Representation of the norm of ideals by quadratic forms with congruence conditions

Abstract

Using a correspondence between the narrow ray class group modulo $m$ of a quadratic field and a certain set of equivalence classes of binary quadratic forms proved by Furuta and Kubota, we find a quadratic form $f$ and a pair of integers $(x_1,y_1)$ such that the norm of all integral ideals $\mathfrak{a}$ in a ray class is represented by $f(mx+x_1,my+y_1)$ with some integers $(x,y)$.