An analogue of Girstmair's formula in function fields
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909215408259072 |
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| author | Shiomi, Daisuke |
| author_facet | Shiomi, Daisuke |
| contents | Suppose that $p$ is an odd prime and $g>1$ is a primitive root modulo $p$. Let $M$ be a number field contained in the $p$-th cyclotomic field. Girstmair found a surprising relation between the relative class number of $M$ and the digits of $1/p$ in base $g$. In this paper, we consider an analogue of Girstmair's formula in function fields. Suppose that $P \in \mathbb{F}_q[T]$ is monic irreducible and $G \in \mathbb{F}_q[T]$ is a primitive root modulo $P$. Let $L$ be an extension field of $\mathbb{F}_q(T)$ contained in the $P$-th cyclotomic function field. The goal of this paper is to give relations between the plus and minus parts of the divisor class number of $L$ and the digits of $1/P$ in base $G$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_01067 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | An analogue of Girstmair's formula in function fields Shiomi, Daisuke Number Theory Suppose that $p$ is an odd prime and $g>1$ is a primitive root modulo $p$. Let $M$ be a number field contained in the $p$-th cyclotomic field. Girstmair found a surprising relation between the relative class number of $M$ and the digits of $1/p$ in base $g$. In this paper, we consider an analogue of Girstmair's formula in function fields. Suppose that $P \in \mathbb{F}_q[T]$ is monic irreducible and $G \in \mathbb{F}_q[T]$ is a primitive root modulo $P$. Let $L$ be an extension field of $\mathbb{F}_q(T)$ contained in the $P$-th cyclotomic function field. The goal of this paper is to give relations between the plus and minus parts of the divisor class number of $L$ and the digits of $1/P$ in base $G$. |
| title | An analogue of Girstmair's formula in function fields |
| topic | Number Theory |
| url | https://arxiv.org/abs/2406.01067 |