#### Translator’s note

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 \mathbb{F}_q. The result that we obtain was inspired by Ihara [2, ch. V] (see also ).

# 1

Let p be a prime number, \mathbb{F}_p the field \mathbb{Z}/(p), and \overline{\mathbb{F}}_p an algebraic closure of \mathbb{F}_p. For every power q of p, let \mathbb{F}_q be the subfield of q elements of \overline{\mathbb{F}}_p. For every algebraic extension k of \mathbb{F}_p, we denote by W_0(k) the discrete valuation Henselian ring essentially of finite type over \mathbb{Z}, absolutely unramified, with residue field k; let W(k) be the ring of Witt vectors over k, i.e. the completion of W_0(k). Let W=W(\overline{F}_p), and let \varphi be an embedding of W into the field \mathbb{C} of complex numbers. We denote by \mathbb{Z}(1) the subgroup 2\pi i\mathbb{Z} of \mathbb{C}. The exponential map defines an isomorphism between \mathbb{Z}(1)\otimes\mathbb{Z}_\ell and \mathbb{Z}_\ell(1)(\mathbb{C})=\varprojlim\mu_{\ell^n}(\mathbb{C}).

We denote by A^* the dual abelian variety of an abelian variety A. For every field k, we denote by \overline{k} the algebraic closure of k.

# 2

Let A be an abelian variety of dimension g, defined over a field k of characteristic p. Recall that A is said to be ordinary if any of the following equivalent conditions are satisfied:

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

If k=\mathbb{F}_q, and if F is the Frobenius endomorphism of A, and \mathrm{Pc}_A(F;x) is its characteristic polynomial, then these conditions are then equivalent to:

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

# 3

Let A be an ordinary abelian variety over \overline{\mathbb{F}}_p. We denote by \widetilde{A} the canonical Serre–Tate covering  of A over W. Recall that \widetilde{A} depends functorially on A, and is characterised by the fact that the p-divisible group T_p(\widetilde{A}) over W attached to \widetilde{A}  is the product of the p-divisible groups (uniquely determined, by §2.iii) that cover, respectively, the neutral component and the largest étale quotient of T_p(A). The canonical covering \widetilde{A} is again the unique covering of A such that every endomorphism of A lifts to \widetilde{A}. We denote by T(A) the integer homology of the complex abelian variety A_\mathbb{C} induced by \widetilde{A} and \varphi by the extension of scalars of W to \mathbb{C}: T(A) = H_1(\widetilde{A}\otimes_\varphi\mathbb{C}). We know that \widetilde{A} descends uniquely to W_0(\overline{F}_p), and so A_\mathbb{C} depends only on A and on the restriction of \varphi to W_0(\overline{F}_p). The free \mathbb{Z}-module T(A) is of rank 2\dim(A); it is functorial in A. Furthermore, if \ell\neq p is a prime number, then we have, functorially, that T(A)\otimes\mathbb{Z}_\ell = T_\ell(A). \tag{3.1}

The canonical covering of the dual abelian variety A^* of A is the dual of \widetilde{A}, and so (A_\mathbb{C})^*=A_\mathbb{C}^*, and T(A) and T(A^*) are in perfect duality with values in \mathbb{Z}(1): T(A)\otimes T(A^*) \to \mathbb{Z}(1) \tag{3.2} (it is necessary to use \mathbb{Z}(1) instead of \mathbb{Z} in order to obtain a theory that is invariant under complex conjugation). The pairings (3.2) are compatible, via (3.1), with the pairings T_\ell(A)\otimes T_\ell(A^*) \to \mathbb{Z}_\ell(1); a morphism \xi\colon A\to A^* defines a polarisation of A if and only if \xi_\mathbb{C}\colon A_\mathbb{C}\to A_\mathbb{C}^* defines a polarisation of A_\mathbb{C}. Set \begin{aligned} T'_p(A) &= \operatorname{Hom}(\mathbb{Q}_p/\mathbb{Z}_p,A(\overline{F}_p)) \\T''_p(A) &= \operatorname{Hom}_{\mathbb{Z}_p}(T'_p(A^*),\mathbb{Z}(1)\otimes\mathbb{Z}_p) \end{aligned} These \mathbb{Z}_p-modules are covariant functors in A.

