Hilbert class field of the imaginary quadratic field ${\mathbb{Q}}(\sqrt{-D})$. In Section \[sect:class\], we study when the value of $\tilde\Psi_m$ at a CM point $z$ associated to a quartic CM number field $K$ with totally real subfield $F$ generates the ‘CM class field’ $H_{\tilde K}$ of the reflex field $\tilde K$ (see Section 6 for precise definitions).

In particular, we show that it generates this CM class field when the CM value $\tilde\Psi_m({\mathcal{CM}}(K))$, the evaluation of $\tilde\Psi_m$ at the CM cycle ${\mathcal{CM}}(K)$ corresponding to $K$, is not an odd power in $F$ (Corollary \[clcor1.3\]). We also prove the following theorem (see Theorem \[cltheo6.6\] for details).

Let $F=\mathbb Q(\sqrt p)$ be a fixed real quadratic field with $p \equiv 1 \mod 4$ prime. Then there is a constant $d >0$ such that for any CM quartic field $K$ of discriminant $d_K=p^2 q$ with $q \equiv 1 \mod 4$ prime, and a CM point $z$ in the Hilbert modular surface $X=\hbox{SL}_2({\mathcal O}_F)
\backslash \mathbb H^2$ of CM type $(K, \Phi)$ by ${\mathcal O}_K$, one has:

1.  $M(\tilde\Psi_1(z), \tilde\Psi_2(z))$ is an unramified abelian extension of $M$, where $M$ is the smallest Galois extension of $\mathbb Q$ containing $K$.

2.  Let $H_{\tilde K}$ be the 