Let $A$, $B$ be abelian varieties over $\mathbb{Q}$, with corresponding Neron models $\mathcal{A}$, $\mathcal{B}$ over $X=Spec{\mathbb{Z}}$. Let $p$ be an odd prime of good reduction for both $A$ and $B$ such that we have an isomorphism $A[p]\cong B[p]$ over $\mathbb{Q}$. Keerthi Sampath has explained to me how this implies that we get an isomorphism $\mathcal{A}[p]\cong\mathcal{B}[p]$ of etale group schemes over $U=Spec{\mathbb{Z}[\frac{1}{p}]}=X-Spec{\mathbb{F}_p}$.

(a) I want to know if the same hypotheses as above also implies that we get an isomorphism $\mathcal{A}^0[p]\cong\mathcal{B}^0[p]$ on the identity components over $U$.

(b) Is it true that $\mathcal{A}^0[p]$ (the $p$-torsion on $\mathcal{A}^0$) is the same as $\mathcal{A}[p]^0$ (the identity component of $\mathcal{A}[p]$)?

`$\mathcal{A}[p]^0$`

in (b)? If $G$ is a group scheme of finite type over a field, $G^0$ is a well-defined open subgroup scheme of $G$. Over a more general base scheme $S$, all you can do is define $G^0$ as a subfunctor of $G$: a point of, say, $G(S)$ is in $G^0$ iff its restriction over every point $s$ is in $G_s^0$. This functor is representable if $G$ is smooth (e.g. Néron models) but not in general (think of $\mu_p$ over $\mathbb{Z}_p$). $\endgroup$ – Laurent Moret-Bailly Jun 5 '11 at 6:38