Octahedral Galois representations arising from Q-curves of degree 2

الأصل · EN

Generically, one can attach to a Q-curve C octahedral representations Gal(Qbar/Q) --> GL(2,Fbar₃) coming from the Galois action on the 3-torsion of those abelian varieties of GL₂-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,F₉). 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,F₉), and the characterization can be given in terms of a quartic polynomial defining the S₄-extension of Q attached to the octahedral representation.