Let $p$ be a prime and $F$ be an algebraic closure of the field with p elements. I will consider abelian varieties over F up to prime-to-$p$ isogeny. Principal polarizations will be $Q$-homogeneous principal polarizations in this category.
Let $A$ be a superspecial abelian variety over F, of dimension $g>1$ and endowed with a principal polarization $L$. Let $f$ : $A(p)\rightarrow A(p)$ be an automorphism of the p-divisible group of $A$, preserving the polarization form of $A(p)$ (that I still call $L$) up to a unit of $Z_{p}$. It is known (I think) that one can always find an abelian variety $A'$ over $F$, a principal polarization $L'$ of $A'$, and a quasi-isogeny of principally polarized abelian varieties $f_{ab}$ : $(A,L)\rightarrow(A',L')$ such that the $p$-divisible group of $(A',L')$ is $(A(p),L)$, and such that $f_{ab}$ induces the original map $f$ on $p$-divisible groups. (If someone has a reference for this, it would be helpful).
I have two questions:
1) Is is true in general that $f_{ab}$ is an isomorphism in the category of abelian varieties up to prime-to-$p$ isogeny? It seems to me that $f_{ab}$ need not to be an iso, even though A' and A are isomorphic.
2) Is it necessary true that the pair $(A,L)$ is isomorphic to the pair $(A',L')$ in the category of homogeneously principally polarized abelian varieties up to prime-to-$p$ isogeny? I think the answer is still no.
Thanks
