Geodesic
Serre’s uniformity problem in the split Cartan case
Yuri Bilu1 ; Pierre Parent2
1 Institut de Mathématiques de Bordeaux<br/> Université Bordeaux I<br/> 33 405 Talence<br/> France
2 Institut de Mathématiques de Bordeaux<br/> Université Bordeaux I<br/>33 405 Talence<br/>France
Annals of mathematics, Tome 173 (2011) no. 1, pp. 569-584.

Voir la notice de l'article provenant de la source Annals of Mathematics website

We prove that there exists an integer $p_{0}$ such that $X_{\mathrm{split}} (p)(\Bbb{Q} )$ is made of cusps and CM-points for any prime ${p>p_0}$. Equivalently, for any non-\rm CM elliptic curve $E$ over $\Bbb{Q}$ and any prime ${p>p_0}$ the image of $\mathrm{Gal} (\overline{\Bbb{Q}} /\Bbb{Q} )$ by the representation induced by the Galois action on the $p$-division points of $E$ is not contained in the normalizer of a split Cartan subgroup. This gives a partial answer to an old question of Serre.
DOI : 10.4007/annals.2011.173.1.13

Yuri Bilu 1 ; Pierre Parent 2

1 Institut de Mathématiques de Bordeaux<br/> Université Bordeaux I<br/> 33 405 Talence<br/> France
2 Institut de Mathématiques de Bordeaux<br/> Université Bordeaux I<br/>33 405 Talence<br/>France 
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Yuri Bilu; Pierre Parent. Serre’s uniformity problem in the split Cartan case. Annals of mathematics, Tome 173 (2011) no. 1, pp. 569-584. doi : 10.4007/annals.2011.173.1.13. http://geodesic.mathdoc.fr/articles/10.4007/annals.2011.173.1.13/

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