Computing abelian varieties over finite fields isogenous to a power

In this paper we give a module-theoretic description of the isomorphism classes of abelian varieties $A$ isogenous to $B^r$, where the characteristic polynomial $g$ of Frobenius of $B$ is an ordinary square-free $q$-Weil polynomial, for a power $q$ of a prime $p$, or a square-free $p$-Weil polynomial with no real roots. Under some extra assumptions on the polynomial $g$ we give an explicit description of all the isomorphism classes which can be computed in terms of fractional ideals of an order in a finite product of number fields. In the ordinary case, we also give a module-theoretic description of the polarizations of $A$.


Introduction
It is well known that abelian varieties of dimension g over the complex numbers can be functorially described by full lattices L ⊂ C g and that such a description becomes an equivalence of categories when we only consider the lattices L such that the associated torus C g /L admits a Riemann form. When we move to the wilder realm of positive characteristic we cannot have such a functorial description due to the existence of objects like supersingular elliptic curves whose endomorphisms form a quaternionic algebra which does not admit a 2dimensional representation, as pointed out by Serre. Nevertheless, when we are working over a finite field F q , with q a power of a prime p, we have analogous descriptions if we restrict ourselves to some subcategories of the category of abelian varieties over finite fields. More precisely, Deligne proved in [Del69] that there is an equivalence between the category of ordinary abelian varieties over F q and the category of finitely generated free Z-modules with an endomorphism satisfying some easy-to-state axioms. This description has been extended by Centeleghe and Stix in [CS15] for abelian varieties over the prime field F p whose characteristic polynomial of Frobenius does not have real roots. In the ordinary case, Howe has extended this equivalence to include the notions of dual variety and polarizations, see [How95].
In [Mar18b] we have used such descriptions to produce algorithms to compute the isomorphism classes of abelian varieties with square-free characteristic polynomial of Frobenius and, when applicable, the polarizations and the corresponding automorphism groups. The algorithms make use of the fact that the target category of Deligne's and Centeleghe-Stix functors is equivalent to a category of fractional ideals of a certain order in the étale algebra Q[x]/(h), where h is the characteristic polynomial.
In the present paper we extend such a description to the case when the characteristic polynomial h is a power of a square-free polynomial, say h = g r . Instead of fractional ideals we will have to consider lattices in K r with an Rmodules structure, where K = Q[x]/(g ) and R = Z[x, y]/(g (x), x y − q). In the ordinary case we translate the notion of a polarization to this context.
When the order R is Bass there is a classification of such modules, see [Bas63] and [LW85], and we can explicitly compute representatives of the isomorphism classes of the abelian varieties.
There are other categorical descriptions, which we do not make use of, of the category of abelian varieties isogenous to a power of elliptic curves, see the Appendix in [Lau02], [Kan11] and [JKP + 17].
The paper is structured as follows. In Section 2 we recall the notion of an order and a fractional ideal, with a focus on Bass orders. In Section 3 we describe the categorical equivalences that we are going to use in Section 4, where we focus on the case of abelian varieties with characteristic polynomial of the form h = g r , with g square-free. In Section 5 we translate the notion of a polarization into our module-theoretic language. Finally, in Section 6 we apply our description and present the results of some computations.

Acknowledgments
The author would like to thank Jonas Bergström for helpful discussions and Rachel Newton and Christophe Ritzhentaler for comments on a previous version of the paper, which is part of the author's Ph.D thesis [Mar18a].

Conventions
All rings considered are commutative and unital. All morphisms between abelian varieties A and B over a field k are also defined over k, unless otherwise specified. In particular, we write Hom(A, B ) for Hom k (A, B ). Also, an abelian variety A is simple if it is so over the field of definition.

