Period, index, and potential Sha

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.