Fixed-point subgroups of GL₃(𝑞)

1.1 Motivation

Let X/𝐐 be a smooth, projective algebraic variety and ℓ a rational prime. Then there are ℓ-adic representations

on the étale cohomology groups of X, and the image of such a representation can have interesting consequences for the arithmetic of the variety X. For example, when X is an elliptic curve and ρℓ is the ℓ-adic representation on the Tate module, then the image of ρℓ gives information on the ℓ-power torsion structure of X⁢(𝐐). A concrete instance of this stems from a question of Lang, answered by Katz [10]. Namely, if X is an elliptic curve over 𝐐 such that, for all but finitely many primes p, the numbers #⁢Xp⁢(𝐅p) are divisible by ℓn (where Xp denotes the reduction of X modulo a good primes p), then it is true that at least one of the curves X′ in the isogeny class of X has #⁢X′⁢(𝐐) divisible by ℓn. By translating to Galois representations, this result amounts to a classification of subgroups G of GL2⁢(𝐙ℓ) such that det⁡(1-g)≡0(modℓn) for all g∈G. One can ask for a similar classification of subgroups of symplectic similitude groups with a view towards higher-dimensional abelian varieties with divisibilities on their number of points modulo p; we provided such classifications in dimensions 4 and 6 in [3, 5, 4, 6] for the groups GSp4⁢(𝐅ℓ) and GSp6⁢(𝐅ℓ).

This raises a natural question: If k is a finite field, can one classify the irreducible subgroups of GLn⁢(k) such that every element has a fixed point? (By “irreducible subgroup” we mean a subgroup G⊂GLn⁢(k) that acts irreducibly on the underlying vector space kn.) Let us call a subgroup G of GLn⁢(k) a fixed-point subgroup if every element fixes a point in its natural representation.

By an exercise of Serre [14, Exercise 1], there are no irreducible fixed-point subgroups of GL2⁢(k). One of the main results of [10] is that there are no irreducible fixed-point subgroups of GSp4⁢(𝐅ℓ), where 𝐅ℓ is the field of ℓ elements. In [3, 5, 4], we classified the fixed-point subgroups of GSp6⁢(𝐅ℓ) and showed that none are irreducible. However, we recall an example of [4], originally communicated to us by Serre [15]. If L3⁢(2) is the simple group of order 168, then the Steinberg representation

is absolutely irreducible, and 𝖲𝗍⁢(L3⁢(2)) is a fixed-point subgroup of Sp8⁢(𝐅2). As an application of this observation, if A is an abelian fourfold defined over a number field K such that the image of the modulo-2 representation

coincides with 𝖲𝗍⁢(L3⁢(2)), then A has the property that, for all but finitely many primes 𝔭, the number of points #⁢A¯𝔭⁢(𝐅𝔭) on the reduction modulo 𝔭 of A is even, while no member of the isogeny class of A has an even number of K-rational torsion points. An interesting related question is whether such a fourfold can be realized over 𝐐.

Leaving the case of abelian varieties and symplectic groups, we focus on three-dimensional representations, which arise naturally in an arithmetic context as well. Using [7] as motivation, one can consider modular forms for congruence subgroups Γ0⁢(N) of SL3⁢(𝐙). Given a cuspidal eigenform f∈H3⁢(Γ0⁢(N),𝐂), let 𝐐f denote the number field generated by the Hecke eigenvalues of f. Let λ∈𝐐f be a prime dividing ℓ. Then we have attached to f the λ-adic Galois representation

(See [7, § 3] for an explicit example of how such compatible families of representations arise.) The residual representation ρλ¯ then provides a natural setting for studying subgroups of GL3⁢(k), where k is a finite field. In the aforementioned example, if im⁡ρλ¯ is a fixed-point subgroup of GL3⁢(k), then we get additional information on congruence properties of the number of points on the variety modulo p, for all but finitely many p, by the Chebatorev density theorem. For this reason, and the ones mentioned above with respect to abelian varieties, the fixed-point subgroups of linear groups have special arithmetic interest.

