  In this paper we advance the theory of O'Neil's period-index obstruction map
and derive consequences for the arithmetic of genus one curves over global
fields. Our first result implies that for every pair of positive integers (P,I)
with P dividing I and I dividing P^2, there exists a number field K and a genus
one curve C over K with period P and index I. Second, let E 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 over K
with period P, index P^2, and Jacobian E. We deduce strong consequences on the
structure of Sharevich-Tate groups under field extension.
