On octahedral extensions of ℚ and quadratic ℚ -curves

Julio Fernández

Journal de théorie des nombres de Bordeaux (2003)

  • Volume: 15, Issue: 1, page 125-131
  • ISSN: 1246-7405

Abstract

top
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.

How to cite

top

Fernández, Julio. "On octahedral extensions of $\mathbb {Q}$ and quadratic $\mathbb {Q}$-curves." Journal de théorie des nombres de Bordeaux 15.1 (2003): 125-131. <http://eudml.org/doc/249076>.

@article{Fernández2003,
abstract = {We give a necessary condition for a surjective representation Gal$(\overline\{\mathbb \{Q\}\} /\mathbb \{Q\}) \rightarrow \mathrm \{PGL\}_2(\mathbb \{F\}_3)$ to arise from the $3$-torsion of a $\mathbb \{Q\}$-curve. We pay a special attention to the case of quadratic $\mathbb \{Q\}$-curves.},
author = {Fernández, Julio},
journal = {Journal de théorie des nombres de Bordeaux},
keywords = {elliptic curve; trace form; discriminant; Witt invariant},
language = {eng},
number = {1},
pages = {125-131},
publisher = {Université Bordeaux I},
title = {On octahedral extensions of $\mathbb \{Q\}$ and quadratic $\mathbb \{Q\}$-curves},
url = {http://eudml.org/doc/249076},
volume = {15},
year = {2003},
}

TY - JOUR
AU - Fernández, Julio
TI - On octahedral extensions of $\mathbb {Q}$ and quadratic $\mathbb {Q}$-curves
JO - Journal de théorie des nombres de Bordeaux
PY - 2003
PB - Université Bordeaux I
VL - 15
IS - 1
SP - 125
EP - 131
AB - We give a necessary condition for a surjective representation Gal$(\overline{\mathbb {Q}} /\mathbb {Q}) \rightarrow \mathrm {PGL}_2(\mathbb {F}_3)$ to arise from the $3$-torsion of a $\mathbb {Q}$-curve. We pay a special attention to the case of quadratic $\mathbb {Q}$-curves.
LA - eng
KW - elliptic curve; trace form; discriminant; Witt invariant
UR - http://eudml.org/doc/249076
ER -

References

top
  1. [1] J.S. Ellenberg, C. Skinner, On the modularity of Q-curves. Duke Math. J.109 (2001), no. 1, 97-122. Zbl1009.11038MR1844206
  2. [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. Zbl1044.11042MR1940236
  3. [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. Zbl0837.11061MR1340079
  4. [4] J. Quer, Liftings of projective 2-dimensional Galois representations and embedding problems. J. Algebra171 (1995), no. 2, 541-566. Zbl0845.12003MR1315912
  5. [5] J. Quer, Q-curves and abelian varieties of GL2 -type. Proc. London Math. Soc. (3) 81 (2000), no. 2, 285-317. Zbl1035.11026MR1770611
  6. [6] J. Quer, Fields of definition of Q-curves. J. Théor Nombres Bordeaux13 (2001), no. 1, 275-285. Zbl1046.11044MR1838087
  7. [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. MR1212980
  8. [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. Zbl0366.10022MR450201
  9. [9] J.-P. Serre, L'invariant de Witt de la forme Tr(x2). Comment. Math. Helv.59 (1984), no. 4, 651-676. Zbl0565.12014MR780081
  10. [10] N. Vila, On stem extensions of Sn as Galois group over number fields. J. Algebra116 (1988), 251-260. Zbl0662.12011MR944159

NotesEmbed ?

top

You must be logged in to post comments.