论文标题
椭圆曲线的非主持分裂场
Non-monogenic Division Fields of Elliptic Curves
论文作者
论文摘要
For various positive integers $n$, we show the existence of infinite families of elliptic curves over $\mathbb{Q}$ with $n$-division fields, $\mathbb{Q}(E[n])$, that are not monogenic, i.e., the ring of integers does not admit a power integral basis.我们明确对其中一些家庭进行参数。此外，我们表明，没有CM的每个$ E/\ Mathbb {Q} $都有无限的许多非单性分区领域。我们的主要技术结合了对杜克和托斯获得的Frobenius的全球描述，以及基于Dedekind的思想的简单算法。
For various positive integers $n$, we show the existence of infinite families of elliptic curves over $\mathbb{Q}$ with $n$-division fields, $\mathbb{Q}(E[n])$, that are not monogenic, i.e., the ring of integers does not admit a power integral basis. We parametrize some of these families explicitly. Moreover, we show that every $E/\mathbb{Q}$ without CM has infinitely many non-monogenic division fields. Our main technique combines a global description of the Frobenius obtained by Duke and Tóth with a simple algorithm based on ideas of Dedekind.
