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Let $S$ be a smooth quasi-projective scheme over $\mathbb{Z}$, and $A$ an abelian scheme over an open subset $U \subset S$. Suppose that $S\backslash U$ has codimension at least $2$ and that for every point $s \in S$, the ramification index at the stalk $\mathcal{O}_{S,s}$ is $1$, so that the hypotheses for the Faltings-Chai and Vasiu-Zink criteria hold at each stalk. 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 $B$?

Take a generic point $\eta$ of an irreducible component of $S \backslash U$. The stalk $\mathcal{O}_{S,\eta}$ has a map $f:\textrm{Spec}(\mathcal{O}_{S,\eta}) \to S$ and we have an abelian scheme $f^*A$ defined outside the maximal point of $\mathcal{O}_{S,\eta}$. We are in a position to apply either Faltings-Chai or Vasiu-Zink to extend $f^*A$ uniquely to an abelian scheme $A'_{\eta}$ over the stalk. We can then spread this out to an abelian scheme $A'_{U_{\eta}}$ over some open $U_{\eta}$ containing $\eta$.

For any other point $\zeta$ so that $\eta \in \overline{\zeta}$, we know $\zeta \in U \cap U_{\eta}$ and if $g: \textrm{Spec}(\mathcal{O}_{S,\zeta}) \to S$ is the natural map then $g^*A \cong g^*A'_{U_{\eta}}$ (abusing notation slightly). We can spread this out to an isomorphism of abelian schemes $g_{\zeta} : A |_{W_{\zeta}} \to A'_{U_{\eta}}|_{W_{\zeta}}$ over some open $W_{\zeta}$ around $\zeta$.

As long as the cocycle conditions are satisfied, we would be able to glue $A$ and $A_{U_{\eta}}'$, after potentially shrinking $U_{\eta}$, to an abelian scheme over a strictly larger open $U \cup U_{\eta}$. The logic behind the previous paragraph ensures that $\eta$ remains in $U_{\eta}$ after shrinking. Now a Zorn's lemma type argument allows us to conclude (1).

The cocycle conditions hold at least generically (after taking generic fibre above $S$), so by rigidity of abelian schemes, I suspect that they hold, and the question (1) is indeed true.

As for (2), the line bundle $L$ on $A$ which induces the principal polarization can be extended over $B$ to some $\overline{L}$, and we obtain a morphism $\phi_{\overline{B}}:B \to B^{\vee}$. I suspect that relative ampleness can be verified pointwise on $S$, and supposing that $\overline{L}$ is a polarization, it must be principal since it is so generically.

Let $S$ be a smooth quasi-projective scheme over $\mathbb{Z}$, and $A$ an abelian scheme over an open subset $U \subset S$. Suppose that $S\backslash U$ has codimension at least $2$ and that for every point $s \in S$, the ramification index at the stalk $\mathcal{O}_{S,s}$ is $1$, so that the hypotheses for the Faltings-Chai and Vasiu-Zink criteria hold at each stalk. 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 $B$?

Take a generic point $\eta$ of an irreducible component of $S \backslash U$. The stalk $\mathcal{O}_{S,\eta}$ has a map $f:\textrm{Spec}(\mathcal{O}_{S,\eta}) \to S$ and we have an abelian scheme $f^*A$ defined outside the maximal point of $\mathcal{O}_{S,\eta}$. We are in a position to apply either Faltings-Chai or Vasiu-Zink to extend $f^*A$ uniquely to an abelian scheme $A'_{\eta}$ over the stalk. We can then spread this out to an abelian scheme $A'_{U_{\eta}}$ over some open $U_{\eta}$ containing $\eta$.

For any other point $\zeta$ so that $\eta \in \overline{\zeta}$, we know $\zeta \in U \cap U_{\eta}$ and if $g: \textrm{Spec}(\mathcal{O}_{S,\zeta}) \to S$ is the natural map then $g^*A \cong g^*A'_{U_{\eta}}$ (abusing notation slightly). We can spread this out to an isomorphism of abelian schemes $g_{\zeta} : A |_{W_{\zeta}} \to A'_{U_{\eta}}|_{W_{\zeta}}$ over some open $W_{\zeta}$ around $\zeta$.

