Functional Equations for Zeta Functions of Groups and Rings

ANNALS OF MATHEMATICS

Functional equations for zeta functions of groups and rings

By Christopher Voll

SECOND SERIES, VOL. 172, NO. 2 September, 2010

anmaah Annals of Mathematics, 172 (2010), 1181–1218

Functional equations for zeta functions of groups and rings

By CHRISTOPHER VOLL

Abstract

We introduce a new method to compute explicit formulae for various zeta functions associated to groups and rings. The speciﬁc form of these formulae enables us to deduce local functional equations. More precisely, we prove local functional equations for the subring zeta functions associated to rings, the subgroup, conjugacy and representation zeta functions of ﬁnitely generated, torsion-free nilpotent (or ᐀-)groups, and the normal zeta functions of ᐀-groups of class 2. We deduce our theorems from a “blueprint result” on certain p-adic integrals which generalises work of Denef and others on Igusa’s local zeta function. The Malcev correspondence and a Kirillov-type theory developed by Howe are used to “linearise” the problems of counting subgroups and representations in ᐀-groups, respectively.

1. Introduction

Zeta functions of groups were introduced by Grunewald, Segal and Smith in the 1980s as a tool to study the subgroup growth of finitely generated groups. In [14] the zeta function of a finitely generated group G was defined as the Dirichlet series X s (1) G.s/ G H ; D j W j H G Ä where s is a complex variable and the sum ranges over the finite index subgroups of G. Grunewald, Segal and Smith derived results on the zeta functions of finitely generated, torsion-free nilpotent (or ᐀-)groups G, and went on to consider variants of (1). These include a ᐀-group’s normal zeta function counting only normal subgroups of finite index, and the conjugacy zeta function counting subgroups up to conjugacy. They also developed an analogous theory for rings. (In the current paper, by a ring we mean a finitely generated abelian group with a bi-additive product.) The ideal zeta function of a ring, for instance, generalises the classical

1181 1182 CHRISTOPHERVOLL

Dedekind zeta function of a number field. Much of the subsequent developments in the theory of zeta functions of groups and rings are documented in the monograph [25] and in the report [9]. Only comparatively recently, Hrushovski and Martin ([17]) started to investi- gate representation zeta functions of ᐀-groups, enumerating (twist-isoclasses of) finite-dimensional complex representations. All the zeta functions mentioned so far have the property that they satisfy an Euler product decomposition into local factors, indexed by the primes. For example, for a ᐀-group G we have Y G.s/ G;p.s/; D p prime P s where G;p.s/ G H enumerates finite p-power index subgroups. D H pG j W j All these local zeta functionsÄ are known to be rational functions in the parame- s ter p with integer coefficients ([14], [17]). In many cases they exhibit a remark- able symmetry: For instance, it had been observed (cf., e.g., [9]) that the local factors of all zeta functions of ᐀-groups G for which explicit formulae are known satisfy a local functional equation of the form a b cs (2) G;p.s/ p p 1 . 1/ p G;p.s/; j ! D for almost all primes p and suitable integers a; b; c depending only on the Hirsch length of G. Here p p 1 denotes a formal inversion of the local parameter p, ! which we shall now explain. The prime example is the case of G ޚn. It is known from [14, Prop. 1.1] D that n 1 Y G.s/ .s i/; D i 0 Q 1 D where .s/ p prime 1 p s is the Riemann zeta function. The functional equation D n 1 n 1 Y 1 n .n/ ns Y 1 . 1/ p 2 1 p .i s/ D 1 pi s i 0 i 0 D D of the local factor at the prime p is easily seen to hold for all primes. Here, as well as in all other cases in which explicit formulae are known to date, the local zeta s functions G;p.s/ are in fact rational functions in p and p, and the left-hand side of (2) denotes the rational function obtained by formally inverting both of these two parameters. This “uniformity” in the prime p, however, is not typical: Results of du Sautoy and Grunewald ([8]) show that the dependence of the local (normal) zeta functions of ᐀-groups on the primes will, in general, reflect the variation of the number of ކp-points of certain algebraic varieties defined over ކp, which may be far from polynomial in the prime p (see also [6] and [7]). In [32] we produced examples of normal zeta functions of ᐀-groups of nilpotency class 2 (or ᐀2-groups) which exhibit functional equations similar to (2) and which are not FUNCTIONALEQUATIONSFORZETAFUNCTIONSOFGROUPSANDRINGS 1183 uniform. Indeed, the local factors of these zeta functions are rational functions in s p , whose coefficients involve the numbers bV .p/ of ކp-rational points of certain smooth projective varieties V over ކp which are, in general, not polynomials in p. By the Weil conjectures, these numbers may be expressed as alternating sums of Frobenius eigenvalues. The operation p p 1 is performed by inverting these ! eigenvalues. In the special (“uniform”) case that the bV .p/ are in fact polynomials in p, this specialises to an inversion of the prime p. Only a single example of a representation zeta function of a ᐀-group seems to have appeared in print so far: In [17] Hrushovski and Martin derive a formula for the representation zeta function of the discrete Heisenberg group in terms of the Riemann zeta function and its inverse (cf. Example 1.2). According to du Sautoy and Segal, to find an explanation for the phenomenon of local functional equations for zeta functions of groups and rings is “one of the most intriguing open problems in this area” ([11, p. 274]). In the current paper we prove that local functional equations hold for (almost all factors of) the following zeta functions: (A) zeta functions of rings (and, as a corollary, of ᐀-groups), (B) conjugacy zeta functions of ᐀-groups,

(C) normal zeta functions of ᐀2-groups and (D) representation zeta functions of ᐀-groups. By proving (A) and (C) we solve Problems 5.1 and 5.2 posed in [9]. In its given generality, (C) is best possible: It is known that the normal zeta functions of nilpotent groups of class 3 may or may not satisfy local functional equations (cf. [13]). To determine the exact scope of this intriguing symmetry for ideal zeta functions of rings remains a challenging open problem. We achieve our results by showing that all of the above-mentioned zeta functions may be expressed in terms of certain p-adic integrals, generalising Igusa’s local zeta function. Given a nonconstant polynomial f .y/ ޚŒy1; : : : ; ym, its 2 associated Igusa local zeta function is the p-adic integral Z f .y/ s dy ; m ޚp j j j j where s is a complex variable, stands for the p-adic absolute value and dy j j m j j denotes the (additive) Haar measure on ޚp , the afﬁne m-space over the p-adic integers ޚp. This p-adic integral is closely related to the Poincare´ series counting p-adic points on the hypersurface deﬁned by f (cf. [4]). We prove functional equations for these integrals by generalising results by Denef and others on Igusa’s local zeta function. The integrals considered in the present paper are quite different from the “cone integrals” introduced by du Sautoy and Grunewald in [8] (see Section 1.2 for further details). 1184 CHRISTOPHERVOLL

1.1. Detailed statement of results. Let L be a ring. Its (subring) zeta function is defined to be the Dirichlet series X s L.s/ L H ; D j W j H L Ä where the sum ranges over all subrings H of finite index in L, and s is a complex variable. This zeta function decomposes naturally as an Euler product, indexed by the primes: Y L.s/ L;p.s/; D p prime where L;p.s/ L ޚp .s/. Grunewald, Segal and Smith proved in [14] that each D ˝ s local factor is a rational function in p with integral coefficients. Our first main theorem is

THEOREM A. Let L be a ring of torsion-free rank n. Then there are smooth projective varieties Vt , t 1; : : : ; m , defined over ޑ, and rational functions 2 f g Wt .X; Y / ޑ.X; Y / such that for almost all primes p the following hold. 2 (a) Denoting by bt .p/ the number of ކp-rational points of Vt , the reduction mod p of Vt , we have m X s (3) L;p.s/ bt .p/Wt .p; p /: D t 1 D 1 dim.Vt / (b) Setting bt .p / p bt .p/ the following functional equation holds: WD n n .2/ ns (4) L;p.s/ p p 1 . 1/ p L;p.s/: j ! D The novelty of this result is that it allows us to deduce the equations (4). In [8], du Sautoy and Grunewald gave formulae akin to (3) for, inter alia, the local factors of L.s/. Their proof depends on a representation of local zeta functions through certain p-adic integrals called “cone integrals” which in general will not satisfy functional equations like (4). See [13, Ch. 4] for a discussion of functional equations for cone integrals. Given a ring L we cannot, in general, pin down the primes p which have to be excluded in Theorem A. On the other hand, any prime p will be amongst the primes for which Theorem A, applied to the ring pL, makes no assertion. We also note that the definition in part (b) of Theorem A is consistent with our explanation above of the operation p p 1. Indeed, by the Weil conjectures, ! the numbers bt .p/ may be expressed as alternating sums of Frobenius eigenvalues. These complex numbers satisfy certain symmetries which are reflected by the functional equations for the Weil zeta functions of the varieties Vt . Had the expressions 1 bt .p / been defined as the numbers obtained from inverting these eigenvalues, 1 dim.Vt / the identities bt .p / p bt .p/ would follow from these symmetries (see D the remarks preceding Theorem 2.3 for details). FUNCTIONALEQUATIONSFORZETAFUNCTIONSOFGROUPSANDRINGS 1185

Example 1.1. Theorem A applies in particular to the zeta functions of “simple” Lie algebras over ޚ such as sl.d; ޚ/. The only such Lie algebra for which a functional equation as in (4) had been previously established is L sl.2; ޚ/ D (cf. [12]1): 1 2s 3 3s Y 1 3s L.s/ .s/.s 1/.2s 1/.2s 2/.1 3 2 2 / .1 p /: D C p 2 6D In fact, sl.2; ޚ/ seems to be the only nonsoluble Lie ring whose subring zeta function has been computed explicitly. Note that the functional equation fails for p 2. A similar phenomenon may occur if one studies the zeta function of a D ޚp-algebra: In [21] Klopsch computes the zeta function of a maximal ޚp-order in a central simple ޑp-division algebra of index 2. The fact that this zeta function does not satisfy a functional equation of the form (4) reflects the fact that it is not the “generic” local factor of the zeta function of a ring. Klopsch and the present author have unified and generalised the above examples. In [22] we gave a formula for the zeta function of an arbitrary 3-dimensional ޚp-Lie algebra, based on the proof of Theorem A. An important corollary of Theorem A is to the theory of zeta functions of finitely generated, torsion-free nilpotent (or ᐀-) groups. In [14, Th. 4.1] it was shown that, given a ᐀-group G of Hirsch length n, there is a Lie ring L L.G/, D lying as a full ޚ-lattice in an n-dimensional Lie algebra ᏸ.G/ over ޑ such that for almost all primes p

(5) G;p.s/ L;p.s/: D Thus we obtain

COROLLARY 1.1. Let G be a ᐀-group of Hirsch length n. For all but finitely many primes p n n .2/ ns G;p.s/ p p 1 . 1/ p G;p.s/: j ! D Theorem A is itself an instance of an application of a “reciprocity” result (Corollary 2.3 to Theorem 2.3) establishing certain functional equations for a family of p-adic integrals. Theorem 2.3 may be viewed as a generalisation of Stan- ley’s “reciprocity theorem for linear homogeneous diophantine equations” ([29, Th. 4.6.14]), which we now briefly explain. Given a set of simultaneous linear homogeneous diophantine equations in n indeterminates, say, one may encode their nonnegative (positive) solutions in a rational generating function E.x/ (E.x/, respectively), where x .x1; : : : ; xn/ is a vector of formal variables. More precisely, D one defines X X E.x/ x˛ and E.x/ x˛; WD WD ˛ ގn Ꮿ ˛ ގn Ꮿ 2 0 \ 2 \

