Analytic Properties of Zeta Functions and Subgroup Growth.

Marcus du Sautoy^{∗}and Fritz Grunewald^{†}.

September 10, 1999

1 Introduction

It has become somewhat of a cottage industry over the last fifteen years to understand the rate of growth of the number of subgroups of finite index in a group G. Although the story began much before, the recent activity grew out of a paper by Dan Segal in the proceedings of Groups St. Andrews 1985 [20]. The story so far has been well-documented in Lubotzky’s survey paper in the proceedings of the subsequent Groups Galway 1993 [16].

In [13] the second author, Segal and Smith introduced the zeta function of a group as a tool for understanding this growth of subgroups. Let a_{n}(G) be the number of subgroups of index n in the finitely generated group G and s_{n}(G) = a_{1}(G) + ⋯ + a_{n}(G) be the number of subgroups of index n or less. The zeta function is defined as the Dirichlet series with coefficients a_{n}(G) and has a natural interpretation as a non-commutative generalization of the Dedekind zeta function of a number field:

ζ_{G}(s) = ∑_{n=1}^{∞} a_{n}(G)n^{−s}  (1)
= ∑_{H≤G} |G : H|^{−s}.

∗DPMMS, 16 Mill Lane, Cambridge UK. dusautoy@dpmms.cam.ac.uk

†Math. Institut, Heinrich Heine Universität, D-40225 Düsseldorf, Germany.

fritz@math.uni-duesseldorf.de

For example, without such a tool it would be difficult to prove that the number of subgroups in the rank two free abelian group G = ℤ^{2} grows as follows:

s_{n}(ℤ^{2}) ∼ (π^{2}/12) n^{2}

as n tends to infinity. (Here f(n) ∼ g(n) means f(n)/g(n) tends to 1 as n tends to infinity.) This is a consequence of the expression for the zeta function of the free abelian group:

ζ_{ℤ}^{d}(s) = ζ(s)...ζ(s − d + 1)

where ζ(s) is the Riemann zeta function.

The zeta function (1) defines an analytic function on some right half of the complex plane ℜ(s) > α(G) precisely when the coefficients a_{n}(G) are bounded by a polynomial. A characterization of such groups, groups of polynomial subgroup growth, was provided by Lubotzky, Mann and Segal [17]. They are the finitely generated groups which have a subgroup of finite index that is soluble of finite rank.

In this paper we consider the analytic behaviour of the zeta function of the subclass of finitely generated nilpotent groups. This class of groups has the added bonus that their zeta functions satisfy an Euler product (see [13]):

ζ_{G}(s) = ∏_{p} ζ_{G,p}(s)

where the local factors for each prime p are defined as:

ζ_{G,p}(s) = ∑_{n=0}^{∞} a_{p^{n}}(G)p^{−ns}.

It was also proved in [13] that if the nilpotent group is torsion-free then these local factors are all rational functions in p^{−s}. However the proof gave little understanding of how these rational functions varied as p varied and was not sufficient to understand the global behaviour of ζ_{G}(s).

In this paper we introduce some new methods to understand the zeta function of a group and prove the following:

Theorem 1 (1) Let G be a finitely generated nilpotent infinite group. Then there exist a rational number α(G) ∈ ℚ, a non-negative integer b ∈ ℕ_{0} and some real number c ∈ ℝ such that

s_{n}(G) ∼ c · n^{α(G)} (log n)^{b} .

(2) The abscissa of convergence α(G) of ζ_{G}(s) is a rational number and ζ_{G}(s) can be meromorphically continued to ℜ(s) > α(G) − δ for some δ > 0.

Part (1) follows from (2) by a suitable application of the Tauberian Theorem. Whether the abscissa of convergence was a rational number was raised as one of the major open problems in the field in Lubotzky’s survey article [16].

Note that the integer b + 1 is the multiplicity of the pole at s = α(G). In [10] several examples are given where this multiplicity is greater than one. For example, the zeta function of the discrete Heisenberg group

G = ( \begin{matrix}1 & ℤ & ℤ \\ 0 & 1 & ℤ \\ 0 & 0 & 1 \end{matrix} )

has the following expression:

ζ_{G}(s) = ζ(s)ζ(s − 1)ζ(2s − 2)ζ(2s − 3) · ζ(3s − 3)^{−1}.

The double pole at s = 2 implies that the growth of subgroups is:

s_{n}(G) ∼ \frac{ζ(2)^{2}}{2ζ(3)}n^{2} log n.

This was first observed in Smith’s thesis [21]. This example has meromorphic continuation to the whole complex plane. In [8] it is shown that this is also true for any finite extension of a free abelian group. In general, though, these functions have natural boundaries as discussed in [10]. However we have introduced in a separate paper [7] the concept of the ghost zeta function which does tend to have meromorphic continuation.

The proof of (2) depends on showing a more general result which holds for any zeta function which can be defined as an Euler product over primes p of cone integrals over ℚ.

Definition 2 (1) Call a formula ψ(x) in the first order language for the valued field ℚ_{p} a cone condition over ℚ if there exist polynomials f_{i}(x), g_{i}(x)(i = 1, ..., l) over ℚ in the variables x = x_{1}, ..., x_{m} such that ψ(x) is a conjunction of formulas

v(f_{i}(x)) ≤ v(g_{i}(x))

for i = 1, ..., l.

(2) Given a cone condition ψ(x) over ℚ and polynomials f_{0} and g_{0} with coefficients in ℚ, we call an integral

Z_{D}(s, p) = ∫_{V_{p}={x∈ℤ_{p}^{m}:ψ(x) is valid}} |f_{0}(x)|^{s} |g_{0}(x)| |dx|

a cone integral defined over ℚ, where |dx| is the normalized additive Haar measure on ℤ_{p}^{m} and D = {f_{0}, g_{0}, f_{1}, g_{1}, ..., f_{l}, g_{l}} is called the cone integral data.

(3) We say that a function Z(s) is defined as an Euler product of cone integrals over ℚ with cone integral data D if

Z(s) = Z_{D}(s) = ∏_{p prime} (a_{p,0}^{−1} · Z_{D}(s, p))

where a_{p,0} = Z_{D}(∞, p) is the constant coefficient of Z_{D}(s, p), i.e. we normalize the local factors to have constant coefficient 1.

It is clear from the defintion of a cone integral that a_{p,0} is non-zero so the normalization in (3) is well defined.

In section 5 we show that for a nilpotent group G, ζ_{G}(s) = Z_{D}(s − d) where Z_{D}(s) is defined as an Euler product of cone integrals over ℚ and d is the Hirsch length of G. (The Hirsch length is the number of infinite cyclic factors in a composition series for G.)

We adapt some ideas of Denef introduced in [3] to give an explicit expression for a cone integral valid for almost all primes p in terms of the resolution of singularities (Y, h) of the polynomial F (x) = ∏_{i=0}^{l} f_{i}(x)g_{i}(x). In particular we show:

Theorem 3 Let (Y, h) be a resolution over ℚ for F (x) = ∏_{i=0}^{l} f_{i}(x)g_{i}(x) and let E_{i}, i ∈ T, be the irreducible components defined over ℚ of the reduced scheme (h^{−1}(D))_{red} where D =Spec(\frac{ℚ[x]}{(F)}) . Then there exist rational functions P_{I}(x, y) ∈ ℚ(x, y) for each I ⊂ T with the property that for almost all primes p

Z_{D}(s, p) = ∑_{I⊂T} c_{p,I}P_{I}(p, p^{−s})  (2)

where

c_{p,I} = card\{a ∈ \overline{Y} (𝔽_{p}) : a ∈ \overline{E_{i}} if and only if i ∈ I\}

and \overline{Y} means the reduction mod p of the scheme Y.

We can then use the Lang-Weil estimates for the number of points on varieties modp to identify the abscissa of convergence of the global zeta function Z_{D}(s).

However just knowing the shape of the zeta function from the expression (2) is not sufficient to infer that the Euler product of these expressions can be meromorphically continued beyond its region of convergence. For example

∏_{p prime} (1 + \frac{p^{−1}}{(1 − p^{−s})})  (3)

converges for ℜ(s) > 0 but has ℜ(s) = 0 as a natural boundary. We give instead a subtler expression for the cone integrals. Rather than a sum over the subsets of T, the indexing set of the irreducible components, this second expression is a sum over the open simplicial pieces of a natural polyhedral cone that one associates to the cone condition ψ.

Theorem 4 There exists a closed polyhedral cone \overline{D} in ℝ_{≥0}^{t} where t = cardT whose open simplicial pieces we denote by R_{k} where k ∈ {0, 1, ..., w} . We arrange that R_{0} = (0, ..., 0) and R_{1}, ..., R_{q} are the one-dimensional pieces. For each k ∈ {0, 1, ..., w} let M_{k} ⊂ {1, ..., q} denote those one-dimensional pieces in the closure \overline{R_{k}} of R_{k}. Then there exist positive integers A_{j}, B_{j} for j ∈ {1, ..., q} such that

Z_{D}(s, p) = ∑_{k=0}^{w}(p − 1)^{I_{k}}p^{−m}c_{p,I_{k}} ∏_{j∈M_{k}} \frac{p^{−(A_{j}s+B_{j})}}{(1 − p^{−(A_{j}s+B_{j})})}  (4)

where c_{p,I_{k}} is defined in Theorem 3 and I_{k} is the subset of T defined so that i ∈ T \I_{k} if and only if the ith coordinate is zero for all elements of R_{k}.

This expression (4) motivates the name cone integral. An explicit expression is given for the integers A_{j} and B_{j} in terms of the numerical data of the resolution. This is contained in the proof of this Theorem which appears in section 3.

With this combinatorial expression to hand, we can show that the pathologies of examples like (3) do not arise. In particular, we show that the abscissa of convergence of the global zeta function is determined by the terms in the expression (4) corresponding to the one dimensional edges R_{1}, ..., R_{q}. We then show how to use Artin L-functions to analytically continue a function like

∏_{p prime} (1 + c_{p,I_{k}} \frac{p^{−s}}{(1 − p^{−s})})

beyond its region of convergence. We can then use the Tauberian Theorem to estimate the growth of the coefficients in the Dirichlet series expressing Z_{D}(s). In particular we prove the following:

Theorem 5 Let Z_{D}(s) be defined as an Euler product of cone integrals over ℚ. Then Z(s) is expressible as a Dirichlet series ∑_{n=1}^{∞} a_{n}n^{−s} with nonnegative coefficients a_{n}.

(1) The abscissa of convergence α of Z(s) is a rational number and Z(s) has a meromorphic continuation to ℜ(s) > α − δ for some δ > 0.

(2) Let the pole at s = α have order w. Then for every β with β + α > 0, there exists some real number c ∈ ℝ such that

a_{1} + ⋯ + a_{n}n^{β} ∼ c · n^{β+α} (log n)^{w−1} .

