Dirichlet series associated with some groups and Lie algebras

Dirichlet series associated with some groups and Lie algebras

群，

Lie

Dirichlet

July 2015

Fumitake HYODO

Dirichlet series associated with some groups and Lie algebras

群，

Lie

Dirichlet

July 2015

Waseda University

Graduate School of Fundamental Science and Engineering, Major in Pure and Applied Mathematics,

Research on Number Theory

Fumitake HYODO

Acknowledgment

I wish to express my sincere gratitude to my advisor Professor Kiichiro Hashimoto for his advices and encouragements. Especially, he made corrections in my first paper very carefully, and taught me the fact zeta functions of free abelian groups are equal to formal Dirichlet series associated with certain Hecke rings. Acting on the hint, I achieved the third results. I would like to thank Hajime Koba for helping me write my third paper in English. My thanks also go to Atsuhiko Mizusawa for helping me write this thesis. I would like to thank Professor Keiich Komastu, Hiroshi Sakata, Tsuchiya Takuya for their encouragement. Finally, it is my pleasure to thank my family for their supports and encouragements.

Fumitake HYODO

Major in Pure and Applied Mathematics

Graduate School of Fundamental Science and Engineering Waseda University

3-4-1, Okubo, Shinjuku-ku Tokyo, 169-8555

JAPAN

Contents

Introduction 1

1 Zeta functions of groups 3

1.1 Definition of zeta functions of groups . . . 3

1.2 The group (Zⁿ,Z^m;f) . . . 4

1.3 Expressions by matrices . . . 6

1.4 Classification of isomorphism classes of{(Zⁿ,Z^m;A)} . . . 7

2 The class T₂ 11 2.1 Classification of isomorphism classes ofT₂ . . . 11

2.2 Explicit forms of zeta functions . . . 12

2.3 Calculation of the zeta function of G(α)×Z^m−1 . . . 16

3 The class T₃ 19 3.1 Classification of isomorphism classes ofT₃ . . . 19

3.2 Basic notions and facts for counting subgroups . . . 20

3.3 The casem̸=m^′ . . . 22

3.4 The casem=m^′ . . . 26

4 Hecke rings associated to Lie algebras 29 4.1 Definition of Hecke rings associated to Lie algebras . . . 29

4.2 Hecke series and isomorphism zeta functions associated to Lie algebras . . . 31

5 The Hecke ring of the Heisenberg Lie algebra 33 5.1 The classical Hecke ring . . . 33

5.2 The Automorphism group of the Heisenberg Lie algebra . . . 34

5.3 An action on Z/p^lZ⊕Z/p^l+kZ . . . 36

5.4 A standard basis and the degree map . . . 37

5.5 The endomorphism θ ofR(Γ,∆) . . . 39

5.6 A formal power series ofR(Γ,∆) . . . 43

List of papers by Fumitake Hyodo 51

Introduction

This thesis concerns with Dirichlet series associated with groups and Lie algebras. Dirichlet series is very important in studying the number theory. The Riemann zeta function is the simplest Dirichlet series. The Birch and Swinnerton-Dyer conjecture is the one about Dirichlet series associated to Elliptic curves. There are even more Dirichlet series which played crucial role in the number theory. In Chapter 1, Chapter 2 and Chapter 3, we study zeta functions of groups and their isomorphism classes. Zeta functions of groups were introduced, by F. J.

Grunewald, D. Segal, and G. C. Smith [2]: Given a finitely generated group G and a positive integern,a_n(G) denotes the number of subgroups of indexn. The zeta function ofGis defined as the Dirichlet series associated to the sequence {an(G)}n≥1, and this function is denoted by ζ_G i.e.,

ζ_G(s) :=

∞

∑

n=1

an(G)n^−s= ∑

H:[G:H]<∞

[G:H]^−s. For each prime p,ζ_G,p(s) is also defined as

ζ_G,p(s) :=

∞

∑

k=0

a_pk(G)p^−ks.

Torsion-free finitely generated nilpotent groups are called T-groups. It is known that zeta functions ofT-groups have various good properties. F.J. Grunewald, D. Segal, and G.C. Smith [2] proved that ifGis aT-group, the zeta function ofGhas a non-empty domain of convergence and can be expressed as an infinite product:

ζ_G(s) =∏

p

ζ_G,p(s),

and for each prime number p, ζ_G,p(s) has a rationality. Moreover, C.Voll [4] showed that for almost all p,ζ_G,p(s) has a functional equation.

We are interested in finding a class of T-groups in which zeta functions determine the isomorphism classes. It is known that for the class of allT-groups, the isomorphism classes are not determined by zeta functions (cf. [1], Prop. B and [3], Lem. 1.2). In these chapters, we give a class ofT-groups in which zeta functions determine the isomorphism classes.

LetM_n(Z^m) denote the set of alln×nmatrices with entries fromZ^m. Forn, m∈Z_≥0 and A∈M_n(Z^m), aT-group (Zⁿ,Z^m;A) is defined as follows:

1. As a set we have (Zⁿ,Z^m;A) =Zⁿ×Z^m.

2. The composition of (a,b),(a^′,b^′)∈Zⁿ×Z^m is defined by

(a,b)(a^′,b^′) := (a+a^′,b+b^′+ ^taAa^′).

These groups are studied in Chapter 1 explicitly. For n ∈ Z_≥0, we define Tn to be the class {(Zⁿ,Z^m;A) | m ∈ Z_≥0, A ∈ M_n(Z^m)}. In these three chapters, we consider the problem as follows:

Let n be a nonnegative integer and G, G^′ ∈ T_n. If ζ_G(s) = ζ_G^′(s), then are G and G^′ isomorphic?

If n = 0 or n = 1, it is a trivial problem. In Chapter 2, it is shown that zeta functions determine the isomorphism classes ofT₂ by calculating explicit forms of the zeta functions. In Chapter 3 we prove that zeta functions determine the isomorphism classes ofT₃ by comparing a_l(G) witha_l(G^′) for somelinstead of evaluating their explicit forms. (Recall that two sequence {an}n≥1 and {bn}n≥1 are equal if and only if their Dirichlet series are the same.) In the case n≥4, the result does not hold (cf.[1], Proposition B, or [5], Example 4.) As far as the author knows, it seems that there are few results on our problem except for the above two results.

