Hasse Zeta Functions of Non-commutative Rings

208, 304᎐342 Ž1998. JA987489

JOURNAL OF ALGEBRA ARTICLE NO.

Hasse Zeta Functions of Non-commutative Rings Takako Fukaya* Tokyo Institute of Technology, Oh-okayama, Meguro-ku, Tokyo, 152, Japan Communicated by Alexander Lubotzky Received June 16, 1997

Contents. 1. 2. 3. 4. 5.

Introduction. General results. Examples. Con¨ ergence of zeta functions and Gelfand᎐Kirillo¨ dimensions. Questions.

1. INTRODUCTION 1.1. In this paper, we define ‘‘Hasse zeta functions’’ of non-commutative finitely generated rings over the ring Z of integers and study their properties. We compute the zeta functions of Weyl algebras, universal enveloping algebras of certain Lie algebras, certain group rings, and some other non-commutative rings. Recall that for a commutative finitely generated ring A over Z, the Hasse zeta function of A is defined by

␨A Ž s . s

ys y1

Ł Ž1 y N Ž ᒊ . .

,

ᒊgᑪ Ž A .

where ᑪ Ž A. denotes the set of maximal ideals of A, and N Ž ᒊ . s 噛Ž Arᒊ . the order of the finite field Arᒊ. Now let A be a Žnot necessarily commutative. finitely generated ring over Z. We define the Hasse zeta function ␨AŽ s . of A as

␨A Ž s . s

ys y1

Ł Ž1 y N Ž ᒊ . .

,

ᒊgᑪ Ž A .

* Current address: Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8914, Japan. 304 0021-8693r98 $25.00 Copyright 䊚 1998 by Academic Press All rights of reproduction in any form reserved.

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HASSE ZETA FUNCTIONS

where ᑪ Ž A. denotes the set of two-sided ideals of A such that Arᒊ is isomorphic to the matrix ring Mr Ž k . for some finite field k and for some integer r G 1, and N Ž ᒊ . s 噛Ž k .. It can happen that ␨AŽ s . diverges, though ␨AŽ s . converges for many interesting types of A. We say ␨AŽ s . converges if it absolutely converges as an infinite product for s whose real part is sufficiently large, and we say ␨AŽ s . diverges otherwise. ‘‘For what type of ring A, does ␨AŽ s . converge?’’ ‘‘If it converges, what is the inf of the real part of s at which ␨AŽ s . absolutely converges?’’ These are very difficult problems, and we will discuss them in Section 4 Žsee also Theorem 1.4 below.. In wKu1, Ku2x, N. Kurokawa suggested that the Hasse zeta function of a nice category C should be defined as an Euler product over simple objects in C . Our ␨AŽ s . can be expressed in the form

␨A Ž s . s

ys y1

Ł Ž1 y N Ž M . .

,

M

where M runs over the isomorphism classes of finite simple A-modules and N Ž M . s 噛 End AŽ M .. For each integer r G 1, we define the partial product ␨A, r Ž s . in ␨AŽ s . by

␨A , r Ž s . s

ys y1

Ł Ž1 y N Ž ᒊ . .

,

ᒊgᑪ r Ž A .

where ᑪ r Ž A. denotes the subset of ᑪ Ž A. consisting of ᒊ such that Arᒊ , Mr Ž k . for some finite field k. We obtain the following results. THEOREM 1.2.

Let A be a finitely generated ring o¨ er Z, and let r G 1.

Ž1. The function ␨A, r Ž s . absolutely con¨ erges if ReŽ s . is sufficiently large. Ž2. Let Fq be a finite field. If A is an Fq-algebra, then ␨A, r Ž s . is a rational function of qys with coefficients in the rational number field Q. The proof of Theorem 1.2 is by showing that ␨A, r Ž s . coincides with the Hasse zeta function of a scheme ᑭ A, r of finite type over Z called the space of r dimensional irreducible representations of A wL-Mx. This ᑭ A, r is an Fq-scheme in the case A is an Fq-algebra. In Section 2, we prove Theorem 1.2 and other general results on our Hasse zeta functions. 1.3. In Section 3, we compute ␨AŽ s . for several types of non-commutative rings A ŽExamples 3.1᎐3.8.. For example, in the case R is a commutative finitely generated ring over Z and A is the Weyl algebra Rw T, drdT x, we will show ␨A Ž s . s ␨ R Ž s y 2 .

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TAKAKO FUKAYA

ŽExample 3.1.. For this A, if R s Z, ᑭ A, r / ⭋ if and only if r is a prime number. Similarly, for most A in Section 3, the set  r; ᑭ A, r is not empty4 is not a finite set, but the product ␨AŽ s . of ␨A, r Ž s . for all r has a surprisingly simple form though each ␨A, r Ž s . is not so simple. It seems that such properties of ␨AŽ s . are not explained by known commutative arithmetic algebraic geometry for each ᑭ A, r . In one example Example 3.5, ␨AŽ s . is not a function of familiar type. This A is the universal enveloping algebra of the Lie algebra sl 2 ŽZ.. In this case,

␨A Ž s . s ␨ Ž s y 3 . =

Ł

Ł

Ž 1 y pyŽ sy1. .

Ž py1 .r2

p:odd prime

Ž 1 y pys . y

Ž py1 .r2

,

p:odd prime

where ␨ Ž s . is Riemann’s zeta function. This function ␨AŽ s . does not have an analytic continuation to the whole s-plane ŽProposition 3.5.1.. In Section 4, we study the convergence of zeta functions. We will prove the following result Ža finer result is given in Theorem 4.3.. THEOREM 1.4. Let H be a finitely generated free abelian group and let G be a group ha¨ ing an exact sequence 1 ª H ª G ª Z ª 1. Let R be a non-zero commutati¨ e finitely generated ring o¨ er Z, and let A be the group ring Rw G x. Then, the function ␨AŽ s . con¨ erges if and only if G has a nilpotent subgroup of finite index. By a theorem of Gromov wGrx, for a finitely generated group G and for a field k, G has a nilpotent subgroup of finite index if and only if the Gelfand᎐Kirillov dimension of k w G x is finite. Thus Theorem 1.4 suggests a strong relationship between the convergence of zeta functions and Gelfand᎐Kirillov dimensions of rings. Concerning this relationship, we formulate Conjectures 4.11 and 4.14 and give some evidence. In Section 5, we present several unsolved questions. In wHex, Hey defined the zeta functions of non-commutative rings in the case A is finitely generated as a Z-module. Our zeta function differs from her zeta function Žour ␨AŽ s . depends only on the Morita equivalence class of A, that is, only on the category of A-module, but her zeta function does not have this property. and our zeta functions work for arbitrary finitely generated rings over Z. I am very grateful to Professor Kazuya Kato who suggested I study this subject, gave a lot of essential advice, and encouraged me. I am very grateful also to Professor Nobushige Kurokawa who suggested to count

HASSE ZETA FUNCTIONS

307

numbers of solutions of equations in matrix rings over finite fields and to find a good theory of Hasse zeta functions of non-commutative rings based on it. I also thank Akiko Baba who discussed this subject with me. Notes. In this paper, all rings are assumed to have a unit 1, and all ring homomorphisms are assumed to respect 1. Fields are assumed to be commutative, though rings are not assumed to be commutative. For a ring A, an A-module means a left A-module on which 1 acts as the identity. As usual, Z, Q, R, C denote the ring of integers, the rational number field, the real number field, and the complex number field, respectively.

2. GENERAL RESULTS In this section we prove Theorem 1.2, and the following propositions: PROPOSITION 2.1.

Let R be a commutati¨ e finitely generated ring o¨ er Z.

Ž1. The function ␨ R, 1Ž s . coincides with the classical Hasse zeta function of the commutati¨ e ring R and

␨R , r Ž s . s 1

if r ) 1.

Ž2. Let A be an R-algebra which is of finitely generated as an R-module. Then the function ␨A, r Ž s . is identically 1 for almost all r. In particular Ž by Theorem 1.2Ž1.. the function ␨AŽ s . con¨ erges if ReŽ s . is sufficiently large. Ž3. Let A be an Azumaya algebra o¨ er R. Then

␨A Ž s . s ␨ R Ž s . . PROPOSITION 2.2. Ž1.

Let A be a finitely generated ring o¨ er Z.

Let n G 1, and let B be the matrix ring MnŽ A.. Then

␨ B , r Ž s . s ␨A , r r n Ž s .

if n di¨ ides r ,

and ␨ B, r Ž s . is identically 1 otherwise. Ž2. Let B be a finitely generated ring o¨ er Z and assume B is Morita equi¨ alent to A. Then ␨ B Ž s . con¨ erges if and only if ␨AŽ s . con¨ erges. If they con¨ erge, we ha¨ e

␨ B Ž s . s ␨A Ž s . . Ž3. Let B be the polynomial ring Aw T1 , . . . , Tn x in n ¨ ariables T1 , . . . , Tn which commute with each other and with any element of A. Then for any r G 1,

␨ B , r Ž s . s ␨A , r Ž s y n . .

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TAKAKO FUKAYA

2.3. We will prove Theorem 1.2. Let A and r be as in Section 1. We introduce ᑭ A, r , the space of r dimensional irreducible representations of A, which was studied for example in wL-Mx. ŽEarlier works on this subject are introduced in wL-Mx.. Let ᑬ A, r be the functor from the category of commutative rings over Z to the category of sets defined by ᑬ A , r Ž R . s Hom Z Ž A, Mr Ž R . . . Then ᑬ A, r is represented by a scheme of finite type over Z which we denote by the same letter ᑬ A, r . To see this let m be the number of generators of A. The scheme in question is the spectrum of the quotient of the polynomial ring in mr 2 variables over Z, each of which represents each entry of the image of the generators of A, by relations deduced from the relations of the generators of A. The projective general linear group scheme PGLr over Z acts on ᑬ A, r . The action is given for h g ᑬ A, r Ž R . s and g g GLr Ž R ., by h ¬ ghŽy. gy1 . Now let ᑬ A, r be the subfunctor of ᑬ A, r defined by ᑬ As , r Ž R . s  h g ᑬ A , r Ž R . ; the image of h : A ª Mr Ž R . generates Mr Ž R . over R 4 . s Then ᑬ A, r is represented by an open subscheme of ᑬ A, r which is stable s under the action of PGLr . We denote this open subscheme also by ᑬ A, r. s The scheme ᑭ A, r is defined as the quotient of ᑬ A, r by the action of PGLr . We use the following properties of ᑭ A, r : It is a scheme of finite type over Z, and concerning the K-rational points for any algebraically s Ž . Ž . closed field K, the canonical map from PGLr Ž K ._ ᑬ A, r K to ᑭ A, r K is Ž . bijective. An element of ᑭ A, r K corresponds bijectively to the isomorphism class of an r-dimensional irreducible representation of A over K. s Because the existence of such a good quotient ᑭ A, r of ᑬ A, r is a Ž w x delicate problem we have to use work of Seshadri Ses on geometric invariant theory in which the base is an arbitrary noetherian ring Žnot necessarily a field, cf. wMu, Introduction to the 2nd ed.x., we give in detail the construction of ᑭ A, r . Let ⌽ be the affine ring of ᑬ A, r , and let ⌿ be the invariant part of ⌽ under the action of PGLr . By wSes, Theorem 3x, ⌿ is a finitely generated ring over Z. The scheme ᑭ A, r is defined as an open subscheme of SpecŽ ⌿ . as follows: Regard an Ž r, r .-matrix as a vector of dimension r 2 . An element of A defines an element of Mr Ž ⌽ ., and it is regarded as a vector of dimension r 2 over ⌽. For a family sŽ1., . . . , sŽ r 2 . of r 2 elements of A, by regarding each sŽ i . as a vector of dimension r 2 over ⌽ and regarding Ž sŽ1., . . . , sŽ r 2 .. as an Ž r 2 , r 2 .-matrix over ⌽, let f s g ⌽ be the determinant of this matrix. Then f s is invariant under the action of PGLr , and hence is an element of ⌿. ŽThis follows from the fact

