Quadratic Chabauty for Atkin-Lehner quotients of modular curves via weakly holomorphic modular forms: Hodge Filtrations

Abstract

For Atkin-Lehner quotients $X_0^+(N)$, of prime level and of genus at least 2, we provide an algorithm for computing one of the main objects in the quadratic Chabauty algorithm in terms of weakly holomorphic modular forms associated to the curve. In particular, the algorithm computes a Hodge filtration on a certain unipotent vector bundle with connection related to $X_0^+(N)$, which is crucial in computing the $p$-adic height which is used to define the finite set of $p$-adic points containing the rational points on $X_0^+(N)$. This improves the current Hodge filtration algorithm by replacing the input of an explicit plane model of the curve with weakly holomorphic modular forms to produce a faster computation. We implement our algorithm on the genus 7 modular curve $X_0^+(193)$, and discover congruences between iterated integrals of weight 2 cusp forms in the plus eigenspace for the Atkin-Lehner involution and single integrals of weight 2 cusp forms in the minus eigenspace for the Atkin-Lehner involution.