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Octahedral Galois Representations Arising From $\mathbf{Q}$-Curves of Degree 2

Published online by Cambridge University Press:  20 November 2018

J. Fernández ,
J-C. Lario  and
A. Rio
J. Fernández
Affiliation:
Departament de Matemàtica Aplicada II, Universitat Politècnica de Catalunya, Pau Gargallo 5, E-08028 Barcelona
J-C. Lario
Affiliation:
Departament de Matemàtica Aplicada II, Universitat Politècnica de Catalunya, Pau Gargallo 5, E-08028 Barcelona
A. Rio
Affiliation:
Departament de Matemàtica Aplicada II, Universitat Politècnica de Catalunya, Pau Gargallo 5, E-08028 Barcelona
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Abstract

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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 $\bar{\rho }$.

Keywords

11G0511G1011R32
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 54 , Issue 6 , 01 December 2002 , pp. 1202 - 1228
Copyright
Copyright © Canadian Mathematical Society 2002

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