One of the key problems in the area was to link zeta functions of groups up to questions in some branch of more classical number theory. We restate Theorem 3 explicitly for groups as it provides just such a path from zeta functions of groups to the more classical question of counting points mod p on a variety. The path is quite explicit. We define in section 5 a polynomial F_{G} over ℚ associated to each nilpotent group G.

Theorem 6 Let G be a finitely generated nilpotent group. Let (Y, h) be a resolution over ℚ for the polynomial F_{G}. Let E_{i}, i ∈ T be the irreducible components of the reduced scheme (h^{−1}(D))_{red} associated to h^{−1}(D) where D =Spec(\frac{ℚ[x]}{(F_{G})}). Then there exist rational functions P_{I}(x, y) ∈ ℚ(x, y) for each I ⊂ T with the property that for almost all primes p

ζ_{G,p}(s) = ∑_{I⊂T} c_{p,I}P_{I}(p, p^{−s})

where

c_{p,I} = card\{a ∈ \overline{Y} (𝔽_{p}) : a ∈ \overline{E_{i}} if and only if i ∈ I\}

and \overline{Y} means the reduction mod p of the scheme Y.

The behaviour of the local factors as we vary p is one of the other major problems in the field. For example in the Heisenberg group with entries from a quadratic number field, the behaviour of the local factors depends on how p behaves in the number field [13]. Our explicit formula however takes the subject away from the behaviour of primes in number fields to the problem of counting points mod p on a variety, a question which is in general wild and far from the uniformity predicted by all previous examples (see [13] and [6]). In a future paper [12] we explain an example constructed by the first author of a class two nilpotent group of Hirsch length 9 whose zeta function depends on counting points on the elliptic curve y^{2} = x^{3} − x mod p.

Given a nilpotent group G it is possible to construct and analyse the polynomial F_{G} in question. For example, in the free abelian group or the Heisenberg group, the polynomial does not require any resolution of singularities as D in this case only involves normal crossings. Hence the E_{i}, i ∈ T in this case are just the irreducible components of the algebraic set F_{G} = 0. However this is not true in general. For example the class two nilpotent group defined using the elliptic curve mentioned above has an F_{G} whose singularities are not normal crossings and which therefore require some resolution.

We have put the emphasis in this introduction on applying these cone integrals to the question of counting subgroups in nilpotent groups, however our results extend in a number of other directions.

(1) Variants of our zeta functions have been considered which count only subgroups with some added feature, for example normal subgroups. Define

a_{n}^{◁}(G) = card{H : H is normal subgroup of G and |G : H| = n}
ζ_{G}^{◁}(s) = ∑ a_{n}^{◁}(G)n^{−s}.

Our Theorems hold for this normal zeta function and many of the other variants.

(2) Let L be a ring additively isomorphic to ℤ^{d}. Define

a_{n}(L) = card{H : H is a subring of L and |L : H| = n}
a_{n}^{◁}(L) = card{H : H is an ideal of L and |L : H| = n}.

Zeta functions of L were also defined in [13] as the Dirichlet series

ζ_{L}(s) = ∑ a_{n}(L)n^{−s}
ζ_{L}^{◁}(s) = ∑ a_{n}^{◁}(L)n^{−s}

It was pointed out in [13] that these zeta functions have an Euler product, as for the case of nilpotent groups:

ζ_{L}(s) = ∏_{p prime} ζ_{L⊗ℤ_{p}}(s)

ζ _{L}^{◁}(s) = ∏_{p prime} ζ_{L⊗ℤ_{p}}^{◁}(s)

Unlike for groups, there is no need to make an assumption of nilpotency in the case of rings. We can therefore consider examples like L = 𝔰𝔩_{2}(ℤ) or the ℤ-points of any simple Lie algebra of classical type. We get then the following:

Theorem 7 Let L be a ring additively isomorphic to ℤ^{d}. Then there exists some rational number α(L) ∈ ℚ, a non-negative integer b ∈ ℕ_{0} and some real number c ∈ ℝ such that ζ_{L}(s) has abscissa of convergence α(L) and

s_{n}(L) = a_{1}(L) + ⋯ + a_{n}(L) ∼ c · n^{α(L)} (log n)^{b} .

There is a similar theorem for the invariant a_{n}^{◁}(L) counting ideals.

We actually prove this Theorem as part of our proof of Theorem 1. We make use of the fact that for a nilpotent group G there is a Lie algebra L(G) defined over ℤ with the property that for almost all primes p

ζ_{G,p}(s) = ζ_{L(G),p}(s).

This fact was established in [13].

In [9] the example of ζ_{𝔰𝔩_{2}(ℤ)}(s) is calculated using some previous work of Ishai Ilani [14]. There is a rational function P (T) such that

ζ_{𝔰l_{2}(ℤ)}(s) = P (2^{−s}) · ζ(s)ζ(s − 1)ζ(2s − 2)ζ(2s − 1)ζ(3s − 1)^{−1}.

Note that this example has a single pole at s = 2 (provided the uncalculated local factor P (T) at p = 2 does not have a pole at s = 2 which is unlikely since it would mean that most additive lattices in ℤ_{2}^{3} were subalgebras in 𝔰𝔩_{2}(ℤ_{2})). This means then that the subalgebra growth, in contrast to the 3dimensional Heisenberg Lie algebra, is s_{n}(𝔰l_{2}(ℤ)) ∼ c · n^{2} where c = P (2^{−2}) · ζ(2)^{2}ζ(3)/2ζ(5).

(3) Let G be a linear algebraic group over ℚ. Let ρ : G → GL_{n} be a ℚrational representation. Define the ‘local zeta function of the algebraic group G at the representation ρ and the prime p ’ to be

Z_{G,ρ,p}(s) = ∫_{G^{+}} |det ρ(g)|^{s} μ_{G}(g)

where G^{+} = ρ^{−1} (ρ (G (ℚ_{p})) ∩ M_{n} (ℤ_{p})) and μ_{G} denotes the right Haar measure on G(ℚ_{p}) normalized such that μ_{G} (G (ℤ_{p})) = 1.

We define the ‘global zeta function of G at the representation ρ’ to be Euler product

Z_{G,ρ}(s) = ∏_{p} Z_{G,ρ,p}(s).

Such zeta functions were first studied by Hey and Tamagawa in the case that G = GL_{l+1} where Z_{G,ρ}(s) = ζ(s)...ζ(s−l). Note that this zeta function is precisely the zeta function ζ_{ℤ^{l+1}}(s). More generally in [13] the zeta functions Z_{G,ρ}(s) are shown to count subgroups H in a nilpotent group Γ with the property that the profinite completions are isomorphic, i.e. Ĥ ≅ Γ̂. In this case the algebraic group G is the automorphism group of Γ. A result of Bryant and Groves shows that any algebraic group can be realised modulo a unipotent group as the automorphism group of a nilpotent group. In [6] an explicit expression is given for the local factors of a class of nilpotent groups in terms of the combinatorics of the building of the algebraic group. The local zeta functions Z_{G,ρ,p}(s) can be expressed in terms of cone integrals. Hence our results apply to these zeta functions.

Although our results imply we can meromorphically continue the zeta function Z_{G,ρ}(s) past its abscissa of convergence, this zeta function in general has a natural boundary, except for the case of G = GL_{l+1} (see [10]).

However we have discovered a procedure which produces something we call the ghost zeta function associated to Z_{G,ρ}(s) which turns out often to have a meromorphic continuation to the whole complex plane (see [7]).

(4) Let g(n, c, d) be the number of finite nilpotent groups of size n of class bounded by c and generated by at most d elements. In [11] the zeta function ζ_{N (c,d)} = ∑_{n=1}^{∞} g(n, c, d)n^{−s} is shown to be expressible as the Euler product of p-adic cone integrals. Hence the results of this paper imply that asymptotically g(n, c, d) behaves as follows:

g(1, c, d) + ⋯ + g(n, c, d) ∼ c · n^{α} (log n)^{b}

where α ∈ ℚ, b ∈ ℕ_{0} and c ∈ ℝ. The details are explained in [11]. (5) The Igusa zeta function of a polynomial f(x) is defined as

Z(s) = ∫_{ℤ_{p}^{m}} |f(x)|^{s} |dx| .

Hence it is a particular example of a cone integral where the cone condition ψ is empty. The global zeta function that one can define as the Euler product of these Igusa zeta functions (normalized to have constant coefficient 1) is a special case of our analysis.

Notation

ℕ denotes the set {1, 2, ...} .

ℕ_{0} denotes the set {0, 1, 2, ...} .

ℝ_{>0} denotes the set {s ∈ ℝ : s > 0} .

ℝ_{≥0} denotes the set {s ∈ ℝ : s ≥ 0} .

ℤ_{p}^{∗} denotes the units of ℤ_{p}.

f(n) ∼ g(n) means f(n)/g(n) tends to 1 as n tends to infinity.

Acknowledgements We should like to thank Jürgen Elstrodt for discussions concerning the Tauberian Theorem. The first author would like to thank the Royal Society, the Max-Planck-Institute in Bonn and the Heinrich Heine Universität in Düsseldorf for support and hospitality during the preparation of this paper.

2 An explicit formula for cone integrals

In this section we give a proof of Theorem 3. We recall from the introduction the definition of a cone integral:

Definition 8 (1) Call a formula ψ(x) in the first order language for the valued field ℚ_{p} a cone condition over ℚ if there exist polynomials f_{i}(x), g_{i}(x)(i = 1, ..., l) over ℚ in the variables x = x_{1}, ..., x_{m} such that ψ(x) is a conjunction of formulas

v(f_{i}(x)) ≤ v(g_{i}(x))

for i = 1, ..., l.

(2) Given a cone condition ψ(x) over ℚ and polynomials f_{0} and g_{0} with coefficients in ℚ, we call an integral

Z_{D}(s, p) = ∫_{V_{p}={x∈ℤ_{p}^{m}:ψ(x) is valid}} |f_{0}(x)|^{s} |g_{0}(x)| |dx|

a cone integral defined over ℚ, where |dx| is the normalized additive Haar measure on ℤ_{p}^{m} and D = {f_{0}, g_{0}, f_{1}, g_{1}, ..., f_{l}, g_{l}} is called the cone integral data.

We are going to use resolution of singularities to get an explicit formula for such cone integrals valid for almost all primes p. We follow section 5 of [3].

Definition 9 A resolution (Y, h) for a polynomial F over ℚ consists of a closed integral subscheme Y of P_{X_{ℚ}}^{k} (where X_{ℚ} = Specℚ[x] and P_{X_{ℚ}}^{k} denotes projective k-space over the scheme X_{ℚ}) and the morphism h : Y → X which is the restriction to Y of the projection morphism P_{X_{ℚ}}^{k} → X_{ℚ}, such that

(i) Y is smooth over Specℚ ;

(ii) the restriction h : Y \h^{−1}(D) → X\D is an isomorphism (where D = Spec (\frac{ℚ[x]}{(F)}) ⊂ X_{ℚ}) ; and

(iii) the reduced scheme (h^{−1}(D))_{red} associated to h^{−1}(D) has only normal crossings (as a subscheme of Y).

