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| Natura: | Preprint |
| Pubblicazione: |
2025
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| Accesso online: | https://arxiv.org/abs/2512.08250 |
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| _version_ | 1866915662320893952 |
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| author | Cho, Peter Jaehyun Yoo, Jinjoo |
| author_facet | Cho, Peter Jaehyun Yoo, Jinjoo |
| contents | Let $\ell$ be an odd prime, $q$ an odd prime power such that $q \not\equiv 0 \pmod \ell$, and $m$ the order of $q$ in $\F_\ell^\times$. We propose an explicit $L$-polynomial of hyperelliptic function field $K:=\F_q(T, \sqrt[\ell]{T^2+aT+b})$ with $a, b \in \F_q$ and $a^2-4b \ne 0$. Using our formula, we obtain the explicit closed formula for the class number of $K$, where $m$ is even or $m=\frac{\ell-1}{2}$.As an application, we compute the average class numbers for hyperelliptic function fields with genus up to $3$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_08250 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The $L$-polynomial of hyperelliptic function fields and its applications Cho, Peter Jaehyun Yoo, Jinjoo Number Theory 11R29, 11G30, 11M06, 11G20 Let $\ell$ be an odd prime, $q$ an odd prime power such that $q \not\equiv 0 \pmod \ell$, and $m$ the order of $q$ in $\F_\ell^\times$. We propose an explicit $L$-polynomial of hyperelliptic function field $K:=\F_q(T, \sqrt[\ell]{T^2+aT+b})$ with $a, b \in \F_q$ and $a^2-4b \ne 0$. Using our formula, we obtain the explicit closed formula for the class number of $K$, where $m$ is even or $m=\frac{\ell-1}{2}$.As an application, we compute the average class numbers for hyperelliptic function fields with genus up to $3$. |
| title | The $L$-polynomial of hyperelliptic function fields and its applications |
| topic | Number Theory 11R29, 11G30, 11M06, 11G20 |
| url | https://arxiv.org/abs/2512.08250 |