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.

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