Octahedral Galois Representations Arising from Q-curves of Degree 2

Generically, one can attach to a Q-curve C octahedral representations ρ : Gal(Q/Q) −→ GL 2 (F 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 ρ having image into GL 2 (F 9). Going the other way, we can ask which mod 3 octahedral representations ρ of Gal(Q/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 corresponding to the projective representation ρ.