Assume that X and Y are abelian schemes (or even abelian varieties) over a base T. If $\ S --> T$ is a PD nilpotent thickening (i.e. the ideal of $\ S$ in $ \ T$ is a nilpotent divided power ideal) and S is of characteristic $\ p>0$ . If $\ X\_{0}$ and $ \ Y\_{0}$ are reductions of $\ X$ and $\ Y$ to $\ S$. If $\ f\_{0} : X\_{0} ---> Y\_{0} $ is an isomorphism over S, is this true that this isomorphism (if it can be lifted!) lifts to an isomorphism $\ f : X ---> Y$ ? one may also assume that $ \ H¹\_ {cris} (f\_{0})$ preserves the Hodge filtration. 



lifting the isomorphisms between abelian schemes over PD thickenings
No, this is very far from being true.


For a counterexample, let $S = Spec(\mathbb{Z}/p)$ and $T = Spec(\mathbb{Z}/p^2)$. The versal deformation space of an elliptic curve $E$ over $S$ is isomorphic to $Spec(\mathbb{Z}\_p[[x]])$ so lifts of $E$ to $T$ are parametrized by the set of homomorphisms of local algebras $Hom(\mathbb{Z}\_p[[x]], \mathbb{Z}/p^2) = p\mathbb{Z}/p^2$. So there do exist lifts for which the identity map of $E$ does not lift to an isomorphism.


