Local heuristics and an exact formula for abelian varieties of odd prime dimension over finite fields
Consider a -Weil polynomial of degree . Using an equidistribution assumption that is too strong to be true, we define and compute a product of local relative densities of matrices in with characteristic polynomial when is an odd prime. This infinite product is closely related to a ratio of class numbers. When we conjecture that the product gives the size of an isogeny class of principally polarized abelian threefolds.
Key words and phrases:abelian varieties, finite fields, matrix groups
2010 Mathematics Subject Classification:14K02
This paper is a direct generalization of the work of Achter-Williams  to abelian varieties of odd prime dimension, and is guided in philosophy by both Gekeler  and Katz . We begin by considering abelian varieties over a finite field , where is a power of a prime. To each such variety , we can associate a characteristic polynomial of Frobenius . A theorem of Tate  tells us that two varieties are isogenous if and only if their characteristic polynomials are equal.
Let denote the moduli space of principally polarized abelian varieties of dimension and define to be those members of with characteristic polynomial where each variety is weighted by the size of its automorphism group. Computing amounts to determining the number of (isomorphism classes of) abelian varieties over in a particular isogeny class.
The Frobenius endomorphism gives an automorphism of the Tate module (for not dividing ), so it has a representation as an element of the matrix group . For each , we define a term measuring the relative frequency of as the characteristic polynomial for an element of , as well as an archimedean term .
Since (as much as possible) Frobenius elements are equidistributed in , it seems possible that the product of these local densities could at least estimate the value of . This, of course, is a ridiculous tactic, as the Frobenius elements are only equidistributed when . However, as in , we again show that this local data does apparently control isogeny class size.
Our main result is as follows. Consider a particular class of -Weil polynomials of degree (see the next section for details). Let be the splitting field of over and be its maximal totally real subfield, with class numbers and respectively. A theorem of Everett Howe in the preprint  implies that
2. Abelian varieties and Weil polynomials
Let be an abelian variety of dimension over a finite field of elements, and let be the characteristic polynomial of its Frobenius endomorphism. Then is a -Weil polynomial of degree with complex roots with for every , ordered such that for .
Each such polynomial corresponds to a (possibly empty) isogeny class of abelian varieties of dimension over . Following , we will assume:
(ordinary) the middle coefficient of is relatively prime to ;
(principally polarizable) there exists a principally polarized abelian variety of dimension with characteristic polynomial ;
(cyclic) the polynomial is irreducible over , and is Galois, cyclic, and unramified at ;
(maximal) for a (complex) root of , with complex conjugate , then , a priori an order in , is actually the maximal order .
Assumptions (W.1), (W.2), and (W.4) are identical to those in . The condition (W.3) is similar but phrased differently; we want to assume is abelian and Galois. In [1, (W.3)], the authors only assumed that is Galois, but as was a number field of degree 4 this also guaranteed it to be abelian. In the present work, assuming that is a cyclic Galois number field is equivalent to requiring to be an abelian Galois number field as there is a unique abelian group of order 2 times any odd prime (namely the cyclic group ).
Note that is a CM field, and as such it comes equipped with an intrinsic complex conjugation . Also, the isomorphism class of is independent of the choice of .
Under these conditions, ; then is the minimal polynomial of and is the maximal totally real subfield of . Note that and define the conductor of , as the index . We will denote the discriminants of the polynomials and as and , respectively, while will represent the discriminant of an order . Note that and .
In the following technical lemma, we give explicit forms for and for any positive integer .
Let be a -Weil polynomial of degree with . Then
Recall that the roots of are of the form for . Then the proof of part (1) of the lemma proceeds by induction on using elementary methods and is omitted here. A direct computation of the discriminant of , which has roots for , proves part (2). ∎
These explicit forms will be helpful in proving the following lemma, as well as for defining local factors in section 4.
The index of in is .
From , we have and Then
Without loss of generality, choose . Then and all of its Galois conjugates are of the form for . Therefore,
Applying and from Proposition 2.2, we find , and the lemma follows. ∎
If , then .
Lastly, we prove that is the maximal order of , but we begin with a technical lemma.
Any element of can be written in the form for some .
It is straightforward to show that the set
forms a basis for . Then can be written where for all and .
We can rewrite by iteratively adding and subtracting terms of the form or for . The process terminates when . We omit the details of this process for the sake of brevity, but record the result below.
For , define , and notice that . If then
where , , and
(A similar formula exists for .)
Then for all , each of and can be written in the form for some , and so , an integer linear combination of terms of this form, can also be written in this form. ∎
The order is the maximal order .