As long as the cocycle conditions are satisfied, we would be able to glue $A$ and $A_{U_{\eta}}'$, after potentially shrinking $U_{\eta}$, to an abelian scheme over a strictly larger open $U \cup U_{\eta}$. The logic behind the previous paragraph ensures that $\eta$ remains in $U_{\eta}$ after shrinking. Now a Zorn's lemma type argument allows us to conclude (1).

The cocycle conditions hold at least generically (after taking generic fibre above $S$), so by rigidity of abelian schemes, I suspect that they hold, and the question (1) is indeed true.

As for (2), the line bundle $L$ on $A$ which induces the principal polarization can be extended over $B$ to some $\overline{L}$, and we obtain a morphism $\phi_{\overline{L}}:B \to B^{\vee}$. I suspect that relative ampleness can be verified pointwise on $S$, and supposing that $\overline{L}$ is a polarization, it must be principal since it is so generically.

Let $S$ be a smooth quasi-projective scheme over $\mathbb{Z}$, and $A$ an abelian scheme over an open subset $U \subset S$ so that $U$ is surjective over $\mathbb{Z}$. 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?

Let $S$ be a smooth quasi-projective scheme over $\mathbb{Z}$, and $A$ an abelian scheme over an open subset $U \subset S$. Suppose that $S\backslash U$ has codimension at least $2$ and that for every point $s \in S$, the ramification index at the stalk $\mathcal{O}_{S,s}$ is $1$, so that the hypotheses for the Faltings-Chai and Vasiu-Zink criteria hold at each stalk. 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 $B$?

Take a generic point $\eta$ of an irreducible component of $S \backslash U$. The stalk $\mathcal{O}_{S,\eta}$ has a map $f:\textrm{Spec}(\mathcal{O}_{S,\eta}) \to S$ and we have an abelian scheme $f^*A$ defined outside the maximal point of $\mathcal{O}_{S,\eta}$. We are in a position to apply either Faltings-Chai or Vasiu-Zink to extend $f^*A$ uniquely to an abelian scheme $A'_{\eta}$ over the stalk. We can then spread this out to an abelian scheme $A'_{U_{\eta}}$ over some open $U_{\eta}$ containing $\eta$.

For any other point $\zeta$ so that $\eta \in \overline{\zeta}$, we know $\zeta \in U \cap U_{\eta}$ and if $g: \textrm{Spec}(\mathcal{O}_{S,\zeta}) \to S$ is the natural map then $g^*A \cong g^*A'_{U_{\eta}}$ (abusing notation slightly). We can spread this out to an isomorphism of abelian schemes $g_{\zeta} : A |_{W_{\zeta}} \to A'_{U_{\eta}}|_{W_{\zeta}}$ over some open $W_{\zeta}$ around $\zeta$.

As long as the cocycle conditions are satisfied, we would be able to glue $A$ and $A_{U_{\eta}}'$, after potentially shrinking $U_{\eta}$, to an abelian scheme over a strictly larger open $U \cup U_{\eta}$. The logic behind the previous paragraph ensures that $\eta$ remains in $U_{\eta}$ after shrinking. Now a Zorn's lemma type argument allows us to conclude (1).

The cocycle conditions hold at least generically (after taking generic fibre above $S$), so by rigidity of abelian schemes, I suspect that they hold, and the question (1) is indeed true.

As for (2), the line bundle $L$ on $A$ which induces the principal polarization can be extended over $B$ to some $\overline{L}$, and we obtain a morphism $\phi_{\overline{B}}:B \to B^{\vee}$. I suspect that relative ampleness can be verified pointwise on $S$, and supposing that $\overline{L}$ is a polarization, it must be principal since it is so generically.

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?

Let $S$ be a smooth quasi-projective scheme over $\mathbb{Z}$, and $A$ an abelian scheme over an open subset $U \subset S$ so that $U$ is surjective over $\mathbb{Z}$. 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?