It is well known that as the negative discriminant $-D$ goes to infinity, the number of quadratic forms of discriminant $-D$ belonging to the principal genus also goes to infinity. Can we say anything about the asymptotic equidistribution of the corresponding CM points on the classical modular curve? I know that Duke '88 proved equidistribution for all CM points and that this has since been refined in various ways, but I couldn't find any literature addressing this question.



Equidistribution of CM points in the principal genus
This is known, and follows from Theorem 2 in Harcos and Michel's paper *The subconvexity problem for Rankin-Selberg $L$-functions and equidistribution of Heegner points. II* (Invent. math., vol. **163**, 2006, pp. 581--655). This result states, more generally, that there exists an absolute positive constant $\epsilon > 0$ such that the incomplete Galois orbits of CM points for a subgroup of index $\leq D^{\epsilon}$ in the full Galois group are still equidistributed in the Poincare measure. (Much as in Michel's earlier theorem from the first paper of the same title, which was about a $p$-adic equidistribution of incomplete Galois orbits of singular moduli supersingular at $p$.) 


Just note that the number of genera is negligibly small: certainly $\ll\_{\epsilon} D^{\epsilon}$ for all $\epsilon >0 $. 


