Cubic and Quartic Points on Modular Curves Using Generalised Symmetric Chabauty

Answering a question of Zureick-Brown, we determine the cubic points on the modular curves X₀(N) for N ∈ \53,57,61,65,67,73\ as well as the quartic points on X₀(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 X₀(65), we rigorously compute the full rational Mordell–Weil group of its Jacobian.