those abelian varieties of GL_2-type whose building block is C. When C is defined over a quadratic field and has an isogeny of degree 2 to its Galois conjugate, there exist such representations having image into GL(2,F_9). Going the other way, we can ask which mod 3 octahedral representations of Gal(Qbar/Q) arise from Q-curves in the above sense. We characterize those arising from quadratic Q-curves of degree 2. The approach makes use of Galois embedding techniques in GL(2,F_9), and the characterization can be given in terms of a quartic polynomial defining the S_4-extension of Q attached to the octahedral representation.
Keywords
Cite
@article{arxiv.math/0106272,
title = {Octahedral Galois representations arising from Q-curves of degree 2},
author = {Julio Fernández-González and Joan-Carles Lario and Anna Rio},
journal= {arXiv preprint arXiv:math/0106272},
year = {2007}
}