By definition of the canonical covering, the p-divisible group T_p(\widetilde{A}) is the sum of the constant proétale group T'_p(A) and the Cartier dual of T'_p(A^*). For every morphism u\colon A\to B, the induced morphism u\colon T_p(\widetilde{A})\to T_p(\widetilde{B}) can be identified with the sum of u|T'_p(A)\colon T'_p(A)\to T'_p(B) and the Cartier transpose of u^t|T'_p(B^*)\colon T'_p(B^*)\to T'_p(A^*). Over \mathbb{C}, we canonically have that \mathbb{Z}(1)/(p^n)\sim\mu_{p^n}, whence an isomorphism of functors: T_{(p)}(A) = T(A)\otimes\mathbb{Z}_p = T'_p(A)\oplus T''_p(A). \tag{3.3}

# 4

Recall that, if \varphi\colon X\to Y is an isogeny between complex abelian varieties, then the exact homotopy sequence reduces to a short exact sequence: 0 \to H_1(X) \to H_1(Y) \to \operatorname{Ker}(\varphi) \to 0. The abelian varieties that are quotients of X by a finite subgroup, and these finite subgroups of X, correspond bijectively with the sub-lattice of H_1(X)\otimes\mathbb{Q} containing H_1(X).

Let A be an ordinary abelian variety over \overline{\mathbb{F}}_p. If n is an integer coprime to p, then the subschemes of finite groups of order n of A, of \widetilde{A}, and of A_\mathbb{C}, correspond bijectively, and also correspond to lattices R containing T(A) such that [R:T(A)]=n.

Set V'_p=T'_p(A)\otimes\mathbb{Q}_p and V''_p(A)=T''_p(A)\otimes\mathbb{Q}_p. The subschemes of finite groups of order p^k of A are products of a étale subgroup and an infinitesimal subgroup. The étale subgroups of order p^k of A correspond to those of subgroups of order p^k of A_\mathbb{C} such that the lattice R corresponding to T(A) is contained inside T_{(p)}(A)+V'_p(A). By duality, the infinitesimal subgroups of A correspond to the lattices R containing T(A) that are p-isogenous to T(A),, i.e. such that [R:T(A)] is a power of p and is contained in T_{(p)}(A)+V''_p(A).

All told, the finite subgroups of A^p, or the abelian varieties that are quotients of A, correspond bijectively to the lattices R containing T(A) such that R\otimes\mathbb{Z}_p = (R\otimes\mathbb{Z}_p \cap V'_p) + (R\otimes\mathbb{Z}_p \cap V''_p). \tag{4.1}

# 5

In particular, A^{(p)}, the quotient of A by the largest infinitesimal subgroup of A that is annihilated by p (for ordinary A), is defined by the lattice T(A)^{(p)} containing T(A) that is p-isogenous to T(A), and such that T(A)^{(p)}\otimes\mathbb{Z}_p = T'_p(A) + \frac1p T''_p(A).

# 6

Let A be an abelian variety over \mathbb{F}_q, and F\colon x\mapsto x^q its Frobenius endomorphism. Recall that A is uniquely determined by the pair (\overline{A},F) induced by (A,F) by extension of scalars from \mathbb{F}_q to \overline{\mathbb{F}}_q; the endomorphism F of \overline{A} factors as the relative Frobenius morphism F_\mathrm{r}^{(q)}\colon\overline{A}\to\overline{A}^{(q)} followed by an isomorphism F'\colon\overline{A}^{(q)}\to\overline{A}. If A is ordinary, then we denote by T(A) the \mathbb{Z}-module T(\overline{A}) endowed with the endomorphism F induced by the Frobenius endomorphism of A. By §5, the above, and (3.3), the lattices T(A) and F(T(A)) are p-isogenous, and we have that F(T'_p(A)) = T'_p(A), \tag{6.1} F(T''_p(A)) = qT''_p(A). \tag{6.2}

