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Let $K\subset\mathbb{C}$ be a number field. There is a surjective map $\phi:\mathbb{P}^1(K)\to Cl(K)$ from the field to the class group, sending $[\alpha:\beta]$ to the class of the ideal $(\alpha,\beta)$. What does the fibre over an ideal class ``look like''?

I'm specifically interested in $K$ imaginary quadratic and whether or not there are circles in each of the fibres. For instance, $\mathbb{Q}=\mathbb{R}\cap K$ is contained in the fibre over the principal class.

Are there any elementary observations to make concerning $\phi$? Or references? Thanks.

Here are some pictures for reference, the two classes in $\mathbb{Q}(\sqrt{-5})$

http://math.colorado.edu/~rohi1040/randomshare/sqrt5princlass.pdf

http://math.colorado.edu/~rohi1040/randomshare/sqrt5otherclass.pdf

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As noted in one of the answers below, $SL_2(\mathcal{O}_K)$ acts on $\mathbb{P}^1(K)$ as fractional linear transformations and the fibres of $\phi$ are the $SL_2(\mathcal{O}_K)$-orbits. So, for instance, $\phi^{-1}(1)\cong SL_2(\mathcal{O}_K)/Stab(\infty)$. However, I don't think the fibres are isomorphic, and I don't know how to pick out the conjugacy classes of point-stabilizers (maximal elementary parabolic subgroups of $SL_2(\mathcal{O}_K)$?).

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The fibers are precisely the orbits of the natural action of $SL_2(\mathcal{O}_K)$ on $\mathbb{P}^1(K)$. See this note by Keith Conrad for details.

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  • $\begingroup$ Thanks for the response. That's why I'm interested in them (and how I made the pictures above). I'm looking for geodesic surfaces in the Bianchi orbifolds ($H^3/PSL_2(\mathcal{O}_K)$, $K$ imaginary quadratic) that live in one cusp (or in subsets of the cusps, etc.) $\endgroup$ – yoyo Jun 17 '16 at 22:25

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