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arXiv · 2609.33027

Escape and Non-Escape of Mass for Monogenic Binary Cubic Forms

Petros Ploumidis
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Abstract

We study the distribution of monogenic integral binary cubic forms $f(x,y)$ with bounded discriminant and bounded monogenizer inside a fundamental domain for the action of $\mathrm{GL}_2(\mathbb{Z})$ on the space of real binary cubic forms. We show that when the monogenizer for $f$ is $[\pm1:0]$ (i.e., $f(\pm1,0)=1$), the corresponding set is fully concentrated in the cusp. We also prove a complementary result, namely, that when the monogenizers $[\pm1:0]$ (with height bounded in terms of the discriminant bound) are excluded, the resulting set has no escape of mass to the cusp. However, we surprisingly show that despite this non-escape of mass, the points in the fundamental domain are not equidistributed.

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Petros Ploumidis. 2026-09-26. Escape and Non-Escape of Mass for Monogenic Binary Cubic Forms. https://arxiv.org/abs/2609.33027

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