1 Introduction

The Number of Monodromy Representations of Abelian Varieties of Low -Rank

Abstract.

Let be an abelian variety of dimension and -rank over an algebraically closed field of characteristic . We compute the number of homomorphisms from to , where is any power of . We show that for fixed , , and , the number of such representations is polynomial in , and give an explicit formula for this polynomial. We show that the set of such homomorphisms forms a constructible set, and use the geometry of this space to deduce information about the coefficients and degree of the polynomial.

In the last section we prove a divisibility theorem about the number of homomorphisms from certain semidirect products of profinite groups into finite groups. As a corollary, we deduce that when ,

is a Laurent polynomial in .

Acknowledgments

This paper is derived from my Ph.D. thesis, and as such there are far too many people who have impacted this work than can be listed here. It is a pleasure to thank Ted Chinburg, my thesis advisor, for suggesting this problem and for countless discussions and guidance. Thanks also to David Harbater, Zach Scherr, and Ching-Li Chai for helpful discussions and answering my questions. I am grateful to Professor Fernando Rodriguez-Villegas for hosting me for a very productive week at the International Centre for Theoretical Physics. It was he who observed Corollary 3, and introduced me to both his theorem with Cameron Gordon [5] and the theorem of Frobenius, [3] from which one easily deduces Corollary 3. He also made me aware of Proposition 2. Bob Guralnick very helpfully referred me to both his paper with Sethuraman [6] and earlier work of Gerstenhaber [4]. Additional thanks are due to Nir Avni for comments on the draft and Sebastian Sewerin for help reading [3]. Many others have had a great influence on my mathematical development, which no doubt manifests itself in some of the work that follows. There are too many to list here, but I have attempted to acknowledge many of these people in the thesis document.[2]

1. Introduction

In Mixed Hodge Polynomials of Character Varieties, [7] Hausel and Rodriguez-Villegas computed the number of homomorphisms from the fundamental group of a Riemann surface of genus into . By van Kampen’s theorem, this is equivalent to counting -tuples of matrices,

where denotes the commutator . They also consider a twisted version of the problem, computing

where is a primitive root of .

The salient feature of both these counts is that, for fixed and , and are both polynomial functions of . Another curious feature is that is divisible by the order of . This phenomenon is explained in [5].

In the twisted case, acts scheme-theoretically freely on , and the GIT quotient is the moduli space of twisted homomorphisms from to . Via a theorem of N. Katz, (in the appendix to [7]), the polynomial point-counting formula is equal to the -polynomial of , which encodes information about the weight and Hodge filtrations on the cohomology of . Baraglia and Hekmati [1] have more recently extended these methods to the moduli of untwisted representations.

The results in this paper may be thought of as an analogue of the combinatorial results in [7] and [1], now applied to varieties of positive characteristic. The most natural analogue would be consider a curve in place of a Riemann surface. However, in positive characteristic, the étale fundamental group is a much more subtle object. In particular, there is not a single curve of genus greater than for which an explicit presentation of the fundamental group is known. [12]

However, these fundamental groups have well-understood abelianization; the abelianization of the fundamental group of a curve is the fundamental group of the Jacobian variety, which is the dual of the Jacobian. This paper treats representations of fundamental groups of abelian varieties.

Let be an abelian variety of dimension . The -torsion points of form a vector space over of dimension . We call the -rank of . Let . The étale fundamental group of is given by

where

is the profinite completion of the group of integers and

is the group of integers completed away from the prime . To lighten the notation, we will henceforth write instead of .

Let be a power of . A homomorphism is determined by the image of the topological generators, so specifying a homomorphism is equivalent to choosing an ordered -tuple of pairwise commuting matrices, such that the first have order not divisible by .

Theorems 1 and 2 explicitly compute the number of homomorphisms , where is a power of and the -rank of is either or , respectively. Theorem 3 computes the number of homomorphisms up to conjugation in the -rank case. All three counting formulas are polynomial in , depending on and but not on the characteristic.

Section 3 considers the space of all such representation, and relates the geometry of this space to certain features of the polynomial formulas in the previous section. We show that the counting polynomial has integer coefficients, and give a lower bound for the degree, which we show to be exact when is in elliptic curve.

