Let $F$ be a number field with ring of integers $O_F$, and suppose that $E/O_F$ is an abelian scheme. If $p$ is a prime of ordinary reduction of $E$, to what extent is an element of $Pic^0(E)$ determined by its restriction to $p$-power torsion subgroup schemes of $E$? This question is motivated by problems concerning the Galois structure of certain torsors obtained by dividing rational points on $E$. I shall discuss an answer that involves a new construction of the $p$-adic height pairing associated to $E$. This is joint work in progress with F. Castella and M. Ciperiani.