## Barcelona Fall Workshop on Number Theory II

### Quadratic Chabauty for nonsplit Cartan modular curves

#### Presenters

Samuel Le Fourn (Grenoble)

#### Abstract

In a famous recent paper, Balakrishnan, Dogra, Müller, Tuitman and Vonk managed to prove that the modular curve $X_{ns}^+(13)$ did not have non-CM rational points, through so-called quadratic Chabauty method. This method requires sophisticated conditions to effectively be applied (in particular, complete knowledge of the rank of the rational points of the jacobian). In this talk, I will explain a joint work with Netan Dogra allowing to weaken the hypotheses of quadratic Chabauty and to apply it unconditionnally in the context of nonsplit Cartan modular curves, leading in itself to an explicit bound on the number of their rational points.