# CM liftings of abelian varieties and liftings of Frobenius

It is well-known that if $A$ is an ordinary abelian variety over a finite perfect field $k$ of characteristic $p>0$ and $W=W(k)$ is the ring of Witt vectors over $k$, then the canonical lifting $A_{can}$ of $A$ to $W$ is characterized by the fact that every endomorphism $f$ of $A$ lifts to and endomorphism of $A_{can}$. In other words, the natural map $End(A_{can})----> End(A)$ is bijection.

Now is there a characterization of CM liftings of abelian varities (not necessarily ordinary) through liftings of endomorphisms? in particular is there a characterization of CM liftings based on liftings of Frobenius?

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In your discussion of canonical lifts, do you intend $A$ to be ordinary? (Not that it necessarily matters for your actual question ... .) –  Emerton Nov 9 '11 at 20:49
yes, your are right! I forgot to write ordinary. But in my question I really want to deal with general $A$ not necessarily ordinary. –  Cyrus Nov 9 '11 at 20:55
Your question seems unlikely to have a positive answer. Supersingular elliptic curves have CM liftings, and also non CM liftings, but if $k$ is sufficiently large then Frobenius will be a power of $p$. –  ulrich Nov 10 '11 at 4:52

It is possible to do when p-rank is coprime to p. In this case, a frobenius lift gives a full endomorphism algebra.

I also would recommend book "CM liftings" (B. Conrad, C-L. Chai, F. Oort) http://math.stanford.edu/~conrad/

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