It is worth pointing out that our result settles the problem whether zeta functions of groups determine the isomorphism classes of the class T_n for each n.

In Chapter 4 we define Hecke rings and Hecke series, and isomorphism zeta functions of Lie algebras. The definition of isomorphism zeta functions by the author is slightly different from the one by [2]. The classical Hecke ring was introduced by [10] as a ring of operators acting on the space of modular forms. [12] defined abstract Hecke rings. By using the definition of [12], we define our Hecke rings. And then we relate isomorphism zeta functions to series associated with our Hecke rings via the degree maps.

Chapter 5 studies the Hecke series of the Heisenberg Lie algebra over Z_p. We relate the series to the classical Hecke series defined by E. Hecke, and prove that the series has a property similar to the rationality theorem of the classical Hecke series. And then, our results recover the rationality theorem of the classical Hecke series and the isomorphism zeta functions. It is known that various Hecke series satisfy the rationality theorems. The rationality theorem of Hecke series associated with Sp(n,Z) was established by Shimura [11] for the case n = 2, and Andrianov [8] for any n ≥3. The rationality theorem of a series with respect to SL_n(Z) was derived by Tamagawa [13]. Dulinski [9] demonstrated the rationality theorem of a Hecke series of the Jacobi group. We emphasize that the coefficients of our series are not always commutative each other. As far as the author knows, it seems that there are few papers on the rationality theorems for Hecke series whose coefficient ring are noncommutative. Andrianov [8], Hecke [10], Shimura [11] and Tamagawa [13] dealt with commutative Hecke rings and Hecke series with commutative coefficient rings. Dulinski [9] dealt with a noncommutative Hecke ring, but Dulinski [9] dealt with a Hecke series whose coefficient ring is commutative. It is also worth pointing out that the property of our Hecke series is similar to rationality theorems, but it does not mean the rationality of our Hecke series.

2

Chapter 1

Zeta functions of groups

In this chapter, we study zeta functions of groups that were introduced in [2] by F. J. Grunewald, D. Segal, and G. C. Smith.

We shall use the notation

• Z(G) := the center of Gfor an arbitrary group G,

• [G, G] := the commutator subgroup ofG for an arbitrary group G,

• h(G) := the Hirsch length ofGfor a T-group G,

• r(G) := the Z-ranks ofGfor a finitely generated abelian group G.

(Recall that the Hirsch length of a T-group is defined by the sum of the Z-ranks of its central factors.)

1.1 Definition of zeta functions of groups

Given a finitely generated groupG, let a_n(G) be the number of subgroups of index n, for each positive integer n. The zeta function of G is defined as the Dirichlet series associated to the sequence {an(G)}n, and this function is denoted by ζ_G i.e.,

ζ_G(s) :=

∞

∑

n=1

a_n(G)n^−s= ∑

H:[G:H]<∞

[G:H]^−s. For each prime p,ζG,p(s) is also defined as

ζ_G,p(s) :=

∞

∑

k=0

a_pk(G)p^−ks.

Example 1.1.1. If G=Z, the subgroup of each index of Gis unique, so that ζZ(s) is equal to the Riemann zeta function denoted by ζ(s), and ζ_Z,p(s) is its Euler p-factor, i.e., (1−p^−s)⁻¹.

Example 1.1.2. ([2])If G=Z^m, we have

ζ_Z^m(s) =ζ(s)ζ(s−1)· · ·ζ(s−m+ 1).

Next example is the case where Gis a non-abelian group.

Example 1.1.3. ([6],Theorem 2.22) Define a subgroup U₃(Z) of GL₃(Z) as follows:

U3(Z) :=











1 a b 0 1 c 0 0 1



 | a, b, c∈Z





 ,

and put G=U3(Z)×Z^m−1 for a positive integer m. Then we have

ζ_U3(Z)×Z^m−1(s) = ζ_Zm+1(s)ζ(2s−m−1)ζ(2s−m−2) ζ(3s−m−2) .

The above group U₃(Z) is usually called the discrete Heisenberg group.

In general, zeta functions are expected to have good properties such as natural Euler product expansions, meromorphic extensions to the whole complex plane, and functional equations.

It was shown that ifGis torsion-free, finitely generated, and nilpotent, the zeta function of Gsatisfies following good properties (cf.[2],[4]):

1. {an(G)}nhas a polynomial growth i.e., there existsc∈Z_>0 such that a_n(G)< n^c. Hence ζ_G(s) has a non-empty domain of convergence.

2. {an(G)}n is multiplicative, hence ζ_G(s) =∏

pζ_G,p(s).

3. For allp, there exists a rational functionf_p(X)∈Q(X) such thatζ_G,p(s) =f_p(p^−s), hence ζ_G,p(s) can be continued to a meromorphic function on the whole complex plane.

4. For all but finitely many primesp,ζ_G,p(s) satisfies a local functional equation (cf.[4]).

Now a fundamental question that arises here is to ask, what kind of equivalence class of G is determined by the zeta function ζ_G(s) ? In particular, doesζ_G(s) =ζ_G^′(s) implies G∼=G^′ ? Unfortunately, the latter question has a negative answer .(cf.[1], PropositionB, or [5], Example 4.)

1.2 The group ( Z

, Z

; f )

The group we study in this book is given for a pair of nonnegative integers n, m∈Z_≥0 and a bilinear map f : Zⁿ×Zⁿ→Z^m, and we shall denoted by (Zⁿ,Z^m;f).

4

Definition 1.2.1. Letn, m andf be as above. Then we define the group(Zⁿ,Z^m;f)as follows:

1. As a set we have (Zⁿ,Z^m;f) =Zⁿ×Z^m.

2. The composition of (a,b),(a^′,b^′)∈Zⁿ×Z^m is defined by

(a,b)(a^′,b^′) := (a+a^′,b+b^′+f(a,a^′)).

