发表机构
University of Zagreb Faculty of Science（萨格勒布大学理学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文确定了五次数域上椭圆曲线所有可能的挠群，除已知无限多出现的群外，恰好新增三个群，并改进了基于Hecke筛和尖点除子类算术的方法。
AI 中文摘要
我们确定了所有在五次数域上的椭圆曲线的挠群中出现的群。除了由Derickx和Sutherland确定的那些无限多次出现的群之外，恰好有三个群出现：$\mathbb{Z}/28\mathbb{Z}$、$\mathbb{Z}/30\mathbb{Z}$和$\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/18\mathbb{Z}$。在配对$(K,E)$的同构意义下，第一个和第三个各由一条椭圆曲线实现，第二个由两条曲线实现，这两条曲线在一个公共的五次域上是$2$-同源的。实现$\mathbb{Z}/28\mathbb{Z}$和$\mathbb{Z}/30\mathbb{Z}$的曲线由van Hoeij发现，而群$\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/18\mathbb{Z}$是新的，并且$5$是出现非循环零星挠群的最小次数。这些方法改进了Derickx和Najman开发的方法，基于Hecke筛和尖点除子类的算术。
英文摘要
We determine all the groups that appear as the torsion group of an elliptic curve over a quintic number field. Apart from the groups that already occur infinitely often, which were determined by Derickx and Sutherland, exactly three groups occur: $\mathbb{Z}/28\mathbb{Z}$, $\mathbb{Z}/30\mathbb{Z}$ and $\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/18\mathbb{Z}$. Up to isomorphism of the pair $(K,E)$, the first and the third are each realised by a single elliptic curve and the second by two curves, which are $2$-isogenous over a common quintic field. The curves realising $\mathbb{Z}/28\mathbb{Z}$ and $\mathbb{Z}/30\mathbb{Z}$ were found by van Hoeij, while the group $\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/18\mathbb{Z}$ is new, and $5$ is the smallest degree in which a non-cyclic sporadic torsion group occurs. The methods improve on those developed by Derickx and Najman, and are based on Hecke sieves and the arithmetic of cuspidal divisor classes.
Comments27 pages
