Boundary CM points and class groups of small exponent

Let $\mathcal{F}$ denote the fundamental domain for ${\text{SL}}_2(\mathbb{Z})$ on the upper half plane $\mathcal{H}$. William Duke showed that as fundamental discriminants $D\to -\mathrm{\infty }$, the sets ${\mathrm{C}\mathrm{M}}_D$ (CM points of discriminant $D$) are equidistributed in $\mathcal{F}$. In this paper, we investigate the behavior of CM points on the boundary of $\mathcal{F}$. We prove that such CM points are equidistributed on the boundary, and also give a complete characterization of when every ${\mathrm{C}\mathrm{M}}_D$ point lies on the boundary. Along the way, we also (conditionally) give a complete classification of negative discriminants with class group of small exponent.