AI 中文总结
本文在去除 GRH 假设后，证明无有理复乘的数域上椭圆曲线的阿贝尔除子域素级的性质等价于无有理复乘，推广了相关研究结果。
AI 中文摘要
Enrique González-Jiménez 和 Álvaro Lozano-Robledo 证明，对定义在有理数域上的椭圆曲线 E/ℚ，其 n 除子域 ℚ(E[n])/ℚ 为阿贝尔时的级 n 存在一致界。在假设 GRH（广义黎曼假设）下，Allen 和 Genao 将该结果部分推广至素级，且允许无有理复乘的任意数域。本文去除 GRH 假设，证明 Allen 和 Genao 确立的性质实际等价于无有理复乘。
英文摘要
Enrique González-Jiménez and Álvaro Lozano-Robledo proved that there exists a uniform bound on the levels $n$ for which the $n$-division field $\mathbb{Q}\left(E\left[n\right]\right) / \mathbb{Q}$ of an elliptic curve $E/\mathbb{Q}$ defined over $\mathbb{Q}$ is abelian. Assuming GRH (Generalized Riemann Hypothesis), Allen and Genao partially generalized this result by restricting to prime levels while allowing arbitrary number fields without rationally defined CM. In this paper, we remove the GRH assumption and prove that the property established by Allen and Genao is in fact equivalent to the absence of rationally defined CM.
Comments6 pages. The e-mail address of Hansol Kim has been corrected to jawlang@ajou.ac.kr
