Cubic and quartic points on modular curves using generalised symmetric Chabauty
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| Format: | Preprint |
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2021
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| _version_ | 1866929582339260416 |
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| author | Box, Josha Gajović, Stevan Goodman, Pip |
| author_facet | Box, Josha Gajović, Stevan Goodman, Pip |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2102_08236 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Cubic and quartic points on modular curves using generalised symmetric Chabauty Box, Josha Gajović, Stevan Goodman, Pip Number Theory 11G30, 11G35, 14H40, 14G05 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. |
| title | Cubic and quartic points on modular curves using generalised symmetric Chabauty |
| topic | Number Theory 11G30, 11G35, 14H40, 14G05 |
| url | https://arxiv.org/abs/2102.08236 |