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

Let K be an imaginary quadratic field and $O$ be an order in K. We construct class fields associated with form class groups which are isomorphic to certain $O$-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 $x^2+n{y}^2$ 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.