单基因二元三次形式的逃逸与非逃逸质量
Escape and Non-Escape of Mass for Monogenic Binary Cubic Forms
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中文总结 AI 辅助
本文研究单基因整二元三次形式在基本域中的分布，证明当单基因子为特定值时质量集中于尖点，排除后无逃逸但非等分布。
中文摘要 AI 辅助
我们研究了在 $\nmathrm{GL}_2(\nmathbb{Z})$ 作用于实二元三次形式空间的基本域内，具有有界判别式和有界单基因子的单基因整二元三次形式 $f(x,y)$ 的分布。我们证明了当 $f$ 的单基因子为 $[\npm1:0]$（即 $f(\npm1,0)=1$）时，相应的集合完全集中于尖点。我们还证明了一个互补的结果，即当单基因子 $[\npm1:0]$（其高度以判别式界为界）被排除时，所得集合没有质量逃逸到尖点。然而，我们令人惊讶地表明，尽管没有质量逃逸，基本域中的点并不是等分布的。
英文摘要
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.