Let E_{i}, i ∈ T, be the irreducible components of (h^{−1}(D))_{red}. For i ∈ T, let N_{i} be the multiplicity of E_{i} in the divisor of F ∘ h on Y and let ν_{i} − 1 be the multiplicity of E_{i} in the divisor of h^{∗}(dx_{1} ∧ ... ∧ dx_{m}). The (N_{i}, ν_{i}) i ∈ T are called the numerical data of the resolution (Y, h) for F.

Let us recall some necessary facts about reduction of varieties modp. When X = X_{ℚ} = Specℚ[x] one defines the reduction modp of a closed integral subscheme Y of P_{X_{ℚ}}^{k} as follows : let X̃ = Specℤ[x] and Ỹ be the scheme-theoretic closure of Y in Pk ; X̃ . Then the reduction modp of Y is the scheme Ỹ ×_{ℤ}Spec𝔽_{p} and we denote it by \overline{Y}. Let h̃ : Ỹ → X̃ be the restriction to Ỹ of the projection morphism Pk ; X̃ → X̃_{ℚ} and \overline{h} : \overline{Y} → \overline{X} be obtained from h̃ by base extension.

Definition 10 A resolution (Y, h) for F over ℚ has good reduction modp if (1) \overline{Y} is smooth over Spec𝔽_{p};

(2) \overline{E_{i}} is smooth over Spec𝔽_{p}, for each i ∈ T, and ⋃_{i∈T} \overline{E_{i}} has only normal crossings as a subscheme of \overline{Y}; and

(3) \overline{E_{i}} and \overline{E_{j}} have no common irreducible components, when i ≠ j.

Note that a resolution over ℚ has good reduction for almost all primes p (see Theorem 2.4 of [3]).

Let (Y^{o}, h^{o}) be a resolution for the polynomial F = f_{0}.g_{0}. ∏_{i=1}^{l} f_{i}g_{i} over ℚ, and p any prime such that (Y^{o}, h^{o}) has good reduction modpℤ_{p} and \overline{f_{0}.g_{0}. ∏_{i=1}^{l} f_{i}g_{i}} ≠ 0. Here \overline{·} means reduction modp . Let (Y, h) be the resolution over ℚ_{p} obtained from (Y^{o}, h^{o}) by base extension.

Let a ∈ \overline{Y} (𝔽_{p}). Since we consider \overline{Y} as a closed subscheme of Ỹ , a is also a closed point of Ỹ . Let T_{a} = \{i ∈ T : a ∈ \overline{E_{i}}\} = \{i ∈ T : a ∈ Ẽ_{i}\} . Let r = cardT_{a} and T_{a} = {i_{1}, ..., i_{r}} . Then in the local ring O_{Ỹ ,a} we can write

F ∘ h̃ = uc_{1}^{N_{i_{1}}}...c_{r}^{N_{ir}}

where c_{i} ∈ O_{Ỹ ,a} generates the ideal of Ẽ_{i_{j}} in O_{Ỹ ,a} and u is a unit in O_{Ỹ ,a}. Since f_{i} and g_{i} divide F we can also write for i = 0, ..., l

f_{i} ∘ h̃ = u_{f_{i}}c_{1}^{N_{i_{1}}(f_{i})}...c_{r}^{N_{ir} (f_{i})}
g_{i} ∘ h̃ = u_{g_{i}}c_{1}^{N_{i_{1}}(g_{i})}...c_{r}^{N_{ir} (g_{i})}

Put

J_{a}(s, p) = ∫_{θ^{−1}(a)∩h^{−1}(V_{p})} |f_{0} ∘ h|^{s} |g_{0} ∘ h| |h^{∗}(dx_{1} ∧ ... ∧ dx_{m})|

where we define θ as follows: let H = {b ∈ Y (ℚ_{p}) : h(b) ∈ ℤ_{p}^{m}}. A point b ∈ H ⊂ Y (ℚ_{p}) can be represented by its coordinates (x_{1}, ..., x_{m}, y_{0}, ..., y_{k}) in ℚ_{p}^{m} × P_{X}^{k}(ℚ_{p}) where (x_{1}, ..., x_{m}) ∈ ℤ_{p}^{m} and y_{0}, ..., y_{k} are homogeneous coordinates which can therefore be chosen such that min_{i=0,...,k} ordy_{i} = 0. The map θ : H → \overline{Y} (𝔽_{p}) is then defined as follows: θ(b) = (\overline{x_{1}}, ..., \overline{x_{m}}, \overline{y_{0}}, ..., \overline{y_{k}}) ∈

\overline{Y} (𝔽_{p}) ⊂ Pk ; \overline{X}(𝔽_{p}).

Then Z_{D}(s, p) = ∑_{a∈\overline{Y} (𝔽_{p})} J_{a}(s, p). Now we have

J_{a}(s, p) = ∫_{θ^{−1}(a)∩h^{−1}(V_{p})} |c_{1}|^{N_{i_{1}}(f_{0})s+N_{i_{1}}(g_{0})+ν_{i_{1}}−1} ... |c_{r}|^{N_{ir} (f_{0})s+N_{ir} (g_{0})+ν_{ir} −1} |dc_{1} ∧ ... ∧ dc_{m}|

Since \overline{c_{1}}, ..., \overline{c_{m}} belong to the maximal ideal of O_{\overline{Y} ,a}, we have that c_{1}(b), ..., c_{m}(b) ∈ pℤ_{p} for all b ∈ θ^{−1}(a). The map

c : θ^{−1}(a) → (pℤ_{p})^{m}
b ↦ (c_{1}(b), ..., c_{m}(b))

is a bijection. Hence

J_{a}(s, p) = ∫_{V′_{p}} |y_{1}|^{N_{i_{1}}(f_{0})s+N_{i_{1}}(g_{0})+ν_{i_{1}}−1} ... |y_{r}|^{N_{ir} (f_{0})s+N_{ir} (g_{0})+ν_{ir} −1} |dy_{1}| ... |dy_{m}|

(5)

where V′_{p} is the set of all y = (y_{1}, ..., y_{m}) ∈ (pℤ_{p})^{m} satisfying for i = 1, ..., l

∑_{j=1}^{r} N_{i_{j}}(f_{i})ord(y_{j}) ≤ ∑_{j=1}^{r} N_{i_{j}}(g_{i})ord(y_{j}).

Let A_{j,a} = N_{i_{j}}(f_{0}) and B_{j,a} = N_{i_{j}}(g_{0}) + ν_{i_{j}} for j = 1, ..., r and A_{j,a} = 0, B_{j,a} = 1 for j > r. Then

J_{a}(s, p) = ∑_{(k_{1},...,k_{m})∈Λ} p^{− ∑_{j=1}^{m} k_{j}(A_{j,a}s+B_{j,a}−1)}(p^{−k_{1}} − p^{−k_{1}−1})...(p^{−k_{m}} − p^{−k_{m}−1})
= (1 − p^{−1})^{m} ∑_{(k_{1},...,k_{m})∈Λ} p^{− ∑_{j=1}^{m} k_{j}(A_{j,a}s+B_{j,a})}

where A_{j,a} ∈ ℕ_{0} and B_{j,a} ∈ ℕ_{0} and

Λ = {(k_{1}, ..., k_{m}) ∈ ℕ^{m} : ∑_{j=1}^{r} N_{i_{j}}(f_{i})k_{j} ≤ ∑_{j=1}^{r} N_{i_{j}}(g_{i})k_{j} for i = 1, ..., l} .

Thus Λ is the intersection of ℕ^{m} and a rational convex polyhedral cone C in ℝ_{>0}^{m}. We can write this cone as a disjoint union of simplicial cones C_{1}, ..., C_{w} of the form:

C_{i} = {α_{1}v_{i1} + ⋯ + α_{m_{i}}v_{im_{i}} : α_{j} ∈ ℝ, α_{j} > 0 for j = 1, ..., m_{i}}

where {v_{i1}, ..., v_{im_{i}}} is a linearly independent set of vectors in ℝ^{m}. Then Λ can be written as the disjoint union of the following sets

Λ_{i} = {l_{1}v_{i1} + ⋯ + l_{m_{i}}v_{im_{i}} : l_{j} ∈ ℕ for j = 1, ..., m_{i}} .

Now v_{jk} = (q_{jk1}, ..., q_{jkm}) ∈ ℝ_{>0}^{m} for k = 1, ..., m_{j}. Hence

J_{a}(s, p) = (1 − p^{−1})^{m} ∑_{j=1}^{w} ∏_{k=1}^{m_{j}} \frac{p^{−(A_{k,a,j}s+B_{k,a,j})}}{1 − p^{−(A_{k,a,j}s+B_{k,a,j})}}

where A_{k,a,j} = ∑_{i=1}^{m} q_{jki}A_{i,a} ∈ ℝ_{>0} and B_{k,a,j} = ∑_{i=1}^{m} q_{jki}B_{i,a} ∈ ℝ_{>0}.

Notice that the above calculations just depended on which components \overline{E_{i}} contained a. If T_{a_{1}} = T_{a_{2}} then J_{a_{1}}(s, p) = J_{a_{2}}(s, p). So for each I ⊂ T let

c_{p,I} = card\{a ∈ \overline{Y} (𝔽_{p}) : a ∈ \overline{E_{i}} if and only if i ∈ I\}

and put A_{k,I,j} = A_{k,a,j} and B_{k,I,j} = B_{k,a,j} for any a ∈ \{a ∈ \overline{Y} (𝔽_{p}) : a ∈ \overline{E_{i}} if and only if i ∈ I} where j = 1, ..., w_{I} and w_{I} is the number of simplicial cones defined by the linear inequalities corresponding to I. Then we have a final formula for Z_{D}(s, p)

Z_{D}(s, p) = (1 − p^{−1})^{m} ∑_{I⊂T} c_{p,I} ∑_{j=1}^{w_{I}} ∏_{k=1}^{m_{j}} \frac{p^{−(A_{k,I,j}s+B_{k,I,j})}}{1 − p^{−(A_{k,I,j}s+B_{k,I,j})}} .  (6)

Note that if A_{k,I,j} = 0 and B_{k,I,j} = 1, which will correspond to a bit of the integral like ∫_{pℤ_{p}} |dy_{m}|, then we get (1 − p^{−1}) · \frac{p^{−1}}{(1−p^{−1})} which is correct.

This completes the proof of Theorem 3. Note that this expression (6) for Z_{D}(s, p) holds for all primes for which the resolution (Y, h) had good reduction.

We could also consider a cone integral defined over ℚ_{p} whose cone data D consisted of polynomials in ℚ_{p}[x]. Our formula (6) would still be valid for such integrals provided that the resolution had good reduction modp.

