Abstract
We prove that there exists an integer p0 such that Xsplit(p)(ℚ) is made of cusps and CM-points for any prime p>p0. Equivalently, for any non-\rm 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.

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