Isomorphism on p-torsion of Neron models 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 $\mathcal{A}[p]$)?
 A: (a) is not going to be true in general (and as Moret-Bailly says, (b) needs some more explanation). 
Here's the "moral" reason (a) isn't true. Let $E$ be an elliptic curve over the rationals. If $E[p^n]$ is unramified at $\ell$ for all $n\geq1$, then $E$ will have good reduction. But if $E[p]$ is unramified then this is not enough -- this is "level-lowering" a la Mazur/Ribet, which was crucial in the proof that STW -> FLT. So if you take, for example, a counterexample to FLT and let $E$ be the associated Frey curve, then $E[p]$ will be unramified at all odd $\ell\not=p$ but $E$ could still have bad reduction at many such $\ell$.
But one does not need a counterexample to FLT to build such a phenomenon. One just needs two elliptic curves $A$ and $B$ with $A$ having good reduction and $B$ having, say, multiplicative reduction at $\ell$, such that $A$ and $B$ are congruent mod $p$. Then the $j$-invariant of $B$ will have valuation at $\ell$ a multiple of $p$, so the number of components will be a multiple of $p$, and this will give you your counterexample (if I've understood the question correctly) because over $\mathbf{Z}_\ell$ the identity component of the special fibre of $B$ is missing some $p$-torsion.
My memory is that Ribet-Stein is full of explicit examples of these phenomena.