The associative law for this composition is easily checked using that f is a bilinear form.

Note also that the identity element of this group is (0,0), and the inverse element of (a,b) is (−a,−b+f(a,a)).

Free abelian groups, the discrete Heisenberg group are isomorphic to (Zⁿ,Z^m;f) for some n, m, f as follows.

Example 1.2.2. If n= 0, then this group is isomorphic to Z^m. Example 1.2.3. If n= 2, m= 1 and f :Z²×Z² →Z is given by

((a c )

, (a^′

c^′ ))

7→ac^′,

then this group is isomorphic to the discrete Heisenberg group.

It is easy to see that the group G= (Zⁿ,Z^m;f) has the following properties.

1. For every k∈Z, (a,b)^k= (ka, kb+ ^k(k−1)₂ f(a,a)), so this group is torsion free.

2. The commutator of (a,b),(a^′,b^′) is

[(a,b),(a^′,b^′)] = (0, f(a,a^′)−f(a^′,a)).

Especially, we have

[(a,b),(0,b^′)] = (0,0).

3. The subgroup N = {(0,b) | b ∈ Z^m} is contained in the center of G, and G/N is isomorphic to Zⁿ. Hence Gis nilpotent.

4. Let{e_i}ⁿ_i=1, {e^′_j}^m_j=1 be the standard basis of Zⁿ, Z^m respectively. Then Gis generated by {(ei,0)},{(0,e^′_j)}. Hence Gis finitely generated.

Thus we see that the group (Zⁿ,Z^m;f) is finitely generated, torsion free, and nilpotent of class 2 and Hirsch length n+m.

1.3 Expressions by matrices

Let M_n(Z^m) be the set of n×n matrices whose entries are elements of the abelian groupZ^m. It has a natural additive group structure. We define actions of the ringsMm(Z) andMn(Z) on this group. For

A=







a₁₁ · · · a_1n ... . .. ... a_n1 · · · a_nn





∈M_n(Z^m), b=





 b₁

... b_n





∈Zⁿ, we define the product Ab by

Ab:=







a₁₁b₁+a₁₂b₂+...+a_1nb_n ...

a_n1b₁+a_n2b₂+...+a_nnb_n







the product ^tbA is defined similarly, where ^tbis the transpose of b. Using these products we put, for a matrixB = (b₁, . . . ,b_n)∈M_n(Z),

AB:= (Ab1, . . . , Abn)∈Mn(Z^m),

tBA:= (^tb₁A, . . . ,^tb_nA)∈M_n(Z^m).

In this way we regard M_n(Z^m) as a right and left M_n(Z)-module. Also for B ∈ M_m(Z), we define:

BA:=







Ba11 · · · Ba1n

... . .. ... Ban1 · · · Bann





∈M_n(Z^m).

ThusM_n(Z^m) is regared as a left M_m(Z)-module as well.

From a bilinear map f :Zⁿ×Zⁿ→Z^m, we define the matrix A by A:=







f(e₁,e₁) · · · f(e₁,e_n) ... . .. ... f(e_n,e₁) · · · f(e_n,e_n)





∈M_n(Z^m).

Then we have f(a,a^′) = ^taAa^′ for every (a,a^′) ∈Zⁿ×Zⁿ.

So we shall often denote the group (Zⁿ,Z^m;f) by (Zⁿ,Z^m;A). With this notation Example 5 can be stated as follows:

Example 1.3.1. If A= (0 1

0 0 )

, then (Z²,Z¹;A) is isomorphic to U3(Z)

The n-th discrete Heisenberg groups H_n (cf.[7] ) is isomorphic to (Z²ⁿ,Z;A) for some A∈ M_2n(Z):

Example 1.3.2. If A =

(O E_n

O O

)

, H_n is isomorphic to (Z²ⁿ,Z;A), where E_n is the identity matrix in M_n(Z).

6

1.4 Classification of isomorphism classes of {( Z

, Z

; A)}

In this section, we classify the set of groups {(Zⁿ,Z^m;A) |A ∈ M_n(Z^m)} up to isomorphism.

First of all the following two isomorphisms are obvious : Proposition 1.4.1.

(1) (Zⁿ,Z^m;^tXAX) ∼= (Zⁿ,Z^m;A) (X ∈GLn(Z))

(a,b) 7→ (Xa,b)

(2) (Zⁿ,Z^m;A) ∼= (Zⁿ,Z^m;Y A) (Y ∈GLm(Z)) (a,b) 7→ (a, Yb)

In what follows we shall fixnandm, and denote our group byG(A) := (Zⁿ,Z^m;A) forA∈ M_n(Z^m). We consider the relations of a system of generators ofG(A). Putx_i = (e_i,0) (1≤i≤ n),y_i = (0,e^′_j) (1≤j≤m), z_ij(A) = [x_i, x_j] (1≤i < j≤n). Thenz_ij(A)’s are generated by y1, ..., ym. Moreover,xi’s andyj’s form a system of generators ofG(A), and satisfy the following relations:

1. [xi, xj] =zij(A) (1≤i < j≤n)

2. [xi, yj] is the identity element (1≤i≤n, 1≤j≤m) 3. [y_i, y_j] is the identity element (1≤i < j≤m).

Let ˜G(A) denote the quotient group of the free group on the words{x1, ..., xn, y1, ..., ym}by the above relations. Then we have:

Lemma 1.4.2. The canonical homomorphism

G(A)˜ −→G(A) is an isomorphism.

Proof. The surjectivity is trivial. By the above relations, we see that eachx∈G(A) is uniquely expressed in the following way

x=

n

∏

i=1

x^a_iⁱ

m

∏

j=1

y^b_j^j, a_i, b_j ∈Z.

Since ˜G(A) has the same property, this morphism is injective.

Proposition 1.4.3. If A−A^′ is symmetric, then the map (Zⁿ,Z^m;A)→(Zⁿ,Z^m;A^′) induced by the identity map of Zⁿ×Z^m is an isomorphism.

