Cubic and quartic points on modular curves using generalised symmetric Chabauty

Josha Box, Stevan Gajovic, Pip Goodman



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.
Originele taal-2English
StatusSubmitted - 16-feb.-2021

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