Class fields arising from the form class groups of order O and level N

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. By utilizing these form class groups, we first derive a congruence relation on special values of a modular function of higher level as an analogue of Kronecker's congruence relation. Furthermore, as an application of such class fields, for a positive integer $n$ we examine primes of the form $x^2+ny^2$ with some additional conditions.Comment: 30 page