On the Rational Cuspidal Divisor Class Groups of Drinfeld Modular Curves $X_0(\mathfrak{p}^r)$
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arXiv
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866929478877315072 |
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| author | Ho, Sheng-Yang Kevin |
| author_facet | Ho, Sheng-Yang Kevin |
| contents | Let $\mathcal{C}(\mathfrak{p}^r)$ be the rational cuspidal divisor class group of the Drinfeld modular curve $X_0(\mathfrak{p}^r)$ for a prime power level $\mathfrak{p}^r\in \mathbb{F}_q[T]$. We relate the rational cuspidal divisors of degree $0$ on $X_0(\mathfrak{p}^r)$ with $Δ$-quotients, where $Δ$ is the Drinfeld discriminant function. As a result, we are able to determine explicitly the structure of $\mathcal{C}(\mathfrak{p}^r)$ for arbitrary prime $\mathfrak{p}\in \mathbb{F}_q[T]$ and $r\geq 2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_09796 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the Rational Cuspidal Divisor Class Groups of Drinfeld Modular Curves $X_0(\mathfrak{p}^r)$ Ho, Sheng-Yang Kevin Number Theory Let $\mathcal{C}(\mathfrak{p}^r)$ be the rational cuspidal divisor class group of the Drinfeld modular curve $X_0(\mathfrak{p}^r)$ for a prime power level $\mathfrak{p}^r\in \mathbb{F}_q[T]$. We relate the rational cuspidal divisors of degree $0$ on $X_0(\mathfrak{p}^r)$ with $Δ$-quotients, where $Δ$ is the Drinfeld discriminant function. As a result, we are able to determine explicitly the structure of $\mathcal{C}(\mathfrak{p}^r)$ for arbitrary prime $\mathfrak{p}\in \mathbb{F}_q[T]$ and $r\geq 2$. |
| title | On the Rational Cuspidal Divisor Class Groups of Drinfeld Modular Curves $X_0(\mathfrak{p}^r)$ |
| topic | Number Theory |
| url | https://arxiv.org/abs/2301.09796 |