Algebraic functions and class number formulas
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866917103319121920 |
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| author | Akkarapakam, Sushmanth J. Morton, Patrick |
| author_facet | Akkarapakam, Sushmanth J. Morton, Patrick |
| contents | A class number formula is proved for extended ring class fields $L_{\mathcal{O},9}$ over imaginary quadratic fields $K_d = \mathbb{Q}(\sqrt{-d})$, in which the prime $p = 3$ splits, by determining the fields generated by the periodic points of a well-chosen algebraic function. The number of periodic points of a given period $n \ge 2$ for this algebraic function equals six times the sum of class numbers of imaginary quadratic orders $\textsf{R}_{-d}$, for which the Artin symbol for a prime ideal divisor $\wp_3$ in $K_d$ of $3$ has order $n$ in the Galois group of $F_d/K_d$, where $F_d$ is the inertia field of $\wp_3$ in $L_{\mathcal{O},9}/K_d$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_00583 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Algebraic functions and class number formulas Akkarapakam, Sushmanth J. Morton, Patrick Number Theory 11G15, 14H05, 11S05, 11S31 A class number formula is proved for extended ring class fields $L_{\mathcal{O},9}$ over imaginary quadratic fields $K_d = \mathbb{Q}(\sqrt{-d})$, in which the prime $p = 3$ splits, by determining the fields generated by the periodic points of a well-chosen algebraic function. The number of periodic points of a given period $n \ge 2$ for this algebraic function equals six times the sum of class numbers of imaginary quadratic orders $\textsf{R}_{-d}$, for which the Artin symbol for a prime ideal divisor $\wp_3$ in $K_d$ of $3$ has order $n$ in the Galois group of $F_d/K_d$, where $F_d$ is the inertia field of $\wp_3$ in $L_{\mathcal{O},9}/K_d$. |
| title | Algebraic functions and class number formulas |
| topic | Number Theory 11G15, 14H05, 11S05, 11S31 |
| url | https://arxiv.org/abs/2511.00583 |