# 7

The functor A\mapsto(T(A),F) is an equivalence of categories between the category of ordinary abelian varieties over \mathbb{F}_q and the category of free \mathbb{Z}-modules T of finite type endowed with an endomorphism F that satisfy the following conditions:

1. F is semi-simple, and its eigenvalues have complex absolute value q^{\frac12},
2. at least half of the roots in \overline{\mathbb{Q}}_p of the characteristic polynomial of F are p-adic units; in other words, if T is of rank d, then the reduction \mod p of the polynomial \mathrm{Pc}_T(F;x) is not divisible by x^{[d/2]+1},
3. there exists an endomorphism V of T such that FV=q.

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

1. the module T\otimes\mathbb{Z}_p admits a decomposition, stable under F, into two sub-\mathbb{Z}_p-modules T'_p and T''_p of equal dimension, and such that F|T'_p is invertible, and F|T''_p is divisible by q.

Proof. A. We first prove that (a)+(b)+(c)\implies(d). If \alpha is a complex eigenvalue of F, then \overline{\alpha} is another, of the same multiplicity, and \alpha\overline{\alpha}=q. If we exclude those that are equal to \pm q^{\frac12}, then the eigenvalues of F in \mathbb{C}, and thus in \overline{\mathbb{Q}}_p, can be grouped into pairs of roots \alpha and q/\alpha. The roots \alpha and q/\alpha can not simultaneously be p-adic units, and so it follows from (b) that \pm q^{\frac12} is not an eigenvalue of F, that half of the eigenvalues of F in \overline{\mathbb{Q}}_p are p-adic units, say \alpha_1,\ldots,\alpha_{d/2}, and that the other half are of the form \beta_1=q/\alpha_1,\ldots,\beta_{d/2}=q/\alpha_{d/2}. Let T_{(p)}=T\otimes\mathbb{Z}_p, V_p=T\otimes\mathbb{Q}_p, V'_p the subspace of V_p given by the kernel of \prod_i(F-\alpha_i), and V''_p the kernel of the endomorphism \varphi=\prod_i(F-\beta_i). We have that V_p=V'_p\oplus V''_p. Let T'_p be the projection from T_{(p)} to V'_p, and let T''_p=T_{(p)}\cap V''_p. Since \varphi annihilates V''_p, and respects T, it sends T'_p to T_{(p)}\cap V'_p\subset T'_p. Also, \det(\varphi|V'_p)=\prod_{i,j}(\alpha_i-\beta_j) is a p-adic unit, and so \varphi(T'_p)=T'_p, and T_{(p)}\cap V'_p=T'_p, and so T_{(p)}=T'_p\oplus T''_p.

B. Full faithfulness. Let A and B be abelian varieties over \mathbb{F}_q, and let \psi be the arrow \psi\colon \operatorname{Hom}(A,B) \to \operatorname{Hom}_F(T(A),T(B)). By the theorem of Tate  and by (3.1), the arrow \psi_\ell\colon \operatorname{Hom}(A,B)\otimes\mathbb{Z}_\ell \to \operatorname{Hom}_F(T(A),T(B))\otimes\mathbb{Z}_\ell is an isomorphism for (\ell,p)=1, and so \psi\otimes\mathbb{Q} is an isomorphism. We know that \operatorname{Hom}(A,B) is torsion free, and so \psi is injective. Let u\colon A\to B be a morphism such that T(u) is divisible by n. The induced morphism u_\mathbb{C}\colon\overline{A}_\mathbb{C}\to\overline{B}_\mathbb{C} is thus divisible by n, and thus so too is \widetilde{u}\colon\widetilde{\overline{A}}\to\widetilde{\overline{B}} at the generic point of W. The kernel of multiplication by n is flat over W; \widetilde{u} thus disappears on this kernel, \widetilde{u} and u are divisible by n, and \psi is bijective.

