Isabel Rendell

Modular forms in quadratic Chabauty for modular curves

Quadratic Chabauty is a method to explicitly compute the rational points on certain modular curves of genus at least 2. The current algorithm, due to Balakrishnan-Dogra-Müller-Tuitman-Vonk, requires an explicit plane model of the curve. The coefficients of such models grow rapidly with the genus and so are inefficient to compute with when the genus is sufficiently large. Therefore, we would like to replace this input with certain modular forms associated to the curve, hence creating a ’model-free’ algorithm. In this talk I will provide an overview of an algorithm to compute the first stage of quadratic Chabauty on Atkin-Lehner quotients of modular curves using weakly holomorphic modular forms. Time permitting, I will mention some joint work with Marius Leonhardt on generalising this approach to normaliser of non-split Cartan modular curves.