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Assume I have an abelian variety $A$ over a finite field $k$ of characteristic $p$. Work of Norman and Oort (1980) says I can lift $A$ to an abelian variety $\mathscr{A}$ over some characteristic zero ring $R$ with a surjective morphism $R \to k$.

In fact, we can lift $A$ to the Witt ring $W(k)$ if $A$ is ordinary (Norman-Oort) or if $A$ is an elliptic curve (lift the cubic equation defining it).

1). In general, we don't expect to lift $A$ to $W(k)$. But can we always lift $A$ to some ring of integers $\mathcal{O}_K$, where $K/W(k)$ is a finite extension? (Such an extension must be totally ramified to ensure a surjection $K \to k$).

2). If not, is there data internal to $A$ that tells us about when such lifts do exist, analogous to $$\text{Ordinary } \implies \text{lifts to $W(k)$}?$$

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    $\begingroup$ Maybe I'm mis-remembering, because I don't deal with abelian varieties much, but isn't the deformation functor formally smooth, i.e. $Def_A \simeq Spf(W(k)[[x_1, \ldots, x_{g^2}]])$ even for non-ordinary ones? I think ordinary allows you to realize the isomorphism as canonically coming from the $p$-divisible group and hence gives you a special "canonical lift" to $W(k)$, but I think they always lift to $W(k)$. $\endgroup$
    – Matt
    Commented Apr 17, 2014 at 21:55
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    $\begingroup$ Smoothness is definitely true if $k$ is algebraically closed. I've been poking around to see if this is true in general and haven't found anything clear. $\endgroup$
    – John Binder
    Commented Apr 17, 2014 at 23:01
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    $\begingroup$ Algebraization of formal deformations is serious, even in the ordinary case (where it is affirmative, by an argument of Serre-Tate at the end of Messing's thesis). Beyond dimension 1 there are many non-algebraizable formal deformations over $W(k)$. The Norman-Oort paper assumes alg. closed $k$ (due Dieudonne modules), but this can be bypassed via a deformation ring argument (giving a lift over a $p$-adic order with residue field $k$, not that its normalization has that residue field!). Later work of Norman & Ogus gave obstructions, and showed a lift always exists over $W(k)[\sqrt{p}]$. $\endgroup$
    – user76758
    Commented Apr 18, 2014 at 1:41
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    $\begingroup$ @FilippoAlbertoEdoardo You're right. I just want a lift as abelian varieties. The only functoriality I'm looking for should be that the actions of the Galois groups (under $G(K) \twoheadrightarrow G(k)$) on the Tate modules should be isomorphic, but this should follow immediately if a lift exists. $\endgroup$
    – John Binder
    Commented Apr 18, 2014 at 14:05
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    $\begingroup$ If $A$ has a polarization of degree prime to $p$, then $A$ lifts to $W(k)$; this is another rather general case. $\endgroup$
    – Tim Dokchitser
    Commented Apr 19, 2014 at 14:29

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