Proof. By the above Lemma, it is sufficient to prove ˜G(A) = ˜G(A^′). For all x,y ∈ Zⁿ×Z^m, the commutator of xand yin (Zⁿ,Z^m;A) is equal to the commutator of them in (Zⁿ,Z^m;A^′).

So the relations of the generators of ˜G(A) and ˜G(A^′) are the same, Hence ˜G(A) = ˜G(A^′).

We define another right-GL_n(Z)-module structure on M_n(Z^m) as follows: for X ∈GL_n(Z) and A∈Mn(Z^m), we put

A ⋆ X :=^tXAX.

Let us define Alt_n(Z^m) and Sym_n(Z^m) to be {A ∈ M_n(Z^m) | ^tA + A = O} and {A ∈ M_n(Z^m) | ^tA−A = O}, respectively. We regard Altn(Z^m) and Sym_n(Z^m) as GL_m(Z) and GL_n(Z)-submodules of M_n(Z^m). It follows from Prop. 1.4.1 and Prop. 1.4.3 that for A, B ∈ M_n(Z^m) ifGL_m(Z)A=GL_m(Z)B orA ⋆ GL_n(Z) =B ⋆ GL_n(Z) or A≡B mod (Sym_n(Z^m)),

then the two groups (Zⁿ,Z^m;A) and (Zⁿ,Z^m;B) are isomorphic. One see easily thatM_n(Z^m)/Sym_n(Z^m) is isomorphic to Alt_n(Z^m) by the map

(A mod (Sym_n(Z^m))7→A−^tA)

as left-GL_m(Z)-modules and right-GL_n(Z)-modules. Forn∈Z_≥0, we defineT_n to be the class {(Zⁿ,Z^m;A)|m∈Z_≥0, A∈M_n(Z^m)}.By considering the double cosetsGL_m(Z)\Altn(Z^m)/GL_n(Z), we classify isomorphism classes of groups T₂ and T₃ in the next two chapters.

In the rest of this section, we prove thatT_nis equal toT_n^′(defined below) up to isomorphism for eachn∈Z_≥0. Letnbe a nonnegative integer, and defineT_n^′ to be the class ofT-groupsG such that

• the nilpotent class ofGis 2,

• r(G/Z(G))≤n,

• r([G, G])≤h(G)−n.

Proposition 1.4.4. For a T-group G, there existsG^′ ∈T_n such that G is isomorphic to G^′ if and only if there exists G^′∈T_n^′ such that G is isomorphic to G^′.

Proof. We put G := (Zⁿ,Z^m;A) for some n, m ∈ Z_≥0, A ∈ M_n(Z^m). By Sect.1.2, G is of nilpotent class 2, and the subgroup N :={(0,b)|b∈Z^m}of Gsatisfies

[G, G]⊂N ⊂Z(G).

Since G/N ∼=Zⁿand N ∼=Z^m, we have

• h(G) =n+m,

• r(G/Z(G))≤r(G/N) =n,

• r([G, G])≤r(N) =m.

Hence r([G, G])≤h(G)−n. ThusG∈T_n^′.

Let us assume G ∈ T^′_n, and let (k, l) := (h(G), r(G/Z(G))). Then there is a Malcev basis {x1, x₂,· · ·, x_k} of G. We put H_i := ⟨xi,· · ·, x_k⟩ (1 ≤i ≤k). Recall that {x1, x₂,· · ·, x_k} is called a Malcev basis of G if{Hi | (1≤i≤k)} is a lower central series of G and H_i/H_i+1 ∼=

8

Z (1 ≤ i ≤ k−1). Since G is of nilpotent class 2, for all elements x, y of G and for all integers k, [x^k, y] = [x, y]^k,where [x, y] means the commutator of x and y. Hence G/Z(G) is torsion free, namely an abelian T-group. Thus we may assume thatx₁, x₂,· · ·, x_l̸∈Z(G) and H_l+1=Z(G). Since [G, G]⊂Z(G) andr([G, G])≤k−n, we may assume that [G, G]⊂Hn+1. Let θ be the canonical isomorphism H_n+1 to Z^k−n. Since the nilpotent class of G is 2, the map f : Zⁿ×Zⁿ −→ Z^k−n being (α₁,· · ·α_n, β₁,· · ·, β_n) 7→ θ(

[x₁^α1· · ·x^α_nⁿ, x₁^β1· · ·x^βnⁿ]) is a bilinear form. Let Abe the matrix corresponding to the bilinear formf, then we construct an isomorphism G → G(Zⁿ,Z^k−n;A) by (

x^α₁¹· · ·x^α_k^k 7→(α₁,· · ·α_k))

,which completes the proof, q.e.d.

Zeta functions of groups does not determine isomorphic classes of T^′₄ (cf.[1], PropositionB, or [5], Example 4). Furthermore, for each G ∈ T^′_n, G×Z is an object of T^′_n+1. Hence in the case n ≥ 4, zeta functions of groups does not determine isomorphic classes of T^′_n. Thus our remaining problem is the casen= 2 or n= 3, which is considerd in the next two chapters.

Chapter 2

The class T ₂

In this chapter we derive explicit forms of zeta functions of groups of T₂, and then we prove they determine isomorphism classes of the class.

2.1 Classification of isomorphism classes of T

First, we state the first main theorem of this book. It gives the classification and the concrete expression of zeta functions ofT₂.

For α ∈Zwe put

G(α) = (

Z²,Z;

(0 α 0 0

)) .

Theorem 2.1.1. (1)For eachA∈M2(Z^m), there exists a uniqueα∈Z_≥0 such that(Z²,Z^m;A) is isomorphic to G(α)×Z^m−1.

(2) The Euler p-factor ofζ_G(α)×Z^m−1(s) is given as ζ_G(α)×Z^m−1_,p(s)

=ζ_Zm+2,p(s) (

1−p^−ρ(s,α)ζp(2s−m−1)ζp(2s−m−2) ζ_p(s)ζ_p(s−1)

) ,

where ρ(s, α) = (s−m−1)(v_p(α) + 1), and v_p(∗) denotes the p-adic additive valuation. (We make the convention that p^−ρ(s,0)≡0.)