Notice that, as we vary p, the only things in this formula which depend on p are the terms c_{p,I}.

We should note that there is one term which is always a constant term in the expression for our final formula (6) corresponding to the subset I = ∅; then w_{∅} = 1, m_{1} = m and A_{k,∅,1} = 0 and B_{k,∅,1} = 1 for k = 1, ..., m. Hence the term corresponding to the subset I = ∅ has the following form:

(1−p^{−1})^{m}c_{p,∅} ∑_{j=1}^{w_{∅}} ∏_{k=1}^{m_{j}} \frac{p^{−(A_{k,∅,j}s+B_{k,∅,j})}}{1 − p^{−(A_{k,∅,j}s+B_{k,∅,j})}} = c_{∅}(1−p^{−1})^{m} \frac{p^{−m}}{(1 − p^{−1})^{m}} = c_{∅}p^{−m}.

(7)

Since the restriction h : Y \h^{−1}(D) → X\D is an isomorphism (where D = Spec (\frac{ℚ[x]}{(F)}) ⊂ X_{ℚ})

c_{p,∅} = card\{a ∈ \overline{Y} (𝔽_{p}) : a ∉ \overline{E_{i}} for all i ∈ T \}
= card\overline{X}(𝔽_{p}) − card\overline{D}(𝔽_{p}).

The term (7) is part of the constant term of the rational function Z_{D}(s, p). The other parts of the constant term come from those I ⊂ T and j ∈ {1, ..., w_{I}} such that A_{k,I,j} = 0 for all k = 1, ..., m_{j}.

3 A second explicit expression for cone integrals

The explicit expression (6) determined in the previous section has a number of advantages. It expresses the function as a sum over the subsets of I which identifies precisely the bits c_{p,I} which depend on p. This form of the sum is also more amenable to Denef and Meuser’s proof that the Igusa local zeta function (where ψ is the empty condition) satisfies a functional equation (see [4]).

However for the analysis of the analytic properties of the global zeta function, as we explained in the introduction, it is preferable to work with a second explicit formula we can establish which writes the cone integrals as a sum over open simplicial pieces of a single cone defined in cardT dimensions where each open simplicial piece of the cone gets a weight according to the size of I and c_{p,I}.

We give a proof in this section of Theorem 4 where all the data in the formula e.g. A_{j} and B_{j} are identified explicitly in terms of the numerical data of the resolution and the underlying cone.

The cone is defined as follows:

\overline{D_{T}} = \{(x_{1}, ..., x_{t}) ∈ ℝ_{≥0}^{t} : ∑_{j=1}^{t} N_{j}(f_{i})x_{j} ≤ ∑_{j=1}^{t} N_{j}(g_{i})x_{j} for i = 1, ..., l\}

where cardT = t and ℝ_{≥0} = {x ∈ ℝ : x ≥ 0} , so this is a closed cone. Denote the lattice points in \overline{D_{T}} by \overline{Δ_{T}} , i.e. \overline{Δ_{T}} = \overline{D_{T}} ∩ℕ_{0}^{t}. We can write this cone as a disjoint union of open simplicial pieces that we shall call R_{k} k = 0, 1, ..., w. We shall assume that R_{0} = (0, ...0) and that the next q pieces are all the open one-dimensional edges in the cone \overline{D_{T}}: for k = 1, ..., q

R_{k} = {αe_{k} = α(q_{k1}, ..., q_{kt}) : α > 0} .

Since these are all the edges, for any k ∈ {0, ..., w} there exists some subset M_{k} ⊂ {1, ..., q} such that

R_{k} = { ∑_{j∈M_{k}} α_{j}e_{j} : α_{j} > 0 for all j ∈ M_{k}} .

Note that m_{k} := cardM_{k} ≤ t.

Define for each k = 1, ..., q the following constants:

A_{k} = ∑_{j=1}^{t} q_{kj}N_{j}(f_{0})  (8)

B_{k} = ∑_{j=1}^{t} q_{kj} (N_{j}(g_{0}) + ν_{j}) .

For each subset I ⊂ T we previously defined a rational convex polyhedral cone C_{I} with lattice points Λ_{I} which we broke down into simplicial cones C_{1}^{I}, ..., C_{w_{I}}^{I} with corresponding lattice points Λ_{1}^{I}, ..., Λ_{w_{I}}^{I}. These were cones in the open positive quadrant ℝ_{>0}^{m}. We are going to define a new cone and lattice points inside C_{T} which we can use to express the same rational function that we associated to C_{I}.

For each I ⊂ T define:

D_{I} = \{(k_{1}, ..., k_{t}) ∈ \overline{D_{T}} : k_{i} > 0 if i ∈ I and k_{i} = 0 if i ∈ T \I\}
Δ_{I} = D_{I} ∩ ℕ_{0}^{t}.

So \overline{D_{T}} = ⋃_{I⊂T} D_{I}, a disjoint union. The reason we chose the notation \overline{D_{T}} is of course because it is the closure then of D_{T} defined as above.

For each I ⊂ T there is then a subset W_{I} ⊂ {0, ..., w} so that

D_{I} = ⋃_{k∈W_{I}} R_{k}.

We now have the following:

Lemma 11

(1 − p^{−1})^{m}c_{p,I} ∑_{j=1}^{w_{I}} ∏_{k=1}^{m_{j}} \frac{p^{−(A_{k,I,j}s+B_{k,I,j})}}{1 − p^{−(A_{k,I,j}s+B_{k,I,j})}}

= c_{p,I} ∑_{k∈W_{I}}(1 − p^{−1})^{|I|}p^{−(m−|I|)} ∏_{j∈M_{k}} \frac{p^{−(A_{j}s+B_{j})}}{1 − p^{−(A_{j}s+B_{j})}} .

Corollary 12 Set c_{p,k} = c_{p,I} and I_{k} = I if k ∈ W_{I}. Then for all primes p for which the resolution has good reduction we get

Z_{D}(s, p) = ∑_{k=0}^{w}(p − 1)^{|I_{k}|}p^{−m}c_{p,k} ∏_{j∈M_{k}} \frac{p^{−(A_{j}s+B_{j})}}{1 − p^{−(A_{j}s+B_{j})}}

Proof We go back to the integral expression (5) for J_{a}(s, p) where a ∈ \{a ∈ \overline{Y} (𝔽_{p}) : a ∈ \overline{E_{i}} if and only if i ∈ I\}. The calculations of the previous section gave:

J_{a}(s, p) = (1 − p^{−1})^{m} ∑_{j=1}^{w_{I}} ∏_{k=1}^{m_{j}} \frac{p^{−(A_{k,I,j}s+B_{k,I,j})}}{1 − p^{−(A_{k,I,j}s+B_{k,I,j})}} .

We can rewrite the expression for J_{a}(s, p) as

J_{a}(s, p) = p^{−(m−|I|)} ∫_{V′_{p}} ∏_{i∈I} |z_{i}|^{N_{i}(f_{0})s+N_{i}(g_{0})+ν_{i}−1} ∏_{i∈I} |dz_{i}|

where V′_{p} is now the set of (z_{i})_{i∈I} ∈ (pℤ_{p})^{|I|} satisfying for j = 1, ..., l

∑_{i∈I} N_{i}(f_{j})ord(z_{i}) ≤ ∑_{i∈I} N_{i}(g_{j})ord(z_{i}).

Looking at the definition of Δ_{I} = D_{I} ∩ ℕ_{0}^{t} ⊂ \overline{D_{T}} this then reduces to

J_{a}(s, p) = p^{−(m−|I|)}(1 − p^{−1})^{|I|} ∑_{(k_{1},...,k_{t})∈Δ_{I}} p^{− ∑_{j=1}^{t} k_{j}(N_{j}(f_{0})s+N_{j}(g_{0})+ν_{j})}
= ∑_{k∈W_{I}} p^{−(m−|I|)}(1 − p^{−1})^{|I|} ∑_{(k_{1},...,k_{t})∈R_{k}∩ℕ_{0}^{t}} p^{− ∑_{j=1}^{t} k_{j}(N_{j}(f_{0})s+N_{j}(g_{0})+ν_{j})}

since D_{I} = ⋃_{k∈W_{I}} R_{k}. Since

R_{k} ∩ ℕ_{0}^{t} = { ∑_{j∈M_{k}} α_{j}e_{j} : α_{j} ∈ ℕ for all j ∈ M_{k}}

by making a change of variable as in the previous section and using the definitions (8) of A_{k} and B_{k} for k = 1, ..., q we get

J_{a}(s, p) = ∑_{k∈W_{I}} p^{−(m−|I|)}(1 − p^{−1})^{|I|} ∏_{j∈M_{k}} \frac{p^{−(A_{j}s+B_{j})}}{1 − p^{−(A_{j}s+B_{j})}} .

This completes the proof of the lemma.

Recall that even if the dimension of the simplicial piece has gone down (i.e. m_{k} < |I| ), we will still get a (1 − p^{−1}) for each variable. For example, the integral ∫_{ℤ_{p}^{2}} |x|^{s}|y|^{s} over v(x) = v(y) is (1 − p^{−1})^{2}(1 − p^{−2(s+1)})^{−1}. The second point to note is that c_{p,I} = 0 for any I ⊂ T for which cardI > m, where recall m is the number of variables in the original integral.

4 Zeta functions defined as Euler products of cone integrals

We now turn to analysing the global behaviour of a product of these cone integrals over all primes p.

We make some normalisation of the cone integrals so that the constant coefficient of the local factors is 1.

Definition 13 Let Z_{D}(s, p) be a cone integral defined over ℚ. Then denote by a_{p,0} the constant coefficient of the power series in p^{−s} representing the rational function Z_{D}(s, p).

We should point out that even for primes p for which the resolution (Y, h) has bad reduction, the cone integral Z_{D}(s, p) is a rational function in p^{−s}. We just don’t get the nice expressions of the previous two sections. The rationality follows from Denef’s proof that definable p-adic integrals are always rational functions in p^{−s} (see [2] ).

Definition 14 We say that a function Z(s) is defined as an Euler product of cone integrals over ℚ with cone integral data D if

Z(s) = Z_{D}(s) = ∏_{p prime} (a_{p,0}^{−1} · Z_{D}(s, p)) .

Theorem 15 A function Z(s) defined as an Euler product of cone integrals over ℚ has a rational abscissa of convergence α ∈ ℚ.

Proof We know that for each prime p, Z_{D}(s, p) is a rational function with rational abscissa of convergence. From our explicit formula for Z_{D}(s, p) which holds for almost all primes it will suffice to prove that for some finite set of primes S

∏_{p∉S} (a_{p,0}^{−1} · ∑_{k=0}^{w}(p − 1)^{|I_{k}|}p^{−m}c_{p,k} ∏_{j∈M_{k}} \frac{p^{−(A_{j}s+B_{j})}}{1 − p^{−(A_{j}s+B_{j})}} )

has a rational abscissa of convergence.