In section 4, we state and prove the following theorem.

Theorem (Theorem 6).

Let be a set of primes (not necessarily finite), and let , where the the inverse limit is taken over all natural numbers not divisible by any prime in . Then for any topologically finitely generated profinite group and finite group ,

where is any semidirect product of and , and is the multiplicative set generated by the elements of . Conversely, if is topologically finitely generated and

for all finite groups , then there exists a with .

Theorem 6 is the profinite analogue of a divisibility theorem by Gordon and Villegas[5]. We deduce as a corollary that when the , is a Laurent polynomial in .

2. Counting Formulas

Our main reference in this section is Macdonald’s Symmetric Functions and Hall Polynomials. [10] Throughout, will be a prime number, and will denote a power of .

2.1. Some Linear Algebra

For any matrix we have an associated -module structure on , where acts by . If and induce isomorphic module structures on , then the isomorphism of -modules defines an element of , so and are conjugate. Note that has order prime to if and only if acts semisimply on . That is, is diagonalizable over . Since elements of order prime to act semisimply, the associated -module is a direct sum of simple modules of the form , where the are irreducible but not necessarily distinct. Matrices of order prime to are thus uniquely characterized, up to conjugation, by their characteristic polynomials.

2.2. Polynomials and their Types

Definition 1.

A type of is a partition of along with a refinement of its conjugate. That is, a partition of along with partitions of the multiplicities of the entries of .

This is a slight generalization of what Macdonald calls a type in [10]. For example, the data , , , , , give a type of . We shall write when is a partition of , and when is a type of . By

we will mean the product taken over all pairs such that and . We index over without multiplicity, but over with multiplicity. So if

then

We associate to an matrix a type by considering the characteristic polynomial of . Factoring into irreducible factors gives a partition of , where the entries of are the degrees of the irreducible factors of , counted with multiplicity. We then take to be the partition consisting of the multiplicity of each distinct degree- factor of .

For example, over , we associate , , to the polynomial .

Recall that the number of irreducible monic polynomials over of degree is

where is the Möbius function. [9]

Definition 2.

Denote by the number of monic polynomials with factorization type .

Note that , but in general . For instance, if , , then

2.3. Counting

We now count the number of homomorphisms from to , where the -rank of is . This is equivalent to counting -tuples of commuting matrices such that each has order prime to . Since the combinatorial arguments below do not make use of the fact that is even, we may state a slightly stronger theorem.

Theorem 1.

The number of ordered -tuples of pairwise-commuting, semisimple, invertible matricies with entries in is

For instance, the number of commuting semisimple pairs is

In , the number of semisimple commuting pairs is

Proof.

We first prove the theorem for . Since semisimple matrices are characterized, up to conjugation, by their characteristic polynomials, we associate to each such conjugacy class a type . There are by definition conjugacy classes of type . Suppose is any matrix of order prime to , with associated type . A matrix commutes with if and only if acts -equivariantly on . That is, writing as a sum of simple modules

the action of is a -automorphism of each . Specifying such an action is given by an element of . The order of the centralizer of a semisimple matrix with type is thus

(1)

The matrix commutes with and also must be semisimple, so acts semisimply on each . A semisimple matrix is determined, up to conjugacy, by its characteristic polynomial, or equivalently by a type and a polynomial of type in . By the same argument used to compute the cardinality of the centralizer of , we see that index of the centralizer of is

The number of all such is therefore

(2)

So, if is semisimple with type , the number of semisimple matrices commuting with is

(3)

Given a semisimple matrix , we have just computed both the number of matrices (1) that commute with and the number of semisimple matrices (3) that commute with . Note that both of these numbers depend only on the type associated to , since the ’s are the entries of , and the ’s are the entries of the corresponding ’s.

From these two computations, we see that the number of pairs of commuting semisimple matrices in is

(4)

where the second sum is taken over polynomials with type . Since

depends only on , we may simplify (4) by replacing the second summation with multiplication by . Thus (4) simplifies to

(5)

We now continue by induction. Suppose the number of pairwise commuting
-tuples of semisimple elements of is

Suppose further that for each possible action of on , the action of on an isotypic summands of the -module has characteristic polynomials of type as above when these isotypic summands are viewed as -vector spaces. (We note that may be different for different summands; indeed, , and the symbol takes on different values in different terms of the above formula).

