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Let $F$ be a totally real number field of class number one, and let $K$ be aCM-field with $F$ as its maximal real subfield. For each positive integer $N$,we construct a class group of certain binary quadratic forms over $F$ which isisomorphic to the ray class group of $K$ modulo $N$. Assuming further that thenarrow class number of $F$ is one, we construct a class field of the reflexfield of $K$ in terms of the singular values of Hilbert modular functions.

Form class groups and class fields of CM-fields