Period, index and potential, III

Abstract

We present three results on the period-index problem for genus-one curves over global fields. Our first result implies that for every pair of positive integers $\left(P,I\right)$ such that $I$ is divisible by $P$ and divides $P^2$, there exists a number field $K$ and a genus-one curve $C_{∕K}$ with period $P$ and index $I$. Second, let $E_{∕K}$ be any elliptic curve over a global field $K$, and let $P>1$ be any integer indivisible by the characteristic of $K$. We construct infinitely many genus-one curves $C_{∕K}$ with period $P$, index $P^2$, and Jacobian $E$. Our third result, on the structure of Shafarevich–Tate groups under field extension, follows as a corollary. Our main tools are Lichtenbaum–Tate duality and the functorial properties of O’Neil’s period-index obstruction map under change of period.