(Cross-posted from <https://math.stackexchange.com/questions/2029407/ray-class-groups-through-binary-quadratic-forms>)


If $d$ is the discriminant of a quadratic number field, then the primitive classes of binary quadratic forms of discriminant $d$ form a group isomorphic to the narrow ideal class group of $\mathbb{Q}(\sqrt{d})$. The classes of forms are equivalence classes under an action by $\mathrm{SL}\_2(\mathbb{Z})$. My question is, is there a collection of subgroups of $\mathrm{SL\_2}(\mathbb{Z})$ for which the sets of classes of forms of discriminant $d$ under action by these groups are in bijection with the narrow ray class groups of $\mathbb{Q}(\sqrt{d})$? 


(I would hope that Gauss's composition operation, which is already well-defined at the level of classes under the action of $\begin{bmatrix} 1 & \mathbb{Z} \\ 0 & 1 \end{bmatrix}$, would provide a group operation that makes these bijections into isomorphisms of ray class groups.) 



Ray class groups through binary quadratic forms
I guess that the answer is no. The binary forms with discriminant 
$\Delta \cdot f^2$ describe ring class fields modulo $f$ (see Cox's book), so by taking the limit as $f \to \infty$ something like the idelic version of the full ring class group will result, and this is what I expect you will get using your weak equivalence. For proving that you cannot get anything beyond ring class fields you probably should look at G. Bruckner, *Charakterisierung der galoisschen Zahlkörper, deren zerlegte Primzahlen durch binäre quadratische Formen gegeben sind*, Math. Nachr. 32 (1966), 317-326, where the proof in the finite case is given.


