Papers › Cubic and quartic points on modular curves using generalised symmetric Chabauty
Cubic and quartic points on modular curves using generalised symmetric Chabauty
Josha Box, Stevan Gajović, Pip Goodman
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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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