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Octahedral Galois representations arising from Q-curves of degree 2

Julio Fernández-González·Joan-Carles Lario·Anna Rio
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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.

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Julio Fernández-González, Joan-Carles Lario, Anna Rio. 2001-06-18. Octahedral Galois representations arising from Q-curves of degree 2. https://arxiv.org/abs/math/0106272

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