(3) If α= 1, we recover the result of Example 3, i.e.,

ζ_G(1)×Z^m−1(s) = ζ_Zm+1(s)ζ(2s−m−1)ζ(2s−m−2) ζ(3s−m−2) .

Let α,α^′ be distinct nonnegative integers, and putG=G(α)×Z^m−1,G^′ =G(α^′)×Z^m−1. By the above theorem, Gis not isomorphic toG^′, and there exists a prime numberpsuch that ζ_G,p(s)̸=ζ_G^′_,p(s). Hence we have:

Theorem 2.1.2. A group G∈T₂ is determined by ζG(s) up to isomorphism.

Proof of Theorem 2.1.1. We prove first the exisitence of α in the statement (1). It is easy to see that, forA∈M₂(Z^m) there existsa∈Z^msuch thatA−

(0 a 0 0 )

is symmetric. Then Proposition 1.4.3 implies that (Z²,Z^m;A) is isomorphic to (Z²,Z^m;

(0 a 0 0 )

). Considering the actions ofGLm(Z) described above, we see that theGLm(Z)-orbit of

(0 a 0 0 )

contains a unique element of the form

(0 αe^′₁ 0 0

)

. (Consider the elementary transformation of the m×1 matrix a.) So by Proposition 1.4.1 we have (Z²,Z^m;

(0 a 0 0 )

)∼= (Z²,Z^m;

(0 αe^′₁ 0 0

)

). Since it is clear that (Z²,Z^m;

(0 αe^′₁ 0 0

)

)∼=G(α)×Z^m−1, it follows that (Z²,Z^m;A)∼=G(α)×Z^m−1.

Next, assuming the statement (2), we prove the uniqueness of αin the first statement (The proof of (2) will be given in the next two sections). It is sufficient to show that, if α, α^′ are distinct nonnegative integers then G(α)×Z^m−1 and G(α^′)×Z^m−1 are not isomorphic. The assumption α ̸=α^′ implies v_p(α)̸=v_p(α^′) for some prime p. From the result of (2), it follows thatζ_G(α)×Z^m−1_,p̸=ζ_G(α^′_)×Z^m−1_,p, which clearly implies thatG(α)×Z^m−1 is not isomorphic to G(α^′)×Z^m−1, q.e.d.

2.2 Explicit forms of zeta functions

We shall prove the statement (2) of Theorem 2.1.1. We can also prove this by the method in [2], section 2, but here, we will give an elementary proof. For the proof, we shall define some symbols.

Let us consider an exact sequence of groups:

1→N →G→g→1.

We choose and fix a subgroup M of N and a subgroup hof g. Let a(M,h) denote the set of subgroups H of G such that H ∩N = M, and that the image of H by the homomorphism G→g is equal to h, i.e., the following diagram

1 −−−−→ N −−−−→ G −−−−→ g −−−−→ 1 x





x





x





1 −−−−→ M −−−−→ H −−−−→ h −−−−→ 1

is commutative, and the second horizontal sequence is exact. It follows that [G : H] = [g : h][N :M] for each H ∈a(M,h). Let a(M,h) denote the cardinality of a(M,h). Then the zeta

12

function of Gis expressed as

ζ_G(s) =∑

M,h

a(M,h)[N :M]^−s[g:h]^−s,

where the sum is taken over all M,h satisfying [N :M]<∞, [g:h]<∞. Similarly the local zeta function at a prime pis expressed as

ζ_G,p(s) =∑

M,h

a(M,h)[N :M]^−s[g:h]^−s,

where the sum is taken over all M,hfor which [N :M], [g:h] arep-th power.

We next assume thatM is a normal subgroup ofG, so that we have the canonical homomor- phism G/M →g. Let Homg(h, G/M) denote the set of all elements in Hom(h, G/M) through which the inclusion map h→gfactors. Then we have:

Lemma 2.2.1. There is an bijective map between a(M,h) and Hom_g(h, G/M).

Proof. For each H ∈ a(M,h), the homomorphism composed by the canonical isomorphism h → H/M and the inclusion map H/M → G/M is an element of Homg(h, G/M). On the other hand, for each element t ∈ Homg(h, G/M), the inverse image of t(h) by the canonical homomorphism G → G/M is an element of a(M,h). Thus we can construct the two maps between a(M,h) and Homg(h, G/M). It is easy to see that the two maps are inverse of each other.

In the following, we consider the case that g is isomorphic to Z^k for some positive integer k, and N is contained in the center of G, hand M are of finite index. We remark that M is a normal subgroup of G, [G, G]⊂N, and his a free abelian group of rank k. Let {h1, ..., h_k} denote a basis ofhand {h˜1, ...h˜_k}denote lifts of h1, ..., h_k toG respectively.

Lemma 2.2.2. A map t:h→G/M is an element of Homg(h, G/M) if and only if it satisfies the following properties.

1. The image of t(hi) under the canonical homomorphism G/M →g is equal to hi. 2. t(h^l₁¹· · ·h^l_k^k) =t(h₁)^l1· · ·t(h_k)^lk for any l₁, ..., l_k∈Z.

3. t(hi) and t(hj) are commutative for each i, j.

Proof. If t ∈ Homg(h, G/M), it is easy to see that t satisfies the above properties. Suppose, conversely, that t satisfies the above properties. Then by properties 2 and 3, we see that t∈Hom(h, G/M). By the property 1, we also see that t∈Homg(h, G/M).

Corollary 2.2.3. Suppose that t is as above, and satisfies property 1 and 2. Then t ∈ Hom_g(h, G/M) if and only if [ ˜h_i,h˜_j]∈M for alli, j.

Proof. Let ˆh_i, ˆh_j denote the images of ˜h_i, ˜h_j by the canonical mapG→G/M. Then [ ˜h_i, ˜h_j]∈M if and only if [ ˆh_i, ˆh_j] = 1. By the above lemma, it is sufficient to prove [ ˆh_i, ˆh_j] = [t(h_i), t(h_j)].

