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arXiv · 2102.08236

Cubic and quartic points on modular curves using generalised symmetric Chabauty

Josha Box·Stevan Gajović·Pip Goodman
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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.

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Josha Box, Stevan Gajović, Pip Goodman. 2021-02-16. Cubic and quartic points on modular curves using generalised symmetric Chabauty. https://arxiv.org/abs/2102.08236

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