1The denominator of P.2 s/ in [9, Eq. (7)] should read 1 21 3s. 1186 CHRISTOPHERVOLL

˛ ˛1 ˛n where ˛ .˛1; : : : ; ˛n/, x x xn and Ꮿ is the cone of all nonnegative real D D 1 solutions to the given set of equations. Stanley’s reciprocity theorem states that, if E.x/ 0, then 6D E.1=x/ . 1/dim ᏯE.x/: D Our generalisation is obtained using (a variant of) Stanley’s result and an explicit formula of the form (3) for the integrals in question (Corollary 2.1 of Theorem 2.2). This formula in turn is inspired by and generalises work of Denef ([3]), Denef and Meuser ([5]) and Veys and Zuńiga-Galindo˜ ([30]) on Igusa’s local zeta function. While Igusa’s local zeta functions associated to homogeneous polynomial mappings may be expressed as integrals over projective space, the p-adic integrals considered in the current paper reduce to integrals over the complete flag variety GLn=B, where B is a Borel subgroup. In a sense this explains the factor n p.2/ in (4). A variant of the problem of counting subgroups consists in counting subgroups only up to conjugacy. Let G be a ᐀-group. The conjugacy zeta function of a ᐀-group G is defined as

cc X s 1 .s/ G H ᏯG.H / G D j W j j j H G Ä where ᏯG.H / is the size of the conjugacy class of H. It is known ([14, remark j j cc on p. 189]) that G .s/ also has an Euler product decomposition into local factors cc s G;p.s/ which are all rational in p . By applying the results of Section 2 we shall prove

THEOREM B. Let G be a ᐀-group of Hirsch length n. For all but finitely many primes p n cc n .2/ ns cc G;p.s/ p p 1 . 1/ p G;p.s/: j ! D As a third application of our rather technical “blueprint result” Theorem 2.2 and its applications, we deduce functional equations for normal zeta functions of ᐀2-groups. The normal zeta function of a ᐀-group G is defined as X .s/ G H s G D j W j H G G where the sum ranges over the normal subgroups H of finite index in G. It also satisfies an Euler product decomposition. We prove

THEOREM C. Let G be a ᐀2-group of Hirsch length n with centre Z.G/ such that G=Z.G/ has torsion-free rank d. For all but ﬁnitely many primes p

n n .2/ .d n/s G;pG .s/ p p 1 . 1/ p C G;pG .s/: j ! D As mentioned above, a functional equation may or may not hold for normal zeta functions of class greater than three. See [13, Ch. 2] for examples and [13, Th. 4.44] for a conjectural form in case it does hold. FUNCTIONALEQUATIONSFORZETAFUNCTIONSOFGROUPSANDRINGS 1187

The fourth and last application in this paper of Theorem 2.2 and its conse- quences is concerned with representation zeta functions of ᐀-groups, which we shall now explain. Given a ᐀-group G, we denote by Rn.G/ the set of n-dimensional irreducible (complex) characters of G. Given 1; 2 Rn.G/, we say that 2 1 and 2 are twist–equivalent if there exists a linear character R1.G/ such 2 that 1 2. The classes of this equivalence relation are called twist-isoclasses. D We say that a character of a representation of G factors through a ﬁnite quotient of G if factors through it. The set Rn.G/ has the structure of a quasi-afﬁne complex algebraic variety whose geometry was analysed by Lubotzky and Magid in [24]. They proved

THEOREM ([24, Th. 6.6]). Let G be a ᐀-group. For every n ގ there is a 2 ﬁnite quotient G.n/ of G such that every n-dimensional irreducible character of G is twist-equivalent to one that factors through G.n/. In particular, the number of twist-isoclasses of irreducible n-dimensional characters is ﬁnite.

Let us call this number an. The representation zeta function of G is deﬁned (cf. [17]) by irr X1 s .s/ ann : G WD n 1 D It follows from the above theorem and [2, (10.33)] that the function n an is 7! multiplicative and thus Y irr.s/ irr .s/; G D G;p p prime where irr X1 sn .s/ apn p : G;p WD n 0 D Example 1.2 ([17, Ex. 8.12], [26, Th. 5]). Let

H x1; x2; y Œx1; x2 y; all other Œ ; trivial D h j D i be the discrete Heisenberg group. Then s X1 .s 1/ Y 1 p irr.s/ .n/n s ; H D D .s/ D 1 p1 s n 1 p prime D where denotes the Euler totient function. By a model-theoretic result of Hrushovski and Martin [17, Th. 8.4], the local representation zeta functions of a ᐀-group are known to be rational functions in s irr p with integer coefﬁcients. By expressing G;p.s/ in terms of p-adic integrals to which Theorem 2.2 is applicable we shall prove

THEOREM D. Let G be a ᐀-group with derived group G ŒG; G of Hirsch 0 D length n. Then for almost all primes p irr n irr G;p.s/ p p 1 p G;p.s/: j ! D 1188 CHRISTOPHERVOLL

1.2. Outline of methodology and related work. We briefly describe how we relate the problems solved in Theorems A, B, C and D to problems about p-adic integrals. To count the subrings of p-power index in a ring L of rank n, we observe that it is enough to keep track of the index of the largest ޚp-subalgebra of L ޚp in n ˝ each given homothety class of lattices in the p-adic vector space ޑp. Using the action of the group GLn.ޚp/ on the set of homothety classes, we show that WD the latter problem reduces to counting polynomial congruences in finite quotients of . This counting problem translates into the problem of computing a p-adic integral in very much the same fashion as the problem of counting polynomial congruences in affine space translates to the problem of computing Igusa’s local zeta function (cf. [4, 1.2]). It proved helpful to think of homothety classes of lattices as the vertices of the Bruhat-Tits building of SLn.ޑp/, and to partition the vertex set into finitely many parts according to their position relative to the “root class” ŒL ޚp. As we remarked above, this approach differs decisively from the ˝ “cone integrals” introduced by du Sautoy and Grunewald in [8]. Their analysis rests on a basis-dependent parametrisation of p-power index subrings of a given ring in terms of upper-triangular matrices over the p-adic integers satisfying certain divisibility conditions (“cone conditions”). To count subgroups up to conjugacy in a ᐀-group G we use the fact that, for almost all primes p, cc .s/ cc .s/; G;p D L;p where L L.G/ is the Lie ring associated to G (cf. the remark preceding Corollary D 1.1), and

cc X s 1 L;p.s/ L ޚp H L ޚp ᏺL ޚp .H / ; D j ˝ W j j ˝ W ˝ j H L ޚp Ä ˝ where H ranges over the subalgebras of L ޚp of finite index and ᏺL ޚp .H / ˝ ˝ is the normaliser of H in L ޚp. We thus have to keep track both of the largest ˝ subring of L ޚp in each given homothety class and of the class’ normaliser. ˝ The index of the latter is given by the index of a system of linear congruences. Enumerating these indices, in turn, may be achieved by counting the elementary divisors of matrices of linear forms, encoding the group’s commutator structure. In Proposition 2.2 we show that, slightly more generally, the generating functions enumerating elementary divisors of matrices of polynomial forms of the same degree may be expressed in terms of p-adic integrals associated to degeneracy loci of these matrices, to which Corollary 2.4 is applicable. In order to count normal subgroups in class-2-nilpotent groups we develop an idea first introduced in [31]. There it was shown that it suffices to evaluate a weight function on the set of homothety classes of lattices in the centre Z.L ޚp/ of the ˝ ޚp-Lie algebra L ޚp, where L is the associated Lie ring. The weight associated ˝ FUNCTIONALEQUATIONSFORZETAFUNCTIONSOFGROUPSANDRINGS 1189 to a vertex in the appropriate affine Bruhat-Tits building corresponding to a given class is again given by the index of a system of linear congruences. As mentioned above, the validity of our main Theorems A and C had been known in many special cases. We refer the reader to the numerous examples col- lected in [13]. In this research monograph du Sautoy and Woodward also present a conjecture on functional equations for cone integrals that would explain functional equations for normal zeta functions of nilpotent of class greater than 2. The key to Theorem D is to use “Kirillov-theory” developed by Howe [16] to translate the problem of counting irreducible representations of G to the problem of counting co-adjoint orbits in the dual of the Lie algebra associated to G by the Malcev correspondence. We use the fact – also established by Howe – that the sizes of co-adjoint orbits may be expressed in terms of the indices of the radicals of certain anti-symmetric forms on the Lie algebra. These may also be described in terms of elementary divisors of matrices encoding the structure of the Lie algebra. Representation zeta functions of nilpotent groups have not been studied until fairly recently, and [17] seems to be the only reference so far on this topic. The idea of using Kirillov-theory to study representation zeta functions of groups, however, has been successfully employed before. Jaikin-Zapirain proved in [19] the rationality of representation zeta functions for certain compact p-adic analytic groups using a Kirillov-type correspondence developed by Howe ([15]) for these groups. In [17] Hrushovski and Martin suggest that Jaikin-Zapirain’s work may be adapted to prove rationality of local representation zeta functions for ᐀-groups, too. Among the variants of the zeta function (1) of a ᐀-group G considered in [14] is also the zeta function G^.s/, enumerating subgroups of finite index whose profinite completion is isomorphic to the profinite completion of G. In [10] du Sautoy and Lubotzky proved a functional equations for the local factors of G^.s/ for a class of ᐀-groups. Their work is based on a reduction of the problem of computing these zeta functions to the problem of computing certain p-adic integrals over the group’s algebraic automorphism group, generalising work of Igusa ([18]). The functional equation for the local factors of these zeta functions arises from a symmetry in the root systems of the associated Weyl groups. An argument of this kind (albeit only for the Weyl groups of type A) is also used in the present paper to deduce Corollary

2.3. We do not know whether the zeta functions G^.s/ (for reasonably large classes of ᐀-groups) may be described by the p-adic integrals studied in the current paper. In [1] Berman extends the approach taken in [10], proving uniformity and local functional equations for these zeta functions for a wider class of nilpotent groups than previously considered. The fact that we have to disregard ﬁnitely many primes in most of our results has two reasons: Firstly, our Theorems 2.1 and 2.2 upon which Theorem 2.3 and its corollaries are based are valid only for primes for which a certain principalisation of ideals has good reduction. Secondly, we are forced to ignore ﬁnitely many primes in order to transfer between the ᐀-group G and its associated Lie algebra. 1190 CHRISTOPHERVOLL

