Geodesic
Octahedral Galois Representations Arising From $\mathbf{Q}$ -Curves of Degree 2
Canadian journal of mathematics, Tome 54 (2002) no. 6, pp. 1202-1228

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Generically, one can attach to a $\mathbf{Q}$ -curve $C$ octahedral representations $\rho $ : $\text{Gal}\left( \mathbf{\bar{Q}}/\mathbf{Q} \right)\,\to \,\text{G}{{\text{L}}_{2}}\left( {{{\mathbf{\bar{F}}}}_{3}} \right)$ coming from the Galois action on the 3-torsion of those abelian varieties of $\text{G}{{\text{L}}_{2}}$ -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 $\rho $ having image into $\text{G}{{\text{L}}_{2}}\left( {{\mathbf{F}}_{9}} \right)$ . Going the other way, we can ask which mod 3 octahedral representations $\rho $ of $\text{Gal}\left( \mathbf{\bar{Q}}/\mathbf{Q} \right)$ arise from $\mathbf{Q}$ -curves in the above sense. We characterize those arising from quadratic $\mathbf{Q}$ -curves of degree 2. The approach makes use of Galois embedding techniques in $\text{G}{{\text{L}}_{2}}\left( {{\mathbf{F}}_{9}} \right)$ , and the characterization can be given in terms of a quartic polynomial defining the ${{S}_{4}}$ -extension of $\mathbf{Q}$ corresponding to the projective representation $\bar{\rho }$ .
DOI : 10.4153/CJM-2002-046-8
Mots-clés : 11G05, 11G10, 11R32 
Fernández, J.; Lario, J-C.; Rio, A. Octahedral Galois Representations Arising From $\mathbf{Q}$ -Curves of Degree 2. Canadian journal of mathematics, Tome 54 (2002) no. 6, pp. 1202-1228. doi: 10.4153/CJM-2002-046-8
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     journal = {Canadian journal of mathematics},
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     year = {2002},
     volume = {54},
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