Orders
Let g be an integral square-free monic polynomial, say of degree n. Let K be the étale Q-algebra Q[x]/(g ). Note that K is a finite product of distinct number fields. An order R in K is a subring of K whose additive group is isomorphic to Z n . Among all orders in K there exists a maximal one with respect to inclusion, which is called the maximal order of K and is denoted O K . An over-order of R is an order S in K containing R. Since the quotient O K /R is finite there are only finitely many over-orders of R. A fractional ideal of R is a finitely generated sub-R-module of K containing a non-zero-divisor. Given two fractional R-ideals I and J , we have that I + J , I ∩ J ,I J , (I : J ) and I t are also fractional R-ideals. Recall that the quotient ideal (I : J ) and the trace dual ideal I t are defined respectively as (I : Observe that the underlying additive subgroup of any fractional ideal I is a free abelian group of rank n, that is I is a lattice in K . Recall, that if I = α 1 Z⊕. . .⊕α n Z then ]. An order R is called Bass if every overorder of R is Gorenstein. Since in this paper we will extensively use the properties of Bass orders we will list here other equivalent definitions.
Proposition 2.1. Let R be an order. The following are equivalent: • R is Bass (every over-order is Gorenstein); • every fractional R-ideal can be generated by 2 elements; • R is a cyclic index order, that is the finite R-module O K /R is cyclic.
The study of such orders started with the paper [Bas63] on Gorenstein rings. There are many sources where one can find a proof of Proposition 2.1 (and other characterizations), for example [LW85, Theorem 2.1]. Since every fractional ideal of a quadratic order can be generated by 2 elements as an abelian group, they are examples of Bass orders.
Given an order R we define the ideal class monoid as and the ideal class group as where the operations are induced by ideal multiplication. We will denote the class of the ideal I by [I ]. Note that ICM(R) ⊇ Pic(R) with equality if and only if R = O K . In general we have that where the disjoint union is taken over the over-orders of R, with equality if and only if R is Bass. In particular, if this is the case, once we have a complete list of over-orders of R, it is easy to compute all the ideal classes of R, using the results from [KP05]. For more about the computation of ICM(R), even in the non-Bass case, we refer to [Mar18c].

Definition 2.2. Let R be an order in K and let B(r ) be the category of torsion-free R-modules M such that M ⊗K is a free K -module of rank r together with R-linear morphisms.
Crucial for our purpose is the fact that, when R is a Bass order, the modules in B(r ) can be written in a canonical form in terms of over-orders of R and fractional ideals.

Theorem 2.3. Let R be a Bass order and let M be in B(r ).
Then there are fractional R-ideals I 1 , . . . , I r with (I 1 : The isomorphism class of M is uniquely determined by the chain of over-orders (I i : I i ) and the isomorphism class [I 1 · · · I r ].
This result was first proved in [Bas62, Theorem 1.7] and then proved with a different method in [BF65,Theorem 8]. It was generalized to Bass rings in [LW85, Theorem 7.1]. As an immediate consequence of Theorem 2.3 we get that M can be written in a canonical form Proof. The statement follows from the fact that Hom In particular, for M = S 1 ⊕ . . . ⊕ S r −1 ⊕ I as above we have If R is a Bass order and M and N are two modules in B(r ), it is easy using Theorem 2.3 to check whether they are isomorphic. If this is the case, it is possible to explicitly construct a matrix A 0 realizing the isomorphism, as the next example shows.
Example 2.5 ([BF65, Lemma 8]). Let I 1 and I 2 be fractional R-ideals with multiplicator rings S 1 and S 2 , respectively, with S 1 ⊆ S 2 . Then by the classification given in Theorem 2.3 we have an R-linear isomorphism We want to exhibit a matrix A 0 realizing the isomorphism. Since I 1 is invertible in S 1 , there are elements c 1 and c 2 in K such that c1I 1 + c 2 I 2 = S 1 . So we can assume that I 1 and I 2 are coprime in S 1 . Thus there are a 1 ∈ I 1 and a 2 ∈ I 2 such that 1 = a 1 + a 2 . Then it is easy to check that the matrix where the action is on column vectors).