1.3. Layout of the paper and notation. In Section 2 we first develop, in Theo- rems 2.1 and 2.2, explicit formulae for certain families of p-adic integrals generalising Igusa’s local zeta function. These two results – which may be understood as close analogues of Theorems 2 and 3 in [5] – form the technical core of the paper. We use them to establish, in Theorem 2.3, an “inversion property” enjoyed by the p-adic integrals considered. In Corollaries 2.3 and 2.4 we exploit this property to deduce functional equations for certain linear combinations of the p-adic integrals in question. The remainder of the paper is dedicated to showing how Theorem 2.2 may be used as a template to describe various kinds of zeta functions. In the second part of Section 2 we give a first application of this idea to the problem of counting elementary divisors of matrices of forms (Proposition 2.2). In the four subsections of Section 3 we prove Theorems A, B, C and D, respectively. We use the following notation. ގ the set 1; 2; : : : of natural numbers f g I i1; : : : ; i < the set I of natural numbers i1 < < i D f l g l I0 the set I 0 for I ގ [ f g Â Œk the set 1; : : : ; k , k ގ f g 2 Œl; k the set l; : : : ; k , k; l ގ a f g 2 b the binomial coefficient for a; b ގ0 a Qb 1 a i 2 b i b X the polynomial i 0 .1 X /=.1 X /, D where a; b ގ0 with a b 2 Note: The q-binomial coefficient or Gaussian a polynomial b q gives the number of a subspaces of dimension b in ކq. n the polynomial n il ::: i2 , I X il X il 1 X i1 X for n ގ, I i1; : : : ; il < Œn 1 n2 D f g Â n Note: I q gives the number of flags of type I in ކq. Sn the symmetric group on n letters M t the transpose of a matrix M ޚp the ring of p-adic integers (p a prime) ޑp the field of p-adic numbers n Œƒ the homothety class ޑp ƒ of a (full) lattice ƒ in ޑp K a finite extension of the field ޑp R the valuation ring of K P the maximal ideal of R K the residue field R=P , of cardinality q x F a number field ıP the “Kronecker delta” which is equal to 1 if the property P holds and equal to 0 otherwise. Given a set f of polynomials and a polynomial g, we write gf for gf f f , and f j 2 g .f/ for the polynomial ideal generated by f. FUNCTIONALEQUATIONSFORZETAFUNCTIONSOFGROUPSANDRINGS 1191

2. Functional equations for some p-adic integrals

2.1. A blueprint result. In this section we study a family of p-adic integrals generalising Igusa’s local zeta functions. We first introduce more notation. Let p be a prime and K be a finite extension of the field ޑp of p-adic numbers. Let R RK denote the valuation ring of K, P PK the maximal ideal of R, and K D D x the residue field R=P . The cardinality of K will be denoted by q. x For x K, let v.x/ vP .x/ ޚ denote the P -adic valuation of x, and 2 D 2 [f1g x q v.x/. For a finite set ᏿ of elements of K, we set ᏿ max s s ᏿ . j j WD k k WD fj j j 2 g Fix k; m; n ގ. For each Œk, let .f/ I be a finite family of finite sets 2 2 2 of polynomials in KŒy1; : : : ; ym, and let x1; : : : ; xn 1 be independent variables. Also, for i Œn 1 we fix nonnegative integers ei . For a set I i1; : : : ; i < 2 D f l g Â Œn 1, Œk, we set 2 [ Y ei g;I .x; y/ x f.y/: D i I i I 2 2 m m Let W R be a subset which is a union of cosets mod P and s .s1; : : : ; s / Â D k be independent complex variables. We then define Z Y s (6) ZW;K;I .s/ g;I .x; y/ dxI dy WD P l W k k j jj j Œk 2 l l where dxI dxi1 dxil is the Haar measure on K normalised so that R j j D j ^ ^l j l has measure 1 (and thus P has measure q ), and dy dy1 dym is the m j j D j ^ ^ j (normalised) Haar measure on K . It is well-known that ZW;K;I .s/ is a rational function in q s , Œk, with integral coefficients. 2 We now assume that the polynomials constituting the sets f are in fact defined over a number field F . We may consider the local zeta functions ZW;K;I .s/ for all non-archimedean completions K of F . In the remainder of this section we shall derive formulae for ZW;K;I .s/, valid for almost all completions K of F under this and further assumptions. They are essentially based on the formulae Denef gave for Igusa’s local zeta function Z Z.s/ f .y/ s dy D Rm j j j j in [3, Th. 3.1], using the concept of resolution of singularities for the hypersurface defined by f . In the case where the single polynomial f is replaced by a finite set of polynomials f, Veys and Zuńiga-Galindo˜ ([30, Th. 2.10]) gave an analogous formula, using instead the concept of principalisation of ideals, which we briefly recall.

THEOREM ([34, Th. 1.0.1]). Let Ᏽ be a sheaf of ideals on a smooth algebraic variety X. There exists a principalisation .Y; h/ of Ᏽ, that is, a sequence

h1 h hr X X0 X1 X Xr Y D D 1192 CHRISTOPHERVOLL of blow-ups h X X 1 of smooth centres C 1 X 1 such that W ! a) The exceptional divisor E of the induced morphism h h h1 X X D ı ı W ! has only simple normal crossings and C has simple normal crossings with E.

b) Setting h hr h1, the total transform h .Ᏽ/ is the ideal of a simple nor- D ı ı mal crossing divisor E. If the subscheme determined by has no components z Ᏽ of codimension one, then E is an ގ-linear combination of the irreducible z components of the divisor Er . Also recall the definition [3, Def. 2.2 (mutatis mutandis)] of a principalisation .Y; h/ with good reduction mod P if Ᏽ and .Y; h/ are defined over a p-adic field K. Note that, given a principalisation .Y; h/ for Ᏽ defined over a number field F , .Y; h/ will have good reduction mod PK (where PK is the maximal ideal in the ring of integers of the completion of F at P ) for almost all maximal ideals P of the ring of integers of F (this is essentially [3, Th. 2.4]). Specifically, let .Y; h/, h Y ށm, be a principalisation of the ideal W ! Y Ᏽ .f/ D Œk; I 2 2 where .f/ denotes the ideal generated by the finite set f of polynomials. We set ᐂ Spec.F Œy=Ᏽ/ and ᐂ Spec.F Œy=.f//. Then, denoting by Et , t T , WD WD 1 2 the irreducible components of .h .ᐂ//red, we have 1 X (7) h .ᐂ/ Nt Et ; D t T 2 1 X (8) h .ᐂ/ Nt Et ; D t T 2 say, for suitable nonnegative integers Nt ;Nt . Note that, for every t T , 2 X Nt Nt : D Œk; I 2 2 In a similar manner, we denote by t 1 the multiplicity of Et in the divisor of h.dy1 dym/. The numbers .Nt ; t /t T; Œk; I will be called the ^ ^ 2 2 2 numerical data of the principalisation .Y; h/.

THEOREM 2.1. Suppose that all the sets f are integral (i.e., contained in RŒy) and do not deﬁne the zero ideal mod PK , and that .Y; h/ has good reduction mod PK . Then 1 I .1 q /j j X U ZW;K;I .s/ cU;W .q/.q 1/ „U;I .q; s/; D qm j j U T Â where

cU;W .q/ a Y.K/ a Eu.K/ u U and h.a/ W D jf 2 x x j 2 x , 2 2 Sgj FUNCTIONALEQUATIONSFORZETAFUNCTIONSOFGROUPSANDRINGS 1193

(where denotes reduction mod P and W .y1;:::; ym/ .y1; : : : ; ym/ W ) S D f j 2 g and (9) P P P P P X i ni u umu s min I i ei ni u Nu mu „U;I .q; s/ q 2 : D f C g U .mu/u U ގj j 2 2 l .ni /i I ގ 2 2 Example 2.1. If l 0 and, for each Œk, I 1, Theorem 2.1 reduces to (a D 2 j j D multivariable version of) Veys’ and Zuńiga-Galindo’s˜ generalisation [30, Th. 2.10] to polynomial mappings of Denef’s explicit formula [3, Th. 3.1] for Igusa’s local zeta function associated to a single polynomial. Notice in particular that in this case P P X X u mu. u Nu s / Y u „U;¿.q; s/ q D D 1 Xu U u U .mu/u U ގj j 2 2 2 P u Nu s for Xu q , where we write Nu for Nu . Also compare Example WD 2.2 for the other “extremal case” „¿;I .q; s/. Proof. The proof is analogous to the one of [3, Th. 3.1] (=[5, Th. 2]), with the concept of resolution of singularities replaced by the concept of principalisation of ideals. We adopt – mutatis mutandis – Denef’s notation and just explain how the proof differs from his. Let a be a closed point of Y , and thus also of Y . Let x mz Ta t T a Et t1; : : : ; tr <, say. Define H b Y.K/ h.b/ R and D f 2 j 2 g D f g D f 2 j 2 g recall the definition of the “reduction mod P ”-map H Y.K/. In the regular W ! x x 1 local ring ᏻY ;a, there exist irreducible elements 1; : : : ; m such that, on .a/, z for all Œk, I, 2 2 Nt Nt f h 1 1 ::: r r and k ı k Dj j j j t 1 t 1 h .dy1 dym/ 1 1 ::: r r d 1 d m : j ^ ^ j Dj j j j j ^ ^ j Setting d d 1 d m we define WD ^ ^ Za;I .s/ 8 9s Z < = Y Y ei Y Nt Y t 1 max xi dxI d WD P l 1.a/ I : j j j j ; j j j jj j Œk 2 i I Œr Œr 2 2 2 2 8 9s Z < = Y Y ei Y Nt Y t 1 max xi y y dxI dy D P l m I j j j j j j j jj j C Œk 2 :i I Œr ; Œr 2 2 2 2 r l .q 1/ X P P P P P C q i ni t t mt s min I i ei ni t Nt mt : m l 2 f C g D q C r .mt /t Ta ގ 2 2 l .ni /i I ގ 2 2 P This suffices as ZW;K;I .s/ a Y.K/ Za;I .s/. D x x h.a/2 W 2 S 1194 CHRISTOPHERVOLL

2 We now make the further assumption that m n2. We identify Kn with D Matn.K/ and assume that the ideals .f/, Œk, I, are B.F /-invariant, where 2 2 B.F / is the group of F -rational points of the Borel subgroup of upper-triangular matrices in G GLn, acting on KŒy11; y12; : : : ; ynn by matrix-multiplication D from the right. Let .Y; h/, h Y G=B be a principalisation of the ideal Ᏽ Q W ! D ;.f/. Denoting, as above, by ᐂ the subvariety of G=B.K/ deﬁned by Ᏽ and by

ᐂ the subvariety deﬁned by .f/ yields numerical data .Nt ; t /t T; Œk; I deﬁned by formulae analogous to (7) and (8) above. We study the integral2 2 2