In this paper, we continue our classification of fixed-point linear groups and determine all fixed-point subgroups of GL3⁢(k), where k is a finite field. Unlike the classifications in [3, 5, 4] for the groups GSp4⁢(𝐅ℓ) and GSp6⁢(𝐅ℓ), the main theorem of this paper allows for k to be an arbitrary finite field of any characteristic.

1.2 The main theorem

We postpone a review of notation until the next section, except to remark that the maximal subgroups of a finite linear group fall into eight geometric classes 𝒞1,…,𝒞8, together with a class 𝒮 of exceptional subgroups; we refer the reader to [1] for the details of the classification.

There are certain subgroups of GL3⁢(q) that are easily identifiable as fixed-point subgroups, for example, those conjugate to

and we do not wish to include them in our classification. We therefore declare a subgroup G of GL3⁢(q) to be a trivial fixed-point group if the semisimplification of the underlying three-dimensional representation contains the trivial representation. Henceforth, we work exclusively with semisimple groups in this paper as these will have the same fixed-point properties as the parabolic groups they lie in. Our main theorem is as follows; see the following sections for all notational definitions.

1.3 Notation and setup

Let k be a finite field of characteristic p (we choose p instead of ℓ for consistency in the group theory literature), and write k=𝐅q, where q=pn. We follow the classification and notational scheme of [1], which is based on Aschbacher’s original classification of the subgroups of the finite classical groups.

Since GL3⁢(q) is not a fixed-point group itself, any fixed-point subgroup must lie in a maximal subgroup of GL3⁢(q) and hence in one of the eight geometric classes 𝒞1,…,𝒞8, or the exceptional class 𝒮. We use the standard notation from finite group theory, basing much of our notational scheme on that of [1]. In particular, we set

1. Altn: the alternating group on n letters,

2. Symn: the symmetric group on n letters,

3. Cn: the cyclic group of order n,

4. Eq: the elementary abelian group of exponent p and rank n,

5. Am+n: if A is elementary abelian, then Am+n has elementary abelian normal subgroup Am and quotient An,

6. p+1+2⁢n: the extra-special p group of order p1+2⁢n and exponent p,

7. d: the center of SL3⁢(q),

8. Z⁢(q): the center (scalar matrices) of GL3⁢(q),

9. Ln⁢(q): the projective special linear group PSLn⁢(q),

10. A≀B: the wreath product of A and B, where B↪Perm⁡(A×⋯×A),

11. N⁢.⁢Q denotes a non-split extension of Q by N,

12. N⁢:⁢Q denotes a split extension of Q by N,

13. N.Q denotes an arbitrary extension of Q by N.

Our strategy for proving Theorem 1.1 is roughly as follows. Given a subgroup G of GL3⁢(q), we intersect with SL3⁢(q) and use the classification of maximal subgroups of SL3⁢(q) outlined in [1, Chapter 2] to determine the fixed-point subgroups of SL3⁢(q). We then lift back to GL3⁢(q) to find the maximal fixed-point subgroups. The issue is that we may encounter novel subgroups – maximal subgroups M of GL3⁢(q) such that M∩SL3⁢(q) is not maximal in SL3⁢(q). We will address any novelties as they arise. Toward that end, we record the maximal subgroups of SL3⁢(q) in Table 1 below; see [1, Table 8.3] for complete details on the subgroup structure of SL3⁢(q).

We also note that the papers [9, 13] provide a classification of the ternary linear groups over finite fields, from which one could recover [1, Table 8.3]; however, we prefer to begin with the classification scheme of [1] due to the modern notation and language.

The groups in class 𝒞1 are the parabolic subgroups of GL3⁢(q) and we treat them separately in the next section. We then focus the rest of the paper on the irreducible fixed-point subgroups of GL3⁢(q).

2.1 Projective image Alt4

Let q be coprime to 6. Let H be a subgroup of GL2⁢(q) such that P⁢H≃Alt4. There are three inequivalent absolutely irreducible ordinary representations σ1, σ2 and σ3 of SL2⁢(3), with character values as follows (ω denotes a fixed primitive third root of unity):

The representation σ1 is defined over 𝐙, and σ1⁢(SL2⁢(3))⊂SL2⁢(q); σ2⁢(SL2⁢(3)) and σ3⁢(SL2⁢(3)) define subgroups of GL2⁢(q) when q≡1(mod3). In any of the three representations, the only class with a fixed point is the identity.

