AI 中文总结
本文研究X₁(N)的孤立j-不变量的有限性，探讨其与领域内其他一致性问题的关联，得到Q中孤立j-不变量的新有限性结果，并将其应用于改进无复乘有理j-不变量椭圆曲线的挠群多项式界。
AI 中文摘要
对模曲线X₁(N)上的孤立点进行刻画，是对固定次数的所有点进行分类的关键障碍。这些点不属于无限参数化族，难以通过几何构造得到。本文聚焦于X₁(N)的“孤立j-不变量”集合，即孤立点映射到j-直线所得的值。作者与Ejder、Liu、Odumodu、Viray的前期工作提出，有界次数扩域中是否仅存在有限个孤立j-不变量的问题。本文探讨该问题与该领域其他一致性问题的关联，给出Q中孤立j-不变量的新有限性结果。作为应用，表明类似方法可对具有有理j-不变量的无复乘（non-CM）椭圆曲线的挠群给出更精确的多项式界。
英文摘要
Characterizing isolated points on the modular curve $X_1(N)$ is a key obstruction to classifying all points of a fixed degree. These points do not lie in infinite parameterized families, making them difficult to obtain through geometric constructions. In this paper, we focus on the collection of "isolated $j$-invariants" for $X_1(N)$, which are the values obtained by mapping isolated points to the $j$-line. Prior work of the author in collaboration with Ejder, Liu, Odumodu, and Viray asks whether there are only finitely many isolated $j$-invariants lying in extensions of bounded degree. Here, we explore how this question relates to other uniformity problems in the field and give new finiteness results for isolated $j$-invariants in $\mathbb{Q}$. As an application, we show similar methods give sharpened polynomial bounds on torsion for non-CM elliptic curves having rational $j$-invariant.
Comments15 pages. Comments welcome!
