On octahedral extensions of ℚ and quadratic ℚ-curves
Journal de théorie des nombres de Bordeaux, Tome 15 (2003) no. 1, pp. 125-131

We give a necessary condition for a surjective representation Gal(ℚ ¯/ℚ)→ PGL 2 (𝔽 3 ) to arise from the 3-torsion of a ℚ-curve. We pay a special attention to the case of quadratic ℚ-curves.

On donne une condition nécessaire pour qu’une représentation surjective Gal(ℚ ¯/ℚ)→ PGL 2 (𝔽 3 ) provienne de la 3-torsion d’une ℚ-courbe. Nous étudions plus particulièrement le cas des ℚ-courbes quadratiques.

@article{JTNB_2003__15_1_125_0,
     author = {Julio Fern\'andez},
     title = {On octahedral extensions of $\mathbb {Q}$ and quadratic $\mathbb {Q}$-curves},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {125--131},
     year = {2003},
     publisher = {Universit\'e Bordeaux I},
     volume = {15},
     number = {1},
     zbl = {1049.11117},
     mrnumber = {2019005},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/item/JTNB_2003__15_1_125_0/}
}
TY  - JOUR
AU  - Julio Fernández
TI  - On octahedral extensions of $\mathbb {Q}$ and quadratic $\mathbb {Q}$-curves
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2003
SP  - 125
EP  - 131
VL  - 15
IS  - 1
PB  - Université Bordeaux I
UR  - https://jtnb.centre-mersenne.org/item/JTNB_2003__15_1_125_0/
LA  - en
ID  - JTNB_2003__15_1_125_0
ER  - 
%0 Journal Article
%A Julio Fernández
%T On octahedral extensions of $\mathbb {Q}$ and quadratic $\mathbb {Q}$-curves
%J Journal de théorie des nombres de Bordeaux
%D 2003
%P 125-131
%V 15
%N 1
%I Université Bordeaux I
%U https://jtnb.centre-mersenne.org/item/JTNB_2003__15_1_125_0/
%G en
%F JTNB_2003__15_1_125_0
Julio Fernández. On octahedral extensions of $\mathbb {Q}$ and quadratic $\mathbb {Q}$-curves. Journal de théorie des nombres de Bordeaux, Tome 15 (2003) no. 1, pp. 125-131. https://jtnb.centre-mersenne.org/item/JTNB_2003__15_1_125_0/

[1] J.S. Ellenberg, C. Skinner, On the modularity of Q-curves. Duke Math. J. 109 (2001), no. 1, 97-122. | Zbl | MR

[2] J. Fernández, J.C. Lario, A. Rio, Octahedral Galois representations arising from Q-curves of degree 2. Canad. J. Math. 54 (2002), 1202-1228. | Zbl | MR

[3] J.C. Lario, A. Rio, An octahedral-elliptic type equality in Br2(k). C. R. Acad. Sci. Paris Sér. I Math. 321 (1995), no. 1, 39-44. | Zbl | MR

[4] J. Quer, Liftings of projective 2-dimensional Galois representations and embedding problems. J. Algebra 171 (1995), no. 2, 541-566. | Zbl | MR

[5] J. Quer, Q-curves and abelian varieties of GL2 -type. Proc. London Math. Soc. (3) 81 (2000), no. 2, 285-317. | Zbl | MR

[6] J. Quer, Fields of definition of Q-curves. J. Théor Nombres Bordeaux 13 (2001), no. 1, 275-285. | Zbl | MR | Numdam

[7] K.A. Ribet, Abelian varieties over Q and modular forms. Algebra and topology 1992 (Taejön), Korea Adv. Inst. Sci. Tech., Taejón, 1992, pp. 53-79. | MR

[8] J.-P. Serre, Modular forms of weight one and Galois representations. Algebraic number fields: L-functions and Galois properties (Proc. Sympos., Univ. Durham, Durham, 1975), Academic Press, London, 1977, pp. 193-268. | Zbl | MR

[9] J.-P. Serre, L'invariant de Witt de la forme Tr(x2). Comment. Math. Helv. 59 (1984), no. 4, 651-676. | Zbl | MR

[10] N. Vila, On stem extensions of Sn as Galois group over number fields. J. Algebra 116 (1988), 251-260. | Zbl | MR