The following Lemma collects some relevant information about the constant a_{p,0} with which we are normalising our integrals:

Lemma 16 (1) a_{p,0} = lim_{s→∞} Z_{D}(s, p).

(2)

a_{p,0} = ∫_{{x∈ℤ_{p}^{m}:ψ(x) holds in ℤ_{p} and f_{0}(x)∈ℤ_{p}^{∗}}} |g_{0}(x)| |dx| .

(3) For each I ⊂ T, let W′_{I} denote the set of those k for which ∑_{j∈M_{k}} A_{j} ≠ 0. Then for almost all primes p

a_{p,0} = ∑_{I⊂T} p^{−m}c_{p,I} ∑_{k∈W_{I}\W′_{I}}(p − 1)^{|I|} ∏_{j∈M_{k}} \frac{p^{−B_{j}}}{1 − p^{−B_{j}}} .

Let T_{0} be the subset of T consisting of those i for which N_{i}(f_{0}) = 0. Then note that W′_{I} = ∅ for I ⊂ T_{0}.

(4) For almost all primes p

1 ≥ a_{p,0} ≥ (1 − p^{−1})^{m}.

(5) There exists M ∈ ℕ such that for almost all primes p

1 ≤ a_{p,0}^{−1} ≤ 1 + Mp^{−1/2}.

Proof Most of this is obvious. For (4) note that for almost all p

ℤ_{p}^{m} ⊇ {x ∈ ℤ_{p}^{m} : ψ(x) holds in ℤ_{p} and f_{0}(x) ∈ℤ_{p}^{∗}} ⊇ (ℤ_{p}^{∗})^{m}

and |g_{0}(x)| ≤ 1 on ℤ_{p}^{m} whilst |g_{0}(x)| = 1 on (ℤ_{p}^{∗})^{m} . Combining this with the description of a_{p,0} in (2) yields the estimate in (4). For (5) we use (1 − p^{−1})^{−1} = 1 + p^{−1/2}(p^{1/2} − p^{−1/2})^{−1} ≤ 1 + p^{−1/2}.

Put W′ = ⋃_{I⊂T} W′_{I}. Then we get:

Z_{D}(s) = ∏_{p} (1 + ∑_{k∈W′} \frac{c_{p,k}}{a_{p,0}} p^{−m}(p − 1)^{|I|} ∏_{j∈M_{k}} \frac{p^{−(A_{j}s+B_{j})}}{1 − p^{−(A_{j}s+B_{j})}} ) . (9)

We explain now a few facts about counting points on the reduction of varieties modp.

We shall be interested in counting points on \overline{X}, the reduction modp of a variety X defined over ℚ which is irreducible over ℚ. Let d be the dimension of X. The reduction \overline{X} need not be irreducible as a variety over 𝔽_{p}.

Let ℚ̂ be the algebraic closure of ℚ and 𝔾 its Galois group. Consider the decomposition

X = X_{1} ∪ ... ∪ X_{n}

of X into irreducible components over ℚ̂. The Galois group 𝔾 acts transitively on the set of components {X_{1}, ..., X_{n}} because X is defined over ℚ. Since the action is transitive all the X_{i} also have dimension d. Let U ≤ 𝔾 be the kernel of this action and put L = ℚ̂^{U}. Then L is a finite Galois extension of ℚ with Galois group G = 𝔾/U and every X_{i} (i = 1, ..., n) is defined over L.

For every prime number p we choose a prime ideal 𝔭 in L which divides p. We write I_{𝔭} ≤ D_{𝔭} ≤ 𝔾 for the corresponding inertia, respectively decomposition group. If p is unramified in L, that is I_{𝔭} = {1}, we denote by Frob_{p} the conjugacy class in G consisting of the Frobenius elements.

We choose a finite set of primes S such that for every p ∉ S the following are satisfied:

(1) the reduction \overline{X} modp of X is smooth; and

(2) p is unramified in L.

We define a function associated to the variety X which will be an essential tool in analysing our zeta functions.

Definition 17 Let X be a smooth projective variety defined over ℚ which is irreducible over ℚ. Define l_{p}(X) to be the number of irreducible components (defined over 𝔽_{p}) of \overline{X}, the reduction modp of X, which are absolutely irreducible. Define

V_{X}(s) = ∏_{p} (1 − l_{p}(X)p^{−s}) .

We prove the following important properties of this function V_{X}(s) :

Lemma 18 (1) The abscissa of convergence of V_{X}(s) is 1.

(2) There is a δ > 0 such that V_{X}(s) has a meromorphic continuation to the half-plane ℜ(s) > 1 − δ.

Proof We apply the set up introduced above. The finite group G acts on {X_{1}, ..., X_{n}} and we write M for the corresponding (complex) permutation module. We obtain a complex finite dimensional representation ρ : G → GL(M). Denote by Tr(ρ(Frob_{p})) the trace of a representation of the conjugacy class Frob_{p} where p ∉ S. We claim that

Tr(ρ(Frob_{p})) = l_{p}(X)  (10)

for all p ∉ S. To prove this we let \overline{X_{1}}, ..., \overline{X_{n}} be the reductions of X_{1}, ..., X_{n} mod𝔭. These are absolutely irreducible, smooth projective varieties defined over the residue field 𝔽_{𝔭} corresponding to the prime ideal 𝔭. Let G_{𝔭} be the Galois group of 𝔽_{𝔭} over 𝔽_{p}. Then G_{𝔭} acts on the set of components \overline{X_{1}}, ..., \overline{X_{n}} and the reduction mod𝔭 is an equivariant map from \{X_{1}, ..., X_{n}\} to \{\overline{X_{1}}, ..., \overline{X_{n}}\} with respect to the natural isomorphism from D_{𝔭} to G_{𝔭}. Hence the number of fixed points of G on {X_{1}, ..., X_{n}} is also equal to l_{p}(X) and formula (10) follows.

We are now ready to prove statement (1) of Lemma 18. Clearly l_{p}(X) is bounded by the number n of absolutely irreducible components of X. This implies that the abscissa of convergence of V_{X}(s) is less than or equal to 1. On the other hand the set of primes p ∉ S such that Frob_{p} = {1} has nonzero arithmetic density by Čebotarov’s theorem (see [18]). This establishes (1).

Finally we prove (2). We define

L_{X}(s) = ∏_{p∉S} det (1 − ρ(Frob_{p}) · p^{−s})^{−1} .

This Euler product is up to finitely many Euler factors equal to the Artin L-function of ρ. It is well known that L_{X}(s) has abscissa if convergence 1 and also has a meromorphic continuation to all of ℂ (see [18]). From formula (10) we infer that

det (1 − ρ(Frob_{p}) · p^{−s}) = 1 − l_{p}(X) · p^{−s} + ∑_{k=2}^{n} a_{k}p^{−ks}

with suitable a_{k}. It follows that

(1 − l_{p}(X) · p^{−s}) (det (1 − ρ(Frob_{p}) · p^{−s})^{−1}) = 1 + ∑_{k=2}^{∞} b_{k}p^{−ks}

with b_{k} ∈ ℂ. Since the eigenvalues of ρ(Frob_{p}) are roots of unity, the b_{k} can be bounded independently of p ∉ S. This proves (2).

Lemma 19 Let X be a smooth projective variety defined over ℚ which is irreducible over ℚ. Let d be the dimension of X. There exists δ ∈ ℝ such that for almost all primes p

|card(\overline{X}(𝔽_{p})) − l_{p}(X)p^{d}| ≤ δp^{d−1/2}

and l_{p}(X) > 0 for a dense set of primes.

Proof Let Y be an smooth projective irreducible variety over 𝔽_{p} which is not absolutely irreducible. Then Y (𝔽_{p}) is empty. To prove this let Y = Y_{1} ∪ ...∪Y_{r} be the decomposition into irreducible components over 𝔽̂_{p}. The Galois group of 𝔽̂_{p} over 𝔽_{p} acts transitively on the set of components Y_{1}, ..., Y_{r}. So an 𝔽_{p}-point of Y would be contained in every component. If there is more than one component then it is a singular point of Y. But Y was assumed to be a smooth variety. Hence no such 𝔽_{p}-point can exist.

Let Y be now a smooth projective, not necessarily irreducible variety over 𝔽_{p} and let Y = Y_{1} ∪ ... ∪ Y_{r} be the decomposition into the disjoint union of the irreducible components over 𝔽_{p}. Let them be ordered such that Y_{1}, ..., Y_{l} are absolutely irreducible and Y_{l+1}, ..., Y_{r} are not. Then the above argument shows that

card(Y (𝔽_{p})) = card(Y_{1}(𝔽_{p})) + ⋯ + card(Y_{l}(𝔽_{p})).

The statement of the lemma follows for all primes p such that reduction modp of X is smooth by the Lang-Weil estimate (see [15]). Note the irreducible components of \overline{X} all have dimension d, the dimension of X, because the irreducible components over 𝔽̂_{p} can be obtained from the irreducible components of X over ℚ̂ by reduction mod𝔭 as explained above.

The fact that l_{p}(X) > 0 for a dense set of primes follows from the proof of the previous Lemma.

We can use Lemma 19 to estimate the size of c_{p,I}. Recall that this is the number of points mod p on E_{I} \ ⋃_{j∈T \I} E_{j} where E_{I} = ⋂_{i∈I} E_{i}.

Let F_{I,k}, k ∈ C_{I}, be the irreducible components over ℚ of E_{I} with maximal dimension d_{I} say. We show later (see Proposition 24) that d_{I} = m − |I|.

Lemma 20 The dimension of F_{I,k} ∩ ⋃_{j∈T \I} E_{j} is strictly smaller than d_{I}. (If F_{j} has dimension zero then this means that F_{I} ∩ ⋃_{j∈T \I} E_{j} is the empty set, which by convention has dimension −∞.)

Proof If not then since F_{I,k} is a variety it means that F_{I,k} ⊂ ⋃_{j∈T \I} E_{j} and hence for some j ∉ I , E_{j} = F_{I,k} ⊂ E_{I}, which can’t happen.

Since the dimension of E_{I} ∩ ⋃_{j∈T \I} E_{j} is strictly smaller than d_{I}, this implies that for some δ′ > 0

|c_{p,I} − ∑_{k∈C_{I}} card\overline{F_{I,k}}(𝔽_{p})| ≤ δ′p^{d_{I}−1}.

But now this together with the Lang-Weil estimate in Lemma 19 implies:

Proposition 21 There exists δ ∈ ℝ such that for almost all primes p

|c_{p,I} − ∑_{k∈C_{I}} l_{p}(F_{I,k})p^{d_{I}}| ≤ δp^{d_{I}−1/2}

and l_{p}(F_{I,k}) > 0 for a dense set of primes p.