Since t satisfies the property 1, the images of ˆh_i and t(h_i) by G/M → g are the same. Since N/M is contained in the center ofG/M, we have [ ˆh_i, ˆh_j] = [t(h_i), t(h_j)].

Corollary 2.2.4. The number of maps t:h→G/M which satisfy1 and2is equal to[N :M]^k. Proof. Let φ denote the map G/M → g induced from the map G → g. Since {h₁, ...h_k} is a basis of h, we have

#{t:h→G/M |t satisf ies1 and2}

= #

k

∏

i=1

φ⁻¹(h_i)

=

k

∏

i=1

#φ⁻¹(h_i)

= [N :M]^k.

We notice that the condition [ ˜hi,h˜j]∈M is independent of t. Therefore we have:

Proposition 2.2.5.

{a(M,h) = [N :M]^k, if [ ˜h_i,h˜_j]∈M for all i, j a(M,h) = 0, otherwise.

Now we apply these results to the computation of ζ_G(s).

Corollary 2.2.6.

ζ_G(s) = ∑

a(M,h)̸=0

[N :M]^k−s[g:h]^−s,

where the sum is extended on M,h such that[N :M], [g:h]are finite. Similarly we have ζ_G,p(s) = ∑

a(M,h)̸=0

[N :M]^k−s[g:h]^−s,

where the sum is taken on M, h such that [N : M], [g : h] are p-th power. Especially if G is abelian, then

ζ_G(s) =ζg(s)ζ_N(s−k), ζ_G,p(s) =ζg,p(s)ζ_N,p(s−k), and

a_n(G) = ∑

lm=n

a_m(g)a_l(N)l^k.

We remark that the validity of the statements in corollary 2.2.3 and proposition 2.2.5 are independent of the choice of {h˜₁, ...h˜_k}.

14

Corollary 2.2.7. Suppose thatk= 2andx1, x2 ∈Gare such that the images ofx1, x2togform a basis ofg. Then in order that a(M,h)̸= 0, it is necessary and sufficient that[x₁, x₂]^[g:h]∈M. Proof. Let x₁, x₂ be the images of x₁,x₂ respectively. Then there exist a, b, c, d∈Zsatisfying ad−bc >0, such that

h₁ :=x^a₁x^b₂, h₂:=x^c₁x^d₂

form a basis of h. Put ˜h₁ = x^a₁x^b₂, h˜₂ = x^c₁x^d₂. Since N is contained in the center, we have [ ˜h₁,h˜₂] = [x₁, x₂]^ad−bc. We also note that [g:h] =ad−bc. Thus we have

[ ˜h₁,h˜₂] = [x₁, x₂]^[g:h].

Corollary 2.2.8. Let the assumptions be as above, and let Nn=N/⟨[x1, x2]ⁿ⟩. Then we have ζ_G(s) =∑

n≥0

a_n(g)n^−sζ_Nn(s−2), ζ_G,p(s) =∑

l≥0

1−p^l+1 1−p p^−lsζ_N

pl,p(s−2).

Proof. By corollary 2.2.6, we have

ζ_G(s) = ∑

h,M:a(M,h)̸=0,[g:h]<∞,[N:M]<∞

[g:h]^−s[N :M]^2−s

= ∑

h:[g:h]<∞

[g:h]^−s ∑

M:a(M,h)̸=0,[N:M]<∞

[N :M]^2−s. By Corollary 2.2.7, we also have

ζ_G(s) =∑

h

[g:h]^−s ∑

[x1,x2]^[g:h]∈M

[N :M]^2−s

=∑

n≥1

∑

[g:h]=n

n^−s ∑

[x1,x2]ⁿ∈M

[N :M]^2−s. Hence we obtain

ζ_G(s) =∑

n≥0

a_n(g)n^−sζ_Nn(s−2) and similary

ζ_G,p(s) =∑

l≥0

a_pl(g)p^−lsζ_N

pl,p(s−2).

By Corollary 2.2.6, we have

a_p^l(g) =a_p^l(Z²) = ∑

0≤i≤l

a_pⁱ(Z)pⁱa_p^l−i(Z)

= ∑

0≤i≤l

pⁱ.

2.3 Calculation of the zeta function of G(α) × Z

Now, we will complete the proof of the theorem 2.1.1 by proving (2). We consider the case : G=⟨x1, x₂, y₁, ..., y_m|[x1, x₂] =y₁^α,[x_i, y_j] = 1,[y_i, y_j] = 1⟩

∼=G(α)×Z^m−1,

N =⟨y1, ..., y_m⟩, g = G/N ∼=Z²

By Corollary 2.2.8, we only have to compute the local factor ζ_N

pl,p(s).

Since [x₁, x₂] =y^α₁ and {y1, ..., y_m} is a basis ofN =∼Z^m, we see thatN_pl=N/⟨[x1, x₂]^pl⟩ is isomorphic to Z^m−1×Z/αp^lZ.

By Corollary 2.2.6, we have then

ζ_Z^m−1×Z/αp^lZ,p(s) =ζ_Z^m−1_,p(s)ζ_Z/αp^l_Z,p(s−m+ 1).

It is easy to see

ζ_Z/αp^l_Z,p(s−m+ 1) =ζ_Z/pl+vp(α)Z,p(s−m+ 1)

= 1−p−(s−m+1)(1+l+vp(α))

1−p^−(s−m+1) , wherevp(∗) is thep-adic additive valuation.

Next, we calculate ζ_G,p(s), i.e., ζ_Z^m−1,p(s−2)∑

l≥0

1−p^l+1

1−p p^−ls1−p−(s−m−1)(1+l+vp(α))

1−p^{−(s−m−1)} . Putt=s−m−1,β = 1 +v_p(α).