The category of abelian varieties over a finite field
Let q be a power of a prime number p and let AV(q) be the category of abelian varieties defined over F q . For A in AV(q) consider the induced action of the Frobenius endomorphism on the l -adic Tate modules T l A, for any prime l = p, and let h A be the corresponding characteristic polynomial.
Recall that for a simple abelian variety B in AV(q) the polynomial h B is a power of an irreducible polynomial, say m a , and the exponent a is uniquely determined by the p-adic factorization of m, see [WM71, Theorem 8].
Using this recipe, we can list all characteristic polynomials h of the Frobenius of abelian varieties over a finite field F q of a given dimension g , for example see [Hal10] for g = 3 and [HS12] for g = 4. By Honda-Tate theory, see [Tat66] and [Hon68], this corresponds to describing all isogeny classes of abelian varieties in AV(q) of a given dimension g . For such a polynomial h, denote by AV(h) the full subcategory of AV(q) whose objects are the abelian varieties in the isogeny class determined by h.
We will restrict our attention to two subcategories of AV(q). Recall that an abelian variety A over F q is called ordinary if exactly half of the roots of h A over Q p are p-adic units. There are many other characterizations of ordinary abelian varieties. For example see [Del69, Section 2]. We will denote the full sub-category of AV(q) consisting of ordinary abelian varieties by AV ord (q). We will also consider the subcategory AV cs (p) of abelian varieties A over the prime field F p such that h A has no real roots, that is h A ( p) = 0. We will give functorial descriptions of AV ord (q) and AV cs (p) in terms of Z-lattices with extra structure. More precisely, consider the following categories: • the category M ord (q) consisting of pairs (T, F ) where T is a free-finitely generated Z-module and F is a Z-linear endomorphism of T such that the action of F ⊗ Q on T ⊗ Z Q is semisimple; the eigenvalues of F ⊗ Q have complex absolute value q; half of the roots of the characteristic polynomial of F ⊗ Q over Q p are units; there exists an endomorphism V of T such that F V = q; • the category M cs (p) consisting of pairs (T, F ) where T is a free-finitely generated Z-module and F is a Z-linear endomorphism of T such that the action of F ⊗ Q on T ⊗ Z Q is semisimple; the eigenvalues of F ⊗ Q have complex absolute value p; the characteristic polynomial of F ⊗ Q has no real roots; there exists an endomorphism V of T such that F V = p.
In both categories, a morphism (T, The main tools to understand the categories AV ord (q) and AV cs (p) are given in the following theorem.

Abelian varieties isogenous to a power
Let h be a characteristic polynomial of an abelian variety in AV ord (q) or AV cs (p). Assume moreover that h = g r for some square-free polynomial g in Z[x]. Put K = Q[x]/(g ) and α = x mod (g ). Denote with R the order Z[α, q/α] in K (with q = p if we are in AV cs (p)). Observe that the order R is Gorenstein, see [CS15, Theorem 11].

where each S i is an over-order of R and [I ] denotes the isomorphism class of a fractional S r -ideal I .
Proof. Denote by M (h) the image of AV(h) via F ord (or F cs ). We will define an equivalence G : M (h) → B(r ). Take A in AV(h) and let (T, F ) be the image of A in M (h) via F ord (or F cs ). The minimal polynomial of the Q-linear endomorphism F ⊗ Q of T ⊗ Q is g . So by definition of M ord (q) (or M cs (p)) we have that F and V induce on T an R-module structure via the isomorphism R ≃ Z[F,V ] given by α → F . Denote this R-module by M and put G ((T, F )) = M . Observe that the action of F on T is faithful, scince it becomes multiplication by q (or by p) after composing with V , and hence M is torsion free. Let's prove that M ⊗ R K is a free K -module of rank r . Since g is square-free, it is a product of distinct irreducible polynomials, say g = g 1 · · · g s . In particular, K is isomorphic to the product of number Let e i be the image in K of the multiplicative unit of K i under this isomorphism, so that 1 K = e 1 + . . . + e s and K e i ≃ K i for each i . Hence Since the action of F ⊗ Q is semisimple, there is a direct sum decomposition T ⊗ Z Q = W 1 ⊕ . . . ⊕ W s such that the action of F ⊗ Q on each W i is simple. This means that, after renumbering, we can assume that the minimal polynomial of F ⊗ Q| W i is g i and that dim Q W i = r deg(g i ).
Since deg(g i ) = dim Q K e i it follows that dim K e i (M ⊗ R K e i ) = r and hence, by taking the direct sum over i , we obtain an isomorphism Therefore M is in B(r ). It is clear by construction that G is a fully faithful and essentially surjective functor. Define F as the composition of the equivalences F ord (or F cs ) and G . In particular F is an equivalence as well and we have concluded the proof of part (a). Part (b) now follows directly from Theorem 2.3.

