We prove that there exists an integer p0 such that X split (p)(Q) is made of cusps and CM-points for any prime p > p0. Equivalently, for any non-CM elliptic curve E over Q and any prime p > p0 the image of Gal(Q/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.

We show how the recent isogeny bounds due to Gaudron and Rémond allow to obtain the triviality of X + 0 (p r )(Q), for r > 1 and p a prime exceeding 2 · 10 11 . This includes the case of the curves X split (p). We then prove, with the help of computer calculations, that the same holds true for p in the range 11 ≤ p ≤ 10 14 , p = 13. The combination of those results completes the qualitative study of such sets of rational points undertook in [4] and [5], with the exception of p = 13.

On se propose de donner une forme effective au théorème de MazurKamienny-Merel sur la torsion des courbes elliptiques sur les corps de nombres.