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arXiv · 2605.26020

Boundary CM points and class groups of small exponent

David Aiken·Erick Ross·Dmitriy Shvydkoy·Hui Xue
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Abstract

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 $D \to -\infty$, the sets $\mathrm{CM}_{D}$ (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 $\mathrm{CM}_D$ point lies on the boundary. Along the way, we also (conditionally) give a complete classification of negative discriminants with class group of small exponent.

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David Aiken, Erick Ross, Dmitriy Shvydkoy, Hui Xue. 2026-05-25. Boundary CM points and class groups of small exponent. https://arxiv.org/abs/2605.26020

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