Local heights on hyperelliptic curves and quadratic Chabauty
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866915295841484800 |
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| author | Betts, L. Alexander Duque-Rosero, Juanita Hashimoto, Sachi Spelier, Pim |
| author_facet | Betts, L. Alexander Duque-Rosero, Juanita Hashimoto, Sachi Spelier, Pim |
| contents | Quadratic Chabauty is a $p$-adic method for determining rational points on curves. Local heights are arithmetic invariants used in the quadratic Chabauty method. We present an algorithm to compute these local heights for hyperelliptic curves at odd primes $\ell \neq p$. This algorithm significantly broadens the applicability of quadratic Chabauty to curves which were previously inaccessible due to the presence of non-trivial local heights. We provide numerous examples, including the first quadratic Chabauty computation for a curve having two primes with non-trivial local heights. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_05228 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Local heights on hyperelliptic curves and quadratic Chabauty Betts, L. Alexander Duque-Rosero, Juanita Hashimoto, Sachi Spelier, Pim Number Theory Algebraic Geometry Quadratic Chabauty is a $p$-adic method for determining rational points on curves. Local heights are arithmetic invariants used in the quadratic Chabauty method. We present an algorithm to compute these local heights for hyperelliptic curves at odd primes $\ell \neq p$. This algorithm significantly broadens the applicability of quadratic Chabauty to curves which were previously inaccessible due to the presence of non-trivial local heights. We provide numerous examples, including the first quadratic Chabauty computation for a curve having two primes with non-trivial local heights. |
| title | Local heights on hyperelliptic curves and quadratic Chabauty |
| topic | Number Theory Algebraic Geometry |
| url | https://arxiv.org/abs/2401.05228 |