Octahedral Galois representations arising from Q-curves of degree 2., by Julio Fernández-González, Joan-Carles Lario, Anna Rio.
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.
Julio Fernández-González, Joan-Carles Lario, Anna Rio. <email@example.com, firstname.lastname@example.org, email@example.com>