309

HASSE ZETA FUNCTIONS

that the determinant of the inner automorphism Mr ª Mr Žas an Ž r 2 , r 2 .matrix. defined by an element of PGLr is 1. In fact, this determinant defines a homomorphism from PGLr to GL1 , but any homomorphism from PGLr to GL1 is trivial.. Let ᑭ A, r be the open subscheme of SpecŽ ⌿ . defined to be the union of SpecŽ ⌿w1rf s x. for all families sŽ1., . . . , sŽ r 2 . of s r 2 elements of A. Note that ᑬ A, r is the union of the open subschemes s SpecŽ ⌽w1rf s x. of SpecŽ ⌽ . for all such families. That is, ᑬ A, r is the inverse Ž . Ž . image of ᑭ A, r under the morphism Spec ⌽ ª Spec ⌿ . By wL-M, 1.17x, s Ž . for any algebraically closed field K and for any x g ᑬ A, r K , the orbit Ž . Ž . PGLr K . x is closed in ᑬ A, r K for the Zariski topology. ŽIn fact, A is assumed to be a group ring in wL-Mx, but the same proof works.. By wSes, s Ž . Theorem 3x, this means that the canonical map from PGLr Ž K ._ ᑬ A, r K Ž . to ᑭ A, r K is bijective. LEMMA 2.3.1. isomorphism.

s Ž . Ž . If k is a finite field, PGLr Ž k ._ ᑬ A, r k ª ᑭ A, r k is an

Proof. Let k be a finite field and let K be its algebraic closure. Then s Ž . Ž . for any x g ᑭ A, r Ž k ., there exists yX g ᑬ A, r K whose image in ᑭ A, r K s Ž . Ž . is x. If there exists y g ᑬ A, r k whose image in ᑭ A, r K is x, then the proof is accomplished. For ␴ g GalŽ Krk ., since yX and ␴ Ž yX . have the same image x in ᑭ A, r Ž K ., there is a unique element sŽ ␴ . of PGLr Ž K . such that ␴ Ž yX . s sŽ ␴ . yX . The map ␴ ¬ sŽ ␴ . is a 1-cocycle. Since H 1 Ž k, PGLr Ž K .. is trivial wSer, Chaps. 10 and 13x, there exists g g PGLr Ž K . such that ␴ Ž g . gy1 s sŽ ␴ . for all ␴ g GalŽ Krk .. Put y s gy1 yX . Then y s Ž . is GalŽ Krk .-invariant and hence belongs to ᑬ A, r k , and the image of y in ᑭ A, r Ž k . is x. 2.4. We will show that ␨A, r Ž s . coincides with the Hasse zeta function of the scheme ᑭ A, r , that is,

␨A , r Ž s . s

⬁

Ł exp Ý p

ns1

噛ᑭ A , r Ž Fp n . n

Ž pys .

n

the product of Weil’s zeta functions of ᑭ A, r mZ Fp wWe1x for all prime numbers p. LEMMA 2.4.1. Let A be a finitely generated ring o¨ er Z, let R be a s Ž . commutati¨ e ring, and let h be an element of ᑬ A, r R . Then h sends the center of A into the ring of scalar matrices in Mr Ž R .. Proof. The image of a central element of A under h belongs to the centralizer of hŽ A. in Mr Ž R .. Since hŽ A. generates Mr Ž R . over R, the image of a central element belongs to the center of Mr Ž R ., the subring of the scalar matrices.

310

TAKAKO FUKAYA

LEMMA 2.4.2. Let B be a subring of Mr ŽFp n . and generate Mr ŽFp n . o¨ er Fp n . Then B is isomorphic to Mr ŽFp m . for some positi¨ e integer m di¨ iding n. Proof. Step 1. We show that B is a semi-simple ring. Let I be the n radical of B. We show I s 0. Let V s Fp[r with the natural action of Mr ŽFp n ., and let V X s  x g V; Ix s 04 . Then V X is an Fp n-subspace of V and stable under the action of B. If V X / V and V X / 0 then B cannot generate Mr ŽFp n ., so V X s V or V X s 0. If V X s V, IV s 0 hence I s 0. It remains to show V X / 0. Since I is a nilpotent ideal, there is an integer e G 0 such that I e V / 0 and I eq 1 V s 0. We have 0 / I e V ; V X . Step 2. We show that B is a simple ring. If not, B is a product B1 = B2 of non-zero rings B1 , B2 . Then V is a direct sum V1 [ V2 of Fp n-sublinear spaces such that B acts on Vi via the projection B ª Bi , for i s 1, 2. Since V1 and V2 are stable under the actions of B, V1 s 0 or V2 s 0 from the condition on B to generate Mr ŽFp n .. Then B1 =  04 or  04 = B2 acts as 0 on V. Hence one of them is zero, that is, B is a simple ring. Step 3. By Step 2, by the Wedderburn᎐Artin Theorem and by the fact that any finite division ring is commutative, B , Mr X Ž Fp m . for some r X and m. It remains to prove m < n and r s r X . Step 4. By Lemma 2.4.1, the center of B which is isomorphic to Fp m is contained in the subring of scalar matrices of Mr ŽFp n . which is isomorphic to Fp n , so m < n. Step 5. We show that r s r X . Consider an Fp n-homomorphism from B mF m Fp n to Mr ŽFp n . which extends the canonical injection from B to p Mr ŽFp n .. This homomorphism is surjective from the assumption on B, and injective because B mF m Fp n , Mr X ŽFp n . is a simple ring, so the kernel of p the homomorphism which is a two-sided ideal of B is 0. So Mr X ŽFp n . is isomorphic to Mr ŽFp n ., and this proves r s r X . LEMMA 2.4.3.

Let P Ž q . s  ᒊ g ᑪ r Ž A. ; N Ž ᒊ . s q4

for q G 1 Ž by fixing r .. Then there is a surjecti¨ e map ␥ from ᑭ A, r ŽFp n . to Dm < n P Ž p m . by h ¬ KerŽ h., and for any ᒊ g P Ž p m . such that m < n, the order of ␥y1 Ž ᒊ . is m. Consequently, we ha¨ e 噛ᑭ A , r Ž Fp n . s

Ý

Ý

m < n ᒊgP Ž p m .

m.

311

HASSE ZETA FUNCTIONS

s Ž . Ž . belongs to Proof. By Lemma 2.4.2, for h g ᑬ A, r Fp n , Ker h m Dm < n P Ž p .. Since the action of PGLr ŽFp n . on h does not change KerŽ h., we have a well defined map ␥ : ᑭ A, r ŽFp n . ª Dm < n P Ž p m .; h ¬ KerŽ h.. The surjectivity is trivial. By the Skolem᎐Noether Theorem, for any ᒊ g P Ž p m ., there is m Žthe number of the embedding from Fp m into Fp n . elements of ᑭ A, r ŽFp n . whose kernel is ᒊ.

2.5. We can deduce the following facts from Lemma 2.4.3, ⬁

Ý 噛ᑭ A , r Ž Fp .

Ž pys .

n

n

s

n

ns1

m Ž pys .

⬁

Ý Ý

Ý

ns1 m < n ᒊgP Ž p m .

s

⬁

⬁

Ý

Ž pys .

bm

Ý

Ý

 ylog Ž1 y Ž pys . m . 4

⬁

Ý

n

Ý

ms1 ᒊgP Ž p m . bs1

s

n

b

ms1 ᒊgP Ž p m .

and hence ⬁

噛ᑭ A , r Ž Fp n .

Ł exp Ý p

n

ns1

s

⬁

Ł exp Ý p

s

Ž pys .

Ý

n

 ylog Ž1 y Ž pys . m . 4

ms1 ᒊgP Ž p m . ys y1

Ł Ž1 y N Ž ᒊ . .

ᒊgᑪ r Ž A .

s ␨A , r Ž s . . 2.6. Now we complete the proof of Theorem 1.2. Proof. Ž1. It is known that the Hasse zeta function of a scheme of finite type over Z absolutely converges if ReŽ s . is sufficiently large. We have shown that if A is a finitely generated ring over Z, the function ␨A, r Ž s . coincides with the Hasse zeta function of the scheme ᑭ A, r of finite type over Z. Ž2. Dwork and Grothendieck proved that the zeta function of a scheme of finite type over Fq is a rational function of qys with rational coefficients. So we show that if A is an Fq-algebra, then ᑭ A, r is a scheme over Fq . s Ž . We define the natural transformation ␸ from the functor ᑬ A, r to Spec Fq

312

TAKAKO FUKAYA

Žwhich we regard as the functor HomŽFq , ... Let R be a commutative ring. The elements of Fq belong to the center of A, so every element of s Ž . ᑬ A, r R sends, by Lemma 2.4.1, the elements of Fq into the subring of the scalar matrices of Mr Ž R . which is isomorphic to R. Thus every element of s Ž . ᑬ A, defines an element of HomŽFq , R .. This defines the natural r R transformation ␸ , and ␸ is invariant under the action of PGLr . Hence ␸ s factors through the quotient ᑭ A, r of ᑬ A, r and this fact shows the scheme ᑭ A, r is over Fq . 2.7. We prove the equation in Section 1

␨A Ž s . s

ys y1

Ł Ž1 y N Ž M . .

,

M

where M runs over the isomorphism classes of finite simple A-modules and N Ž M . s 噛End AŽ M .. This is a consequence of: LEMMA 2.7.1. Let fsŽ A. be the set of all isomorphism classes of finite simple A-modules. Then, the correspondence M ª ᒊ s AnnŽ M . gi¨ es a bijection from fsŽ A. to ᑪ Ž A. satisfying N Ž ᒊ . s 噛ŽEnd AŽ M ... Here AnnŽ M . denotes the annihilator of M in A. Proof. We show that AnnŽ M . g ᑪ Ž A. for M g fsŽ A.. The endomorphism ring End AŽ M . is a finite ring and a division algebra, so End AŽ M . is a finite field k. Let r be the dimension of M as a k-module. Then End k Ž M . , Mr Ž k .. The canonical map A ª End k Ž M .; a ¬ Ž m ¬ am. is surjective by Burnside wF-D, Corollary 1.16x. This shows that ᒊ s AnnŽ M . belongs to ᑪ Ž A. and 噛ŽEnd AŽ M .. s N Ž ᒊ .. This correspondence is oneto-one and onto. The injectivity follows from the fact that if M g fsŽ A., M is a module over ArAnnŽ M . , Mr Ž k . which has only one simple module up to isomorphism. The surjectivity follows from the fact that if ᒊ g ᑪ Ž A. and Arᒊ ( Mr Ž k ., M s k[r with the canonical action of Mr Ž k . is a simple A-module and AnnŽ M . s ᒊ. 2.8. We prove Proposition 2.1. Proof. Ž1. The first assertion is clear. For an ideal ᒊ of R, Rrᒊ is a commutative ring, so it never coincides with Mr Ž k . for any finite field k if r ) 1. This proves the second assertion. Ž2. Let m be the number of generators of A as an R-module. For s Ž . r G 1 and for a field k, an element h of ᑬ A, r k sends R into k by Lemma 2.4.1. Hence the k-subspace of Mr Ž k . generated by hŽ A. is at most m dimensional. Thus the existence of h implies r 2 F m. This shows that if r ) m2 , ᑭ A, r s ⭋ and hence ␨A, r Ž s . s 1. Ž 3 . For a finitely generated R-algebra A , ␨ A Ž s . s Ł ᒊ g ᑪ Ž R. ␨A r ᒊ AŽ s .. Recall that A is an Azumaya algebra over R if and only if it is a finitely generated projective R-module and Arᒊ A are

HASSE ZETA FUNCTIONS

313

central simple algebras for all maximal ideals ᒊ of R. Because any central simple algebra over the finite field Rrᒊ is isomorphic to Mr Ž Rrᒊ . for some integer r G 1, for the ring A in question,

␨A Ž s . s

Ł

ᒊgᑪ Ž R .

