SIEGEL FUNCTIONS, MODULAR CURVES, AND SERRE’S UNIFORMITY PROBLEM

Abstract

Serre’s uniformity problem asks whether there exists a bound $k$ such that for any , the Galois representation associated to the $p$-torsion of an elliptic curve $E/\mathbb{Q}$ is surjective independent of the choice of $E$. Serre showed that if this representation is not surjective, then it has to be contained in either a Borel subgroup, the normalizer of a split Cartan subgroup, the normalizer of a non-split Cartan subgroup, or one of a finite list of “exceptional” subgroups. We will focus on the case when the image is contained in the normalizer of a split Cartan subgroup. In particular, we will show that the only elliptic curves whose Galois representation at $11$ is contained in the normalizer of a split Cartan have complex multiplication. To prove this we compute $X_s^{+}(11)$ using modular units, use the methods of Poonen and Schaefer to compute its Jacobian, and then use the method of Chabauty and Coleman to show that the only points on this curve correspond to CM elliptic curves.