A result of Serre-Tate states that we can canonically lift an ordinary abelian variety over a perfect field $k$ of positive characteristic to an abelian scheme over the ring of Witt vectors of $k$ and that we can also lift an endomorphism of the variety uniquely to an endomorphism of the canonical lifting.
In the case where $k$ is finite, is the canonical lift a variety? Do we know its dimension? I'd be glad to read your answers and the references to those answers as well.
As you just said, the canonical lift is an abelian scheme over the ring of Witt vectors $W(k)$. Now, if $k$ is finite of characteristic $p$, $W(k)$ is the ring of integers of the unramified extension (say $K$) of $\mathbb{Q}_p$ whose residue field is $k$. The canonical lift has a generic fiber which is an abelian variety over $K$ and has the same dimension as the abelian variety over $k$ you started with. I guess there is a bit of ambiguity in the literature where some people refer to the abelian scheme over $W(k)$ as the canonical lift whereas others refer to its generic fiber over the field of fractions of $W(k)$ as the canonical lift. You can recover the former from the latter by taking Néron models.
