  We study the relationship between the period and the index of a principal
homogeneous space over an abelian variety, obtaining results which generalize
work of Cassels and Lichtenbaum on curves of genus one. In addition, we show
that the p-torsion in the Shafarevich-Tate group of a fixed abelian variety
over a number field k grows arbitrarily large when considered over field
extensions l/k of bounded degree. Essential use is made of an abelian variety
version of O'Neil's period-index obstruction.
