发表机构
Nanjing University（南京大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对O'Neil提出的椭圆曲线齐性空间提升的周期-指数障碍阶的问题，本文通过构造显式反例给出否定回答，证明在数域$\mathbb{Q}(ζ_8)$上存在无穷多椭圆曲线及其齐性空间不满足该猜想。
AI 中文摘要
设$E/K$为数域$K$上的椭圆曲线，$[C]\in H^1(K,E(\overline{K}))$为$E$下的齐性空间。假设$C$的周期为$n$、指数为$d$。O'Neil提出问题：是否总能选取$[C]$到$H^1(K,E[n])$的一个提升，使得其周期-指数障碍的阶恰好为$d/n$。我们通过构造一族显式例子给出了否定回答。取$K=\mathbb{Q}(ζ_8)$，我们证明存在无穷多条两两不同构的椭圆曲线$E/K$，使得每条$E$都有无穷多个齐性空间$[C]\in H^1(K,E(\overline{K}))$满足周期为8、指数为16，但$[C]$到$H^1(K,E[8])$的每个提升的周期-指数障碍的阶都恰好为8。
英文摘要
Let $E/K$ be an elliptic curve over a number field $K$, and let $[C]\in H^1(K,E(\overline{K}))$ be a homogeneous space under $E$. Suppose that $C$ has period $n$ and index $d$. O'Neil asked whether one can always choose a lift of $[C]$ to $H^1(K,E[n])$ whose period-index obstruction has order exactly $d/n$. We give a negative answer to this question by constructing an explicit family of examples.Taking $K=\mathbb{Q}(ζ_8)$, we prove that there exist infinitely many pairwise non-isomorphic elliptic curves $E/K$ such that each $E$ admits infinitely many homogeneous spaces $[C]\in H^1(K,E(\overline{K}))$ with period $8$ and index $16$, but the period-index obstruction of every lift of $[C]$ to $H^1(K,E[8])$ has order exactly $8$.
