Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2023
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2311.07837 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866911790697283584 |
|---|---|
| author | Koo, Ja Kyung Shin, Dong Hwa Yoon, Dong Sung |
| author_facet | Koo, Ja Kyung Shin, Dong Hwa Yoon, Dong Sung |
| contents | Let $N$ be a positive integer and $Γ$ be a subgroup of $\mathrm{SL}_2(\mathbb{Z})$ containing $Γ_1(N)$. Let $K$ be an imaginary quadratic field and $\mathcal{O}$ be an order of discriminant $D_\mathcal{O}$ in $K$. Under some assumptions, we show that $Γ$ induces a form class group of discriminant $D_\mathcal{O}$ (or of order $\mathcal{O}$) and level $N$ if and only if there is a certain canonical model of the modular curve for $Γ$ defined over a suitably small number field. In this way we can find an interesting link between two different subjects, which will be useful in the study of certain quadratic Diophantine equations in terms of primes $p$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_07837 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Gauss's form class groups and Shimura's canonical models Koo, Ja Kyung Shin, Dong Hwa Yoon, Dong Sung Number Theory Primary 11R37, Secondary 11E12, 11R65 Let $N$ be a positive integer and $Γ$ be a subgroup of $\mathrm{SL}_2(\mathbb{Z})$ containing $Γ_1(N)$. Let $K$ be an imaginary quadratic field and $\mathcal{O}$ be an order of discriminant $D_\mathcal{O}$ in $K$. Under some assumptions, we show that $Γ$ induces a form class group of discriminant $D_\mathcal{O}$ (or of order $\mathcal{O}$) and level $N$ if and only if there is a certain canonical model of the modular curve for $Γ$ defined over a suitably small number field. In this way we can find an interesting link between two different subjects, which will be useful in the study of certain quadratic Diophantine equations in terms of primes $p$. |
| title | Gauss's form class groups and Shimura's canonical models |
| topic | Number Theory Primary 11R37, Secondary 11E12, 11R65 |
| url | https://arxiv.org/abs/2311.07837 |