Number Theory Seminar

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, to what extent is an element of $Pic^{0}(E)$ determined by its restriction to $p$-power torsion subgroup schemes of $E$? I shall describe an answer to this question that involves a new construction of the $p$-adic height pairing on $$E$. This is joint work in progress with F. Castella and M. Ciperiani.