Let $S$ be a smooth quasi-projective variety over $\mathbb{Z}$, and $A$ an abelian scheme over an open subset $U \subset S$. Then

1. Can $A$ be extended to an abelian scheme over $S$?
2. If $A$ comes equipped with a principal polarization $\lambda$, and $A$ extends to an abelian scheme $B$ over $S$, can we extend $\lambda$ to a principal polarization on $B$?

The Faltings-Chai and Vasiu-Zink criteria tell us how to answer (1) when $S$ is defined over $\mathbb{Q}$, or $\mathbb{Z}_{(p)}$ for a prime $p$, respectively. Then one should be able to spread out an abelian scheme defined over a stalk $\mathcal{O}_{S,s}$ to an abelian scheme over an open neighbourhood of $s$. Glueing these abelian schemes together is then possible since their generic fibres are isomorphic, and we can spread this isomorphism out to some open subset too.

As for (2), any rational map $\lambda: B \to B'$ between abelian schemes over $S$ is actually a morphism. Then, is this morphism also a principal polarization?