Venue
Aula Seminari
Abstract
The study of Galois representations attached to elliptic curves is a very fruitful branch of number theory, leading to the solution of very difficult problems, such as Fermat’s Last Theorem. Given a rational elliptic curve $E$, the representation $\rho_{E,p}$ is described by the action of the absolute Galois group of $\mathbb{Q}$ on the $p$-torsion points of $E$. In 1972 Serre proved that for every rational elliptic curve $E$ without complex multiplication there exists an integer $N_E$ such that, for every prime $p>N_E$, the Galois representation $\rho_{E,p}$ is surjective onto $\operatorname{GL}_2(\mathbb{F}_p)$. In the same article, he asked whether the constant $N_E$ can be taken to be independent of the curve, and this became known as Serre’s Uniformity Question. In this talk, I will discuss the current progress towards an answer to this question, in particular the Runge method for modular curves developed by Bilu and Parent and the recent improvements obtained via this method by Le Fourn and Lemos, as well as explain how to solve some of the questions left open by the latter result.
This is joint work with Davide Lombardo.