3. Conjugacy classes in
3.1. Symplectic groups and conjugacy
The symplectic group is the subgroup of preserving an antisymmetric bilinear form up to a scalar multiple. We choose
but note that different choices of produce isomorphic copies of .
The value is called the multiplier of . All matrices in have the property that there exists a pairing (dictated by the choice of antisymmetric bilinear form) of its eigenvalues such that each pair has product .
In [7, Theorem 1.18], Shinoda parametrizes the set of conjugacy classes of . For our purposes, we do not need their parametrization in full generality, and will describe the relevant portions in our own notation. Let
be a polynomial in . Define the dual of with respect to the multiplier to be
(We will occasionally omit the on the left hand side for notational convenience.) Then we have three types of polynomials:
Root polynomials are polynomials of the form when is not a square, or either of when is a square. (It is easy to check that all root polynomials satisfy )
pairs are pairs of polynomials such that .
polynomials are polynomials such that and is not a root polynomial.
One consequence of these definitions is as follows.
There are no irreducible polynomials of odd degree. That is, for all irreducible polynomials of odd degree and all multipliers ,
Let be an irreducible polynomial with odd and suppose . This implies , but if is non-square then is non-square since is odd, which is a contradiction. If instead is a square, then is a root of , contradicting the fact that is irreducible. ∎
The general theory of conjugacy classes in as well as the additional intricacies of conjugacy in are given (briefly) in [1, Section 3.2], and the reader is encouraged to revisit this section if needed. For our current purposes, recall that to each irreducible factor of the characteristic polynomial of a matrix, we associate a partition of its multiplicity; the characteristic polynomial together with its partition data determine conjugacy in . We also remind the reader that cyclic matrices are those for which the characteristic and minimal polynomials coincide, and thus are those where all partitions are maximal (consist of only one part).
Then characteristic polynomials for conjugacy classes of cyclic matrices in are constructed as follows.
The following characteristic polynomials uniquely determine a conjugacy class of cyclic matrices in .
For square ,
for a choice of and such that any odd parts in the partitions of and have even multiplicity and
For non-square ,
for a choice of and such that any odd parts in the partition of have even multiplicity and
For each exponent defined above, the only allowable partition is .
Shinoda’s parameterization ([7, Theorem 1.18]) also includes sets of (nondegenerate symmetric) bilinear forms with ranks relating to the number of parts of even sizes in the partitions of any root polynomials present in the factorization of the characteristic polynomial. The number of equivalence classes of these bilinear forms detect when a characteristic polynomial (and associated set of partitions) gives rise to two conjugacy classes in , indexed by . We omit this component of Shinoda’s parameterization because all of the characteristic polynomials that we will declare as relevant in the next section give rise to only one conjugacy class.
3.2. Relevant conjugacy classes of
For the remainder of section 3, let be an odd prime.
We want to identify the possible shapes (factorization structures) of characteristic polynomials which correspond to conjugacy classes of cyclic matrices in which respect the assumptions (W.1)-(W.4). Therefore, a relevant characteristic polynomial is one such that
the factorization is as in Theorem 3.2,
the degrees of all irreducible factors are equal, and
the multiplicity of each irreducible factor are equal.
The first condition forces our matrices to be cyclic elements of , and the other two correspond with the requirement that be Galois.
Let denote a monic irreducible degree polynomial in , and assume that if . Then the following are the shapes of all relevant characteristic polynomials.
We exclude the shape by Lemma 3.1. Additionally, we exclude ; since there are an odd number of factors, it must be that one of these linear factors is a root polynomial while the rest are pairs. Then the Galois group of cannot act transitively on the roots of such an .
The seven relevant conjugacy class shapes fall into two categories: regular semisimple and non-semisimple. A regular semisimple conjugacy class contains elements with a squarefree characteristic polynomial (and thus are cyclic by definition). A class is not semisimple when the characteristic polynomial of its elements is not squarefree.
We list the relevant conjugacy classes by the shape of their characteristic polynomial in Tables 3.1 (regular semisimple) and 3.2 (non-semisimple). Each table also contains information about the multiplier and the type of the irreducible factors.
|Char. Pol. Shape||Valid||Polynomial type|
|Char. Pol. Shape||Valid||Polynomial type|
3.3. Centralizer orders
Denote by a conjugacy class of matrices in with characteristic polynomial . For each relevant conjugacy class, we will find its order by instead finding the order of its centralizer. Let be the centralizer in of an element of . Since we need only the size of the centralizer, the choice of this element is arbitrary.
