Torsors under abelian schemes via Picard schemes

Abstract

Given a torsor $P$ under a principally polarised abelian scheme $J$, projective over a Noetherian base, we use the Picard scheme of $P$ to write down an explicit extension of $\mathbb{Z}$ by $J$ giving the class of $P$. As an application, we give a version of Bhatt's period-index result valid over an arbitrary Noetherian base.