We make the following definitions for k ∈ W′:

Z_{k}(s) = \frac{c_{p,k}}{a_{p,0}} p^{−m}(p − 1)^{|I|} ∏_{j∈M_{k}} \frac{p^{−(A_{j}s+B_{j})}}{1 − p^{−(A_{j}s+B_{j})}}
d_{k} = d_{I} and I_{k} = I if k ∈ W_{I}.

If d_{k} ≥ 0 then put

α_{k} = max \{\frac{1 + d_{k} − m + |I_{k}| − ∑_{j∈M_{k}} B_{j}}{∑_{j∈M_{k}} A_{j}} , \frac{−B_{j}}{A_{j}} for j ∈ M_{k} and A_{j} ≠ 0\} .

If d_{k} = −∞, i.e. c_{p,k} = 0 for almost all p, then put α_{k} = −∞.

To analyse the convergence of our Euler product we use the following basic facts:

(A) An infinite product ∏_{n∈J}(1+a_{n}) converges absolutely if and only if the corresponding sum ∑_{n∈J} |a_{n}| converges.

(B) ∑_{p prime} |p^{−s}| converges if and only if {s ∈ ℂ : ℜ(s) > 1}. In fact if P is a dense set of primes we also have that ∑_{p∈P} |p^{−s}| converges if and only if {s ∈ ℂ : ℜ(s) > 1}.

(C) The Lang-Weil estimate in Proposition 21 means that a sum of the form

∑_{p} c_{p,I}r_{p}

converges absolutely if and only

∑_{p} p^{d_{I}}r_{p}

converges absolutely. Note that we are using here that l_{p}(F_{I,k}) is nonzero with positive density.

Lemma 22 For k ∈ W′, α_{k} is the abscissa of convergence of

∏_{p} (1 + Z_{k}(s)) .

Proof First note that each term \frac{p^{−(A_{j} s+B_{j})}}{1−p^{−(A_{j} s+B_{j})}} converges if and only if

ℜ(s) > \frac{−B_{j}}{A_{j}} .

Suppose that for some l ∈ M_{k} for which A_{l} ≠ 0, \frac{−B_{l}}{A_{l}} ≥ \frac{1+d_{k}−m+|I_{k}|−∑_{j∈M_{k}} B_{j}}{∑_{j∈M_{k}} A_{j}} . To prove that α_{k} is the abscissa of convergence it will suffice in this case to show that

∑_{p}(p − 1)^{|I_{k}|} \frac{c_{p,k}}{a_{p,0}} p^{−m} ∏_{j∈M_{k}} \frac{p^{−(A_{j}s+B_{j})}}{1 − p^{−(A_{j}s+B_{j})}}  (11)

converges for ℜ(s) > α_{k}. Since (1 − p^{−(A_{j}s+B_{j})})^{−1} is a positive decreasing sequence as p increases for ℜ(s) > \frac{−B_{j}}{A_{j}} if A_{j} ≠ 0 it suffices to show that

∑_{p}(p − 1)^{|I_{k}|} \frac{c_{p,k}}{a_{p,0}} p^{−m}p^{−(∑_{j∈M_{k}} A_{j}s+B_{j})}  (12)

converges.

From Lemma 16 (5) a_{p,0}^{−1} ≤ 1 + Mp^{−1/2}. Hence (12) converges provided

∑_{p}(p − 1)^{|I_{k}|}c_{p,k}p^{−m}p^{−(∑_{j∈M_{k}} A_{j}s+B_{j})}

converges. This sum converges (using our Lang-Weil argument (C)) if

ℜ(s) > \frac{1 + d_{k} − m + |I_{k}| − ∑_{j∈M_{k}} B_{j}}{∑_{j∈M_{k}} A_{j}} .

This is the case since ℜ(s) > α_{k} ≥ \frac{1+d_{k}−m+|I_{k}|−∑_{j∈M_{k}} B_{j}}{∑_{j∈M_{k}} A_{j}} .

Suppose now that α_{k} = \frac{1+d_{k}−m+|I_{k}|−∑_{j∈M_{k}} B_{j}}{∑_{j∈M_{k}} A_{j}} > \frac{−B_{j}}{A_{j}} for all j ∈ M_{k}.

Now there exist ε > 0, δ_{1}, δ_{2} > 0 such that α_{k} − ε > \frac{−B_{j}}{A_{j}} for all j ∈ M_{k}, and for ℜ(s) > α_{k} − ε and all primes p:

1 < ∏_{j∈M_{k}} (1 − p^{−(A_{k,I,j}s+B_{k,I,j})})^{−1} < 1 + δ_{1}p^{−δ_{2}}.

Hence, combining this with Lemma 16 (5) and our Lang-Weil argument (C), for ℜ(s) > α_{k} − ε the sum (11) converges if and only if the sum

∑_{p}(p − 1)^{|I_{k}|}p^{d_{k}}p^{−m}p^{−(∑_{j∈M_{k}} A_{j}s+B_{j})}

i.e. if and only if ℜ(s) > α_{k}. This completes our proof that α_{k} is the abscissa of convergence of ∏_{p} (1 + Z_{k}(s)) .

Corollary 23 The abscissa of convergence of Z_{D}(s), α_{D}, is equal to max {α_{k} : k ∈ W′} . In particular, α_{D} is a rational number.

The next task is to show that we can meromorphically continue Z_{D}(s) a little beyond ℜ(s) > α_{D} to allow an application of the Tauberian Theorem. The first step is to show that the maximal value of α_{k} as k varies over the set W′ ⊂ W (where recall W indexes the open simplicial pieces R_{k} of \overline{D_{T}} ) is realised only by the one dimensional simplicial edges, i.e. k ∈ {1, ..., q} .

The essential fact that we shall use here is the following:

Proposition 24 For k ∈ W, d_{k} = m − |I_{k}|.

Proof Each E_{i} is of dimension m − 1. The E_{i} intersect with normal crossings hence the irreducible components of ⋂_{i∈I_{k}} E_{i} have dimension d_{k} = m − |I_{k}|. For further details see the proof of Theorem 2.4 of [3] or 17.F of [19].

This already gives us a much simpler description of α_{k} :

Corollary 25 (1) For k ∈ W′

α_{k} = max \{\frac{1 − ∑_{j∈M_{k}} B_{j}}{∑_{j∈M_{k}} A_{j}} , \frac{−B_{j}}{A_{j}} for j ∈ M_{k} and A_{j} ≠ 0\} .

(2) If k ∈ W′ ∩ {1, ..., q} then

α_{k} = \frac{1 − B_{k}}{A_{k}} .

Lemma 26

α_{D} = max {α_{k} : k ∈ W′} = max {α_{k} : k ∈ W′ ∩ {1, ..., q}}

and α_{k} < α_{D} if k > q.

Proof This follows once we can show that for k > q and k ∈ W′

\frac{1 − ∑_{j∈M_{k}} B_{j}}{∑_{j∈M_{k}} A_{j}} < max \{\frac{1 − B_{j}}{A_{j}} : j ∈ M_{k} ∩ W′\}  (13)

Recall the definition of W′ from Lemma 16 that k ∈ W′ if ∑_{j∈M_{k}} A_{j} ≠ 0. We may suppose that if j ∈ M_{k} then j ∈ W′ since

\frac{1 − ∑_{j∈M_{k}} B_{j}}{∑_{j∈M_{k}} A_{j}} ≤ \frac{1 − ∑_{j∈M_{k}∩W′} B_{j}}{∑_{j∈M_{k}∩W′} A_{j}} .

We may suppose without loss of generality that |M_{k}| = 2, \frac{1−B_{2}}{A_{2}} ≤ \frac{1−B_{1}}{A_{1}} and that A_{1} and A_{2} > 0. Then

\frac{1 − (B_{1} + B_{2})}{A_{1} + A_{2}} < \frac{1 − B_{1}}{A_{1}}

if and only if

A_{1} − A_{1}B_{1} − A_{1}B_{2} < A_{1} + A_{2} − A_{1}B_{1} − A_{2}B_{1}.

But our assumptions that \frac{1−B_{2}}{A_{2}} ≤ \frac{1−B_{1}}{A_{1}} and A_{2} > 0 imply this second inequality. This confirms statement (13) and hence the lemma.

Theorem 27 ∏_{p} (1 + ∑_{j=1}^{w} Z_{j}(s)) has a meromorphic continuation to ℜ(s) > α_{D} − δ for some δ > 0.

Proof Define

R = \{k ∈ W′ ∩ \{1, ..., q\} : \frac{1 − B_{j}}{A_{j}} = α_{D} \}
V_{p}(s) = ∏_{k∈R} (1 − c_{p,k}p^{−m+|I_{k}|}p^{−(A_{k}s+B_{k})}) .

We introduce the following notation which will be convenient during the course of the proof. Write ∏_{p} F_{p}(s) ≡ ∏_{p} G_{p}(s) if there exists δ > 0 such that ∑_{p}(F_{p}(s) − G_{p}(s)) converges for ℜ(s) > α_{D} − δ. To prove our lemma it will suffice to prove the following:

(1) ∏_{p} V_{p}(s) is a meromorphic function on ℜ(s) > α_{D} − δ for some δ > 0.

(2) ∏_{p} (1 + ∑_{j=1}^{w} Z_{j}(s)) W_{p}(s)V_{p}(s) ≡ 1, i.e. the Euler product converges on ℜ(s) > α_{D} − δ for some δ > 0.

We prove (2). It will be convenient to note the following fact: suppose that for s > α_{D} − δ, the function X_{p}(s) as p → ∞ is a positive decreasing sequence. If ∏_{p} F_{p}(s) ≡ ∏_{p} G_{p}(s) then ∏_{p} F_{p}(s)X_{p}(s) ≡ ∏_{p} G_{p}(s)X_{p}(s).

By Lemma 26 ∏_{p} (1 + ∑_{j=1}^{w} Z_{j}(s)) ≡ ∏_{p} (1 + ∑_{k∈R_{1}∪R_{0}} Z_{k}(s)) . Recall the definition of Z_{k}(s) for k ∈ {1, ..., q} :

Z_{k}(s) = \frac{c_{p,k}}{a_{p,0}} p^{−m}(p − 1)^{|I_{k}|} \frac{p^{−(A_{k}s+B_{k})}}{1 − p^{−(A_{k}s+B_{k})}} .

Note that ∑_{p} c_{p,k}p^{−m+|I_{k}|} \frac{p^{−2(A_{k}s+B_{k})}}{1−p^{−(A_{k}s+B_{k})}} converges on ℜ(s) > α_{D} − δ for some δ > 0. This follows because (1 − p^{−(A_{k}s+B_{k})})^{−1} is a positive decreasing sequence and \frac{(1−2B_{k})}{2A_{k}} < \frac{1−B_{k}}{A_{k}} . Putting this together with our analysis on the estimate for a_{p,0} means that ∏_{p} (1 + ∑_{k∈R_{1}∪R_{0}} Z_{k}(s)) ≡ ∏_{p} (1 + ∑_{k∈R_{1}∪R_{0}} \overline{Z_{k}(s)}) where

\overline{Z_{k}(s)} = c_{p,k}p^{−m+|I_{k}|}p^{−(A_{k}s+B_{k})}.