ZI .s/ ZW;K;I .s/ WD for W GLn.R/ for almost all completions K of F . Note that the Haar D D measure 0 on the compact topological group coincides with the additive Haar 2 2 measure induced from Rn (and normalised such that .Rn / 1), as D 0 D det n . This will be important in later applications as it implies, for exam- j j D ple, that all the cosets of a finite index subgroup have measure ./= , 0 Ä j W 0j with ./ .1 q 1/ .1 q n/. D THEOREM 2.2. Suppose that, in addition to the above assumptions, none of the ideals .f/ is equal to the zero ideal mod PK , and that .Y; h/ has good reduction mod PK . Then 1 I n .1 q /j jC X U ZI .s/ n cU .q/.q 1/j j„U;I .q; s/; D .2/ q U T Â where each cU .q/ is the number of K-rational points of EU V ©U EV (EU x n [ WD u U Eu) and „U;I .q; s/ is defined as in (9) above. \ 2 Proof. The proof follows closely the spirit of the proof of [5, Th. 3]. In fact, our function ZI .s/ is a close analogue of the function ZbK .s/, defined in [5, p. 1140]. We write as a disjoint union of sets

x x B.ކq/B.ކq/ ; D f 2 j 2 g ކ S ކ ކ Sn, where GLn. q/ Sn B. q/B. q/ is the Bruhat decomposition (here 2 D 2 Sn is identified with the respective permutation matrix in GLn.ކq/). Thus 2 X ZI .s/ Z ;K;I .s/: D Sn 2 There is an obvious map G=B.K/, and, by our invariance assumption on W ! l the ideals .f/, the value of the integrand of ZI .s/ at a point .x; y/ P only 2 depends on x and .y/. By taking the measure ! on G=B.K/ which induces the n n Haar measure on the unit ball R.2/ of each affine chart satisfying !.a P .2// n 1 n C D q .2/ and noting that .B/ .1 q / , we obtain D Z 1 n Y s Z ;K;I .s/ .1 q / g;I .x; y/ dxI d!; l D P V k k j j Œk 2 FUNCTIONALEQUATIONSFORZETAFUNCTIONSOFGROUPSANDRINGS 1195 where V . /. The projective variety G=B may be covered by varieties U , D n isomorphic to affine 2 -space, indexed by the elements of the symmetric group Sn, .n/ such that each V is contained in U and is a union of cosets mod P 2 . Theorem 2.1 may thus be applied to the restriction .Y ; h / of .Y; h/, a principalisation of 1 the ideal defining the restriction of ᐂ to U (Y h .U /, h h Y ), with D D j good reduction mod P . We obtain Z Y s g;I .x; y/ dxI d! l P V k k j j Œk 2 1 l .1 q / X U n cU; .q/.q 1/j j„U;I .q; s/; D .2/ q U T Â where cU; .q/ a Y .K/ a Eu.K/ u U and h.a/ V : D jf 2 x x j 2 x , 2 2 gj The result follows since, if a Y.K/, then h.a/ is in exactly one V . Thus x x P 2 Sn cU; .q/ a Y.K/ a Eu.K/ u U cU .q/. 2 D jf 2 x x j 2 x , 2 gj D We now consider the normalised integrals

ZI .s/ (10) ZfI .s/ : WD .1 q 1/ I ./ j j COROLLARY 2.1. For U T , let bU .q/ denote the number of K-rational Â x points of EU . Then 1 X X U V V (11) ZfI .s/ G=B.ކq/ bU .q/ . 1/j n j.q 1/j j„V;I .q; s/: D j j U T V U Â Â Proof. This follows immediately from the formula given for ZI .s/ in Theorem n 2.2, Deﬁnition (10), the fact that G=B.ކq/ and from the identity Œn 1 q j j D X U V cV .q/ . 1/ bU .q/: D j n j V U T Â Â Before we proceed we consider a very special case.

Example 2.2. Assume that, for all Œk, I 1 and that all f 1 . We 2 j j D D f g write ei for ei . Now,

X P n . 1 P e s / Y Xi „¿ .q; s/ q i i i ; ;I 1 X D l D i .ni /i I ގ i I 2 2 2 P 1 e s with Xi q i . Also note that b¿.q/ G=B.ކq/ and that bU .q/ 0 if D D j j D U ¿. Thus Z .s/ Q Xi . It is trivial to verify the “inversion property” fI i I 1 Xi 6D D 2 I X (12) ZfI .s/ q q 1 . 1/j j ZfJ .s/; j ! D J I Â 1196 CHRISTOPHERVOLL as 1 Y Xi I X Y Xj 1 . 1/j j : 1 X D 1 Xj i I i J I j J 2 Â 2 In the remainder of the current section we shall show that equation (12) holds under the premises of Theorem 2.2. To give meaning to the left-most term in (12) 1 in general, we have to explain what we mean by bU .q / (the other constituents s1 sk of the expression (11) for ZfI .s/ being rational functions in q and q ; : : : ; q ). Recall that by properties of the Weil zeta functions associated to the n U - 2 j j dimensional smooth projective varieties EU it is known that

n 2.. / U / tU;r 2 j j X r X bU .q/ . 1/ ˛U;r;j D r 0 j 1 D D for suitable nonnegative integers tU;r and nonzero complex numbers ˛U;r;j , with the property that, for each U , r, the multisets

( n U ˇ ) n ˇ o q.2/ ˇ ˇ j j ˇ ˛U;2..n/ U / r;j ˇ j ŒtU;2..n/ U / r and ˇ j ŒtU;r 2 j j 2 2 j j ˛U;r;j ˇ 2 coincide (cf., e.g., [5, Proof of Th. 4]). This motivates the deﬁnition

n 2.. / U / tU;r 2 j j 1 ..n/ U / X r X 1 (13) bU .q / q 2 bU .q/ . 1/ ˛ : WD j j D U;r;j r 0 j 1 D D We shall prove

THEOREM 2.3. Under the assumptions of Theorem 2.2, the following “inversion properties” hold:

I X (IP) I Œn 1 ZfI .s/ q q 1 . 1/j j ZfJ .s/: 8 Â W j ! D J I Â Proof. To see what happens to the (rational) functions „V;I .q; s/ in expression (11) if we formally invert the prime power q, we employ a result of Stanley:

PROPOSITION 2.1. Let L .n/, Œs, Œt, be ޚ-linear forms in the 2 2 variables n1; : : : ; nr and X1;:::;Xr ;Y1;:::;Ys independent variables, and set

X Y n Y min Œt L .n/ Z .X; Y/ X Y 2 f g; ı WD n ގr Œr Œs 2 2 2 X Y n Y min Œt L .n/ Z.X; Y/ X Y 2 f g: WD n ގr Œr Œs 2 0 2 2 Then Z .X 1; Y 1/ . 1/r Z.X; Y/: ı D FUNCTIONALEQUATIONSFORZETAFUNCTIONSOFGROUPSANDRINGS 1197

Proof. The proof of [29, Th. 4.6.14] carries through to this slightly more general situation, provided one chooses a triangulation of ގr that reﬁnes a subdivi- sion into rational polyhedral cones eliminating the “min”-terms in the sum deﬁning Z.X; Y/.

COROLLARY 2.2. For all I Œn 1, V T , Â Â V I X (14) „V;I .q; s/ q q 1 . 1/j jCj j „W;J .q; s/: j ! D W V;J I Â Â We record the following simple fact:

LEMMA 2.1. For all U T;J Œn 1, Â Â X U V 1 V X (15) . 1/ .1 q / „W;J .q; s/ j n j j j V U W V Â Â U X U V V q . 1/ .q 1/ „V;J .q; s/: D j j j n j j j V U Â The proof is a simple computation. We can now deduce

n I .2/ . 1/j jq X 1 X U V 1 V ZfI .s/ q q 1 bU .q / . 1/j n j.1 q /j j j ! D G=B.ކq/ j j U T V U XÂ Â „W;J .q; s/ .11/; .14/ W V;J I Â Â I X 1 X U . 1/ G=B.ކq/ q bU .q/ D j j j j j j J I U T Â Â X U V 1 V X . 1/ .1 q / „W;J .q; s/ .13/ j n j j j V U W V Â Â I X 1 X . 1/ G=B.ކq/ bU .q/ D j j j j J I U T Â Â X U V V . 1/ .q 1/ „V;J .q; s/ .15/ j n j j j V U Â I X . 1/j j ZfJ .s/: .11/ D J I Â This completes the proof of Theorem 2.3. n Recall that the polynomials I X were introduced at the end of the introduction. We deﬁne ! X n (16) Z.s/ ZfI .s/: z D I I Œn 1 q 1 Â To prove Theorems A, B and C we shall need 1198 CHRISTOPHERVOLL

COROLLARY 2.3. Under the assumptions of Theorem 2.2, the following functional equation holds: n n 1 .2/ (17) Z.s/ q q 1 . 1/ q Z.s/: z j ! D z Proof. This follows from the proof of [33, Cor. 2]. Note that Theorem 2.2 provides the required analogue of [33, Lemma 6]. Theorem D will follow from Proposition 2.2 of the next section which is in turn a special case of the following straightforward corollary. COROLLARY 2.4. Under the assumptions of Theorem 2.2, for any i Œn 1, 2 n n n (18) Z¿.s/ .1 q /Z i .s/ q q 1 q Z¿.s/ .1 q /Z i .s/ : C f g j ! D C f g 2.2. A ﬁrst application: Counting elementary divisors. We show how the problem of counting elementary divisors of matrices of forms may be reduced to the problem of computing p-adic integrals of the form studied in the previous section, associatede to the polynomialse describing the degeneracye loci of thesee matrices. The main result of this subsection — Proposition 2.2 — will be needed to prove Theorem D in Section 3.4. Again, let K be a p-adic ﬁeld with valuation ring R, whose maximal ideal is denoted by P generated by a uniformizer say. Let ᏾ be an e f -matrix (with e f , say) of polynomials ᏾ij .Y/ RŒY1;:::;Yn. We make the assumption on 2 n ᏾ that, whenever y .y1; : : : ; yn/ R is a vector with y 0, at least one entry D 2 6D of ᏾.y/ is nonzero. For a nonnegative integer N and a vector yP N .R=P N /n 2 we say that ᏾.yP N / has elementary divisor type m (written .᏾.yP N // m) D if m .m1; : : : ; mf /, mi Œ0; N , m1 mf , and there are matrices ˇ D N 2 N Ä Ä 2 GLe.R=P /, GL .R=P / such that 2 f 0 m1 1 N : N ˇ᏾.yP / B :: C mod P : Á @ A mf

For m ގf we set 2 0 ˇ ˇ n N N n N N o ᏺN; ;m ˇ yP .R=P / yP 0; .᏾.yP // m ˇ : ᏾ WD ˇ 2 j 6D D ˇ Note that ᏺN; ;m 0 unless 0 m1 m N (the necessity of m1 0 being ᏾ D D Ä Ä f Ä D a consequence of our assumption on ᏾). Given, in addition, a g h-submatrix ᏿ of ᏾ (WLOG g h) deﬁned by choosing g rows and h columns of ᏾, and an h-tuple n we deﬁne ˇ N ˇ ˇ N N n .᏾.yP // m; ˇ ᏺN;᏾;᏿;m;n ˇ yP .R=P / y 0; D ˇ : WD ˇ 2 j 6D .᏿.yP N // n ˇ D Again, ᏺN; ; ;m;n 0 unless 0 m1 m N and n1 n . We ᏾ ᏿ D D Ä Ä f Ä Ä Ä h suppress the subscripts ᏾ and ᏿ if they are clear from the context. FUNCTIONALEQUATIONSFORZETAFUNCTIONSOFGROUPSANDRINGS 1199

