"Number theory, and in particular diophantine equations (equations with integers), is a theme of mathematics almost as old as civilisation itself, as seen on Babylonian clay tablets from 1800 B.C. More recently, the study of so-called Galois representations of elliptic curves is at the heart of modern arithmetic geometry, and intimately related to the celebrated proof of Fermat's Last Theorem by Wiles.
Our focus is on the possible images of a Galois representation: to each possibility we can associate a modular curve, which is a moduli
space of elliptic curves with representation having that image. The central conjecture regarding these images, called Serre's uniformity conjecture, states that these modular curves have no nontrivial rational points when their genus is large enough, and can be split in three main cases. Two of those cases have been solved in the fundamental works of Mazur, Kamienny, Merel, Bilu, Parent and Rebolledo (Mazur's being crucial to the proof of Wiles), but there was a persistent obstruction to the last one, ""non-split Cartan"".
In this project, we study low degree points on interesting modular curves (or even rational points where this type of obstruction holds), by developing and extending powerful methods including an overdetermined version of Chabauty's method in the symmetric power setting,
and a recent breakthrough called ""quadratic Chabauty method"" for the non-split Cartan modular curves.
Any advance towards the resolution of Serre's uniformity conjecture is expected to have applications in multiple fields ranging from cryptography to arithmetic geometry. Moreover, the tools developed on the way should shed a new light on the general problem of explicitly determining rational points on curves of large genus."