∑

l≥0

1−p^l+1

1−p p^−ls1−p^−t(l+β) 1−p^−t

= 1

(1−p)(1−p^−t)

∑

l≥0

p^−ls(1−p^l+1)(1−p^−t(l+β))

= 1

(1−p)(1−p^−t)

∑

l≥0

p^−ls(1−p^l+1−p^−t(l+β)+p−t(l+β)+l+1)

= 1

(1−p)(1−p^−t)

∑

l≥0

(p^−ls−p^1−l(s−1)−p^{−tβ−l(s+t)}+p1−tβ−l(s+t−1))

= 1

(1−p)(1−p^−t) { 1

1−p^−s − p

1−p^−(s−1) − p^−tβ

1−p^−(s+t) + p^−tβ+1 1−p^−(s+t−1)

}

= 1

(1−p)(1−p^−t)

{ 1−p

(1−p^−s)(1−p^−(s−1)) −p^−tβ 1−p

(1−p^−(s+t))(1−p^−(s+t−1)) }

= ζ_p(t)ζ_p(s)ζ_p(s−1) (

1−p^−tβζ_p(s+t)ζ_p(s+t−1) ζp(s)ζp(s−1)

) . 16

By Corollary 2.2.6, ζ_Z^m−1_,p(s−2)ζp(s−1)ζp(s)ζp(t) =ζ_Z^m+2_,p(s). Hence we have ζ_G,p(s) =ζ_Zm+2,p(s)

(

1−p^−tβζ_p(s+t)ζ_p(s+t−1) ζ_p(s)ζ_p(s−1)

) . This completes the proof of theorem 2.1.1.

Chapter 3

The class T ₃

In this chapter we show that zeta functions of groups determine isomorphism classes of T₃.

3.1 Classification of isomorphism classes of T

Let us confine ourselves to the case n = 3. We denote by M(m,3;Z) the set of all m×3 matrices with entries from Z. M(m,3;Z) is regarded as a left-GLm(Z)-module and a right- GL₃(Z)-module naturally. It is clear that the map τ : Alt₃(Z^m)→M(m,3;Z) being





0 x y

−x 0 z

−y −z 0



7→(

z y x)

is isomorphism of left-GL_m(Z)-modules. ForX∈GL₃(Q), we denote by ˜Xthe adjugate matrix ofX. Let us define a homomorphismρ:GL₃(Q)→GL₃(Q) by (X 7→diag[1,−1,1]^t( ˜X)diag[1,−1,1]).

We remark that ρ|_GL3(Z) is a surjective homomorphism from GL₃(Z) to SL₃(Z). By a direct computation, one can check that the mapsρ and τ are compatible, i.e., for A∈Alt₃(Z^m) and X∈GL₃(Z)

τ(A ⋆ X) =τ(A)ρ(X).

Hence we are reduce to giving a complete system of representatives of GL_m(Z)\M(m,3;Z)/SL₃(Z),

which is settled by the fundamental theorem of elementary divisors.

We denote by G(x,y,z;m) the group



Z³,Z^m;





0 x y 0 0 z 0 0 0







 . For non-negative integers α, β, γ, we put

• G(α, β, γ; 0) :=G(0,0,0; 0) :=Z³.

• G(α, β, γ; 1) :=G(α,0,0; 1) :=G((

α 0 0)

; 1) .

• G(α, β, γ; 2) :=G(α, β,0; 2) :=G

((α 0 0 0 β 0

)

; 2 )

.

• G(α, β, γ; 3) :=G









α 0 0

0 β 0

0 0 γ



; 3



.

• G(α, β, γ;m) :=G(α, β, γ; 3)×Z^m−3 (m≥3).

Let us denote by G_m the class of groups of all G(α, β, γ;m) such that

• αZ⊃βZ⊃γZ (m≥3).

• αZ⊃βZ, γ= 0 (m= 2).

• β =γ = 0 (m= 1).

• α=β =γ = 0 (m= 0).

We also define Gto be the union of the classes G_m for all non-negative integers m. By the above discussion, Gis a system of representatives of isomorphism classes of T₃.

Proposition 3.1.1. Let A ∈M₃(Z^m) and G:= (Z³,Z^m;A). Then there existsG^′ ∈ G_m such thatG is isomorphic to G^′.

In the next three sections, we show thatG forms a complete system of representatives.

3.2 Basic notions and facts for counting subgroups

In this section, we introduce some basic notions and facts to count subgroups of objects of the class G. For a group G, we write H ≤ G (resp. H≤fG) if H is a subgroup (resp. a subgroup of finite index) of G. For the rest of this section, we fix a prime number p and G:=G(α, β, γ;m)∈G_m. One can identify Gwith the group

⟨

x₁, x₂, x₃, y₁,· · · , y_m

[x₁, x₂] = y₁^α, [x₁, x₃] = y₂^β, [x₂, x₃] = y^γ₃,

[x_i, y_j] = 1 (1≤i≤3,1≤j ≤m), [y_i, y_j] = 1 (1≤i, j ≤m),

⟩ .

Hence we have a canonical exact sequence:

1→Z^{m ι}→G→^π Z³ →1.

As in §2.2, for h ≤f Z³ and M ≤f Z^m, we define a_G(h, M) to be the cardinality of the set {H ≤G|π(H) =h, ι⁻¹(H) =M}. We denote it briefly bya(h, M).Note that if a subgroupH of Gsatisfiesπ(H) =handι⁻¹(H) =M, then [G:H] = [Z³ :h][Z^m:M].Let n∈Z_≥0 and let h≤Z³ satisfy that [Z³:h] dividespⁿ. We put

a_pⁿ(G,h) :=∑

M

a(h, M), 20

where the sum is taken over all M satisfying [Z³ :h][Z^m:M] =pⁿ.Then we see a_pⁿ(G) =∑

h

a_pⁿ(G,h),

where the sum is taken over all hsuch that [Z³:h] dividespⁿ. By Prop. 2.2.5, we have:

a_pⁿ(G,h) = ∑

M:a(h,M)̸=0

[Z^m:M]³,

where the sum is taken over allMsatisfying [Z³ :h][Z^m :M] =pⁿ. By the basis{π(x₁), π(x₂), π(x₃)}

ofZ³, we can identify the subgroups of finite index inZ³ withM₃(Z)∩GL3(Q)/GL₃(Z).ForA∈ M₃(Z)∩GL3(Q) and a subgrouphofZ³, ifhis generated by the entries of(

π(x₁) π(x₂) π(x₃)) A, we say that h corresponds to A, or that A corresponds to h. We fix h ≤_f Z³. Let A ∈ M₃(Z)∩GL₃(Q) be a matrix corresponding to h.One can assume that Ais a lower triangular matrix. Let us put

A:=





a 0 0 b d 0 c e f



, and

(h1 h2 h3) :=(

x1 x2 x3) A, i.e.,

h₁ :=x^a₁x^b₂x^c₃, h₂ :=x^d₂x^e₃, h₃ :=x^f₃.