␨A r ᒊ A Ž s . s

Ł

ᒊgᑪ Ž R .

␨R r ᒊ Ž s . s ␨R Ž s . .

2.9. We prove Proposition 2.2. Proof. Ž1. Let M be an A-module. The category of A-modules and the category of MnŽ A.-modules are equivalent by the functor M ¬ M [n where M [n is an MnŽ A.-module on which MnŽ A. acts canonically. This gives a bijection from fsŽ A. to fsŽ MnŽ A.. Ž fsŽ . is as in Lemma 2.7.1.. Furthermore, in this bijection, any finite simple MnŽ A.-module M X satisfies dim k Ž M X . s n dim k Ž M . where M is the corresponding finite simple A-module and k s End M n Ž A.Ž M X . s End AŽ M .. Hence the result follows. Ž2. By Morita’s Theorem, B is isomorphic to the opposite ring of End AŽ P . where P is a finitely generated projective A-module such that A is a direct summand of P [n for some n G 1, and an A-module M corresponds to the B-module Hom AŽ P, M . in the equivalence of categories. An A-module M is finite if and only if Hom AŽ P, M . is finite. Hence the categorical equivalence gives a bijection between fsŽ A. and fsŽ B . preserving N Ž .. Ž3. We may assume n s 1. We apply Lemma 2.4.1 by taking B s Aw T x as A in Lemma 2.4.1, to the central element T of B. We can see that s Ž . the image of T under an element of ᑬ B, r Fp n must be a scalar matrix and all the scalar matrices in Mr ŽFp n . can be taken as the image of T. There are p n scalar matrices in Mr ŽFp n .. Hence 噛ᑭ B , r Ž Fp n . s 噛ᑭ A , r Ž Fp n . ⭈ p n . This shows

␨ B , r Ž s . s ␨A , r Ž s y 1 . .

3. EXAMPLES In this section we compute some examples of zeta functions in non-commutative rings. The method is to count the number of rational points of ᑭ A, r in finite fields.

314

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Notations. For a ring R, R X 1 , . . . , X n4 denotes the non-commutative polynomial ring in n variables. For elements a1 , . . . , a n of a ring A, ² a1 , . . . , a n : denotes the two-sided ideal of A generated by a1 , . . . , a n . EXAMPLE 3.1 ŽWeyl Algebra.. Let R be a commutative finitely generated ring over Z, let a i g R for i s 1, . . . , n, and let A s R  X 1 , Y1 , . . . , X n , Yn 4 rI, where I is the two-sided ideal of A generated by the elements X i Yj y Yj X i y a i ␦ i , j ,

X i X j y X j X i , Yi Yj y Yj Yi Ž 1 F i , j F n . ,

where ␦ i, j is the Kronecker symbol. In the case a i s 1 for all i, A is the nth Weyl algebra. On the other hand, if a i s 0 for all i, A is the usual commutative polynomial ring in 2 n variables. Surprisingly, we have always the same ␨AŽ s . though each ␨A, r Ž s . depends heavily on properties of a1 , . . . , a n . We have: Ž1. The function ␨AŽ s . converges, and

␨A Ž s . s ␨ R Ž s y 2 n . . Ž2. Let p be a prime number, and let m be an integer such that 0 F m F n. Assume that R is a ring over Fp , a i is invertible if 1 F i F m, and a i s 0 if m - i F n. Then

␨A , r Ž s . s ␨ R Ž s y 2 n .

if r s p m

and

␨A , r s 1

otherwise.

Proof. We may assume that R is a finite field, for ␨ R Ž s ., ␨AŽ s ., ␨A, r Ž s . are products of ␨ R r ᒊ Ž s ., ␨A r ᒊ AŽ s ., ␨A r ᒊ A, r Ž s ., respectively, where ᒊ ranges over all maximal ideals of R, and Rrᒊ are finite fields. So assume R is a finite field of characteristic p which we denote by k, and let K be the algebraic closure of k. By Proposition 2.2Ž3., we may assume that k Ž X. a i / 0 for all i. For an extension kX of k, let ᑭ A, be the set of r k X k -rational points of ᑭ A, r as a k-scheme. Then we have: k Ž . n Ž3.1.1. The set ᑭ A, r K is empty unless r s p . Ž3.1.2. As a GalŽ Krk .-set Ži.e., a set endowed with an action of k 2n Ž Ž . Gal Krk .., ᑭ A, , the product of 2 n copies of p n K is isomorphic to K K.

The results Ž1., Ž2. are obtained from Ž3.1.1. and Ž3.1.2., by counting the X k Ž X. k Ž . Ž . order of the set ᑭ A, r k s the Gal Krk -fixed part of ᑭ A, r K , for X each finite extension k of k in K.

315

HASSE ZETA FUNCTIONS

k Ž . The isomorphism in Ž3.1.2. is defined as follows: Let b g ᑭ A, p n K , and for an element x of A whose action in the representation b is a scalar, let bŽ x . g K be the action of x in b. ŽSo bŽ x . g K is defined for any central element x of A by Lemma 2.4.1.. The elements X 1p , . . . , X np , Y1p , . . . , Ynp belong to the center of A. The isomorphism Ž3.1.2. is defined by sending b to Ž bŽ X 1p ., . . . , bŽ X np ., bŽ Y1p ., . . . , bŽ Ynp ... We prove Ž3.1.1. and Ž3.1.2.. Let M be a finite dimensional irreducible representation of A over K. Since X 1 , . . . , X n commute, there is a non-zero element ¨ of M which is an eigenvector of X i , with eigenvalue c i g K, for any i s 1, . . . , n. Then Y1sŽ1. Y2sŽ2. ⭈⭈⭈ YnsŽ n. ¨ Ž0 F sŽ i . - p for i s 1, . . . , n. is a K-basis of M. In fact by applying X i y c i Ž1 F i F n. to these elements several times, we see that these elements are linearly independent over K and the K-linear space spanned by them is stable under the action of A and hence coincides with M. This proves Ž3.1.1. and the fact k Ž . Ž Ž p. that to give b g ᑭ A, p n K is equivalent to giving c1 , . . . , c n , b Y1 , . . . , p p p bŽ Yn ... Since bŽ X i . s c i , we have the isomorphism stated in Ž3.1.2..

COROLLARY 3.2 ŽHeisenberg Lie Algebra.. Let R be a commutati¨ e finitely generated ring o¨ er Z, and let A be the uni¨ ersal en¨ eloping algebra of the following Lie algebra ᒄ o¨ er R: ᒄ is a free R-module of rank 3 with basis x, y, z satisfying w x, y x s z, w x, z x s 0, w y, z x s 0. We ha¨ e that the function ␨AŽ s . con¨ erges, and ␨A Ž s . s ␨ R Ž s y 3 . . Proof. The universal enveloping algebra of the Heisenberg algebra is expressed as RX  X , Y 4 r² XY y YX y Z : , where RX s R w Z x . We apply Example 3.1 by taking RX as R in Example 3.1 to this universal enveloping algebra and apply Proposition 2.2Ž3. to RX . Then the result follows. EXAMPLE 3.3. Let R be a commutative finitely generated ring over Z, and let A be the universal enveloping algebra of the following Lie algebra ᒄ over R: ᒄ is a free R-module of rank 2 with basis h, e satisfying w h, e x s e. We have: Ž1. The function ␨AŽ s . converges, and

␨A Ž s . s ␨ R Ž s y 2 . . Ž2. Let p be a prime number. Assume that R is a ring over Fp . Then

␨A , r Ž s . s ␨ R Ž s y 1 .

if r s 1

␨A , r Ž s . s ␨ R Ž s y 2 . ␨ R Ž s y 1 . ␨A , r Ž s . s 1

otherwise.

y1

if r s p

316

TAKAKO FUKAYA

Proof. We may assume that R is a finite field which we denote by k and let K be the algebraic closure of k. By wS-F, 5.9, Example 1x, we have Žsee Example 3.1 for notations.: k Ž . Ž3.3.1. The set ᑭ A, r K is empty unless r s 1 or r s p. k Ž . Ž3.3.2. As a GalŽ Krk .-set, ᑭ A, Ž . 1 K is isomorphic to K by b ¬ b h . k Ž3.3.3. As a GalŽ Krk .-set, ᑭ A, p Ž K . is isomorphic to Ž y, z . g K = K; y / 04 by b ¬ Ž bŽ e p ., bŽ h p y h...

The results Ž1., Ž2. are obtained from Ž3.3.1., Ž3.3.2., Ž3.3.3. by counting k Ž X. the order of the set ᑭ A, for each finite extension kX of k in K. r k EXAMPLE 3.4. Let R be a commutative finitely generated ring over Z, and let A be the universal enveloping algebra of the following Lie algebra ᒄ over R: ᒄ is a free R-module of rank 3 with basis h, e, f satisfying w h, e x s e, w h, f x s ␣ f, w e, f x s 0, where ␣ is an invertible element of R. We have: Ž1. The function ␨AŽ s . converges, and

␨A Ž s . s ␨ R Ž s y 3 . . Ž2. Let p be a prime number. Assume that R is a ring over Fp . Then

␨A , r Ž s . s ␨ R Ž s y 1 . ␨A , r Ž s . s

Ł

ᒊgᑧ

=

␨A , r Ž s . s

Ł

ᒊgᑧ

X

ž␨

Ł

ᒊgᑧ

ž␨

Rrᒊ

Rrᒊ

X

ž␨

if r s 1.

Ž s y 3. ␨R r ᒊ Ž s y 1.

Rrᒊ

y1

/

2 y2 Ž s y 2. ␨R r ᒊ Ž s y 1. /

Ž s y 3. ␨R r ᒊ Ž s y 2.

y3

if r s p 3

␨R r ᒊ Ž s y 1. ␨R r ᒊ Ž s .

y1

/

if r s p 2

␨A , r Ž s . s 1

otherwise,

where ᑧ s  ᒊ g ᑪ Ž R . ; the canonical image of ␣ in Rrᒊ belongs to Fp 4 , ᑧX s  ᒊ g ᑪ Ž R . ; the canonical image of ␣ in Rrᒊ does not belong to Fp 4 .

317

HASSE ZETA FUNCTIONS

Proof. We may assume that R is a finite field which we denote by k, and let K be the algebraic closure of k, and let K U be the multiplicative group of K. By wS-F, 5.9, Example 3x, we have: Ž1. The case ␣ belongs to Fp . In this case h p y h, e p , f p belong to the center of A, and the followings hold. k Ž . Ž3.4.1. The set ᑭ A, r K is empty unless r s 1 or r s p. k Ž . Ž3.4.2. As a GalŽ Krk .-set, ᑭ A, 1 K is isomorphic to K by b ¬ bŽ h.. k Ž . Ž x, y, z . g Ž3.4.3. As a GalŽ Krk .-set, ᑭ A, p K is isomorphic to p. p. p Ž . Ž .4 Ž Ž Ž Ž .. K = K = K; x, y / 0, 0 by b ¬ b e , b f , b h y h . 2 Ž2. The case of ␣ does not belong to Fp . In this case, e p , f p , t s h p p py 1 p p Ž h y h. belong to the center of A, and the followy h y Ž␣ y ␣ . ings hold. k Ž . 2 Ž3.4.4. The set ᑭ A, r K is empty unless r s 1 or r s p or r s p . k Ž . Ž3.4.5. As a GalŽ Krk .-set, ᑭ A, 1 K is isomorphic to K by b ¬ bŽ h.. k Ž . Ž3.4.6. As a GalŽ Krk .-set, ᑭ A, is isomorphic to K U = p K U K @ K = K. The isomorphism is defined as follows. For an element b of k Ž . Ž p . and bŽ f p . is zero. In the case bŽ f p . s 0, h p y h ᑭ A, p K , one of b e acts as a scalar in b, and b is sent to Ž bŽ e p ., bŽ h p y h.. in the left K U = K. In the case bŽ e p . s 0, h p y ␣ py1 h acts as a scalar, and b is sent to Ž bŽ f p ., bŽ h p y ␣ py1 h.. in the right K U = K. U U k Ž3.4.7. As a GalŽ Krk .-set, ᑭ A, Ž . p 2 K is isomorphic to K = K U p p = K by b ¬ Ž bŽ e ., bŽ f ., bŽ t ... The results Ž1., Ž2. are obtained from Ž3.4.1. ᎐ Ž3.4.7. by counting the k Ž X. order of the set ᑭ A, for each finite extension kX of k in K. r k The Examples 3.2᎐3.4 may give the feeling that the zeta function of the universal enveloping algebra of a Lie algebra has always a simple form. The following Example 3.5 shows this is not the case. EXAMPLE 3.5 Ž sl 2 ŽZ... Let A be the universal enveloping algebra of sl 2 ŽZ.. We have: Ž1. The function ␨AŽ s . converges, and

␨A Ž s . s ␨ Ž s y 3 . =

Ł

Ł

Ž 1 y pyŽ sy1. .

p:odd prime

p:odd prime

Ž 1 y pys . y

Ž py1 .r2

.