Hence we have

∏_{p} (1 + ∑_{j=1}^{w} Z_{j}(s)) ≡ ∏_{p} (1 + ∑_{k∈R} \overline{Z_{k}(s)}) .  (14)

Now

∏_{p} V_{p}(s) ≡ ∏_{p} (1 − ∑_{k∈R} c_{p,k}p^{−m+|I_{k}|}p^{−(A_{k}s+B_{k})}) = ∏_{p} (1 − ∑_{k∈R} \overline{Z_{k}(s)})

(15)

since \frac{(1−B_{k_{1}}−...−B_{k_{i}})}{A_{k_{1}}+...+A_{k_{i}}} < \frac{1−B_{k}}{A_{k}} for i > 1. Now from (14) and (15) we get that

∏_{p} (1 + ∑_{j=1}^{w} Z_{j}(s)) V_{p}(s) ≡ ∏_{p} (1 + ∑_{j=1}^{w} Z_{j}(s)) (1 − ∑_{k∈R} \overline{Z_{k}(s)})
≡ ∏_{p} (1 + ∑_{k∈R} \overline{Z_{k}(s)}) (1 − ∑_{k∈R} \overline{Z_{k}(s)}) .

Here we are using that ∑_{j=1}^{w} Z_{j}(s) and ∑_{k∈R} c_{p,k}p^{−m+|I_{k}|}p^{−(A_{k}s+B_{k})} are both decreasing positive sequences. Finally the fact that \frac{(1−B_{k_{1}}−B_{k_{2}})}{A_{k_{1}}+A_{k_{2}}} < \frac{1−B_{k_{i}}}{A_{k_{i}}} implies that

∏_{p} (1 + ∑_{k∈R} \overline{Z_{k}(s)}) (1 − ∑_{k∈R} \overline{Z_{k}(s)}) ≡ 1.

Hence we have proved (2).

For the proof of (1) note that by Proposition 21

∏_{p prime} (1 − c_{p,k}p^{−m+|I_{k}|}p^{−(A_{k}s+B_{k})}) ≡ ∏_{p prime} (1 − ∑_{j∈C_{I}} l_{p}(F_{I_{k},j})p^{d_{k}}p^{−m+|I_{k}|}p^{−(A_{k}s+B_{k})})
≡ ∏_{j∈C_{I}} ∏_{p prime} (1 − l_{p}(F_{I_{k},j})p^{−(A_{k}s+B_{k})})

Hence the fact that ∏_{p} V_{p}(s) is meromorphic on ℜ(s) > α_{D} − δ for some δ > 0 follows from Lemma 18. This completes the proof of Theorem 27.

To prove our main Theorem about the growth of the coefficients in a Dirichlet series defined as the Euler product of cone integrals over ℚ we now can apply the Tauberian Theorem. We take the following statement from page 62 of [1]:

Theorem 28 Suppose F (s) = \frac{1}{s} ∫_{0}^{∞} e^{−st}dA(t) for ℜ(s) > a > 0 , A(t) > 0, and that for ℜ(s) > a

F (s) = g(s)(s − a)^{−w} + h(s)

where g and h are holomorphic in a neighbourhood of a with g(a) ≠ 0 and w > 0. Then

A(t) ∼ \frac{g(a)}{Γ(w)} · e^{at}t^{w−1}.

In our case we take A(t) = ∑_{log n<t} a_{n} and f(s) = ∑_{n=1}^{∞} a_{n}n^{−s} = sF (s) then substituting t = log x we get:

Theorem 29 Let the Dirichlet series f(s) = ∑_{n=1}^{∞} a_{n}n^{−s} with non-negative coefficients be convergent for ℜ(s) > a > 0. Assume in a neighbourhood of a, f(s) = g(s)(s−a)^{−w}+h(s), holds where g(s), h(s) are holomorphic functions, g(a) ≠ 0 and w > 0. Then for x tending to infinity, we have

∑_{n≤x} a_{n} ∼ (\frac{g(a)}{aΓ(w)}) · x^{a}(log x)^{w−1}.

We collect together the results of this section in:

Corollary 30 Let Z(s) = ∑_{n=1}^{∞} a_{n}n^{−s} be defined as an Euler product of cone integrals over ℚ.

(1) The abscissa of convergence α of Z(s) is a rational number and Z(s) has a meromorphic continuation to ℜ(s) > α − δ for some δ > 0.

(2) Let the pole at s = α have order w. Then for every β with β + α > 0, there exists some real number c ∈ ℝ such that

a_{1} + ⋯ + a_{n}n^{β} ∼ c · n^{β+α} (log n)^{w−1} .

Note the we do not know anything about the possibility of other poles in the region α = ℜ(s) > α − δ other than the one at s = α. This is because we have used Artin L-functions to do our meromorphic continuation. However it is conjectured that these Artin L -functions actually have this one pole and no others (see [18]).

5 Nilpotent Groups

In this section we show that the zeta functions of nilpotent groups are Euler products of cone integrals over ℚ.

To simplify our analysis of subgroups and normal subgroups we use the following notation: ζ_{G}^{≤}(s) = ζ_{G}(s).

The following result established in [13] means we can concentrate on counting in rings rather than groups:

Theorem 31 Let G be a finitely generated nilpotent group. Then there is a Lie algebra L(G) over ℤ constructed as the image under log of some subgroup of finite index in G such that for ∗ ∈ {≤, ◁} and almost all primes p

ζ_{G,p}^{∗}(s) = ζ_{L(G),p}^{∗}(s).

The finitely many primes will not worry us thanks to the following result, proved first for torsion-free nilpotent groups in [13] and more generally by the first author in [5]:

Theorem 32 Let G be a finitely generated group of finite rank then, for ∗ ∈ {≤, ◁} and for each prime p, ζ_{G,p}^{∗}(s) is a rational function with rational abscissa of convergence α^{∗}(G)_{p} ∈ ℚ.

Recall that a group G has finite rank if there is a bound on the number of generators of finitely generated subgroups; for example this holds if G is a finitely generated nilpotent group or more generally a polycyclic group.

The proof of this rationality of the local factors depended on expressing them as “definable” p-adic integrals. We recall the description of these integrals in the case of a ring L which is additively isomorphic to ℤ^{d}. Fix a basis for L which identifies it with ℤ^{d}. The multiplication in L is given by a bilinear mapping

β : ℤ^{d} × ℤ^{d} → ℤ^{d},

which extends to a bilinear map ℤ_{p}^{d} × ℤ_{p}^{d} → ℤ_{p}^{d} for each prime p, giving the structure of the ℤ_{p}-algebra L_{p} = L ⊗ ℤ_{p}.

Define

V_{p}^{≤} = \{ \begin{matrix}(m_{kl}) ∈ Tr_{d} (ℤ_{p}) : for 1 ≤ i, j ≤ d, ∃Y_{ij}^{1}, ..., Y_{ij}^{d} ∈ ℤ_{p} \\ such that β(m_{i}, m_{j}) = ∑_{k=1}^{d} Y_{ij}^{k}m_{k} \end{matrix} \}

and

V_{p}^{◁} = \{ \begin{matrix}(m_{kl}) ∈ Tr_{d} (ℤ_{p}) : for 1 ≤ i, j ≤ d, ∃Y_{ij}^{1}, ..., Y_{ij}^{d} ∈ ℤ_{p} \\ such that β(m_{i}, e_{j}) = ∑_{k=1}^{d} Y_{ij}^{k}m_{k} \end{matrix} \}

where Tr_{d} (ℤ_{p}) denotes the set of upper triangular matrices. Each subset consists of matrices whose rows m_{i} generate a lattice in ℤ_{p}^{d} which is either a subalgebra or an ideal. The d tuple e_{j} denotes the standard unit vector with 1 in the jth entry and zeros elsewhere and corresponds to the jth basis element.

Proposition 33 ([13] Proposition 3.1) For ∗ ∈ {≤, ◁} and each prime p

ζ_{L,p}^{∗} = (1 − p^{−1})^{−d} ∫_{V_{p}^{∗}} |m_{11}|^{s−1} ... |m_{dd}|^{s−d} |dx|

where v is the valuation on ℤ_{p}, |m| = p^{−v(m)} and |dx| is the normalized additive Haar measure on ℤ_{p}^{d(d+1)/2} ≡ Tr_{d} (ℤ_{p}) .

The proof of the rationality of these p-adic integrals relied on observing that V_{p}^{∗} are definable subsets in the language of fields. One can then apply a Theorem of Denef’s [2] which establishes the rationality of definable p-adic integrals. Denef’s proof relies on an application of Macintyre’s quantifier elimination for the theory of ℚ_{p} which simplifies in a generally mysterious way the description of definable subsets like V_{p}^{∗}. However in our case it is possible to do this elimination by hand since it involves solving linear equations. This removes the necessity for the model theoretic black box in the proof of the rationality.

Theorem 34 Let L be a ring additively isomorphic to R^{d} where R = ℤ or ℤ_{p}. For ∗ ∈ {≤, ◁} there exist homogeneous polynomials g_{ijk}^{∗}(X) ∈ R[X_{rs} : 1 ≤ r ≤ s ≤ d] (i, j, k ∈ {1, ..., d}) of degree i for ∗ = ◁ and i + 1 for ∗ =≤ such that

V_{p}^{∗} = {(m_{rs}) ∈ Tr_{d} (ℤ_{p}) : v(m_{11}...m_{kk}) ≤ v(g_{ijk}^{∗}(m_{rs})) for i, j, k ∈ {1, ..., d}} .

Proof We can express the defining conditions for V_{p}^{∗} in matrix form which makes things quite transparent. Let C_{j} denote the matrix whose rows are c_{i} = β(e_{i}, e_{j}).

M ∈ V_{p}^{◁} if we can solve for each 1 ≤ i, j ≤ d the equation

m_{i}C_{j} = (Y_{ij}^{1}, ..., Y_{ij}^{d}) M

with (Y_{ij}^{1}, ..., Y_{ij}^{d}) ∈ ℤ_{p}^{d}. Let M′ denote the adjoint matrix and

M^{♮} = M′diag(m_{22}^{−1}...m_{dd}^{−1}, ..., m_{dd}^{−1}, 1).

Then since the matrix M is upper triangular, the ikth entry of M^{♮} is a homogeneous polynomial of degree k − 1 in the variables m_{rs} with 1 ≤ r ≤ s ≤ k − 1. Then we can rewrite the above equation as:

m_{i}C_{j}M^{♮} = (m_{11}Y_{ij}^{1}, ..., m_{11}...m_{dd}Y_{ij}^{d}) .