Then π(h₁), π(h₂), π(h₃) is a basis ofh. By Prop. 2.2.5, we see that forM ≤Z^m,a(h, M)̸= 0 if and only if

[h₁, h₂],[h₁, h₃],[h₂, h₃]∈M.

Let us putp^k:=pⁿ/[Z³ :h]. Then we have

a_pⁿ(G,h) =p^3ka_pk(Z^m/N(h)),

whereN(h) := ⟨[h1, h2],[h1, h3],[h2, h3]⟩.By a direct computation, we have:

([h₂, h₃] [h₁, h₃] [h₁, h₂])

=(

[x₂, x₃] [x₁, x₃] [x₁, x₂]) ρ(A), i.e.,

([h₂, h₃] [h₁, h₃] [h₁, h₂])

=(

y₃^γ y^β₂ y^α₁ )

ρ(A), whereρ is defined in the previous section. It is easy to see that

ρ(A) =





df bf be−cd

0 af ae

0 0 ad



.

Note that the above formula is well-defined although m <3. For example, if m = 2, then y3

is not defined, but γ = 0. Hence y^γ₃ is the identity. Thus we don’t have to consider y₃ in the above formula. Put

(z₁ z₂ z₃) :=(

y^p₃^vp(γ) y₂^pvp(β) y₁^pvp(α)

)ρ(A),

where v_p(∗) means the p-adic valuation. Since [Z^m : M] is p-th power, M contains p^kZ^m for somek. Thusa(h, M)̸= 0 if and only if

z1, z2, z3 ∈M.

Therefore we have the following formulas:

Lemma 3.2.1. Let us put G_p :=G(

p^vp(α), p^vp(β), p^vp(γ);m) . Then a_pⁿ(G) =a_pⁿ(G_p) (n≥0), where v_p(∗) means the p-adic valuation.

Lemma 3.2.2. Let n∈Z_≥0, and hbe a subgroup of Z³ whose index dividespⁿ, and let

A:=





a 0 0 b d 0 c e f





be a matrix corresponding to h. Putp^k:=pⁿ/[Z³ :h] and

ρ_G(A) :=















diag[γ, β, α]ρ(A) (m≥3).

diag[β, α]

(af ae 0 ad

)

(m= 2).

αad (m= 1).

Then

a_pⁿ(G,h) =p^3k







a_pk(Z^m−3×Z³/ρ_G(A)Z³) (m≥3).

a_p^k(Z²/ρ_G(A)Z²) (m= 2).

a_pk(Z/ρ_G(A)Z) (m= 1).

3.3 The case m ̸= m

In this section we shall prove:

Proposition 3.3.1. Let G∈G_m and G^′∈G_m^′. Ifm̸=m^′, then for some prime p, ζ_G,p(s)̸=ζ_G^′_,p(s).

By the above proposition, we have:

Corollary 3.3.2. Let G∈G_m and G^′∈G_m^′. Ifζ_G(s) =ζ_G^′(s), then m=m^′. 22

To prove Prop. 3.3.1, it is sufficient to prove:

Proposition 3.3.3. Let G and G^′ be as in Prop. 3.3.1. Then for some prime number p, (a_p(G), a_p2(G))̸= (ap(G^′), a_p2(G^′)).

We fix a prime numberpandG:=G(α, β, γ;m)∈G_m. Now we evaluatea_p(G) anda_p2(G).

Proposition 3.3.4. The following formulas hold:

a_p(G) =















ap(Z^m+3) (p|α).

a_p(Z^m+2) (p̸ |α, p|β).

a_p(Z^m+1) (p̸ |α, p̸ |β, p|γ).

a_p(Z^m) (p̸ |α, p̸ |β, p̸ |γ).

Proof. We see that

ap(G) =ap(G,Z³) + ∑

h:[Z³:h]=p

ap(G,h).

Assume [Z³ :h] = p. For a subgroup M of Z^m, if [Z³ :h][Z^m :M] = p, then Z^m =M. Hence a(h, M)̸= 0. Thus

∑

h:[Z³:h]=p

a_p(G,h) =a_p(Z³).

Assumeh=Z³. We evaluatea_p(G,Z³). For a subgroupMofZ^msatisfying [Z³ :h][Z^m :M] =p, a(h, M)̸= 0 if and only if

y₁^α, y^β₂, y₃^γ∈M, Since [Z^m :M] =p,

pZ^m ⊂M.

PutN :=⟨y^α₁, y^β₂, y₃^γ⟩+pZ^m. Then by the definition of a_p(G,Z³), a_p(G,Z³) =p³a_p(Z^m/N).

Z^m/N is isomorphic to















F^m_p (p|α).

F^m−1_p (p̸ |α, p|β).

F^m−2_p (p̸ |α, p̸ |β, p|γ).

F^m−3_p (p̸ |α, p̸ |β, p̸ |γ).

Since a_p(F^k_p) =a_p(Z^k) for k∈Z_≥0,

a_p(G) =a_p(Z³) +p³















ap(Z^m) (p|α).

a_p(Z^m−1) (p̸ |α, p|β).

a_p(Z^m−2) (p̸ |α, p̸ |β, p|γ).

a_p(Z^m−3) (p̸ |α, p̸ |β, p̸ |γ).