Serre's uniformity problem in the split Cartan case

Abstract

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.