Question

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.

Answer 1

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.