# Canonical liftings of endomorphisms of ordinary abelian varieties

I am looking for a reference to the following well known" fact.

Let $k$ be a perfect field of prime characteristic $p$ and $W(k)$ its ring of Witt vectors. Let $A_0$ be an ordinary abelian variety over $k$ and let $A$ be an abelian scheme over $W(k)$ that is the canonical (Serre--Tate) lifting of $A_0$. Then every endomorphism $u_0$ of $A_0$ lifts to an endomorphism of $A$. In other words, the natural map $End(A) \to End(A_0)$ is bijective.

My problem is that I need it for infinite $k$. (I know a couple of references that deal with finite $k$.)

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have you asked frans oort? –  roy smith Aug 28 '11 at 2:54
@ roy smith: Yes - he gave me several references that deal with the case of finite $k$. –  Yuri Zarhin Aug 28 '11 at 18:10

A reference which proves a more general result is Theorem 1 of the appendix to the paper by Mehta and Srinivas "Varieties in positive characteristic with trivial tangent bundle." With an appendix by Srinivas and M. V. Nori. Compositio Math. 64 (1987), no. 2, 191–212.

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An almost reference (some assembly required) is N. Katz: Serre-Tate local moduli, Springer Lecture notes in Mathematics 868. By Lemma 1.1.2 an endomorphism lifts if and only if it lifts on the $p$-divisible group and for the canonical lift the endomorphism lifts trivially. I haven't looked at Drinfeld's original article to see if that is more suitable.

Addendum: There is no denying that my reference suffers in comparison with the two others given. I would like to point out however that Drinfeld's argument is very beautiful and arguably the slickest approach to canonical liftings (and Serre-Tate coordinates in general).

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The canonical reference is Messing, LNM 264 1972, Chapter V, 3.4, p174.

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