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arXiv · 2308.11250

Class fields and form class groups for solving certain quadratic Diophantine equations

Ho Yun Jung·Ja Kyung Koo·Dong Hwa Shin·Dong Sung Yoon
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Abstract

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.

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Ho Yun Jung, Ja Kyung Koo, Dong Hwa Shin, Dong Sung Yoon. 2023-08-22. Class fields and form class groups for solving certain quadratic Diophantine equations. https://arxiv.org/abs/2308.11250

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