Ž py1 .r2

318

TAKAKO FUKAYA

Ž2. For any odd prime number p, we have

␨A r p A , r Ž s . s ␨A r p A , r Ž s . s

1 yŽ sy3.

1yp

1

if r s 1, . . . , p y 1

1 y pys

 Ž 1 y pyŽ sy1. . Ž 1 y pys . 4

␨A r p A , r Ž s . s 1

Ž py1 .r2

if r s p

otherwise.

For p s 2, we have

␨A r2 A , r Ž s . s

1

if r s 1

1 y 2yŽ sy2.

␨A r2 A , r Ž s . s Ž 1 y 2yŽ sy2. . ⭈ ␨A r2 A , r Ž s . s 1

1

if r s 2

1 y 2yŽ sy3. otherwise.

Proof. We take the basis h, e, f of sl 2 ŽZ. as hs

ž

1 0

0 , y1

/

es

ž

0 0

1 , 0

/

fs

ž

0 1

0 . 0

/

Then, w h, e x s 2 e, w h, f x s y2 f, w e, f x s h. Ž1. The case where p is an odd prime. Let k s Fp , and let K be the algebraic closure of k. Then e p , f p , h p y h, Ž h y 1. 2 q 4 ef belong to the center of ArpA. By Rudakov and Shafarevich wR-Sx, we have: Ž3.5.1. The set ᑭ A, r Ž K . is empty unless r s 1, . . . , p. Ž3.5.2. The set ᑭ A, r Ž K . is a one point set if r s 1, . . . , p y 1. Ž3.5.3. As a GalŽ Krk .-set, ᑭ A, p Ž K . is isomorphic to Z y S where 2

Z s Ž x, y, z, t . g K = K = K = K ; 4 xy s t Ž t Ž py1.r2 y 1 . y z 2

½

S s  Ž 0, 0, 0, t . g K = K = K = K ; t Ž py1.r2 s 1 4

5

Ž ).

by b ¬ Ž bŽ e p ., bŽ f p ., bŽ h p y h., bŽŽ h y 1. 2 q 4 ef ... The parts of Ž1. and Ž2. concerning the Euler factor at an odd prime number p are obtained from Ž3.5.1. ᎐ Ž3.5.3., by counting the order of the set ᑭ A, r Ž kX . for each finite extension kX of k in K. I explain how to compute the order of the set ᑭ A, p Ž kX .. Let kX s Fq a finite extension of k s Fp . We compute the order of the GalŽ KrkX .-fixed part of Z.

319

HASSE ZETA FUNCTIONS

Ži. The case where both sides of the equation in Ž). are zero. In this case 2 q y 1 pairs Ž x, y . g kX = kX satisfy 4 xy s 0. Put g Ž t . s t Ž py1.r2 y 1. If g Ž t . / 0, we put u s zrg Ž t .. Then we see that a pair Ž z, t . corresponds to a unique u g kX y kU by z s g Ž t . u, t s u 2 . Hence there are q y Ž p y 1. pairs of Ž z, t . g kX = kX for which the right hand side of Ž). is 0 and g Ž t . / 0. If g Ž t . s 0, t Ž py1.r2 s 1 and z s 0. There are Ž2 q y 1.Ž q y Ž p y 1.r2. solutions of Eq. Ž).. Žii. The case where both sides of the equation in Ž). are non-zero. By Ži., among the whole q 2 elements of Ž z, t . g kX = kX , q y Ž p y 1.r2 satisfy the condition that the right hand side of Ž). is zero. So the rest q 2 y q q Ž p y 1.r2 pairs of Ž z, t . make the right hand side of Ž). not equal to zero. For each Ž z, t ., q y 1 pairs of Ž x, y . g kX = kX satisfy Ž).. Hence there are Ž q y 1.Ž q 2 y q q Ž p y 1.r2. solutions of Eq. Ž).. So since the order of the set S is Ž p y 1.r2, the order of the set ᑭ A, p Ž kX . is Ž2 q y 1.Ž q y Ž p y 1.r2 q Ž q y 1.Ž q 2 y q q Ž p y 1.r2. y Ž p y 1.r2 s q 3 y Ž q q 1. ⭈ Ž p y 1.r2. Hence the result follows. Ž2. The case where p s 2. In this case, we have w h, e x s 0, w h, f x s 0, w e, f x s h in sl 2 ŽF2 .. So Ar2 A is expressed as F2 w h x  X , Y 4 r² XY y YX y h: , where X corresponds to e and Y corresponds to f. Hence we apply Example 3.1. Note

␨A r2 A , r Ž s . s

Ł ␨Ž A r2 A.rŽ ᒊ ., r Ž s . , ᒊ

where ᒊ ranges over all maximal ideals of F2 w h x. Ži. By Example 3.1, only one maximal ideal ᒊ s Ž h. of F2 w h x contributes to ␨A r2 A, 1Ž s .. We have

␨A r2 A , 1 Ž s . s ␨ F2 w hxrŽ h. Ž s y 2 . s

1 1 y 2yŽ sy2.

.

Žii. By Example 3.1, all of the maximal ideals ᒊ / Ž h. of F2 w h x contribute to ␨A r2 A, 2 Ž s .. We have

␨A r2 A , 2 Ž s . s

Ł

ᒊ/ Ž h .

␨ F2 w hxr ᒊ Ž s y 2 . s

1 y 2yŽ sy2. 1 y 2yŽ sy3.

.

320

TAKAKO FUKAYA

Let A be the uni¨ ersal en¨ eloping algebra of sl 2 ŽZ..

PROPOSITION 3.5.1. Then

Ž1. The function ␨AŽ s . absolutely con¨ erges if and only if ReŽ s . ) 4. Ž2. The function ␨AŽ s . is analytic in ReŽ s . ) 2. Ž3. The function ␨AŽ s . has the natural boundary ReŽ s . s 2. This is a consequence of the following result, Proposition 3.5.2 of N. Kurokawa. The key point of the proof of Ž3. is that any point on the line ReŽ s . s 2 is a limit point of poles of ␨AX Ž s . ␨AŽ s .y1 , where ␨AX Ž s . means the derivative of ␨AŽ s ., in the region ReŽ s . ) 2. For the proof of Ž3.5.1., the case ␹ s 1 of Ž3.5.2. is enough, but we consider here all Dirichlet characters expecting future applications to zeta functions of other rings. PROPOSITION 3.5.2. Let hŽT . be an element of Cw T x of degree d G 1, and let ␹ be a Dirichlet character Ž which is not necessarily primiti¨ e .. Then we ha¨ e: Ž1.

The function ⌽ Ž s, ␹ . s

Ž . Ł Ž 1 y ␹ Ž p . pys . h p

p:prime

absolutely con¨ erges in ReŽ s . ) d q 1. Ž2. The function ⌽ Ž s, ␹ . is analytic in ReŽ s . ) d. Ž3. The function ⌽ Ž s, ␹ . has the natural boundary ReŽ s . s d. We give the proofs of Proposition 3.5.1 and Proposition 3.5.2 at the end of this section. EXAMPLE 3.6 ŽQuantum Plane.. Let R be a commutative finitely generated ring over Z, and let A s R  X , Y 4 r² XY y aYX : , where a is an invertible element of R. We have: Ž1. The function ␨AŽ s . converges, and

␨A Ž s . s ␨ R Ž s y 2 . . Ž2. Let 2

␨A , r Ž s . s ␨ R Ž s y 1 . ␨ R Ž s . =

␨A , r Ž s . s

Ł

ᒊgᑧ r

Ł

ᒊgᑧ r

ž␨

ž␨

Rrᒊ

Rrᒊ

y1

Ž s y 2. ␨R r ᒊ Ž s y 1.

Ž s y 2. ␨R r ᒊ Ž s y 1.

y2

y2

␨R r ᒊ Ž s .

␨R r ᒊ Ž s .

/

/

if r s 1 if r / 1,

HASSE ZETA FUNCTIONS

321

where ᑧ r s  ᒊ g ᑪ Ž R . ; the canonical image of a in Rrᒊ is a primitive r th root of 1 4 . Proof. We assume that R is a finite field which we denote by k, and let K be the algebraic closure of k. Let l be the order of a. If l s 1, A s k w X, Y x and hence we are reduced to Proposition 2.2Ž3.. Assume l / 1. Note that X l and Y l belong to the center of A. We have: k Ž . Ž3.6.1. The set ᑭ A, r K is empty unless r s 1 or r s l. k Ž . Ž3.6.2. As a GalŽ Krk .-set, ᑭ A, Ž y, z . g K 1 K is isomorphic to 4 Ž . Ž .. =K; y s 0 or z s 0 by b ¬ b X , b Y . k Ž . Ž y, z . g K Ž3.6.3. As a GalŽ Krk .-set, ᑭ A, l K is isomorphic to l. l .. 4 Ž Ž Ž =K; y / 0, z / 0 by b ¬ b X , b Y .

The results Ž1., Ž2. are obtained from Ž3.6.1., Ž3.6.2., and Ž3.6.3. by k Ž X. counting the order of the set ᑭ A, for each finite extension kX of k r k in K. We prove Ž3.6.1. ᎐ Ž3.6.3.. Let ¨ be a non-zero element of M which is an eigenvector of X, with eigenvalue c g K. If c s 0, since KerŽ X : M ª M . is stable under the action of Y, we can take ¨ to be an eigenvector also of Y. Then K¨ is stable under the action of A, and hence M s K¨ , and M is determined by the eigenvalue of Y. In the case where Y has 0 as an eigenvalue, we can proceed in the same way. So we assume that c / 0 and Y does not have 0 as an eigenvalue. Then Y i ¨ for i s 0, . . . , l y 1 is a K-basis of M. In fact as X multiplies Y i ¨ by a i c, we see that these elements are linearly independent over K and the K-linear space spanned by them is stable under the action of A and hence coincides with M. k Ž . Ž Ž l ... These prove Ž3.6.1. and the fact b g ᑭ A, l K is determined by c, b Y X l When we fix the value of bŽ Y ., if two eigenvalues c, c of X with eigenvectors ¨ , ¨ X , respectively, satisfy c l s cX l, then the irreducible representations of A generated by ¨ and that by ¨ X are isomorphic. In fact since cX s a m c for some 0 F m - l, the isomorphism is defined by sending X ¨ to Y lym ¨ . Relation Ž3.6.3. follows from these things. Remark 3.6.1. As an example, let b G 1, c G 1 be integers which are relatively prime, let a s brc, and let R s Zw1rŽ bc .x, A s R X, Y 4 r² XY y aYX :. Then for r G 2, the following holds by Example 3.6Ž2.: ␨A, r Ž s . is the product of Ž1 y pyŽ sy2. .y1 Ž1 y pyŽ sy1. . 2 Ž1 y pys .y1 over all prime divisors p of b r y c r which do not divide b i y c i for 0 - i - r, and ᑭ A, r / ⭋ if and only if such p exists. By Birkhoff and Vandiver wB-V, Theorem 5x, such p exists for any r G 3 except the case  b, c4 s  2, 14 and

322

TAKAKO FUKAYA

r s 6, and the case b s c s 1. When we regard the work wB-Vx as the determination of the set  r; ᑭ A, r / ⭋4 for a special non-commutative finitely generated ring A over Z, the following general question arises: Is there a characterization of a set of integers which coincides with  r; ᑭ A, r / ⭋4 for some finitely generated ring A over Z? EXAMPLE 3.7 ŽHeisenberg Group.. Let G be the Heisenberg group which is defined by generators x, y, z with relations xyxy1 yy1 s z, xz s zx, yz s zy, and let A be the group ring Zw G x. We have: Ž1. The function ␨AŽ s . converges, and

␨A Ž s . s ␨ Ž s y 3 . ␨ Ž s y 2 . Ž2.

y3

3

␨ Ž s y 1. ␨ Ž s .

y1

.