Let be a regular semisimple conjugacy class with characteristic polynomial having one of the shapes listed in Table 3.1. Then we have
Since each class is regular and semisimple, their centralizers are tori. For example, consider the case where has the shape . Then has roots in in a single orbit under the action of Galois. Since , it must be that the elements of have multiplier for . Thus, elements of the centralizer of are those where roots of the characteristic polynomial are elements of with an -norm lying in . There are elements of with such a norm for a fixed . Since we have choices for the multiplier,
The centralizer sizes for the other cases are computed similarly. ∎
In the case where is not semisimple, the process for determining the order of its centralizer is significantly more challenging. In these cases, we must construct an explicit matrix which is a representative of , and verify that has the correct characteristic polynomial, that is cyclic, and that . Then we must find an explicit matrix which is a generic member of the centralizer of that (also in ) and use it to count the number of possible elements of .
For any particular , this process is possible. (As an example, the centralizer orders of the non-semisimple classes for are given in Proposition 7.5.) However, we have not yet constructed representatives for all three non-semisimple classes for a general (odd prime) and thus do not have formulae for their centralizer orders. We hope, in future work, to address this gap.
4. Local factors for
In this section we define local factors for each finite rational prime and one for the archimedean prime, . For all , this local factor is given by the density of elements of with a fixed multiplier and characteristic polynomial with respect to the “average” frequency. We also define and based on the same notions. These definitions are in direct analogue with those of , and are thus philosophically guided by  as well.
Suppose is a rational prime and consider a principally polarized abelian variety of dimension . The Frobenius endomorphism of acts as an automorphism of , and scales by a factor of the symplectic pairing on induced by the polarization. Thus, we can consider as an element of (the set of elements of with multiplier ).
There are possible characteristic polynomials for an element of , and so the average frequency of a particular polynomial occurring as the characteristic polynomial of an element of the group (with respect to all such polynomials) is given by
Then for primes unramified in , we define as
(See (5.1) for a definition for all .)
The definition of is similar to but more intricate than (4.1). Under our assumptions, is an ordinary abelian variety of (odd prime) dimension with characteristic polynomial of Frobenius
Thus, as in , we have a canonical decomposition of the -torsion group scheme into étale and toric components . By ordinarity, and , and the -power Frobenius acts invertibly on . This action of on both the étale and toric components of has characteristic polynomial , and must preserve the decomposition of . Let be the multiplier of (so is a root of ). Thus, we set
Lastly, we define an archimedean term, which is related to the Sato-Tate measure. As stated in , the Sato-Tate measure on abelian varieties conjecturally explains the distribution of Frobenius elements, and is a pushforward of Haar measure on the space of “Frobenius angles”, . The Weyl integration formula [10, p218, 7.8B] gives the Sato-Tate measure on abelian varieties of dimension explicitly as
Fixing a particular , the set of angles gives rise to a -Weil polynomial. We use the induced measure on the space of all such polynomials to define the archimedean term . To derive this induced measure, we first write the polynomial in terms of its roots and in terms of its coefficients as
and perform a change of variables. Thus, we find our induced measure on the space of all -Weil polynomials to be
Note that there are approximately principally polarized abelian varieties over , so can be thought of as an archimedean predictor for . Then we define
(We note that definition (4.3) holds for any , not just odd prime .)
5. Polynomials and primes in
Our goal in this section is to relate the polynomial to (a representative of) one of the conjugacy classes defined in section 3.2. There are two lenses through which we can consider such a correspondence, as outlined in [1, Section 5]. Regardless of the perspective, we will use the factorization of to determine a cyclic element of whose semisimplification is conjugate to , the image of the action of on . Then we define
Note that is precisely . Also note that by Corollary 2.4, so the factorization of is determined by the splitting of in . That is, if , then for primes of where the residue degree of equals the degree of the irreducible polynomial .
Because of Condition (W.3), is a finite Galois extension with . Then the residue degrees and ramification degrees for all , and in particular . (Notice that this restriction is precisely how we identified relevant class shapes in Section 3.2.) Let and denote the decomposition and inertia groups, respectively, of a rational prime . Lastly, if , then (the residue field of ) is cyclic. Let be the element which induces the generator of this group, and call it the Frobenius endomorphism of over .
Let so that complex conjugation is given by . We classify the splitting of rational primes of by enumerating the possibilities for and .
In Table 5.1, we enumerated all possibilities for pairs of subgroups . In every case, there are distinct irreducible factors of , each with degree and multiplicity . In all cases, this factorization pattern exactly determines the conjugacy class for which is a representative. ∎
6. Local factors for
Let be the character group of the Galois group of . For , let be the fixed field of . For a rational prime , define
Let and define
Recall that is a generator of and let be a generator of . Then and , so A quick computation shows and for all other . We have for all odd , and so to compute for odd , we only need to know and
The values of and are determined by and , as given in Table 6.1.