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| Format: | Preprint |
| Veröffentlicht: |
2023
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| Schlagworte: | |
| Online-Zugang: | https://arxiv.org/abs/2307.04864 |
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| _version_ | 1866912138174398464 |
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| author | Derickx, Maarten Najman, Filip |
| author_facet | Derickx, Maarten Najman, Filip |
| contents | Let $X_Δ(N)$ be an intermediate modular curve of level $N$, meaning that there exist (possibly trivial) morphisms $X_1(N)\rightarrow X_Δ(N) \rightarrow X_0(N)$. For all such intermediate modular curves, we give an explicit description of all primes $p$ such that $X_Δ(N)_{\overline{\mathbb F}_p}$ is either hyperelliptic or trigonal. Furthermore we also determine all primes $p$ such that $X_Δ(N)_{\mathbb F_p}$ is trigonal.
This is done by first using the Castelnuovo-Severi inequality to establish a bound $N_0$ such that if $X_0(N)_{{\overline{\mathbb F}_p}}$ is hyperelliptic or trigonal, then $N \leq N_0$. To deal with the remaining small values of $N$, we develop a method based on the careful study of the canonical ideal to determine, for a fixed curve $X_Δ(N)$, all the primes $p$ such that the $X_Δ(N)_{ {\overline{\mathbb F}_p}}$ is trigonal or hyperelliptic.
Furthermore, using similar methods, we show that $X_Δ(N)_{{\overline{\mathbb F}_p}}$ is not a smooth plane quintic, for any $N$ and any $p$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_04864 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Hyperelliptic and trigonal modular curves in characteristic $p$ Derickx, Maarten Najman, Filip Number Theory Let $X_Δ(N)$ be an intermediate modular curve of level $N$, meaning that there exist (possibly trivial) morphisms $X_1(N)\rightarrow X_Δ(N) \rightarrow X_0(N)$. For all such intermediate modular curves, we give an explicit description of all primes $p$ such that $X_Δ(N)_{\overline{\mathbb F}_p}$ is either hyperelliptic or trigonal. Furthermore we also determine all primes $p$ such that $X_Δ(N)_{\mathbb F_p}$ is trigonal. This is done by first using the Castelnuovo-Severi inequality to establish a bound $N_0$ such that if $X_0(N)_{{\overline{\mathbb F}_p}}$ is hyperelliptic or trigonal, then $N \leq N_0$. To deal with the remaining small values of $N$, we develop a method based on the careful study of the canonical ideal to determine, for a fixed curve $X_Δ(N)$, all the primes $p$ such that the $X_Δ(N)_{ {\overline{\mathbb F}_p}}$ is trigonal or hyperelliptic. Furthermore, using similar methods, we show that $X_Δ(N)_{{\overline{\mathbb F}_p}}$ is not a smooth plane quintic, for any $N$ and any $p$. |
| title | Hyperelliptic and trigonal modular curves in characteristic $p$ |
| topic | Number Theory |
| url | https://arxiv.org/abs/2307.04864 |