Since commutes with each , , then acts on by acting on each isotypic summand, which by induction are -modules, or -vector spaces of dimension . There are

conjugacy classes of semisimple elements of . By (1), each conjugacy class has

elements, for a total of

possible -actions on . So by induction, the number of pairwise-commuting -tuples of semisimple matrices in is

Canceling factors of , the above formula simplifies to the statement of the theorem. ∎

Observation.

From the proof, we see that the formula in Theorem 1 is polynomial in . This polynomial depends on and , but not on the characteristic.

The next theorem computes when the -rank of is .

Remark.

The polynomial behavior of the formula in Theorem 1 is strictly a same-characteristic phenomenon, which fails even in the simplest case in characteristic . Consider the set . When , there are infinitely many for which , and for such . But for those such that , . When , we may write with . Then for those such that , the -adic valuation of is , and . When , .

Theorem 2.

The number of ordered -tuples of invertible, pairwise-commuting matrices such that the are all semisimple is

Proof.

As in the inductive step of the theorem, we assume the number of pairwise commuting -tuples of semisimple elements of is

As before, for each possible action of on , the action of on an isotypic summands of the -module has characteristic polynomials of type as above when these isotypic summands are viewed as -vector spaces.

Thus it suffices to count the number of pairs which act on each isotypic summand of the -module . By assumption, these summands are of the form .

Let be the orbit, under the conjugation action in , of a matrix of order prime-to-. Given any , we will denote by its restriction to the isotypic summand in question, and similarly will denote the restriction of . For any , the number of nonsingular matrices that commute with is the order of the centralizer of . The number of elements in is the index of the centralizer of . Thus there are precisely pairs such that and commute and . So to compute the number of pairs with and prime to , we need only count the number of possible conjugacy classes . These are in bijection with characteristic polynomials, of which there are .

Thus the number of pairwise commuting -tuples of invertible matrices
, of which all but possibly the last have order prime to , is

where

After cancellation we arrive at the statement of the theorem. ∎

Theorem 3.

The number of conjugacy classes of -tuples of pairwise-commuting, invertible, semisimple matrices is

Proof.

Observe that stabilizes if and only if it commutes with each . Writing for the order of the stabilizer of and applying the orbit-stabilizer lemma,

where the last sum is taken over -tuples pairwise commuting semisimple matrices with all semisimple. The value of this sum was computed in Theorem 2. ∎

3. The Space of Representations

In this section, will be an abelian variety with -rank , as before. However, all statements apply if the -rank of is , mutatis mutandi. All schemes below are reduced. Recall that a subset of affine (or projective) space is said to be constructible if it may be expressed as a Boolean combination of Zariski-closed sets. [11]

Fix , and let . Recall that a representation of is given by a choice of invertible matrices , , such that these matrices are pairwise commuting and each have order relatively prime to . Assigning a matrix to each generator, we may view as a subset of .

We will now show that for fixed and , the set of all representations may be considered a constructible set.

Proposition 1.

is constructible.

Proof.

The requirement that the matrices pairwise commute defines a closed affine subscheme. Invertibility is an open condition, so the variety of pairwise commuting, invertible matrices is quasi-affine. We may further specify that the matrices are (simultaneously) diagonalizable with the statement that there exists an invertible matrix (with coordinates , ) so that is diagonal for each . This last statement requires an existential quantifier, and thus does not necessarily define a subscheme, but it is a first order statement in the sense of model theory, and hence the space of representations is a definable subset of . Since the theory of algebraically closed fields admits quantifier elimination, [11] every definable set is in fact constructible, i.e. a Boolean combination of closed subschemes.∎

Alternatively, since we are considering simultaneously diagonalizable matrices, one may write the diagonalization of each as an element of a torus, and then define a morphism , . By Chevalley’s Theorem, [11] the image of this morphism, which is equal to , is constructible.∎

Since is constructible, we may write