We have

␨A , r Ž s . s ␨S rŽ s y 2 . ␨S rŽ s y 1 .

y2

␨S rŽ s . ,

where Sr s Zw ␨r , 1rr x with ␨r a primitive r th root of 1. Proof. The group ring of Heisenberg group is expressed as R  X , Xy1 , Y , Yy1 4 r² XY y ZYX : , where R s Zw Z, Zy1 x. By the similar argument as in Example 3.6, we have

␨A , r Ž s . s

Ł

ᒊgᑧ r

␨R r ᒊ Ž s y 2. ␨R r ᒊ Ž s y 1.

y2

␨R r ᒊ Ž s .

for all r G 1 where ᑧ r s  ᒊ g ᑪ Ž R . ; the canonical image of Z in Rrᒊ is a primitive r th root of 1 4 . The set ᑧ r is identified with the set of all maximal ideals of Sr . Furthermore

␨R Ž s . s ␨ Ž s y 1. ␨ Ž s .

y1

.

So the results Ž1., Ž2. are obtained. Finally we consider an example of A such that ␨AŽ s . diverges. For Example 3.8, I have computed ␨A, r Ž s . only in the case r s 2. EXAMPLE 3.8. Let R be a commutative finitely generated ring over Z, assume that R / 0, and let Z s R X, Y 4 . We have: Ž1. The function ␨AŽ s . diverges. Ž2. We have

␨A , 2 Ž s . s ␨ R Ž s y 5 . ␨ R Ž s y 4 .

y1

.

HASSE ZETA FUNCTIONS

323

Proof. Ž1. We may assume that R is a finite field k. For each integer r G 1, let Mr be the r dimensional k-vector space with basis eŽ i . Ž1 F i F r ., and define the actions of X, Y on Mr as follows: X sends eŽ i . to eŽ i q 1. if i / r, and sends eŽ r . to 0. Y sends eŽ i . to eŽ i y 1. if i / 1, and sends eŽ1. to 0. So we have an action of A on Mr . To prove that the function ␨AŽ s . diverges, it is enough to show that all the A-modules Mr Ž r G 1. are simple A-modules such that End AŽ Mr . s k. In fact any non-zero k-subspaces of Mr which is stable under the action of X has the form Ý iG r X keŽ i . for some 1 F r X F r, and any non-zero k-subspace of Mr which is stable under the action of Y has the form Ý iF r Y keŽ i . for some 1 F r Y F r. This shows that Mr has no non-zero A-submodule except Mr itself, that is, Mr is a simple A-module. Let h g End AŽ Mr .. Since the kernel of X : Mr ª Mr is a one dimension k-vector space generated by eŽ r ., we have hŽ eŽ r .. s aeŽ r . for some a g k. Since eŽ r . generates Mr over A Žfor the A-module Mr is simple., we have hŽ x . s ax for any x g Mr . Hence End AŽ Mr . s k. Ž2. We may assume that R is a finite field k with q elements. It is sufficient to prove that the number of isomorphism classes of A-modules M satisfying the condition Ž). below is q 5 y q 4 . Ž). M is a finite simple A-module of dimension 2 over k such that End AŽ M . s k. Let B s k w X x. As a B-module, such an M is isomorphic to BrŽ f . for a unique monic polynomial f of degree 2 over k. We consider the following three cases individually. Ž3.8.1. Ž3.8.2. Ž3.8.3.

f s Ž X y a.Ž X y b . with a, b g k, a / b. f s Ž X y a. 2 with a g k. f is irreducible.

Ž3.8.1. In this case there are q Ž q y 1.r2 polynomials f. An action of Y on M s BrŽ f . gives M an A-module structure satisfying Ž). if and only if Y stabilizes none of the k-subspaces k Ž X y a. modŽ f . and k Ž X y b . modŽ f . of BrŽ f .. There are Ž q 2 y q . 2 actions of Y satisfying this condition. Since two actions of Y give isomorphic A-module structures on BrŽ f . if and only if they are conjugate by some element of Ž BrŽ f ..U , each A-isomorphism class includes 噛ŽŽ BrŽ f ..U rkU . s q y 1 different actions of Y. Thus in this case we have q Ž q y 1.r2 ⭈ Ž q 2 y q . 2 ⭈ 1rŽ q y 1. s Ž q 3 Ž q y 1. 2 .r2 isomorphism classes of A-modules satisfying the condition Ž).. Ž3.8.2. In this case there are q polynomials f. An action of Y on M s BrŽ f . gives M an A-modules structure satisfying Ž). if and only if

324

TAKAKO FUKAYA

the action of Y does not stabilize the k-subspace k Ž X y a. modŽ f .. There are q 3 Ž q y 1. such actions, and each A-isomorphism class includes 噛ŽŽ BrŽ f ..U rkU . s q different actions of Y. Thus in this case we get q ⭈ q 3 Ž q y 1. ⭈ qy1 s q 3 Ž q y 1. isomorphism classes of A-modules satisfying the condition Ž).. Ž3.8.3. In this case there ae q Ž q y 1.r2 polynomials f. An action of Y on M s BrŽ f . gives M an A-module structure satisfying Ž). if and only if the action of Y does not commute with the action of B. There are q 4 y q 2 such actions, and each A-isomorphism class includes 噛ŽŽ BrŽ f ..U rkU . s q q 1 different actions of Y. Thus in this case we get q Ž q y 1.r2 ⭈ Ž q 4 y q 2 . ⭈ Ž q q 1.y1 s q 3 Ž q y 1. 2r2 isomorphism classes of A-modules satisfying Ž).. So the number of all isomorphism classes of A-modules which satisfy Ž). is q 3 Ž q y 1. 2r2 q q 3 Ž q y 1. q q 3 Ž q y 1. 2r2 s q 5 y q 4 . 3.9. Here we give the proof of Proposition 3.5.2 due to N. Kurokawa. The main line of the proof is based on the method in wL-Wx Žcf. also wKu3x.. Ž3.9.1. In preparation for the proof of Proposition 3.5.2, we give here an expression of log ⌽ Ž s, ␹ . by using Dirichlet L-functions of powers of ␹ . Let ␸ Ž s, ␹ . s Ý p:prime ␹ Ž p . pys . By The Mobius inversion theorem, we ¨ have ⬁ ␮ Ž n. ␸ Ž s, ␹ . s Ý log L Ž ns, ␹ n . , Ž i. n ns1 where ␮ Ž n. is the Mobius ␮-function. Write hŽT . s Ý dks 0 c k T k . Then ¨ log ⌽ Ž s, ␹ . s

Ý

h Ž p . log Ž 1 y ␹ Ž p . pys .

p:prime

sy

m

c k ␹ Ž p . p kym s

⬁

d

Ý Ý

Ý

d

sy

⬁

ck

Ý Ý ks0 ms1 d

sy

⬁

m

␸ Ž ms y k, ␹ m .

ks0 ms1

␮ Ž n.

⬁

ck

Ý Ý

m

Ž ii .

m

ks0 p :prime ms1

Ý ns1

n

log L Ž n Ž ms y k . , ␹ m n .

Ž by Ž i . . d

sy

⬁

Ý ck Ý ks0

Ns1

1 N

žÝ

n< N

␮ Ž n . log L Ž Ns y nk, ␹ N . . Ž iii .

/

325

HASSE ZETA FUNCTIONS

Ž3.9.2. We prove Ž1. of Proposition 3.5.2. We show the function log ⌽ Ž s, ␹ . absolutely converges in ReŽ s . ) d q 1. We consider the expression Žii. of log ⌽ Ž x, ␹ .. Let ␴ s ReŽ s . ) d q 1. Then for each k s 0, . . . , d, m

␹ Ž p . p kym s

⬁

Ý

Ý

m

p prime ms1 ⬁

F

Ý

p dym ␴

Ý

p:prime ms1

pd

s

Ý p:prime

F2

p␴ y 1

Ý

p dy ␴

ž

p:prime

F2

from the inequality

1 ␴

p y1

F

2 p␴

/

⬁

Ý n dy ␴ - ⬁. ns1

Ž3.9.3. We prove Ž2. of Proposition 3.5.2. It suffices to show that the function log ⌽ Ž s, ␹ . is analytic in ReŽ s . ) d. Since log L Ž s, ␹ . ; ␹ Ž 2 . 2ys

as Re Ž s . ª ⬁

we have N

log L Ž Ns y nk, ␹ N . ; ␹ Ž 2 . 2yNsqn k

as N ª ⬁.

Here and in what follows, ; means that the ratio converges to 1. By Žiii., this implies that d

log ⌽ Ž s, ␹ . q

N

Ý ck Ý ks0

d

;y

Ý ck ks0

X

½

Ns1 ⬁

1 N 1

Ý X

NsN q1

N

žÝ

␮ Ž n . log L Ž Ns y nk, ␹ N .

žÝ

␮ Ž n . ␹ Ž 2 . 2yNsqn k

n< N N

n< N

/

/5

for a sufficiently large integer N X . Ž iv . So we show that Živ. absolutely converges in ReŽ s . ) d. Let c s max k < c k < and let ␴ s ReŽ s . ) d. Then

Ý < ck < k

⬁

1

Ý X

NsN q1

N

Ý 2yN ␴qn k F cd

n< N

s cd

N 2yN ␴qNd

⬁

Ý X

NsN q1 ⬁

N 1

Ý X

NsN q1

Ž 2 ␴yd .

N

- ⬁.

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TAKAKO FUKAYA

Ž3.9.4. We prove Ž3. of Proposition 3.5.2 in Subsections 3.9.4᎐3.9.7. By Žiii. we have ⌽X ⌽

⬁

d

Ž s, ␹ . s y

LX

Ý c k Ý Ý ␮ Ž n . L Ž Ns y nk, ␹ N .

ks0

Ns1

ž

n< N

/

.

Ž v.

We show that every point on ReŽ s . s d is a limit point of poles of the function Ž ⌽Xr⌽ .Ž s, ␹ .. To show this, it is sufficient to prove that for any real number u and for any real number ␦ ) 0, the function Ž ⌽Xr⌽ .Ž s, ␹ . has a pole in the region S d , u , ␦ s  s g C; d - Re Ž s . - d q ␦ , u - Im Ž s . - u q ␦ 4 . By the principle of reflection, it is enough to prove this only for u ) 0. Ž3.9.5. The following facts are known Žsee wTix.: Žvi. Let N ŽT, ␹ . be the number of the zeros of LŽ s, ␹ . in the region

 s g C; 0 - Re Ž s . - 1, 0 - Im Ž s . - T 4 . Then NŽT , ␹ . ;

T 2␲

as T ª ⬁.

log T

Žvii. Let ␳ be a zero of LŽ s, ␹ ., and let multŽ ␳ , ␹ . be the multiplicity of ␳ . Then mult Ž ␳ , ␹ . - C log
pª⬁

pŽ u q ␦ . 2␲

log Ž p Ž u q ␦ . .

pu 2␲

log Ž pu . s

uq␦ u

) 1,

327

HASSE ZETA FUNCTIONS

we have by Žvi. the relation ; in N Ž p Ž u q ␦ . , ␹ . y N Ž pu, ␹ . ; )

pŽ u q ␦ .

log Ž p Ž u q ␦ . . y

2␲ p␦ 2␲

log Ž pu . ª ⬁

pu 2␲

log Ž pu .

as p ª ⬁.

