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] (see also [3]).
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:
- A has p^g points of order dividing p 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 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:
- 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 [4] 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} [5] 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:
- F is semi-simple, and its eigenvalues have complex absolute value q^{\frac12},
- 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},
- 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:
- 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 [7] 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 [1] (see also [6]), 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:
- 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 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).