Welcome to MathOverflow

Question

1

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?

edited Nov 9 2011 at 20:55

asked Nov 9 2011 at 20:44

Cyrus
269●1●4

	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 2011 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 2011 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 2011 at 4:52

Alexey Zaytsev · Answer 1 · 2012-10-21 12:18:55Z UTC

2

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/

answered Oct 21 at 12:18

Alexey Zaytsev
153●5

CM liftings of abelian varieties and liftings of Frobenius

Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)

1 Answer

You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you.