Hence if p is a sufficiently large prime number satisfying Žxii., it satisfies all Žviii. ᎐ Žxii.. Ž3.9.7. Take a prime number p which satisfies Žviii. ᎐ Žxii.. Then by Žviii., there exists a complex number ␳ such that 0 - ReŽ ␳ . - 1 and pu - ImŽ ␳ . - pŽ u q ␦ ., LŽ ␳ , ␹ . s 0. By the principle of reflection, we may assume that 1r2 F ReŽ ␳ . - 1. We will prove that ␻ s ␳rp q d is a pole of Ž ⌽Xr⌽ .Ž s, ␹ . and is contained in the region S d, u, ␦ . The fact ␻ is contained in S d, u, ␦ is deduced from d-

Re Ž ␳ . p

qd-

1 p

qd-dq␦

Ž by Ž ix . .

X and from our assumption on ImŽ ␳ .. The point ␻ is a pole of yc d Ž Lr L.Ž ps y pd, ␹ .. It is deduced from

L Ž p␻ y pd, ␹ . s 0

by p␻ y pd s ␳ .

Now we prove that ␻ is a pole of the function Ž ⌽Xr⌽ .Ž s, ␹ .. That is, the X .Ž ps y pd, ␹ . does not vanish by the influpole ␻ of the function Ž LrL X .Ž Ns y nk, ␹ N . which ence of the other poles of the functions Ž LrL X appear in Žv.. We show the residue Res ss ␻ ŽŽ ⌽ r⌽ .Ž s, ␹ .. / 0. This is deduced as Res ss ␻

ž

⌽X ⌽

⬁

d

Ž s, ␹ . s y

/

Ý

ck

ks0

Ý

Ns1 n < N ⬁

d

sy

Ý ␮ Ž n . Res ss ␻

Ý ck Ý Ý ␮ Ž n. ␮Ž N .

Ý 0-N-2 p

L

Ž Ns y nk, ␹ N .

N

mult Ž N␻ y Nd, ␹ N . N

The last equality Žxiii. is obtained as follows: If N G 2 p, Re Ž N␻ y nk . s

N p

/

mult Ž N␻ y nk, ␹ N .

Ns1 n < N

ks0

s yc d

ž

LX

Re Ž ␳ . q Ž Nd y nk . G 1

.

Ž xiii .

328

TAKAKO FUKAYA

for Ž Nrp.ReŽ ␳ . G 1 and Nd y nk G 0. If 0 - N - 2 p and Ž k, n. / Ž d, N ., Re Ž N␻ y nk . s

N p

Re Ž ␳ . q Ž Nd y nk . ) 1

for ReŽ ␳ . ) 0 and Nd y nk G 1. Since LŽ s, ␹ N . is holomorphic except possibly at s s 1 and does not have zero in ReŽ s . G 1, these prove Žxiii.. Finally Res ss ␻

ž

⌽X ⌽

Ž s, ␹ . s yc d

/

␮Ž N .

Ý

mult Ž N␻ y Nd, ␹ N . N

0-N-2 p



s yc d y

q

mul Ž p␻ y pd, ␹ p . p

␮Ž N .

Ý

mult Ž N␻ y Nd, ␹ N . N

0-N-2 p N/p

0

. Ž xiv .

Here p␻ y pd s ␳ and we have 1 F mult Ž ␳ , ␹ p . s mult Ž ␳ , ␹ . - C log Im Ž ␳ .

Ž by Ž xii . .

Ž by Ž vii. and Ž x . .

- C log Ž p Ž u q ␦ . . -p

Ž by Ž xi . . .

Hence, the denominator of multŽ p␻ y pd, ␹ p .rp is p whereas the denominators of multŽ N␻ y Nd, ␹ N .rN for 0 - N - 2 p, N / p are prime to p. So Žxiv. does not coincide with 0. 3.10. We prove Proposition 3.5.1. Proof. We apply Proposition 3.5.2 by taking hŽT . s ŽT y 1.r2 and taking the following Dirichlet character ␹ : ␹ Ž n. s 1 if n is congruent to 1 modulo 2 and ␹ Ž n. s 0 if n is congruent to 0 modulo 2. The degree of hŽT . is 1. By Proposition 3.5.2, we obtain that in Ž1. of Example 3.5, the second Žresp. third. term absolutely converges in ReŽ s . ) 3 Žresp. ReŽ s . ) 2., is analytic in ReŽ s . ) 2 Žresp. ReŽ s . ) 1., and has the natural boundary ReŽ s . s 2 Žresp. ReŽ s . s 1.. Since the first term ␨ Ž s y 3. absolutely converges if and only if ReŽ s . ) 4 and is analytic on the whole s-plane, we obtain Proposition 3.5.1.

329

HASSE ZETA FUNCTIONS

4. CONVERGENCE OF ZETA FUNCTIONS AND GELFAND᎐KIRILLOV DIMENSIONS It is known that for a commutative finitely generated ring R over Z, ␨ R Ž s . absolutely converges if and only if ReŽ s . ) dimŽ R ., where dimŽ R . denotes the Krull dimension of R. In this section, we consider the relationship between convergence of zeta functions of non-commutative rings and Gelfand᎐Kirillov dimensions of rings. 4.1. We introduce the definition of the Gelfand᎐Kirillov dimension. The Gelfand᎐Kirillov dimension is defined usually for algebras over fields. For a finitely generated algebra A over a field k, the Gelfand᎐Kirillov dimension GKdimŽ A. g  t g R; t G 04 j  "⬁4 of A is defined as follows: Let S be a finite set of the generators of A over k. Then GKdim Ž A . s lim sup nª⬁

ž

log dim k Ž Vn Ž S . . log n

/

,

where VnŽ S . is the k-subspace Ý njs0 kS j of A Ž S j s  x 1 ⭈⭈⭈ x j ; x 1 , . . . , x j g S4.. This is independent of the choice of S. If A is commutative, GKdimŽ A. coincides with the Krull dimension dimŽ A.. ŽSee wM-R, Chap. 8x.. For a commutative ring R and a finitely generated R-algebra A, we define GKdim Ž A; R . s sup  GKdim Ž A mR k Ž ᒍ . . q dim Ž Rrᒍ . ; ᒍ g Spec Ž R . 4 , where k Ž ᒍ . denotes the residue field of ᒍ. For a finitely generated ring A over Z, we define the Gelfand᎐Kirillov dimension of A by GKdim Ž A . s GKdim Ž A; Z. . GKdimŽ A. s dimŽ A. holds again for a commutative finitely generated ring A over Z. 4.2. By Gromov wGrx, for a finitely generated group G and for a field k, GKdimŽ k w G x. is finite if and only if G has a nilpotent subgroup of finite index. From this we can deduce that for a finitely generated group G and for a commutative finitely generated ring R / 0 over Z, G has a nilpotent subgroup of finite index if and only if GKdimŽ Rw G x. is finite. ŽTo see this, we have to show that if G is a finitely generated group having a nilpotent subgroup of finite index, then GKdimŽ Rw G x mZ k Ž ᒍ .. is bounded by a number which is independent of ᒍ g SpecŽZ.. This fact is a consequence of wBax.. ŽWe say the Gelfand᎐Kirillov dimension is finite also in the case it is y⬁..

330

TAKAKO FUKAYA

THEOREM 4.3. Let H be a free abelian group of finite rank n, and G be a group ha¨ ing an exact sequence 1 ª H ª G ª Z ª 1. Let R be a commutati¨ e finitely generated ring o¨ er Z, and let A be the group ring Rw G x. Then we ha¨ e: Ž1. If R / 0, the function ␨AŽ s . con¨ erges if and only if G has a nilpotent subgroup of finite index. Ž2. If G has a nilpotent subgroup of finite index, there exist integers cŽ0., . . . , cŽ n q 1. such that cŽ n q 1. s 1 and such that nq1

␨A Ž s . s

Ł ␨R Ž s y i . c i . Ž .

is0

In particular, the function ␨AŽ s . absolutely con¨ erges if and only if ReŽ s . ) dimŽ R . q n q 1. Ž3. If G is nilpotent,

␨ A Ž s . s ␨ RX Ž s . , y1 y1 x where RX s Rw X 1 , Xy1 1 , . . . , X n , X n , X nq1 , X nq1 .

By 4.2, Theorem 4.3Ž1. implies COROLLARY 4.4.

Let A be as in Theorem 4.3. The function ␨AŽ s . con-

¨ erges if and only if the Gelfand᎐Kirillo¨ dimension of A is finite.

We prove Theorem 4.3 in Subsections 4.5᎐4.8. 4.5. Here we give the preparations for the proof of Theorem 4.3. PROPOSITION 4.5.1. Let B be a commutati¨ e finitely generated ring o¨ er Z, and let f : B ª B be an automorphism of B. Let A be the skew Laurent N polynomial ring  Ý isyN a i T i ; N G 0, a i g B4 in which the multiplication is defined by the rule Ta s f Ž a.T Ž a g B .. Let p be a prime number, and let K be the algebraic closure of Fp . We put ᑭ A s Ł r G 1 ᑭ A, r , and ᑭ B s SpecŽ B ., so ᑭ B Ž K . s HomŽ B, K . the set of ring homomorphisms from B into K. Then as sets endowed with actions of GalŽ KrFp . we ha¨ e ᑭ B Ž K . rf = K U , ᑭ A Ž K . , where ᑭ B Ž K .rf is the quotient of ᑭ B Ž K . by the action of f, and K U is the multiplicati¨ e group of K. The isomorphism is defined as follows: For h g ᑭ B Ž K . s HomŽ B, K . and c g K U , the image of the pair Ž h, c . in ᑭ AŽ K . is the class of the following irreducible representation V o¨ er K. Let m be the minimal integer G 1 such that hf m s h. Ž Such m exists since the images of hf i for all i g Z are a finite field k which is independent of i and there is only

HASSE ZETA FUNCTIONS

331

a finite number of homomorphisms from B to k Ž for B is finitely generated o¨ er Z... Then V is the m-dimensional K-¨ ector space with basis e i Ž1 F i F m. on which A acts by ae i s h Ž f i Ž a . . e i

Ž a g B.

Te i s e iy1 if 2 F i F m,

Te1 s ce m .

