Formal subgroups of abelian varieties

Abstract.

In this paper, we generalize the result of [12] in the following sense. Let A be an abelian variety over a number field k, let ? be the Néron model of A over the ring of integers O _k of k. Completing ? along its zero section defines a formal group $\widehat{\mathcal{A}}$ over O _k . We prove that any formal subgroup of the generic fiber of $\widehat{\mathcal{A}}$ whose closure in $\widehat{\mathcal{A}}$ is smooth over an open subset of Spec O _k arises in fact from an abelian subvariety of A. The proof is of a transcendental nature and uses the Arakelovian formalism introduced by Bost [3].