Given complex variables r1; : : : ; rf , s1; : : : ; sh, we define the generating function P P X i Œf .N mi /ri j Œh.N nj /sj P.r; s/ P ; ;K .r; s/ ᏺN;m;nq : D ᏾ ᏿ D 2 2 N ގ0 f2 m ގ ;n ގh 2 0 2 0 We now assume that the matrix ᏾ is in fact defined over a number field F , that its entries are all homogeneous of the same degree and that the above assumption on ᏾ is satisfied for almost all completions K of F for which all ᏾ij .Y/ RK ŒY. 2 We consider such a “good” completion K and drop the subscript K. For i Œf 0, 2 let denote the set of i-minors of ᏾. The polynomials define the .rk i 1/- i i Ä locus (or i-th degeneracy locus) of ᏾.Y/. Similarly, let j , j Œh0, denote the 2 set of j -minors of ᏿. Let

k max i Œf 0 . / .0/ and l max j Œh0 .j / .0/ : WD f 2 j i 6D g WD f 2 j 6D g Note firstly that 0 1 , secondly that, by our assumption on ᏾, k 1, 0 D D f g thirdly that 0 l k and fourthly that P.r; s/ is really a function in the variables Ä Ä r1; : : : ; rk; s1; : : : ; sl : P P X Œk.N m /r Œl.N n/s (19) P.r; s/ ᏺN;m;nq D 2 2 N ގ0 m ގk2;n ގl 2 0 2 0 where we set, given m .m1; : : : ; m / and n .n1; : : : ; n /, D k D l ᏺN;m;n ᏺ : WD N;.m1;:::;mk ;N;:::;N /;.n1;:::;nl ;N;:::;N / For I 1 and W GLn.R/ as above, consider the p-adic integral Â f g D D (20) ZI .r;er; s; s; t/ Z z I t Y 1 I 1 r 1 r xj j .y / xj j 1.y / 1.y / e WD P I j j k [ k k k j j Œk 2 Y 1 I 1 s 1 s .y / xj j 1.y / 1.y / e dxI dy ; k [ k k k j jj j Œl 2 where y1 denotes the first column of the matrix y . 2 I Remark 2.1. Whilst artificial, the formulation of ZI as an integral over P j j rather than over P I Rn P n serves to make it fit the “blueprint” Theorem 2.2 j j n provided in the previous section. k l We now set, for m .m1; : : : ; m / ގ , n .n1; : : : ; n / ގ and N ގ, D k 2 0 D l 2 0 2 n 1 N 1 N o N;m;n .x; y/ P v.x/ N; .᏾.y P // m; .᏿.y P // n D 2 j D D D and ZI .r; s; t/ ZI .r; r; s; s; t/. Note that, by definition, N;m;n 0 unless WD D 0 m1 m N and n1 n N . By definition of the polynomials Ä Ä Ä k Ä Ä Ä l Ä 1200 CHRISTOPHERVOLL

, , we have

(21) Z¿.r; s; t/ ./ .and thus, by (10), Z¿.r; s; t/ 1/; D D P P X tN r m sn (22) Z 1 .r; s; t/ N;m;nq : f g D N ގ m ގk2;n ގl 2 0 2 0

Theorem 2.2 is applicable to ZI , I 1 , together with a principalisation Â f g .Y; h/, h Y G=B, of the ideal Ᏽ Q . / Qe . /. Indeed, B.F /- W ! D Œk Œl invariance is a consequence of the fact that the2 entries of ᏾2 were all assumed to be homogeneous of the same degree. The following crucial lemma relates the numbers N;m;n with the data ᏺN;m;n we would like to capture:

LEMMA 2.2. For N ގ, 2 n 1 1 q N.n 1/ (23) ᏺN;m;n N;m;nq : D ./ C n n 1 Recall that 1 q 1 .1 q /=.1 q /. D

Proof. Let 1 ;N denote the group f g 1 N P n 1 where i stands for a matrix in i GLi .R/, for an arbitrary matrix with entries D in R, and P N for a matrix of the appropriate size with entries in P N , respectively. Then the set n o .x; y/ P v.x/ N; .᏾.y1P N // m; .᏿.y1P N // n 2 j D D D may be written as a disjoint union of the 1 ;N sets j W f g j n 1 N 1 N o .x; y/ P 1 ;N v.x/ N; .᏾.y P // m; .᏿.y P // n 2 f g j D D D where runs through a complete set of coset representatives of = 1 ;N . The 1 Nf g measure of each of these sets is either zero or equals .1 q /q . 1 ;N /. f g The latter happens ᏺN;0 m;n times, where ˇn oˇ ᏺ ˇ y1 ސn 1.R=P N / y1 0; .᏾.y1// m; .᏿.y1// n ˇ : N;0 m;n D ˇ 2 j 6D D D ˇ 1 N Clearly ᏺN;m;n .1 q /q ᏺ . Using the identity D N;0 m;n ! n N.n 1/ ./=. 1 ;N / q f g D 1 1 q FUNCTIONALEQUATIONSFORZETAFUNCTIONSOFGROUPSANDRINGS 1201 we obtain ./ ᏺ .1 q 1/q N . / ᏺ N;m;n N;0 m;n 1 ;N N;m;n n N.n 1/ D f g D 1 q C 1 q as claimed. Lemma 2.2 yields X X P.r; s/ 1 P.r; s/ Z¿. r; s; r s n 1/ .21/ D C P P X Œk.N m /r Œl.N n/s ᏺN;m;nq .19/ D 2 2 N ގ m ގk2;n ގl 2 0 2 0 n 1 e 1 q D ./ P P P P X N.n 1 r s/ m r ns N;m;nq C C C .23/ N ގ m ގf2;n ގh 2 0 2 0 n 1 1 q X X Z 1 . r; s; r s n 1/ .22/ D ./ f g C n X X .1 q /Z 1 . r; s; r s n 1/: .10/ D f g C From Corollary 2.4 we deduce

PROPOSITION 2.2. For all but ﬁnitely many completions K of F , the following functional equation holdse: n P᏾;᏿;K .r; s/ q q 1 q P᏾;᏿;K .r; s/: j ! D 3. Applications to zeta functions of groups and rings

3.1. Zeta functions of rings. In this section we prove Theorem A. Let L be a ring of torsion-free rank n. In fact, without loss of generality we may assume that L is additively isomorphic to ޚn. Let p be a prime. Multiplication in L is n n n n n n a bi-additive mapping ˇ ޚ ޚ ޚ , which extends to ˇp ޚ ޚ ޚ , W ! W p p ! p inducing a ޚp-algebra structure on Lp L ޚp. We shall give a formula for the WD ˝ local zeta functions X s L;p.s/ Lp H ; D j W j H Lp Ä where H runs over the subalgebras of ﬁnite index in Lp, valid for almost all primes p, in terms of the p-adic integrals studied in Section 2. More precisely, we shall show that .s/ is expressible in terms of functions Z.s/, deﬁned as L;p z in (16), to which Corollary 2.3 is applicable. 1202 CHRISTOPHERVOLL

Write L ޚl1 ޚln. We consider the n n-matrix of ޚ-linear forms D ˚ ˚ ᏾.y/ .Lij .y// Matn.ޚŒy/; D 2 P k k where Lij .y/ k Œn ij yk, encoding the structure constants ij of L with WD 2 P k respect to the chosen basis; that is, li lj k Œn ij lk. Let Ci denote the matrix D 2 of the linear map given by right-multiplication with the generator li . A full sublattice ƒ in .Lp; / corresponds to a coset M , where n C D D GLn.ޚp/ and the rows of the matrix M .mij / GLn.ޑp/ Matn.ޚp/, the D 2 \ set of integral n n-matrices with nonzero determinant, encode the coordinates of generators for ƒ with respect to the chosen basis. Denote by Mi the i-th row of M . It is not hard to check (cf. the proof of [8, Th. 5.5]) that ƒ is a ޚp-subalgebra of Lp if and only if X (24) i; j Œn Mi Cr mjr M k Œn ޚ : 8 2 W 2 h kj 2 i p r Œn 2 Rather than trying to analyse the restrictions condition (24) imposes on the entries of suitable upper-triangular representatives of the coset M as in [8], we base our analysis on the following two basic observations. The ﬁrst point is that every homothety class of lattices ƒ in .Lp; / contains C a largest subalgebra ƒ0, and the subalgebras in this class are exactly the multiples m p ƒ0, m ގ0. We thus have 2 ns 1 X s (25) L;p.s/ .1 p / Lp ƒ0 ; D j W j Œƒ where ƒ0 denotes the largest subalgebra in the homothety class Œƒ Œƒ0. D The second observation is that it is easy to check condition (24) if M happens to be a diagonal matrix. With respect to the given basis, M may not admit a diagonal representative. By the elementary divisor theorem, however, it does contain a representative of the form M D˛ 1, where ˛ , D 2 P P r0 I r I r ril ril D D.I; r0/ p diag.p 2 ; : : : ; p 2 ; : : : ; p ; : : : ; p ; 1; : : : ; 1/ D D „ ƒ‚ … i1 „ ƒ‚ … il

l for a set I i1; : : : ; i < Œn 1 and a vector .r0; ri ; : : : ; ri / r0 ގ0 ގ D f l g Â 1 l DW 2 (both depending only on M ). We say that ƒ has type .I; r0/ and call ƒ maximal (in its homothety class) if r0 0. We say that the homothety class Œƒ has type l D .I; r/, r .ri ; : : : ; ri / ގ , – written .Œƒ/ .I; r/ – if its maximal element has D 1 l 2 D type .I; .0; ri1 ; : : : ; ril //. By slight abuse of notation we may also say that a lattice ƒ has type I if .Œƒ/ .I; r/ for some integral vector r, and that a homothety D class Œƒ has type I if any of its elements does. In this case we write .Œƒ/ I . j D We shall denote by ˛ the j -th column of the matrix ˛ and by Dii the i-th diagonal FUNCTIONALEQUATIONSFORZETAFUNCTIONSOFGROUPSANDRINGS 1203 entry of D. Note that ˛ is only unique up to right-multiplication by an element of 80 ::: 19 ˆ i1 > ˆ : : > ˆB ri1 : : C> ˆB p i2 i1 : : C> <ˆB C=> r r r : : I;r Stab .D/ B p i1 i2 p i2 :: : C WD D B C C ˆB : : C> ˆB : : C> ˆ@ : : il il 1 A> ˆ > : ri1 ril ri2 ril ril ; p CC p CC ::: p n il where stands for a matrix in , for an arbitrary matrix with entries in ޚp, and r r p for a matrix with entries in p ޚp of the appropriate sizes, respectively. Thus there is a 1 1-correspondence between lattice classes Œƒ of type .I; .ri ; : : : ; ri // 1 l and cosets ˛I;r. Furthermore (26) ! n P I r.n / Œƒ .Œƒ/ .I; r/ I;r ./=.I;r/ p 2 ; jf j D gj D j W j D D I 1 p where, as usual, denotes the Haar measure on normalised so that ./ 1 n D .1 p / : : : .1 p /. As we have noted above it coincides with the additive n2 Haar measure on Matn.ޚp/ ޚ , normalised so that .Matn.ޚp// 1. Š p D It is now straightforward to check that (24) is equivalent to < (27) i Œn D᏾ .˛/D 0 mod Dii ; 8 2 W .i/ Á where ᏾< .˛/ ˛ 1᏾.˛i /.˛ 1/t : It is easy to verify that condition (27) is equiv- .i/ WD alent to P P P P < r0 r r r r i; r; s Œn .᏾ .˛//rs p s I r I i> I 0 mod p I 8 2 W .i/ C Ä 2 C Ä 2 C 2 Á 2 which may in turn be reformulated as (28) 8 9 X I ; 2 „ 2 Ä 2 Ä 2 ƒ‚ 2 … m.Œƒ/ DW n < o where virs.˛/ min vp .᏾ .˛// i; r; s . WD ./ j Ä Remark 3.1. Whilst it might seem more natural to replace the inequalities in this deﬁnition of virs.˛/ by equalities, the present formulation is preferable as it allows us to translate the counting problem into the language developed in Section 2. Note that the right-hand side of (28) depends only on the homothety class of ƒ and is neatly separated in terms that depend on the type .I; r/ of Œƒ and terms that depend solely on ˛. The subalgebra ƒ0 is characterised by equality in (28). As we shall see, this formulation of the “subalgebra condition” (24) therefore enables 1204 CHRISTOPHERVOLL us to express the p-th local zeta function of L in terms of p-adic integrals on I .pޚp/ , I Œn 1. j j Â By (25) it sufﬁces to compute

