# Reference for a theorem of Tate on the endomorphism rings of AVs over finite fields

Let $k$ be a finite field, and $A$, $B$ abelian varieties over $k$. Let $T_p(A)$, $T_p(B)$ be the (contravariant) Dieudonn\'e modules associated to the p-divisible groups attached to $A$ and $B$, respectively. The theorem of Tate in the question is that the natural map

${\rm Hom}_k(A,B)\otimes\mathbf{Z}_p\longrightarrow{\rm Hom}(T_p(B),T_p(A))$

is an isomorphism.

It seems to me that the standard reference for the proof of this theorem is a paper of Tate titled Endomorphisms of abelian varieties over finite fields. II that should have appeared 50 years ago in Inventiones but, I am right, never did.

Is there another reference in the literature for the proof of this theorem?

Thanks.

• Well, may be I should have said 40 years go – Tommaso Centeleghe May 24 '11 at 9:30
• Minor point: when $A$ and $B$ are not the same variety, one usually uses $\operatorname{Hom}$ instead of $\operatorname{End}$ – S. Carnahan May 24 '11 at 10:03
• Thanks. You are perfectly right. I guess I was thinking to the case A=B... I'll fix it. – Tommaso Centeleghe May 24 '11 at 11:08

## 3 Answers

I think the result appears in:

Waterhouse, W. C.; Milne, J. S.: Abelian varieties over finite fields. 1969 Number Theory Institute (Proc. Sympos. Pure Math., Vol. XX, State Univ. New York, Stony Brook, N.Y., 1969), pp. 53–64. Amer. Math. Soc., Providence, R.I., 1971

I guess $p=\operatorname{char}(k)$. For another (unified) proof of Tate's theorem (that works for primes $\ell\ne p$ and $\ell=p$) see arXiv:0711.1615 [math.AG]; MR2484084 (2010a:11117).

• Welcome to MathOverflow, Professor Zarhin. I've taken the liberty of adding an explicit link to your Arxiv preprint. – David Loeffler May 24 '11 at 15:40
• Thanks for the comment on the characteristic of $k$ being $p$. And thanks as well for the interesting reference! – Tommaso Centeleghe May 24 '11 at 18:49
• As I was browsing through the preprint, it took me a few moments to realize why the author wasn't calling it 'Zarhin's trick'. – Keerthi Madapusi Pera May 27 '11 at 14:46

Is it not treated by CP Ramanujan in an appendix to Mumford's book on abelian varieties ?

• I don't know. I will take a look though, thanks. – Tommaso Centeleghe May 27 '11 at 10:29
• @Dalawat: The appendix in Mumford's book treats the $\ell \neq p$ case (Galois modules), but the OP's question seems to be about the $p$-divisible group case. – SGP May 27 '11 at 11:52