Let g_{ijk}^{◁}(m_{rs}) denote the kth entry of the d-tuple m_{i}C_{j}M^{♮} which is a homogeneous polynomial of degree k in m_{rs}. (In fact we can see that it is homogenous of degree 1 in m_{is} ( s = 1, ..., d) and degree k − 1 in m_{rs} with 1 ≤ r ≤ s ≤ k − 1.) Then V_{p}^{◁} has the description detailed in the statement of the Theorem.

For M ∈ V_{p}^{≤} we are required to solve for each 1 ≤ i, j ≤ d the equation

m_{i} (∑_{l=j}^{d} m_{jl}C_{l}) M^{♮} = (m_{11}Y_{ij}^{1}, ..., m_{11}...m_{dd}Y_{ij}^{d})

with (Y_{ij}^{1}, ..., Y_{ij}^{d}) ∈ ℤ_{p}^{d}. Let g_{ijk}^{≤}(m_{rs}) denote the kth entry of the d-tuple

m_{i} (∑_{l=j}^{d} m_{jl}C_{l}) M^{♮} which is a homogeneous polynomial of degree k + 1 in m_{rs}. Again with these polynomials, V_{p}^{≤} has the description detailed in the statement of the Theorem.

So our Theorem has the following corollary that the local zeta functions of the ring L can be represented by cone integrals over ℚ:

Corollary 35 Let ψ^{∗} be the cone condition defining V_{p}^{∗}. Set f_{0} = m_{11}...m_{dd} and g_{0} = m_{11}^{d−1}...m_{d−1,d−1}. Let D^{∗} be the associated cone integral data. Then

ζ_{L,p}^{∗}(s) = (1 − p^{−1})^{−d}Z_{D^{∗}}(s − d, p).

We know that the constant term of Z_{D^{∗}}(s − d, p) must be (1 − p^{−1})^{d} since ζ_{L,p}^{∗}(s) has constant term 1. So the global zeta function ζ_{G}^{∗}(s) is the Euler product of the cone integrals. Hence we get as a corollary of the work we have done on the zeta functions defined as Euler products of cone integrals the following:

Theorem 36 (1) Let G be a finitely generated nilpotent infinite group. Then there exist finitely many varieties E_{i}, i ∈ T defined over ℚ sitting in some ℚ-scheme Y and rational functions P_{I}(x, y) ∈ ℚ[x, y] for each I ⊂ T with the property that for almost all primes p

ζ_{G,p}^{∗}(s) = ∑_{I⊂T} c_{p,I}P_{I}(p, p^{−s})

where

c_{p,I} = card\{a ∈ \overline{Y} (𝔽_{p}) : a ∈ \overline{E_{i}} if and only if i ∈ I\}

and \overline{Y} means the reduction mod p of the scheme Y.

(2) The abscissa of convergence α^{∗}(G) of ζ_{G}^{∗}(s) is a rational number and ζ_{G}^{∗}(s) has a meromorphic continuation to ℜ(s) > α^{∗}(G) − δ for some δ > 0.

(3) There exist a non-negative integer b ∈ ℕ_{0} and some real number c ∈ ℝ such that

s_{n}(G) ∼ c · n^{α^{∗}(G)}(log n)^{b}.

Recall that in (1) the irreducible varieties arise from the resolution of singularities of F, the product of all the polynomials defining the cone integral. Let us identify this F = F_{L}^{∗} explicitly for the cone integrals defining the zeta function of the Lie algebra L = L(G) of G. The polynomial depends on a choice of basis for L which identifies it with ℤ^{d}. Let multiplication in L be given by the bilinear mapping β : ℤ^{d} × ℤ^{d} → ℤ^{d}. Let C_{j} denote the matrix whose rows are c_{i} = β(e_{i}, e_{j}). Let M = (m_{ij}) be a d×d upper triangular matrix and M^{♮} = M′diag(m_{22}^{−1}...m_{dd}^{−1}, ..., m_{dd}^{−1}, 1) where M′ denotes the adjoint matrix of M.

Definition 37 (1) F_{L}^{◁}(m_{ij}) is defined to be the product of m_{11}^{(d^{2}+1)d}...m_{ii}^{(d^{2}+1)(d−i+1)}...m_{dd}^{(d^{2}+1)} with all the entries in the d matrices MC_{j}M^{♮} for j = 1, ..., d.

(2) F_{L}^{≤}(m_{ij}) is defined to be the product of m_{11}^{(d^{2}+1)d}...m_{ii}^{(d^{2}+1)(d−i+1)}...m_{dd}^{(d^{2}+1)} with all the entries in the d matrices M (∑_{k=j}^{d} m_{jk}C_{k}) M^{♮} for j = 1, ..., d.

The expression m_{11}^{2d}...m_{ii}^{2(d−i+1)}...m_{dd}^{2} arises from the product of the two monomials in the integrand f_{0} = m_{11}...m_{dd} and g_{0} = m_{11}^{d−1}...m_{dd}^{d} and the monomials f_{ijk} = m_{11}...m_{kk} for i, j, k ∈ {1, ..., d} .

Definition 38 For a finitely generated nilpotent group G we define the polynomials F_{G}^{∗} = F_{L(G)}^{∗} for ∗ ∈ {≤, ◁} , where L(G) is the associated Lie algebra of G.

Again there is some choice involved in the definition of the Lie algebra L(G) as we have written it as it is the image under log of some subgroup of finite index in G. However up to finitely many primes, the explicit formulas associated to the cone integrals will be the same for polynomials F_{G}^{∗} = F_{L(G)}^{∗} arising from different choices of Lie algebra and basis for the Lie algebra.

Although we know that the constant term of the cone integrals is (1−p^{−1})^{d} it is instructive to see what the analysis of the resolution corresponding to F_{L}^{∗}(m_{ij}) tells us. The constant term corresponds to when the integrand is constant which occurs when f_{0} = m_{11}...m_{dd} is a unit, i.e. for those basis for the whole group G. Let T_{0} ⊂ T, (where recall T is the set indexing the irreducible components E_{i}), be defined as follows: if i ∈ T_{0} then N_{i}(f_{0}) ≠ 0. Note that this also implies that N_{i}(g_{0}) ≠ 0 since f_{0} = m_{11}...m_{dd} and g_{0} = m_{11}^{d−1}...m_{d−1d−1}. So those I ⊂ T with I ∩ T_{0} non-empty are precisely those that don’t contribute to the constant term. So the constant term comes from those I ⊂ T \T_{0}. We analyse now the data in our formula associated to such I. If I ⊂ T \T_{0} then for i ∈ I, N_{i}(f_{0}) = 0 and since f_{ijk} = m_{11}...m_{kk} this also implies that N_{i}(f_{ijk}) = 0. Hence there are no conditions on our subset since the conditions are

∑_{l=1}^{r} N_{i_{l}}(f_{ijk})ord(y_{l}) ≤ ∑_{l=1}^{r} N_{i_{l}}(g_{ijk})ord(y_{l}).

This implies then that w_{I} = 1, m_{1} = m and A_{k,I,1} = 0 and B_{k,I,1} = 1 for k = 1, ..., m as before in the case that I = ∅ considered at the end of section 2. Hence the constant term is

p^{−m} ∑_{I⊂T \T_{0}} c_{I}.

We know of course what the constant term should be for this integral, namely (1 − p^{−1})^{d}. Hence we get that

∑_{I⊂T \T_{0}} c_{I} = p^{m−d}(p − 1)^{d}.

This is the same size as the number of points over 𝔽_{p} in affine space of dimension m minus the hyperplanes defined by m_{ii} = 0. This means that the number of points has not changed outside of the union of hyperplanes m_{ii} = 0 after we have done the resolution. Does this mean that our variety is non-singular outside this union of hyperplanes?

We collect together here the present knowledge we have about the possible values for α^{∗}(G). Let G be a torsion-free, finitely nilpotent group. Let h denote the Hirsch length of the nilpotent group G and h_{ab} denote the Hirsch length of the abelianisation, h(G/G′) .

(1) h_{ab} ≤ α^{◁}(G) ≤ h.

(2) (3 − 2\sqrt{2})h − \frac{1}{2} ≤ α^{≤}(G) ≤ h. The lower bound, which for large h (h ≥ 17) exceeds h/6, is due to Segal and a proof will appear in [12].

(3) The paper [13] contains a number of examples of zeta functions of class two groups calculated by Geoff Smith. There are some other examples calculated by Dermott Grenham which are recorded in [10]. In all these examples α^{≤}(G) = α^{◁}(G) = h_{ab}. However this reflects the paucity and small nature of the nilpotent groups so far considered rather than a general feature. It is true that for class two nilpotent groups one can prove that α^{◁}(G) = h_{ab} (see [12]). However in the same paper [12] we shall show that the zeta function counting all subgroups in the free class two nilpotent group F_{2}^{3} on three generators has abscissa of convergence at α^{≤}(F_{2}^{3}) = 7/2. In particular this is the first example of a group for which the abscissa of convergence is not an integer.

At present we do not know anything about the order of the poles of ζ_{G}(s). Our analysis above might help in giving a bound in terms of the Hirsch length since the pole at the abscissa of convergence has order given by the number of times we need to multiply by the Artin L-function. This is determined by the number of irreducible components of E_{I} as I ⊂ T.

If the nilpotent group is not torsion-free then we do not know very much about the abscissa of convergence. Extending even the free abelian group by a finite group can have quite a dramatic effect on the movement of the poles of ζ_{G}(s). As an example compare the infinite cyclic group ℤ where ζ_{ℤ}(s) = ζ(s) and the infinite dihedral group D_{∞} where ζ_{D_{∞}}(s) = 2^{−s}ζ(s) + ζ(s − 1). So extending by a finite group (here C_{2}) therefore has quite a subtle effect on the lattice of subgroups even to the extent of changing the rate of polynomial growth.

In [8] explicit examples calculated by John McDermott are recorded for the wallpaper groups which show the effect of extending ℤ^{2} by a finite group. It reveals how sensitive the zeta function is as the nature of the poles varies dramatically.

References

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[10] M.P.F. du Sautoy, Natural boundaries for zeta functions of groups, preprint.

[11] M.P.F. du Sautoy, Counting finite p-groups and nilpotent groups, preprint.

[12] M.P.F. du Sautoy and F.J. Grunewald, Counting subgroups of finite index in nilpotent groups of class two. In preparation.

[13] F.J. Grunewald, D. Segal and G.C. Smith, Subgroups of finite index in nilpotent groups, Invent. Math., 93 (1988), 185-223.

[14] I. Ilani, Zeta functions related to the group SL_{2}(ℤ_{p}). To appear in the Israel J. of Math.

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[21] G. C. Smith, Zeta functions of torsion free finitely generated nilpotent groups, Ph.D. thesis, (Manchester (UMIST), 1983).