# Ordinary abelian varieties over a finite field

*1969*

#### Translator’s note

*This page is a translation into English of the following:*

Deligne, P. “Variétés abéliennes ordinaires sur un corps fini.” *Inventiones Math.* **8** (1969), 238–243. publications.ias.edu/node/352

*The translator (Tim Hosgood) takes full responsibility for any errors introduced, and claims no rights to any of the mathematical content herein.*

Version: `f36a214`

We give here a down-to-earth description of the category of ordinary abelian varieties over a finite field

# 1

Let

We denote by

# 2

Let *ordinary* if any of the following equivalent conditions are satisfied:

A hasp^g points of order dividingp with values in\overline{k} .- The “Hasse-Witte matrix”
F^*\colon H^1(A^{(p)},{\mathscr{O}}_{A^{(p)}}) \to H^1(A,{\mathscr{O}}_A) is invertible. - The neutral component of the group scheme
A_p that is the kernel of multiplication byp is of multiplicative type (and thus geometrically isomorphic to a power of\mu_p ).

If

- At least half of the roots of
\mathrm{Pc}_A(F;X) in\overline{\mathbb{Q}}_p arep -adic units. In other words, ifn=\dim A , then the reduction\mod p of the polynomial\mathrm{Pc}_A(F;x) is not divisible byx^{n+1} .

# 3

Let

The canonical covering of the dual abelian variety

By definition of the canonical covering, the

# 4

Recall that, if

Let

Set

All told, the finite subgroups of

# 5

In particular,

# 6

Let

# 7

The functor

F is semi-simple, and its eigenvalues have complex absolute valueq^{\frac12} ,- at least half of the roots in
\overline{\mathbb{Q}}_p of the characteristic polynomial ofF arep -adic units; in other words, ifT is of rankd , then the reduction\mod p of the polynomial\mathrm{Pc}_T(F;x) is not divisible byx^{[d/2]+1} , - there exists an endomorphism
V ofT such thatFV=q .

If condition (a) is satisfied, then conditions (b) and (c) are equivalent to the following:

- the module
T\otimes\mathbb{Z}_p admits a decomposition, stable underF , into two sub-\mathbb{Z}_p -modulesT'_p andT''_p of equal dimension, and such thatF|T'_p is invertible, andF|T''_p is divisible byq .

*Proof*. A. We first prove that (a)+(b)+(c)

B. *Full faithfulness.*
Let

C. *Necessity.*
The fact that

D. *Isogenies.*
Let

E. *Surjectivity.*
The functor

# 8

Let

The complex structure on

- The endomorphism
F is\mathbb{C} -linear. - If
v is the valuation of the algebraic closure\overline{\mathbb{Q}} of\mathbb{Q} in\mathbb{C} that extends the valuation ofW_0(\mathbb{F}_p) , then the valuations of theg eigenvalues of this endomorphism are strictly positive.

*Proof*. Condition (i) is evident, and condition (ii) follows from the fact that the action of

# Bibliography

*J. Math. Soc. Jap.*

**20**(1968), 83–95.

*On congruence monodromy problems*. University of Tokyo, 1968.

**1**.

*J. Math. Soc. Jap.*

**20**(1968), 107–121.

*Elliptic curves and formal groups*. Woods Hole Summer Institute, 1964.

*Séminaire Bourbaki*.

**10**(1966-67).

*Séminaire Bourbaki*.

**11**(1968-69).

*Inventiones Math.*

**2**(1966), 134–144.