< X s X X s (29) A .s/ Lp ƒ0 Lp ƒ0 : WD j W j D j W j Œƒ I Œn 1 .Œƒ/ I Â „ D ƒ‚ … A<.s/ DW I

For a fixed I i1; : : : ; i < Œn 1, we set D f l g Â ᏺ< Œƒ .Œƒ/ .I; r/; m.Œƒ/ m I;r;m WD jf j D D gj with m.Œƒ/ defined as in (28). Thus P P < X s I r X < sn. I r m/ (30) AI .s/ p 2 ᏺI;r;mp 2 D l r .ri ;:::;ri / ގ m ގ0 D 1 l 2 2 X P X p s I r. n/ ᏺ< psnm: D 2 C I;r;m r ގl m ގ0 2 2 As in the proof of Proposition 2.2 we shall design a p-adic integral to de- < scribe the generating functions AI .s/. Consider the p-adic integral ZI .s/ 2 D ZW;ޑ ;I .s1; : : : ; sn/ defined as in Section 2 with k n, m n , W and p D D D set

< 3 (31) firs.y/ .᏾ .y// i; r; s ; .i; r; s/ Œn ; Df ./ j Ä g 2 ( ) ! Y [ Y ı r ı s ı

< n l o .x; y/ .pޚp/ vp.x/ r; m.x; y/ m ; I;r;m D 2 j D D FUNCTIONALEQUATIONSFORZETAFUNCTIONSOFGROUPSANDRINGS 1205 with m.x; y/ WD X X X X 3 min r; vp.x/ vp.x/ vp.x/ virs.y/ .i; r; s/ Œn : C C C j 2 I s I r I i> I 2 Ä 2 Ä 2 2 < < Again, a lemma is needed to relate the numbers I;r;m to the data ᏺI;r;m we are trying to understand.

LEMMA 3.1. < < I;r;m (33) ᏺI;r;m P D 1 l I r .1 p / p 2 .I;r/ n I 1 P p < I r..n / 1/ p C : D .1 p 1/l ./ I;r;m 2 Proof. The set

n l o .x; y/ .pޚp/ vp.x/ r; m.x; y/ m 2 j D D may be written as a disjoint union of the I;r sets j W j n l o .x; y/ .pޚp/ I;r vp.x/ r; m.x; y/ m 2 j D D where runs through a complete set of coset representatives for =I;r. The mea- P 1 l r sure of each of these sets is either zero or equal to .1 p / p I .I;r/. The 2 latter happens if and only if I;r corresponds to a lattice of type .I; r/ such that m.Œƒ/ m. This proves the ﬁrst equality. The second equality is an immediate D consequence of (26).

Lemma 3.1 allows us to express the generating functions (30) in terms of the p-adic integrals (32). Indeed,

X P X A<.s/ p s I r. n/ ᏺ< psnm .30/ I D 2 C I;r;m r ގl m ގ0 2 2 n I 1 P p X I r.s. n/ .n / 1/ X < snm p C p .33/ D .1 p 1/l ./ 2 I;r;m r m n I p 1 Z<..s. n/ .n / 1/ ; sn/ .32/ 1 l I I D .1 p / ./ C 2 ! n < ZI ..s. n/ .n / 1/ I ; sn/: .10/ D I 1 C 2 p Theorem A follows now from equations (25) and (29), and Corollary 2.3. e 1206 CHRISTOPHERVOLL

3.2. Conjugacy zeta functions of nilpotent groups. In this section we prove Theorem B. As mentioned in the introduction, it suffices to compute cc X s 1 (34) .s/ Lp H Lp ᏺL .H / L;p D j W j j W p j H Lp Ä for almost all primes p, where L L.G/, Lp L ޚp and D D ˝ ᏺL .H / l Lp ŒH; l H p D f 2 j Ä g is the normaliser of H in Lp. Remark 3.2. It was pointed out, e.g. in [35, 3.8], that, whilst (34) might be used to define local “conjugacy” zeta functions for arbitrary rings, they might not encode the solutions to any actual counting problem unless Lp L ޚp for a D ˝ nilpotent Lie ring L. It is for this reason that we formulate Theorem B in terms of ᐀-groups G, bearing in mind that its proof is immediately reduced to a computation in the Lie ring L.G/ that draws upon neither the nilpotency nor the Lie property of L.G/. Along the lines of the “first observation” in the proof of Theorem A we use the fact that, for a subring H of Lp, the normaliser ᏺLp .H / is an invariant of the homothety class ŒH of H . We may thus write ᏺLp .ŒH / for ᏺLp .H /. Resuming the notation of the proof of Theorem A we have 2 (35) Lp ᏺL .ŒH / ƒ0 pƒ0 p ƒ0 p m (H p ƒ0 m ގ0 ). In addition to recording the index Lp ƒ0 as we run 2 f j 2 g n j W j over homothety classes Œƒ of lattices in ޑp, we now have to control the index Lp ᏺL .Œƒ/ . We shall see that this index, too, might be expressed in terms j W p j of congruences involving the type .I; r/ and coset ˛I;r determining Œƒ Œƒ0, D similar to the congruences (27). To compute their index requires us to control the elementary divisors of certain matrices. A similar problem was considered in Section 2.2; the problem we will have to solve in Section 3.3 is also of this kind (though slightly simpler). We resume the notation from Section 3.1. From (35) we deduce cc ns 1 X s 1 .s/ .1 p / Lp ƒ0 Lp ᏺL .Œƒ/ : L;p D j W j j W p j Œƒ n Using a fixed basis to identify Lp with ޚ we may express the condition x p 2 ᏺLp .ŒH / in a similar fashion to (24). If M is any matrix whose rows Mk encode n the coordinates of generators for any element of Œƒ, a vector x .x1; : : : ; xn/ ޚ D 2 p is in ᏺLp .Œƒ/ if and only if X (36) i Œn x Cr mir M k Œn ޚ : 8 2 W 2 h kj 2 i p r Œn 2 Using the correspondence between lattice classes and pairs ..I; r/; ˛I;r/, condition (36) may be reformulated as cc (37) i Œn x᏾ .˛/D 0 mod Dii ; 8 2 W .i/ Á FUNCTIONALEQUATIONSFORZETAFUNCTIONSOFGROUPSANDRINGS 1207 where ᏾cc .˛/ ᏾.˛i /.˛ 1/t and D D.I; r/. We note that the scalar pr0 in D .i/ WD D cancels in (37); we may thus assume r0 0. Condition (37) is then equivalent to D P P (38) i Œn x᏾cc .˛/Dp

᏾cc.˛/ .᏾cc .˛/ ::: ᏾cc .˛//; D .1/ j j .n/ P P cc ri1 ri1 I r I r D .I; r/ diag.D; : : : ; D; p D; : : : ; p D; : : : ; p 2 D; : : : ; p 2 D/; D „ ƒ‚ … „ ƒ‚ … „ ƒ‚ … i1 i2 i1 n il (where “diag” refers to the diagonal n2 n2-matrix built from n scalar multiples of the diagonal n n-matrix D), (38) may in turn be reformulated as P (39) x᏾cc.˛/Dcc.I; r/ 0 mod p I r : Á 2 n To keep track of the index of ᏺLp .Œƒ/, the full sublattice of Lp ޚp of solutions cc n Š cc to (39), we introduce an invariant .Œƒ/ ގ0 as follows. We say that .Œƒ/ n 2 D m .m1; : : : ; mn/ ގ if D 2 0 cc cc the matrix ᏾ .˛/D .I; r/ has elementary divisor type m .m1;:::; mn/ zcc D zcc z (i.e., there are matrices ˇ n, n2 such that ˇ᏾ .˛/D .I; r/ m mn 2 2 D diag.p 1 ; : : : ; p / 0 and mi ގ0 , m1 mn) and z z j z 2 [ f1g z Ä Ä z ˚P m .m1; : : : ; mn/ is deﬁned by mi min r; mi for each i Œn. I D D 2 z 2 P P . r mi / The index Lp ᏺL .Œƒ/ equals p i Œn I . As in Section 3.1 it is j W p j 2 2 helpful to write

cc X s 1 A .s/ Lp ƒ0 Lp ᏺL .Œƒ/ WD j W j j W p j Œƒ X X s 1 Lp ƒ0 Lp ᏺL .Œƒ/ : D j W j j W p j I Œn 1 .Œƒ/ I Â „ D ƒ‚ … Acc.s/ DW I For a ﬁxed I i1; : : : ; i < Œn 1, we set D f l g Â ᏺcc Œƒ .Œƒ/ .I; r/; m.Œƒ/ m; cc.Œƒ/ m I;r;m;m WD jf j D D D gj with m.Œƒ/ deﬁned as in (28). We thus have (40) X P X P P P Acc.s/ p s I r ᏺcc p sn. I r m/ i Œn. I r mi / I D 2 I;r;m;m 2 2 2 r ގl m ގ0 2 m2ގn 2 0 X P X P p I r. s. n/ n/ ᏺcc psnm i Œn mi : D 2 C I;r;m;m C 2 r ގl m ގ0 2 m2ގn 2 0 1208 CHRISTOPHERVOLL

