On a remark of Serre
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866914514059919360 |
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| author | Ahrens, Katie Grantham, Jon |
| author_facet | Ahrens, Katie Grantham, Jon |
| contents | Inspired by a remark of Serre, we extend the search for primes $p$ such that the maximum Hasse bound for the number of points on an elliptic curve over $\mathbb{F}_{p^5}$ is not achieved. We then give a list of all $q<10^{70}$ such that the Hasse bound is not achieved over $\mathbb{F}_{q}$. We explore the heuristics for how many such numbers should exist in each case. Finally, look at similar criteria for genus $2$ and $3$ curves. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_11048 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On a remark of Serre Ahrens, Katie Grantham, Jon Number Theory 11G20, 14H52, 11Y55, and 11A41 Inspired by a remark of Serre, we extend the search for primes $p$ such that the maximum Hasse bound for the number of points on an elliptic curve over $\mathbb{F}_{p^5}$ is not achieved. We then give a list of all $q<10^{70}$ such that the Hasse bound is not achieved over $\mathbb{F}_{q}$. We explore the heuristics for how many such numbers should exist in each case. Finally, look at similar criteria for genus $2$ and $3$ curves. |
| title | On a remark of Serre |
| topic | Number Theory 11G20, 14H52, 11Y55, and 11A41 |
| url | https://arxiv.org/abs/2508.11048 |