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

TL;DR

This paper constructs class fields linked to form class groups in imaginary quadratic fields, enabling the explicit identification of primes of specific quadratic forms and establishing congruence relations for modular function values.

Contribution

It introduces a method to construct class fields from form class groups using Shimura's theory, with applications to prime representation and modular function congruences.

Findings

Identified primes of the form x^2 + ny^2 with conditions on x and y.

Derived a congruence relation on special values of higher-level modular functions.

Connected form class groups to class fields in imaginary quadratic fields.

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.