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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2601.16975 |
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| _version_ | 1866913125174870016 |
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| author | Bruin, Nils Creutz, Brendan |
| author_facet | Bruin, Nils Creutz, Brendan |
| contents | We describe a method to show a plane quartic over a number field has no rational points. The method can be adapted to show that a curve does not have divisors of degree 1 or 2 and can be generalized to arbitrary smooth projective curves. Our approach significantly improves on the applicability over previous 2-cover descent methods by not requiring the computation of the full $S$-unit group of the étale algebras involved. We illustrate the practicality with several examples, including examples where we determine plane quartics to be of index 2 or 4 when the maximum local index is strictly smaller. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_16975 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Explicit Brauer-Manin obstructions on plane quartics Bruin, Nils Creutz, Brendan Number Theory We describe a method to show a plane quartic over a number field has no rational points. The method can be adapted to show that a curve does not have divisors of degree 1 or 2 and can be generalized to arbitrary smooth projective curves. Our approach significantly improves on the applicability over previous 2-cover descent methods by not requiring the computation of the full $S$-unit group of the étale algebras involved. We illustrate the practicality with several examples, including examples where we determine plane quartics to be of index 2 or 4 when the maximum local index is strictly smaller. |
| title | Explicit Brauer-Manin obstructions on plane quartics |
| topic | Number Theory |
| url | https://arxiv.org/abs/2601.16975 |