Counterexamples to O'Neil's Period-Index Problem on Elliptic Curves

Abstract

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 $K=\mathbb{Q}({\zeta }_8)$, 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 $8$.