Now we consider the special cases of modular characteristic. If q is even, then any group H such that P⁢H=Alt4 is not irreducible in GL2⁢(q) [11, Lemma 6.1]. If q is a power of 3, then the isomorphism 2.Alt4≃SL2⁢(3) shows that 2.Alt4 occurs naturally as a subfield subgroup of GL2⁢(q). The same counting argument of Lemma 2.5 shows that more than half the elements of H do not have a fixed point, and hence H cannot give rise to a non-trivial fixed-point subgroup of GL3⁢(q).

2.2 Projective image Alt5

Let q be coprime to 30. Then there are two inequivalent ordinary absolutely irreducible representations σ1 and σ2 of SL2⁢(5) with the following character data:

In both representations, the only element with a fixed point is the identity.

In modular characteristic, if q is even, then the isomorphism

shows that Alt5 occurs as a subfield subgroup of SL2⁢(q) (once q>4). The same counting argument of Lemma 2.5 shows that more than half the elements of H do not have a fixed point, and hence H cannot give rise to a non-trivial fixed-point subgroup of GL3⁢(q). The same argument applies when q is a power of 5 via the isomorphism 2.Alt5≃SL2⁢(5).

If q is a power of 3, then 2.Alt5 only occurs as a subgroup of GL2⁢(q) when q is an even power of 3 since it is required that 5∈(𝐅q×)2. And if q is an even power of 3, then 𝐅q contains 𝐅9, so it suffices to work in GL2⁢(9). In GL2⁢(9), the group 2.Alt5 has 15 elements without a fixed point; hence ker⁡ψ=2.Alt5, and so H2 is trivial.

2.3 Projective image Sym4

Let q be coprime to 6. We consider the four groups 2i.Sym4 for i=1,2,3,4 separately. The group 21.Sym4 has no faithful irreducible degree-2 ordinary representations, and we do not consider unfaithful representations in this analysis for fixed-point subgroups.

The group 22.Sym4 has two faithful irreducible ordinary degree-2 representations σ1, σ2 with the following character data:

In the representations σ1 and σ2, the group 22.Sym4 has no non-trivial elements with a fixed point.

The group 23.Sym4 has two faithful irreducible ordinary degree-2 representations σ1, σ2 with the following character data:

In both representations, there are exactly 35 elements without a fixed point.

The group 24.Sym4 has two unfaithful irreducible degree-2 representations, and we do not consider unfaithful representations in this analysis.

We finish this section by considering the groups 22.Sym4 and 23.Sym4 in modular characteristic. If q is even, then neither 22.Sym4 nor 23.Sym4 is irreducible [11, Lemma 6.1]. If q is a power of 3, then the isomorphism 23.Sym4≃GL2⁢(3) shows that H1 occurs as a subfield subgroup of GL2⁢(q). The same counting argument of Lemma 2.5 shows that more than half the elements of H do not have a fixed point, and hence H cannot give rise to a non-trivial fixed-point subgroup of GL3⁢(q). Finally, 22.Sym4 contains SL2⁢(3) as index-2 subgroup, and the full group 22.Sym4 is contained in GL2⁢(9). Again, the same counting argument of Lemma 2.5 shows that there are no non-trivial fixed-point subgroups in this case.

3.1 Subgroups of type 𝒞2

The maximal subgroup of GL3⁢(q) of type 𝒞2 is isomorphic to GL1⁢(q)≀Sym3 as long as q≥5. If G is a subgroup of GL1⁢(q)≀Sym3, then G fits into a split short exact sequence

where G0 is a subgroup of GL1⁢(q)3 and P is subgroup of Sym3. If G is a fixed-point subgroup of GL1⁢(q)≀Sym3, then so is G0. By Lemma 2.1, either G0 fixes a line or G0≃C2×C2.