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

  Published:2002-12-01
 Printed: Dec 2002
  • J. Fern├índez
  • J-C. Lario
  • A. Rio
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Abstract

Generically, one can attach to a $\mathbf{Q}$-curve $C$ octahedral representations $\rho\colon\Gal(\bar{\mathbf{Q}}/\mathbf{Q})\rightarrow\GL_2(\bar\mathbf{F}_3)$ 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 $\rho$ having image into $\GL_2(\mathbf{F}_9)$. Going the other way, we can ask which $\mod 3$ octahedral representations $\rho$ of $\Gal(\bar\mathbf{Q}/\mathbf{Q})$ 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 $\GL_2(\mathbf{F}_9)$, and the characterization can be given in terms of a quartic polynomial defining the $\mathcal{S}_4$-extension of $\mathbf{Q}$ corresponding to the projective representation $\bar{\rho}$.
MSC Classifications: 11G05, 11G10, 11R32 show english descriptions Elliptic curves over global fields [See also 14H52]
Abelian varieties of dimension $> 1$ [See also 14Kxx]
Galois theory
11G05 - Elliptic curves over global fields [See also 14H52]
11G10 - Abelian varieties of dimension $> 1$ [See also 14Kxx]
11R32 - Galois theory
 

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