Class fields and form class groups for solving certain quadratic Diophantine equations
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866910342663110656 |
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| author | Jung, Ho Yun Koo, Ja Kyung Shin, Dong Hwa Yoon, Dong Sung |
| author_facet | Jung, Ho Yun Koo, Ja Kyung Shin, Dong Hwa Yoon, Dong Sung |
| contents | Let $K$ be an imaginary quadratic field and $\mathcal{O}$ be an order in $K$. We construct class fields associated with form class groups which are isomorphic to certain $\mathcal{O}$-ideal class groups in terms of the theory of canonical models due to Shimura. As its applications, by using such class fields, for a positive integer $n$ we first find primes of the form $x^2+ny^2$ with additional conditions on $x$ and $y$. Second, by utilizing these form class groups, we derive a congruence relation on special values of a modular function of higher level as an analogue of Kronecker's congruence relation. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2308_11250 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Class fields and form class groups for solving certain quadratic Diophantine equations Jung, Ho Yun Koo, Ja Kyung Shin, Dong Hwa Yoon, Dong Sung Number Theory Primary 11R37, Secondary 11E12, 11R65 Let $K$ be an imaginary quadratic field and $\mathcal{O}$ be an order in $K$. We construct class fields associated with form class groups which are isomorphic to certain $\mathcal{O}$-ideal class groups in terms of the theory of canonical models due to Shimura. As its applications, by using such class fields, for a positive integer $n$ we first find primes of the form $x^2+ny^2$ with additional conditions on $x$ and $y$. Second, by utilizing these form class groups, we derive a congruence relation on special values of a modular function of higher level as an analogue of Kronecker's congruence relation. |
| title | Class fields and form class groups for solving certain quadratic Diophantine equations |
| topic | Number Theory Primary 11R37, Secondary 11E12, 11R65 |
| url | https://arxiv.org/abs/2308.11250 |