The proof is similar to the argument in Example 3.6. Let V be a finite dimensional irreducible representation of A over K. Let ¨ be a non-zero element of V which is an eigenvector for all elements of B. The eigenvalues give a homomorphism h : B ª K. Let m be the minimal integer G 1 such that hf m s h. Then by the argument as in Example 3.6, we see that B acts on T i ¨ via hfyi, the T i ¨ Ž0 F i - m. form a K-basis of V, and V is determined by h and the action of T m Žwhich is a scalar since it commutes with the actions of B and T .. Proposition 4.5.1 follows from these facts. We omit the proofs of the following elementary lemmas. LEMMA 4.5.2. Let G and H be as in Theorem 4.3. Let f : H ª H be the automorphism of H defined by x ¬ ␣ x ␣y1 where ␣ is an element of G whose image in Z is 1. Then the automorphism f is quasi-unipotent Ž that is, f m is unipotent for some m G 1. if and only if the group G has a nilpotent subgroup of finite index. The automorphism f is unipotent if and only if the group G is nilpotent. LEMMA 4.5.3. Let L be a finitely generated free abelian group and let g : L ª L be a homomorphism such that detŽ g . / 0. Then 噛Ker Ž g : L mZ Ž QrZ. ª L mZ Ž QrZ. . s
332

TAKAKO FUKAYA

where ŽQrZ.Žnon y p . is the subgroup of QrZ consisting of elements whose denominators are prime to p, and L s HomŽ H, Z.. For a finite extension Fq of k in K which has q elements, let X R ŽFq . be the GalŽ KrFq .-fixed part of ᑭ BR Ž K .rf. Then we have X R Ž Fq . s  x mod f ; x g L mZ Ž QrZ. Ž non y p . , Ž q y f i . x s 0 for some i g Z 4 s  x mod f ; x g L mZ Ž QrZ. , Ž q y f i . x s 0 for some i g Z 4 , where the f i are considered as endomorphisms of L mZ ŽQrZ.. 4.7. We prove Ž2., Ž3. of Theorem 4.3. In Subsection 4.7, we assume that G has a nilpotent subgroup of finite index. We define the integers cŽ i . which appear in Theorem 4.3Ž2.. Let ␣ 1 , . . . , ␣ n be all the eigenvalues of f, and let gŽT . s

1 m

m

Ý Ž T y ␣ 1i . ⭈⭈⭈ Ž T y ␣ ni . , is1

where m is an integer ) 0 such that all of ␣ i Ž i s 1, . . . , n. are mth roots of 1. Then g ŽT . g Zw T x. This is because every coefficient of the above polynomial can be expressed as a finite sum in which each term has the i form "Ž1rm.Ý m is1 b for an mth root b of 1, which is 0 if b / 1 and is "1 if b s 1. We put

Ž T y 1 . g Ž T . s c Ž 0 . q c Ž 1 . T q ⭈⭈⭈ qc Ž n q 1 . T nq 1 . By the above argument, all of the cŽ i . Ž i s 0, . . . , n q 1. are integers and cŽ n q 1. s 1. Let the notation be as in Subsection 4.6. LEMMA 4.7.1. There exists an integer r ) 0 such that KerŽ q y f i . s KerŽ q y f iqr . for all i g Z. Proof. By Lemma 4.5.3, 噛KerŽ q y f i . s
333

HASSE ZETA FUNCTIONS

LEMMA 4.7.2. Let r be as in Lemma 4.7.1. Then under the map ␸ : ⌸ 0 F i- r KerŽ q y f i . ª X R ŽFq ., the in¨ erse image of each element of X R ŽFq . is of order r. Proof. The inverse image of any element y of X R ŽFq . in L mZ ŽQr Z.Žnon y p . is expressed as S x s  f j Ž x .; j g Z4 for some element x of L mZ ŽQrZ.Žnon y p . such that ␸ Ž x . s y. The order l Ž x . of the set S x is finite as we have seen in the proof of Lemma 4.7.1. Furthermore we have X X S x ; KerŽ q y f iql . when l Ž x . divides l⬘, and S x l KerŽ q y f iql . s ⭋ when l Ž x . does not divide l⬘. So for any element of X R ŽFq ., the order of the inverse image is l Ž x . = Ž rrl Ž x .. s r. Now we can prove Ž2., Ž3. of Theorem 4.3. By Lemma 4.5.3 and Lemma 4.7.2, 噛X R Ž Fq . s g Ž q . . So 噛ᑭ AR Ž Fq . s Ž q y 1 . g Ž q . s c Ž 0 . q c Ž 1 . q q ⭈⭈⭈ qc Ž n q 1 . q nq 1 . Hence we have nq1

␨A Ž s . s

Ł ␨R Ž s y i . c i . Ž .

is0

Concerning Ž3., if G is nilpotent, all the eigenvalues of f are 1. So in this case we have n

g Ž T . s Ž T y 1. . Hence

Ž T y 1. g Ž T . s Ž T y 1.

nq 1

.

Hence the result follows. 4.8. We prove Ž1. of Theorem 4.3. It remains to prove that if G has no nilpotent subgroup of finite index, then the function ␨AŽ s . diverges. Proof. Since f is not quasi-unipotent, there is an eigenvalue of f which is not a root of 1. Since a non-zero algebraic integer which is not a root of 1 has a conjugate over Q whose absolute value is ) 1, there exists an eigenvalue of f whose absolute value is ) 1 wWe2, Chap. 4, Theorem 8x. Hence by Lemma 4.5.3, the order of KerŽ q y f i . can become arbitrarily large when i varies. We show that, with the notation in Subsection 4.6, X R ŽFq . is an infinite set Žthis shows that the function ␨AŽ s . diverges.. Assume that X R ŽFq . is finite. The inverse image of any element of X R ŽFq .

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under the map Di g Z KerŽ q y f i . ª X R ŽFq . is expressed as S x s  f j Ž x .; j g Z4 for some element x of L mZ ŽQrZ.Žnon y p .. The set S x is a finite set by the fact that S x ; KerŽ q y f i . and Lemma 4.5.3. Hence Di g Z KerŽ q yf i . is a finite set. It contradicts the fact that the order of KerŽ q y f i . can become arbitrarily large when i varies. Hence we accomplish the proof of Theorem 4.3. DEFINITION 4.9. Here we introduce the ‘‘modified Hasse zeta function’’ ␨AU Ž s . of a finitely generated ring A over Z, to formulate Conjectures 4.11 and 4.14 concerning the relationship between the convergence of the zeta function and GKdimŽ A.. Theorem 4.3 suggests the existence of a strong relationship, but there is an example of a finitely generated ring A over Z such that ␨AŽ s . s 1 Židentically. and GKdimŽ A. s ⬁ Žsee Lemma 4.10.2.. For a finitely generated ring A over Z, we define the modified Hasse zeta function ␨AU Ž s . of A as

␨AU Ž s . s

ys y1

Ł Ž1 y N Ž M . .

,

M

where M runs over all isomorphism classes of simple A-modules such that 噛End AŽ M . is finite, and N Ž M . s 噛End AŽ M .. Remark that in this definition, M itself need not be a finite module. 4.10. The function ␨AŽ s . and the function ␨AU Ž s . coincide for many examples: LEMMA 4.10.1. Ž1. Define the properties F Ž i . of finitely generated rings o¨ er Z for i G 0, inducti¨ ely as follows: A is F Ž0. if and only if A is a finite ring. For i ) 0, A is F Ž i . if and only if there is a commutati¨ e finitely generated ring R o¨ er Z such that A has an R-algebra structure and Arᒊ A are F Ž i y 1. for all maximal ideals ᒊ of R. Then, if A is F Ž i . for some i G 0, we ha¨ e

␨A Ž s . s ␨AU Ž s . . Ž2. Let G be a finitely generated group ha¨ ing a polycyclic subgroup of finite index. Let R be a commutati¨ e finitely generated ring o¨ er Z and let A be the group ring Rw G x. Then

␨A Ž s . s ␨AU Ž s . . Ž For example, the group G in Theorem 4.3 is a polycyclic group, and hence ␨AŽ s . s ␨AU Ž s . for the ring A in Theorem 4.3.. Ž3. The rings A in Examples 3.1᎐3.7 satisfy

␨A Ž s . s ␨AU Ž s . .

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Proof. Ž1. Let A be a ring which is F Ž i . for some i G 0 and let M be an A-module such that End AŽ M . is finite. We prove by induction on i that M itself is finite. In the case i s 0, this is clear since a simple A-module is isomorphic to the quotient of the finite ring A by a left ideal of A. Assume i ) 0. The action of R on M is contained in the finite field End AŽ M .. Let ᒊ be the kernel of R ª End AŽ M .. Since the image is finite, Rrᒊ is a finite field, that is, ᒊ is a maximal ideal of R. Since M is regarded as an Arᒊ A-module, M is finite by the hypothesis of the induction. Ž2. By wRox, any simple A-module is finite. Ž3. The rings A in Examples 3.1᎐3.7 satisfy the assumption of Ž1.. In fact the rings in Examples 3.1᎐3.7 are F Ž2.. For each ring A in Examples 3.1᎐3.7, let ᒊ be a maximal ideal of R in Examples 3.1, 3.3, 3.4, 3.6, and 3.7 Žresp. of Z in Example 3.5. Žresp. of RX in Corollary 3.2.. Then there is a finitely generated subring B over Rrᒊ in Examples 3.1, 3.3, 3.4, 3.6, and 3.7 Žresp. over Zrᒊ in Example 3.5. Žresp. over RXrᒊ in Corollary 3.2. contained in the center of the ring Arᒊ A such that Arᒊ A is a finitely generated B-module. So the Arᒊ A are F Ž1.. LEMMA 4.10.2 Ždue to K. Kato.. Let k be a finite field, and let A s k X, Y, U, V 4 r² XY y 1, UV y 1, YX q VU y 1:. Ž1. The function ␨AŽ s . is identically 1. Ž2. The function ␨AU Ž s . di¨ erges. Ž3. GKdimŽ A. s ⬁. s Ž X. Proof. Ž1. We show that the set ᑬ A, r k is empty for any integer r G 1 X s Ž X. Ž . Ž . and any finite field k . Indeed for an element h g ᑬ A, r k , h X h Y s1 and hŽU . hŽ V . s 1, so hŽ Y . hŽ X . s 1 and hŽ V . hŽU . s 1. Hence

1 s h Ž 1 . s h Ž YX q VU . s 1 q 1 s 2, a contradiction. Ž2. For each integer r ) 1, let M Ž r . be the following A-module: Let m s 2 r y 1, and let S be the set of all integers i which are congruent to some power of 2 modulo m. Let M Ž r . be the infinite dimensional k-vector space with basis eŽ i . Ž i g S ., and define the actions of X, Y, U, V on M Ž r . as follows: X sends eŽ i . to eŽ ir2. if i is even, and sends eŽ i . to 0 if i is odd. Y sends eŽ i . to eŽ2 i .. U sends eŽ i . to eŽŽ i q m.r2. if i is odd, and sends eŽ i . to 0 if i is even. V sends eŽ i . to eŽ2 i y m..

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We have XY s 1, UV s 1, YX q VU s 1. So we have an action of A on M Ž r . for each r. We show that the A-module M Ž r . is a simple A-module such that End AŽ M Ž r .. s k for each r, and if r and s are not equal, M Ž r . and M Ž s . are not isomorphic. For each r, let N Ž r . be the r-dimensional k-vector space of M Ž r . generated by eŽ2 i . for i s 0, . . . , r y 1. Then N Ž r . is stable under the action of k X, U 4 , and any non-zero k X, U 4 -submodule of M Ž r . contains N Ž r ., as is seen easily. N Ž r . is simple as a k X, U 4 module. Hence M Ž r . has a unique simple k X, U 4 -submodule which has dimension r over k. Hence r s s if and only if M Ž r . and M Ž s . are isomorphic. If M X is a non-zero A-submodule of M Ž r ., M X contains N Ž r ., and N Ž r . generates M Ž r . over A as is easily seen. Hence M X s M Ž r ., and this shows that M Ž r . is a simple A-module. Let h g End AŽ M Ž r ... Since N Ž r . is the unique simple k X, U 4 -module contained in M Ž r ., h sends N Ž r . into N Ž r .. Since the kernel of X : N Ž r . ª N Ž r . is a one dimensional k-vector space generated by eŽ1., we have hŽ eŽ1.. s aeŽ1. for some a g k. Since eŽ1. generates M Ž r . over A Žfor the A-module M Ž r . is simple., we have hŽ x . s ax for any x g M Ž r .. Hence End AŽ M Ž r .. s k for each r. Ž3. It can be seen easily that the map k Y, V 4 ª End k Ž M Ž r .. is injective for any r. This shows that k Y, V 4 ª A is injective. By wM-R, 8.1.15 and 8.2.2x, this implies GKdimŽ A. G GKdimŽ k Y, V 4. s ⬁. CONJECTURE 4.11. Let A be a finitely generated ring o¨ er Z. The function ␨AU Ž s . con¨ erges if and only if the Gelfand᎐Kirillo¨ dimension of A is finite. DEFINITION 4.12. We define the ‘‘zeta dimension’’ ␨ dimŽ A. g  t g R; t G 04 j  "⬁4 of A by

␨ dim Ž A . s Inf  d g R; ␨AU Ž s . absolutely converges if Re Ž s . ) d 4 . PROPOSITION 4.13. Ž1. For rings A in Examples 3.1᎐3.8, we ha¨ e: Example 3.1.