Corollary 4.2. Assume that R is a Bass order. Then every abelian variety A in
for some abelian varieties B i in AV(g ).  [NR67] and in [Hae90] in the local case and in [HL88] it is described how to go from the local case to the global case. We have not analyzed if those exceptions could arise from orders generated by Weil polynomials, which could potentially extend our description to more isogeny classes.

Polarizations
In this section we will continue using the same notation as in Section 4, but we will restrict to the case when h is ordinary. Our goal is to describe what the polarizations of an abelian variety A in AV(h) correspond to in the category B(r ) via the equivalence F of Theorem 4.1.(b). Note that K is a CM-algebra, that is, there is an involution a → a that acts as complex conjugation after composing with any non-zero homomorphism ϕ : K → C. In particular, we have that α = q/α. Observe that the homomorphisms K → C come in conjugate pairs. We call a choice of half of these homomorphisms, one for each conjugate pair, a CM-type of K . For every R-module M in B(r ), since we can identify M with a sub-R-module of K r , we have an induced action M → M. Moreover, if we consider M as a submodule of K r , we see that the trace Tr K /Q : K → Q induces a non-degenerate bilinear form Tr on M by where we think of all vectors in K r as columns vectors. In analogy to the r = 1 case, when M is a fractional R-ideal, we define the trace dual M t of M to be the dual module with respect to Tr. In particular, if n = deg(h) and we fix a Z-basis where α * i is the dual basis characterized by Tr K /Q (α i α * j ) = 1 if i = j and 0 otherwise. Since the action of F ∨ on T ∨ is "pre-composition with V " and V = F , we see that it will correspond, via G , to the multiplication by α after taking the complex conjugate. More precisely, write M = α 1 Z⊕. . .⊕α nr Z, for α j ∈ K , with n = [K : Q], and consider the Z-linear isomorphism where x T is the transpose of x. Using this identification the pre-composition with α on Hom Z (M , Z) will correspond to multiplication by α on M t .
In order to describe the polarizations we need a particular kind of CM-type which, roughly speaking, detects the complex structure "coming from characteristic p" on a pair (T, F ) in M ord (q). More precisely, put where v p is the p-adic valuation induced by a fixed isomorphism Q p ≃ C. In [Mar18b] we give an algorithm to compute such a Φ. Recall that an element a in K is called totally imaginary if a = −a. For such an a, we say that it is Φ-nonpositive if Im(ϕ(a)) ≤ 0 for every ϕ in Φ.
Observe that in M ord (q), an isogeny λ : Then there exists a unique bilinear form S on T ⊗ Q such that b = Tr K /Q •S and, using [How95, Proposition 4.9], we have that µ is a polarization if and only if the associated S is skew-Hermitian and for every a in K the element S(a, a) is Φ-non-positive. In particular, a good understanding of Aut(M ) will most likely allow us to handle Q, but if r > 1, then Aut(M ) is an infinite non-abelian group, making the situation computationally difficult, even if we were able to produce a (finite) set of generators.
Recall that a polarized abelian variety (A, µ) is called decomposable if there are proper sub-varieties B 1 and B 2 of A, admitting polarizations β 1 and β 2 , respectively, such that (A, λ) ≃ (B 1 × B 2 , β 1 × β 2 ). Proof. The statement follows directly from 5.4 The next example shows that a polarized module (M , Λ) can be decomposable even if there is no way to put Λ into a block diagonal matrix by the action of an element of Aut(M ). Again, the matrix A has been computed using results from [GHR18].