Period, index and potential, III

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 (P, I) 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.