C. Necessity. The fact that (T(A),F) satisfies (a) follows from Weil; condition (d), which implies (b) and (c), follows from §6.

D. Isogenies. Let (T_0,F) satisfy (a) and (d), and let T be a lattice in T_0\otimes\mathbb{Q}, stable under F, that also satisfies (d). Suppose that (T_0,F) is the image of an abelian variety A over \mathbb{F}_q; we will prove that (T,F) comes from an isogenous abelian variety. By T with \frac1k T, which is isomorphic to T, we can suppose that T\supset T_0. Condition (d) implies that T satisfies (4.1), and that T defines a subgroup H of \overline{A}, defined over \mathbb{F}_q, and such that (T,F)=T(A/H).

E. Surjectivity. The functor T induces a functor T_\mathbb{Q} from the category of isogeny classes of ordinary abelian varieties over \mathbb{F}_q to the category of finite-dimensional \mathbb{Q}-vector spaces endowed with an automorphism F that satisfies (a) and (b). By (D), it suffices to prove that this functor T_\mathbb{Q} is essentially surjective. It even suffices to show that every simple object (V,F) in the codomain is in the image. By Honda  (see also ), there exists an abelian variety A over \mathbb{F}_q such that the characteristic polynomial of the Frobenius F_A of A is a power of that of F. The third characterisation in §2 of ordinary abelian varieties shows that A is ordinary. Furthermore, (T(A)\otimes\mathbb{Q},F) is the sum of copies of (V,F), and thus, by (B), the isogeny class of the abelian variety A\otimes\mathbb{Q} is the sum of copies of an abelian variety B that satisfies T(B)\otimes\mathbb{Q}=(V,F).

# 8

Let (T,F) be a pair satisfying the hypotheses of the theorem, 2g the rank of T, A the corresponding abelian variety over \mathbb{F}_q, and A_\mathbb{C} the induced complex abelian variety (§3). We have that T= H_1(A_\mathbb{C}), and so T\otimes\mathbb{R} can be identified with the Lie algebra of A_\mathbb{C}, and is thus endowed with a complex structure. Here, thanks to J.-P. Serre, is how to reconstruct this complex structure in terms of T, F, and the restriction of \varphi to W_0(\mathbb{F}_p):

The complex structure on T\otimes\mathbb{R} defined above is characterised by the following properties:

1. The endomorphism F is \mathbb{C}-linear.
2. If v is the valuation of the algebraic closure \overline{\mathbb{Q}} of \mathbb{Q} in \mathbb{C} that extends the valuation of W_0(\mathbb{F}_p), then the valuations of the g eigenvalues of this endomorphism are strictly positive.

Proof. Condition (i) is evident, and condition (ii) follows from the fact that the action of F on the Lie algebra of A is congruent to zero \mod p. The uniqueness of a structure satisfying (i) and (ii) follows easily from condition (b), satisfied by (T,F).

# Bibliography


T. Honda. “Isogeny classes of abelian varieties over finite fields.” J. Math. Soc. Jap. 20 (1968), 83–95.

Y. Ihara. On congruence monodromy problems. University of Tokyo, 1968. 1.

Y. Ihara. “The congruence monodromy problems.” J. Math. Soc. Jap. 20 (1968), 107–121.

J. Lubin, J.-P. Serre, J. Tate. Elliptic curves and formal groups. Woods Hole Summer Institute, 1964.

J.-P. Serre. “Groups p-divisibles (d’après J. Tate).” Séminaire Bourbaki. 10 (1966-67).

J. Tate. “Classes d’isogénies de variétés abéliennes sur un corps fini (d’après T. Honda).” Séminaire Bourbaki. 11 (1968-69).

J. Tate. “Endomorphisms of abelian varieties over finite fields.” Inventiones Math. 2 (1966), 134–144.