Cubic and quartic points on modular curves using generalised symmetric Chabauty

TL;DR

This paper advances the understanding of cubic and quartic points on specific modular curves by developing a generalized symmetric Chabauty method, lowering prime bounds, and computing the Jacobian's Mordell--Weil group.

Contribution

It introduces a 'partially relative' symmetric Chabauty method, generalizes existing theorems, and applies these to determine points on modular curves, including a full Mordell--Weil group computation.

Findings

Determined cubic points on X_0(N) for specified N values.

Identified quartic points on X_0(65).

Developed a higher order Chabauty theorem.

Abstract

Answering a question of Zureick-Brown, we determine the cubic points on the modular curves $X_0(N)$ for $N \in \{53,57,61,65,67,73\}$ as well as the quartic points on $X_0(65)$. To do so, we develop a "partially relative" symmetric Chabauty method. Our results generalise current symmetric Chabauty theorems, and improve upon them by lowering the involved prime bound. For our curves a number of novelties occur. We prove a "higher order" Chabauty theorem to deal with these cases. Finally, to study the isolated quartic points on $X_0(65)$, we rigorously compute the full rational Mordell--Weil group of its Jacobian.