Serre’s uniformity problem in the split Cartan case

Abstract

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.