cc As in Section 3.1 we shall show that the generating functions AI .s/ may be expressed in terms of the p-adic integrals to which the results of Section 2 may be applied. In order to control the invariant cc.Œƒ/ we need to parametrise the minors of the matrix ᏾cc.y/ in a suitable way. This motivates the following combinatorial definitions. Let Matn. 0; 1 / denote the set of n n-matrices with entries in 0; 1 , and cc f g P f gcc ᏿ S Matn. 0; 1 / Srs j . We introduce a partial order on ᏿ by j;n D f 2 f g j r;s D g j;n saying that, given S;T ᏿cc , T S if, 2 j;n P P P P for all r Œn: T S and, 2 r Œn r Œn P Ä P 2 P Ä P 2 for all s Œn: >s Œn T >s Œn S . 2 2 2 Pictorially speaking, this amounts to saying that the matrix T may be obtained from the matrix S by moving some of the nonzero entries towards “north-east”. cc 2 Given a matrix S ᏿ and a n n -matrix M .M1 ::: Mn/, Mi Matn.ޚp/, 2 j;n D j j 2 a j j -submatrix of M of column-type S is a submatrix obtained by choosing j cc rows of M and the s-th column of Mr if and only if Srs 1. For S ᏿ , we set D 2 j;n (41) fcc .y/ det.N / N a j j -submatrix of ᏾cc.y/ of column-type T S j;S D f j g and define the monomial P Pn P Pn cc Y > 1 S 1 S M .x/ x D C Ä D : S;I D I 2 We set [ fcc .x; y/ M cc .x/fcc .y/: j;I D S;I j;S S ᏿cc 2 j;n Now we define, using the sets of polynomials g;I .x; y/ introduced in (31), Z cc Y s (42) ZI .s; s;s/ g;I .x; y/ l z z D .pޚp/ W k k Œn 2 Y cc Y cc sj cc esj fj;I .x; y/ x fj 1;I .x; y/ e fj 1;I .x; y/ e dxI dy : k [ k k k j jj j j Œn I 2 2 Remark 3.3. Note that M cc .x/ M cc .x/ if T S. We therefore could have T;I j S;I kept definition (41) simpler by replacing “T S” by “S”, without changing the cc integral ZI . The extra complication ensures that the results from Section 2 are applicable. cc We leave it to the reader to verify that, for each j Œn and S ᏿j;n, cc 2 2 the ideal .fj;S .y// is B.ޑp/-invariant. Therefore the Theorems 2.2 and 2.3 and cc cc their corollaries are applicable to the integral ZI .s; s;s/. Note that ZI .s; s;s/ cc z z z z D ZI ..s/ I ; sn; s;s/. We set 2 z z cc l cc .x; y/ .pޚp/ W p.x/ r; m.x; y/ m; .x; y/ m : I;r;m;m D f 2 j D D D g FUNCTIONALEQUATIONSFORZETAFUNCTIONSOFGROUPSANDRINGS 1209

cc Here .x; y/ m .m1; : : : ; mn/ if, for each i Œn, D D 2 X mi min vp.x/; mi ; D z I 2 where m .m1;:::; mn/ is the elementary divisor type of the matrix z D z z cc cc ᏾ .y/D .I; .vp.x// I /: 2 Then cc cc ZI ..s/ I ; sn; s/ ZI ..s/ I ; sn; s; s/ 2 z WD 2 z z X P X P p I sr cc p snm i Œnesi mi : D 2 I;r;m;m 2 r ގl m ގ0 2 m2ގn 2 cc cc As in (33) we want to relate the numbers I;r;m;m with the integers ᏺI;r;m;m. LEMMA 3.2. n I 1 P cc p cc I r..n / 1/ (43) ᏺ p C : I;r;m;m D .1 p 1/l ./ I;r;m;m 2 Proof. Analogous to the proof of Lemma 3.1. Thus cc AI .s/ X P X P p I r. s. n/ n/ ᏺcc psnm i Œn mi .40/ D 2 C I;r;m;m C 2 r ގl m ގ0; 2 m2ގn 2 0 n 1 I p 1 l D.1 p / ./ X P X P p r. s. n/ .n / 1 n/ cc psnm i Œn mi .43/ C C C I;r;m;m C 2 r ގl m;m 2n 1 I p 1 l D.1 p / ./ cc ZI ..s. n/ .n / 1 n/ I ; sn; 1; : : : ; 1/ .42/ C C 2 ! n cc ZI ..s. n/ .n / 1 n/ I ; sn; 1; : : : ; 1/: .10/ D I 1 C C 2 p Theorem B now follows from Corollary 2.3.

3.3. Normal zeta functions of class-2-nilpotent groups. In this section we prove Theorem Ce. Let G be a ᐀2-group as in the statement of the theorem. With- out loss of generality we may assume that G=Z.G/ and Z.G/ are torsion-free abelian groups of rank d and d , respectively (so n d d ), and that G admits 0 D C 0 1210 CHRISTOPHERVOLL a presentation

X k k G g1; : : : ; gd ; h1; : : : ; hd Œgi ; gj hk; ޚ; all other Œ ; trivial : Dh 0 j D ij ij 2 i k Œd 2 (Note that we used additive notation for expressions in the abelian group G0.) Thus we obtain a matrix ᏹ.y/ Lij .y/ Mat .ޚŒy/ WD 2 d P k of linear forms Lij .y/ y , encoding the commutator structure of G. WD k Œd ij k By disregarding at most ﬁnitely2 many further primes we may also assume that p does not divide ᏹ.˛/ whenever ˛ ޚd 0 pޚd 0 . 2 p n p We begin our argument as in [32, 3], albeit with slightly different notation. Note, however, that we do not require the assumption that Z.G/ G . By [32, D 0 Lemma 1] and its corollary, it sufﬁces to compute a functional equation for the generating function X A .s/ A .s/; G D IG I Œd 1 Â 0 where X d s s A .s/ Z.Lp/ ƒ Lp X.ƒ/ : IG D j W j j W j .Œƒ/ I D Here the sum ranges over homothety classes of maximal lattices ƒ of type2 I in the centre Z.Lp/ of the ޚp-algebra

Lp .G=Z.G/ Z.G// ޚp; WD ˚ ˝ and X.ƒ/=ƒ Z.L =ƒ/. We identify Z.L / with ޚd 0 using the basis .h ; : : : ; h /. p p p 1 d 0 D d The index Lp X.ƒ/ is the index in Lp=Z.Lp/ ޚ of a system of linear j W j Š p congruences which we now describe (cf. [31, 2.2]). Let Œƒ be of type .I; r/, l I i1; : : : ; i < Œd 1, r .ri ; : : : ; ri / ގ , corresponding to the coset D f l g Â 0 D 1 l 2 ˛I;r d =I;r as in the proof of Theorem A. We set 2 0 (44) ᏹ .˛/ ᏹ.˛1/ ::: ᏹ.˛d 0 / ; G D j j P P ri1 ri1 I r I r DG.I; r/ diag.1; : : : ; 1; p ; : : : ; p ; : : : ; p 2 ; : : : ; p 2 /: D „ ƒ‚ … „ ƒ‚ … „ ƒ‚ … di1 d.i2 i1/ d.d il / 0 The system of linear congruences under consideration is P (45) xᏹ .˛/D .I; r/ 0 mod p I r : G G Á 2 The solutions to this system form a full lattice in Lp=Z.Lp/. To keep track of its index we deﬁne the invariant .Œƒ/ ގd as follows. We say that .Œƒ/ m 0 2 0 0 D D .m1; : : : ; md / if

2Note that the deﬁnition of a lattice’s type given in [32] differs from the one in the current paper insofar as I is replaced by d I d i i I . 0 D f 0 j 2 g FUNCTIONALEQUATIONSFORZETAFUNCTIONSOFGROUPSANDRINGS 1211

the matrix ᏹ .˛/D .I; r/ has elementary divisor type G G

m .m1;:::; m / z D z z d and P m .m1; : : : ; mn/ is deﬁned by mi min r; mi for each i Œn. I D D f 2 z g 2 The index of the system (45) equals P P j Œd. I r mj / p 2 2 : By deﬁning ᏺ Œƒ .Œƒ/ .I; r/; .Œƒ/ m I;G r;m WD jf j D 0 D gj (note that ᏺ 0 implies m1 0, as p ᏹ .˛/), we obtain a formula for A .s/ I;G r;m 6D D 6 j G IG that is analogous to (30): X P X P P (46) A .s/ p.d s/ I r ᏺ p s j Œd. I r mj / IG D 2 I;G r;m 2 2 r ގl m ގd 2 2 0 P P X I r. s.d / d/ X s j Œd mj p 2 C C ᏺI;G r;mp 2 : D r m

As in Section 3.1 we shall show that the generating function AIG.s/ may be expressed in terms of a p-adic integral to which the results of Section 2 can be applied. We shall need more notation. Given the d dd -matrix M .M1 ::: Md /, Mi Matd .ޚp/, a j j - 0 D j j 0 2 submatrix of M of column-type S . ; : : : ; / ގd 0 , P j , is a 1 d 0 0 i Œd i D 2 2 0 D submatrix obtained by choosing j rows of M and i columns in the “block” Mi for each i Œd . We denote by ᏿ . ; : : : ; / P j the set of 0 j;dG 1 d 0 i Œd i 2 0 D f j 2 0 D g possible such column-types. Given S .i /, T .i / ᏿j;dG , we write T S if, P P D D 2 0 for all i Œd 0, i i . For S ᏿j;dG , we set 2 Ä Ä 2 0 (47) f .y/ det.N / N is a j j -submatrix of ᏹ.y/ of column-type T S : j;SG Df j g P < Œd Q 0 We deﬁne the monomial MS;IG .x/ I x 2 and set D 2 [ f .x; y/ M .x/f .y/: j;IG D S;IG j;SG S ᏿j;dG 2 0 We are now ready to deﬁne Z Y t ZIG.t; s; s/ ZIG..t/ I ; s2; : : : ; sd ; s2;:::; sd / x e e l z D 2 D .pޚp/ W j j I ! 2 ! Y Y sj sj fj;IG .x; y/ x fjG 1;I .x; y/ fjG 1;I .x; y/ e dxI dy : k [ k k k j jj j j Œ2;d I 2 2 1212 CHRISTOPHERVOLL

Remark 3.4. Note that M .x/ M .x/ if T S. We could have kept deﬁ- T;IG j S;IG nition (47) simpler by replacing “T S” by “S”. The extra complication ensures that the results from Section 2 are applicable. Note also that we are not losing anything by omitting the factor for j 1, as f0;IG .x; y/ f1;IG .x; y/ 1 for l D k k D k k D all x .pޚp/ , y W . 2 2

We leave it to the reader to verify that, for each j Œd and S ᏿j;dG , the ideal 2 2 0 .fj;SG .y// is B.ޑp/-invariant. Therefore Theorems 2.2, 2.3 and their corollaries are applicable. We set

l .x; y/ .pޚp/ W vp.x/ r; .x; y/ m ; I;G r;m D f 2 j D 0 D g d P where 0.x; y/ m .m1; : : : ; md / ގ0 if m min I r; m where m D D 2 D f 2 z g z is the elementary divisor type of the matrix ᏹG.y/DG.I; .vp.x// I /. Note that 2 0 implies m1 0. Then I;G r;m 6D D (48) P P X I tr X j Œ2;d sj mj ZIG..t/ I ; s/ ZIG..t/ I ; s; s/ p 2 I;G r;mp 2 : 2 WD 2 D r ގl m ގd 2 2 0 LEMMA 3.3.

d 0 I 1 P p I r..d / 1/ (49) ᏺG G p 0 C : I;r;m D .1 p 1/l ./ I;r;m 2 Proof. Analogous to the proof of Lemma 3.1.

Thus, AIG.s/ P P X I r. s.d / d/ X s j Œd mj p 2 C C ᏺI;G r;mp 2 .46/ D r m

d 0 1 I p 1 l D.1 p / ./ P P X I r. s.d / .d 0 d / 1/ X s j Œd mj p 2 C C C C I;G r;m.p / 2 .49/ r m

d 0 I p 1 Z ..s.d / .d d / 1/ ; s; : : : ; s/ .48/ 1 l I 0 I D.1 p / ./ C C 2 ! d 0 ZfI ..s.d / .d d 0 / 1/ I ; s; : : : ; s/: .10/ D I 1 C C 2 p Theorem C now follows from Corollary 2.3.