␨ dim Ž A . s GKdim Ž A; R . s GKdim Ž A . s dim Ž R . q 2 n. Example 3.2.

␨ dim Ž A . s GKdim Ž A; R . s GKdim Ž A . s dim Ž R . q 3. Example 3.3.

␨ dim Ž A . s GKdim Ž A; R . s GKdim Ž A . s dim Ž R . q 2. Example 3.4.

␨ dim Ž A . s GKdim Ž A; R . s GKdim Ž A . s dim Ž R . q 3.

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Example 3.5.

␨ dim Ž A . s GKdim Ž A . s 4. Example 3.6.

␨ dim Ž A . s GKdim Ž A; R . s dim Ž R . q 2 F GKdim Ž A . - ⬁. Example 3.7.

␨ dim Ž A . s GKdim Ž A; R . s 4. GKdim Ž A . s 5. Example 3.8.

␨ dim Ž A . s GKdim Ž A; R . s GKdim Ž A . s ⬁. Ž2. Let R be a commutati¨ e finitely generated ring o¨ er Z and let A be a finitely generated algebra o¨ er R which is of finite type as an R-module. Then

␨ dim Ž A . s GKdim Ž A; R . s GKdim Ž A . and this number is equal to dimŽ R . if R ª A is injecti¨ e. Ž3. Let R and A be as in Theorem 4.3. Then

␨ dim Ž A . F GKdim Ž A; R . s GKdim Ž A . . If R / 0, the inequality is an equality if and only if G has an abelian subgroup of finite index. Proof. Ž1. In Example 3.5, put R s Z. For the rings A in Examples 3.1᎐3.8, the relations between ␨ dimŽ A. and dimŽ R . follow from our computation of ␨AŽ s . and from Lemma 4.10.1Ž3.. The relations between GKdimŽ A; R . or GKdimŽ A. and dimŽ R . follow from wM-R, 8.2.10x for the rings A in Example 3.1᎐3.5, from wBa; Wa, Theorem 1Žb.x for Example 3.7, and from wM-R, 8.1.15x for Example 3.8. In Example 3.6, GKdimŽ A; R . s dimŽ R . q 2 F GKdimŽ A. is easily proved. It remains to prove GKdimŽ A. - ⬁ in Example 3.6. It is sufficient to prove the following fact: Let F be a field, R a commutative finitely generated ring over F, a g R, and let A s R X, Y 4 r² XY y aYX :. Then GKdimŽ A. F 2 dimŽ R . q 2. To show this, let SX be a finite set of generators of R containing a, let SY be the set  X, Y 4 , and let S s SX j SY . Then S is a finite set of generators of A. For n G 0, an element of SY n is expressed in the form a m Y i X j with i, j, m G 0, i q j s n, m F n2 . Hence VnŽ S . ; Ý iqjF nVn 2 Ž SX .Y i X j. From this we have dim F Ž Vn Ž S . . F

1 2

Ž n q 1 . Ž n q 2 . dim F Ž Vn 2 Ž SX . . .

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This shows GKdim Ž A . F 2 q 2 GKdim Ž R . s 2 dim Ž R . q 2. Ž2. We may assume that R ª A is injective. By wM-R, 8.2.9Žii.x, we have GKdimŽ A; R . s GKdimŽ A. s dimŽ R .. It remains to prove ␨ dimŽ A. s dimŽ R .. Replacing SpecŽ R . by its dense open affine subscheme U, and by applying the induction on dimŽ R . to the complement of U in SpecŽ R ., we are reduced to the following situation. There is a nilpotent two-sided ideal I of A such that ArI is an Azumaya algebra over the center C and C is a finitely generated module over RrŽ R l I .. In this situation, we have dim Ž R . s dim Ž Rr Ž R l I . . s dim Ž C . s ␨ dim Ž C . s ␨ dim Ž ArI . s ␨ dim Ž A . , where the first equation follows from the fact R l I is a nilpotent ideal of R, the fourth equation follows from Proposition 2.1Ž3., and the last equation follows from ␨AŽ s . s ␨A r I Ž s . Žfor I is nilpotent.. Ž3. By Theorem 4.3, ␨ dimŽ A. s dimŽ R . q n q 1. Hence the result follows from wBa; Wa, Theorem 1Žb.x. CONJECTURE 4.14. Let R be a commutati¨ e finitely generated ring o¨ er Z, and let A be a finitely generated R-algebra. Then

␨ dim Ž A . F GKdim Ž A; R . . 4.15. By Proposition 4.13, the Conjectures 4.11 and 4.14 are true for the rings A in Examples 3.1᎐3.8, and also for the rings A as in Proposition 4.13Ž2., Ž3.. Remark 4.16. Ž1. If R1 and R 2 are commutative finitely generated rings over Z with a homomorphism R1 ª R 2 , and if A is a finitely generated R 2-algebra, then the following holds: GKdim Ž A; R1 . G GKdim Ž A; R 2 . . For example, for a commutative finitely generated ring R over Z and for a finitely generated algebra A over R, we have GKdim Ž A; R . F GKdim Ž A . . The inequality can be strict as in Proposition 4.13Ž1., Example 3.7.

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Ž2. The Krull dimensions of non-commutative rings do not behave so well as the Gelfand᎐Kirillov dimensions concerning the convergence of zeta functions. For example, for a ring A as in Theorem 4.3, the Krull dimension of A is always dimŽ R . q n q 1, whether the zeta function converges or not wM-F, 6.9.13x.

5. QUESTIONS Many questions arise concerning zeta functions of non-commutative rings. I present some of them here. Concerning the zeta dimension, the following questions arise: QUESTION 5.1.

Is there a purely algebraic definition of ␨ dimŽ A.?

QUESTION 5.2.

Is ␨ dimŽ A. always an integer if it is not ⬁ or y⬁?

QUESTION 5.3. Let R be a commutati¨ e finitely generated ring o¨ er Z, and ᒄ a Lie algebra o¨ er R which is free of finite rank n as an R-module. Then, for the uni¨ ersal en¨ eloping algebra A of ᒄ, does

␨ dim Ž A . s dim Ž R . q n? Concerning ‘‘what kind of functions appear as zeta functions of noncommutative rings’’: QUESTION 5.4. Is there a purely algebraic characterization of a finitely generated ring o¨ er Z whose zeta function coincides with the zeta function of some commutati¨ e finitely generated ring o¨ er Z? A more specific question is the following: If ᒄ is a Lie algebra o¨ er Z which is of finite type as a Z-module, A is the uni¨ ersal en¨ eloping algebra of ᒄ, and if ᒄ mZ Q is not sol¨ able, does it always happen that ␨AŽ s . does not coincide with the Hasse zeta functions of any commutati¨ e finitely generated rings o¨ er Z? QUESTION 5.5. For a finite field Fq , is there a finitely generated Fq-algebra A such that the function ␨AŽ s . con¨ erges but it is not a rational function of qys ? Ž Remark each function ␨A, r Ž s . for an integer r ) 0 is a rational function of qys by Theorem 1.2.. ŽThe same question about ␨AU Ž s . arises.. Concerning zeros, poles, and values of zeta functions: QUESTION 5.6. Is it possible to formulate Tate conjectures and Beilinson᎐Bloch conjectures which relate the orders of zeros or poles of ␨AŽ s . at integer points to K-groups of A? QUESTION 5.7. What are the arithmetic meanings of ¨ alues of ␨AŽ s . at integer points? Are they related to structures of K-groups of A as in Lichten-

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baum conjectures? Is there a non-commutati¨ e ¨ ersion of Iwasawa theory, which generalizes the relationship between p-adic L-functions and ideal class groups to non-commutati¨ e rings? Are there Euler systems for non-commutati¨ e rings? QUESTION 5.8. Is there a ‘‘Riemann’s hypothesis’’ for non-commutati¨ e rings? If it exists, is there and what is the non-commutati¨ e ¨ ersion of the prime number theorem? In the commutati¨ e case, Deligne pro¨ ed Riemann’s hypothesis for rings o¨ er Fp , de¨ eloping the theory of ‘‘weights.’’ What can we do for non-commutati¨ e Fp-algebras? Concerning the relation with transcendental theory: QUESTION 5.9. How can the ‘‘gamma factor’’ ⌫AŽ s . of a finitely generated ring A o¨ er Z be defined? For example, if A s Z X, Y 4 r² XY y YX y 1: whose zeta function is ␨ Ž s y 2. Ž Example 3.1., it is natural to expect ⌫A Ž s . s ⌫

ž

sy2 2

/

⭈ ␲yŽ sy2.r2 .

Does this function arise from analytic properties of the ring R X, Y 4 r² XY y YX y 1:? Does it arise from some ‘‘Hodge theory’’ of R X, Y 4 r² XY y YX y 1: just as the gamma factors of commutati¨ e rings? In the case A is the uni¨ ersal en¨ eloping algebra of sl 2 ŽZ., can we find a gamma factor ⌫AŽ s . for which the product ⌫AŽ s . ␨AŽ s . has an analytic continuation to the whole s-plane and satisfies a functional equation Ž so this ⌫AŽ s . should also ha¨ e the natural boundary ReŽ s . s 2 and cannot be similar to the usual gamma function.? For the ring of integers in a number field, the adelic method gi¨ es uniform definitions of the gamma factors and the Euler factors at maximal ideals, and gi¨ es the proofs of the analytic continuation and the functional equation. N. Kurokawa imagines that this should be extended to all finitely generated rings o¨ er Z Ž of course e¨ en the commutati¨ e case is not yet done. and he imagines that Wiener measure theory on infinite dimensional Adelic spaces Ž cf. wSax for Wiener measures on p-adic Banach spaces. should be used to ha¨ e uniform definitions of ⌫AŽ s . and ␨AŽ s . Ž or ␨AU Ž s .. as integrals. QUESTION 5.10. If G is a finitely generated group and if ᒄ is a finitely generated Lie algebra o¨ er Z, what is the relationship between the representation theory of G or ᒄ o¨ er C and the shapes of the zeta function of the group ring Zw G x or the en¨ eloping algebra of ᒄ o¨ er Z, respecti¨ ely? Concerning the lack of non-commutative arithmetic geometry: QUESTION 5.11. For each integer r G 1, the function ␨A, r Ž s . comes from the commutati¨ e arithmetic geometry of the scheme ᑭ A, r and there is not

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much relation between the schemes ᑭ A, r , but the function ␨AŽ s . which is the product of the functions ␨A, r Ž s . for all r G 1 has a simple shape in many examples we computed Ž Section 3.. Furthermore in Examples 3.1, 3.4, 3.6, ␨AŽ s . is ¨ ery rigid when we change the parameters defining A through each ␨A, r Ž s . depends hea¨ ily on parameters. Is there non-commutati¨ e arithmetic geometric theory on SpecŽ A. which explains these phenomena? QUESTION 5.12. Is it possible to extend the K-theoretic class field theory of Parshin, Kato᎐Saito, and Bloch to non-commutati¨ e rings, and relate it to ‘‘abelian L-functions’’ for non-commutati¨ e rings? QUESTION 5.13. The zeta functions of commutati¨ e rings are related, ¨ ia Weil conjectures pro¨ ed by Deligne, to ¨ arious cohomology theories: etale cohomology, Betti cohomology, de Rham cohomology, and crystalline cohomology. Are there non-commutati¨ e analogues of these relations? Is the periodic cycle homology related to zeta functions of non-commutati¨ e rings?

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