Any map $A \to B$ of abelian varieties of the same dimension over a global field $K$ induces a map $\mathcal{A} \to \mathcal{B}$ on the corresponding Neron models over $X$ (where $X=Spec{\mathcal{O}\_K}$ for any number field $K$ or $X$ is a complete smooth curve over a perfect field with function field $K$) which further restricts to a map $\mathcal{A}^0 \to \mathcal{B}^0$ on the respective identity components. Thus we have a map 


$${\textrm{Res}}^0:{\textrm{Hom}}\_K(A,B) \cong {\textrm{Hom}}\_X(\mathcal{A},\mathcal{B}) \to {\textrm{Hom}}\_X(\mathcal{A}^0,\mathcal{B}^0)$$


My question is - are there maps from $\mathcal{A}^0$ to $\mathcal{B}^0$ over $X$ that do not arise from any map over $K$ from $A$ to $B$ (equivalently, from any map over $X$ from $\mathcal{A}$ to $\mathcal{B}$)? In other words, is ${\textrm{Res}}^0$ surjective? I'm particularly interested to know if it is possible to have an isogeny of identity components that does not lie in the image of ${\textrm{Res}}^0$ above.



Maps on the identity components of Neron models
If I'm not mistaken, the map $Res^0$ is a bijection. Indeed, restriction induces maps 
$$Hom\_X(\mathcal A,\mathcal B) \to Hom\_X(\mathcal A^0,\mathcal B^0)\to Hom\_K(A,B).$$
The composite is a bijection, by the universal mapping property of Neron models,
and both maps are injective, since $A$ is Zariski dense in $\mathcal A^0$, which in turn
is Zariksi dense in $\mathcal A$. Thus both maps are bijections, and hence $Res^0$ is
a bijection.


