On a Fiber Conjecture of Wan
Fuente:
arXiv
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| Autor principal: | |
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| Formato: | Preprint |
| Publicado: |
2023
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| Materias: | |
| Acceso en línea: | |
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| _version_ | 1866913259688296448 |
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| author | Schmidt, Matthew |
| author_facet | Schmidt, Matthew |
| contents | For a prime $p$ and $p$-power $q$, let $f(x)\in\mathbb{F}_q[x]$ with $\textrm{deg}\ f$ coprime to $p$. As $λ$ varies in $\overline{\mathbb{F}_p^\times}$, Wan has conjectured that the $p$-adic Newton polygon of the corresponding Artin-Schreier curve given by $λf$ is constant. That is, \[ \textrm{NP}(f) = \textrm{NP}(λf). \] In this paper, we prove this conjecture when $λ\in\mathbb{F}_p^\times$ and provide a detailed counterexample showing it is false in general. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_14819 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On a Fiber Conjecture of Wan Schmidt, Matthew Number Theory For a prime $p$ and $p$-power $q$, let $f(x)\in\mathbb{F}_q[x]$ with $\textrm{deg}\ f$ coprime to $p$. As $λ$ varies in $\overline{\mathbb{F}_p^\times}$, Wan has conjectured that the $p$-adic Newton polygon of the corresponding Artin-Schreier curve given by $λf$ is constant. That is, \[ \textrm{NP}(f) = \textrm{NP}(λf). \] In this paper, we prove this conjecture when $λ\in\mathbb{F}_p^\times$ and provide a detailed counterexample showing it is false in general. |
| title | On a Fiber Conjecture of Wan |
| topic | Number Theory |
| url | https://arxiv.org/abs/2311.14819 |