3.4. Representation zeta functions of ᐀-groups. In this section we recall some of Howe’s work [16] on irreducible representations of ᐀-groups and co-adjoint orbits and prove Theorem D. FUNCTIONALEQUATIONSFORZETAFUNCTIONSOFGROUPSANDRINGS 1213

If G is a group and H G is a subgroup we say that H is saturated if gn H Ä 2 implies g H for all g G. Denote by Hs the smallest saturated subgroup of G 2 2 containing H (this is called the isolator of H in [27, Ch. 8, A]). Clearly, if H G, G H Hs if and only if G=H is torsion-free. D Now let G be a ᐀-group. Recall (cf., for example, [27, Ch. 6]) that, by the Malcev correspondence, there is a Lie algebra ᏸG.ޑ/ over ޑ, of dimension equal to h.G/, the Hirsch length of G, and an injective mapping log G ᏸG.ޑ/, such W ! that span .log.G// ᏸG.ޑ/. In general L log.G/ will fail to be a Lie subring ޑ D WD (or even just a lattice). However, by replacing G by a subgroup of finite index, if necessary, we may assume it is a Lie subring ([14, 4]) and even that ŒL; L cŠL, Â where c is the nilpotency class of L (or G), i.e., that L (and G) are elementarily irr exponentiable (e.e.) in Howe’s nomenclature. As we are looking to study G;p.s/ for all but finitely many primes, there is no harm in descending to a subgroup H of finite index in G. Indeed, for all p and all n, there is a 1 1 – correspondence between twist isoclasses of irreducible characters of degree pn and p-admissible n twist isoclasses of degree p of Gbp, the pro-p-completion of G ([17, Lemma 8.5]). However, Gbp Hbp if p G H . Š 6 j j W j For a ᐀-group G, denote by .G.i// the group’s lower central series, defined by G.1/ G, G.i 1/ ŒG.i/; G, i 2. We say that G 1 has nilpotency D C D 6D f g class c (or is step-c-nilpotent) if G.c/ 1 but G.c 1/ 1 . Similarly, we 6D f g C D f g denote by .L.i// the lower central series of the Lie algebra L, defined by L.1/ L, D L.i 1/ ŒL.i/; L, i 2, and we hope that there will be no confusion between C D group commutators and Lie brackets. It is well-known that log induces a bijection between L.i/ and G.i/, i Œc. We write G for G.2/, and L for L.2/. By Z.L/ 2 0 0 we denote the centre of L. For a sublattice M L log.G/, denote its dual Hom.M; ރ / by M and Â D by rM L M ; M the restriction to M . We say that L is rational W ! 7! j 2 on M if rM . / is a torsion element. Clearly is rational on M if and only if is rational on Ms if and only if .nM / 1 for some n ގ. D 2 Recall that the adjoint action Ad of G on L is given by (50) l l Œlog ; l .higher terms/; 7! C C where we omitted Lie terms of degree 3. These may be computed in terms of the Baker-Campbell-Hausdorff-formula. This element F .X; Y / bᏸ X;Y , the 2 f g completion of the free Lie algebra on the symbols X and Y , provides the solution to exp.F .x; y// exp.x/ exp.y/: D The Baker-Campbell-Hausdorff-formula allows us therefore to recover the group structure on the Lie algebra (cf. [20, Ch. 9] and [28, V.3.4 and IV.7]). By (50), the co-adjoint action Ad of G on L is thus given by Ad . /.l/ .l/ .Œlog ; l/ .higher terms/: D Given L we define a bi-additive, anti-symmetric form B L L ރ by 2 W ! setting B ..l1; l2// .Œl1; l2/. We say that a subalgebra P L polarises B D Ä 1214 CHRISTOPHERVOLL if B P P 1 and it is a maximal additive subgroup with respect to this prop- j D erty. Note that any such additive subgroup contains the radical Rad l L D f 2 j .Œl; L/ 1 of B . D g LEMMA ([16, Lemmata 1–4]). Given L, the isotropy subgroup G of 2 under AdG is an e.e. subgroup of G and log G Rad . If L is AdG- D 2 invariant, then 2 defines a one-dimensional character on G. The orbit of L 2 under the co-adjoint action Ad of G is finite if and only if is rational on log Gs0 . If L is rational on log G then there are e.e. polarising subalgebras P for B . 2 s0 They have finite index in L satisfying L P P Rad . j W j D j W j THEOREM ([16, Th. 1(a)]). Let G be an e.e. ᐀-group and set L log G, L D s0 D log G . Let be a finite AdG-orbit in L, and . Let N be the period of s0 2 and assume N to be odd. A finite-dimensional irreducible representation U may be associated to in the following manner: Let P be an e.e. polarising subalgebra for B , set … exp P and …, a linear character on …. Put U G D Q WD j 1=2 WD Ind . Then the dimension of U is , and the character of U is … Q j j D 1 P . All representations of the form V , G=G , V defined 1=2 ˝ 2 s0 moduloj j NG2 exp.N L/, N odd, are realised in this manner. WD COROLLARY 3.1. For almost all primes p,

irr X s=2 1 (51) .s/ L Rad L L ;2 ; G;p D j W j j W j1 Lb0s rational2 of p-power period where L ;2 l L .Œl; L / 1 . D f 2 j s0 D g Proof. For primes p not dividing 2 Gs0 G0 , Howe’s theorem yields that we j W j s=2 count every p-power degree twist-isoclass at least once when we sum G G j W j over the rational characters of Ls0 of p-power period. By further excluding finitely many primes, we may assume that G G L Rad (cf. [14, Lemma j W j D j W j 4.8], in which it is established that log induces an index-preserving correspondence between p-power index subgroups of G and p-power index subalgebras of L away from a finite number of primes). Hereby we overcount every orbit by exactly the index G G ;2 , where G ;2 G .Œlog ; L / 1 is the stabiliser of j W j D f 2 j s0 D g under the coadjoint action of G on the restriction of characters to Ls0 . (Note that this index is always equal to 1 if G is class-2-nilpotent.) Again at the cost of at most finitely many primes we may assume G G ;2 L L ;2 . j W j D j W j Howe’s parametrisation of irreducible representations allows us to prove

PROPOSITION 3.1. Let G be a ᐀-group. Then there are matrices ᏿ ᏾ of homogeneous ޑ-linear forms such that, for almost all primes p, irr .s/ P ; ;ޑ .s=2; : : : ; s=2 1; : : : ; 1/; G;p D ᏾ ᏿ p I FUNCTIONALEQUATIONSFORZETAFUNCTIONSOFGROUPSANDRINGS 1215

where P᏾;᏿;ޑp is the generating function deﬁned in Section 2.2. Proposition 2.2 is applicable for n h.G /. D 0 Theorem D clearly follows from this.

Proof. We aim to express both factors in the summands of (51) in terms of elementary divisors of matrices of linear forms. Recall that L is additively isomorphic to ޚh, where h is the Hirsch length of G. Without loss of generality we may assume that L is saturated in L, and that L Z.L/ is saturated in Z.L/ 0 0 \ (otherwise we disregard ﬁnitely many primes). We ﬁx a ޚ-basis

x1; : : : ; xd ; xd 1; : : : ; xd m; xd m 1; : : : ; xd n f C C „ C C ƒ‚ C…g L0 Z.L/ „ ƒ‚ \ … L0 of L such that

xd 1; : : : ; xd n is a ޚ-basis for L0; f C C g xd m 1; : : : ; xd n is a ޚ-basis for L0 Z.L/: f C C C g \ The Lie bracket induces an anti-symmetric, bi-additive mapping

ˇ L=.L Z.L// L=.L Z.L// L ; W 0 \ 0 \ ! 0 l1.L Z.L//; l2.L Z.L// Œl1; l2: 0 \ 0 \ 7! We may express this map in terms of our chosen basis as follows. For 1 i; j Ä Ä d m, let C X Œxi ; xj ijkxd k; ijk ޚ: D C 2 k Œn 2 Let ᏾ .᏾ij / denote the .d m/ .d m/-matrix of linear forms D C C X ᏾ij .Y/ Y ޑŒY1;:::;Yn: D ijk k 2 k Œn 2 By ᏿ we denote the submatrix of ᏾ consisting of the last m columns of ᏾. By further disregarding ﬁnitely many primes if necessary we may assume that, given n n y ޚ , ᏾.y/ is not zero unless y 0 ކ . We denote by Ci , i Œd m, the 2 p D 2 p 2 C matrices of the additive maps L=.L Z.L// L ; x.L Z.L// Œx; xi with 0 \ ! 0 0 \ 7! respect to these bases. For a given nonnegative integer N ގ0 we identify the set 2 n N o ‰ N Lb the period of equals p p WD 2 0j with ޚn=pN ޚn .ޚ=pN /n .pޚ=pN /n WD n 1216 CHRISTOPHERVOLL

n N n by sending ` .l1; : : : ; ln/ ޚ =p ޚ to D 2 P ! 2i i Œn li bi .b1; : : : ; bn/ exp 2 ; b .b1; : : : ; bn/ Lb: D pN D 2 0

With these identiﬁcations we obtain (52) hX i k .k1; : : : ; kd m/ Rad =Z.L/ G ki xi ; log 1 D C 2 ,8 2 W D t N j Œd m k Cj ` 0 mod p ,8 2 C W Á k ᏾.`/ 0 mod pN , Á and hX i (53) k .k1; : : : ; kd m/ L ;2=Z.L/ G0 ki xi ; log 1 D C 2 , 8 2 W D t N j Œd 1; d m k Cj ` 0 mod p , 8 2 C C W Á k ᏿.`/ 0 mod pN : , Á In order to use the congruence conditions (52) and (53) for an effective computation irr of G;p.s/, we need to enumerate the (p-parts of) the elementary divisors of the matrices ᏾.`/ and ᏿.`/ as ` runs through the sets ‰ N , N ގ. By Corollary 3.1 p 2 we may write

irr X s=2 1 .s/ L Rad L L ;2 G;p D j W j j W j N ގ0 2‰ N 2 p P P X i Œd m.N mi /s=2 j Œm.N nj / ᏺN;m;nq D 2 C 2 N ގ0 d 2m m m ގ C ; n ގ 2 0 2 0 P ; ;ޑ .s=2; : : : ; s=2 1; : : : ; 1/: D ᏾ ᏿ p I Acknowledgements. I started work on this paper when I was a guest at the Max-Planck-Institut fur¨ Mathematik in Bonn, Germany. I continued it as a guest at the Chaire de Mathematiques´ Algebriques´ Discretes` at the EPF Lausanne, Switzer- land. I gratefully acknowledge both institutions’ support and hospitality. I am particularly obliged to Laurent Bartholdi, my host in Lausanne, and to the Fond National Suisse, which jointly funded my stay there. Thanks are due to Benjamin Klopsch for frequent inspiring discussions. I wish to thank Benjamin Martin for making Hrushovski’s and his preprint [17] available to me, and to Willem Veys for alerting me to [30]. Mark Berman’s and Ulrich Voll’s comments on previous versions helped a great deal to improve the presentation of this paper. I am particularly grateful to Mark Berman for his help with proofreading. FUNCTIONALEQUATIONSFORZETAFUNCTIONSOFGROUPSANDRINGS 1217

Note added in proof. Theorem A also conﬁrms, for almost all primes, the conjecture in [23, 7] regarding functional equations for local zeta functions enumerating subrings in ޚn.

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(Received December 18, 2006)

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