Equidistribution of CM points in the principal genus

Question

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.

Answer 1

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 $.

Answer 2

Vesselin Dimitrov gave a nice answer, but let me point out that for the OP's question one does not need the result of Harcos-Michel (2006). Instead, the original work of Duke (1988) or alternatively Duke-Friedlander-Iwaniec (1993) suffices.

Indeed, one only needs to detect cancellation in Weyl sums twisted by genus class characters, i.e. the real characters of the class group of $Q(\sqrt{-D})$. If $\chi$ is such a character, and $g$ is a Maass cusp form of level 1 (or an Eisenstein series of level 1 participating in the spectrum), then one needs a subconvex estimate $L(\tfrac{1}{2},g\otimes\chi)\ll|D|^c$ for some fixed $c<1/2$, with an implied constant depending polynomially on the Laplace eigenvalue of $g$. As $\chi$ is a genus class character, the $L$-value factors as $L(\tfrac{1}{2},g\otimes(\tfrac{\cdot}{D_1}))L(\tfrac{1}{2},g\otimes(\tfrac{\cdot}{D_2}))$, where $D_1$ and $D_2$ are certain fundamental quadratic discriminants satisfying $D_1D_2=-D$. As a result, the problem reduces to a subconvex bound $L(\tfrac{1}{2},g\otimes(\tfrac{\cdot}{D_i}))\ll|D_i|^c$, and this is covered by the earlier results mentioned above when $g$ is a Maass cusp form, and by Burgess's classical estimate when $g$ is an Eisenstein series.

For more details, see these concise notes (along with errata which only concern the equidistribution of closed geodesics on the modular surface).