Serre's uniformity in the split Cartan case

Let $E$ be an elliptic curve over $\bf Q$, without complex multiplication over $\overline{\bf Q}$. For $p$ a prime number, consider the representation ${\mathrm{Gal}}(\overline{\bf Q} /{\bf Q})\to GL (E[p])\simeq GL_2 ({\bf F}_p )$ induced by the Galois action on the group of $p$-torsion points of $E$. A theorem of Serre, published in 1972, asserts that there exists an integer $B_E$ such that the above representation is surjective for $p$ larger than $B_E$. Serre then asked the following question: can $B_E$ be chosen independently of $E$? The classification of maximal subgroups of $GL_2 ({\bf F}_p )$ shows that this boils down to proving the triviality, for large enough $p$, of the sets of rational points of four families of modular curves, namely $X_0 (p)$, $X_{\mathrm{split}} (p)$, $X_{\mathrm{non-split}} (p)$ and $X_{{\frak A}_4} (p)$ (we say that a point of one of those curves is {\it trivial} if it is either a cusp, or the underlying isomorphism class of elliptic curves has complex multiplication over $\overline{\bf Q}$). The (so-called exceptional) case of $X_{{\frak A}_4} (p)$ was ruled out by Serre. The fact that $X_0 (p)({\bf Q} )$ is made of only cusps for $p>163$ is a well-known theorem of Mazur. In this talk we will present a proof that $X_{\mathrm{split}} (p)({\bf Q})$ is trivial for large enough $p$ (joint work with Yuri Bilu).