1 Introduction and statement of main results
1.1 Introduction
Let G be a group, and, for n∈ℕ, write
rn(G)=|{isomorphism classes ofn-dimensional irreducible complexrepresentations ofG}|,cn(G)=|{conjugacy classes ofGof cardinalityn}|.
If G is a topological group, we only consider continuous representations.
We study the bivariate zeta functions of groups associated to unipotent group schemes encoding either the numbers rn(Q) or the numbers cn(Q) of certain finite quotients Q of the infinite groups considered.
We first recall the definitions of (univariate) representation and conjugacy class zeta functions.
Definition 1.1.
Let G be a group and s a complex variable.
If all rn(G) are finite, then the representation zeta function of G is
ζGirr(s)=∑n=1∞rn(G)n-s.
If all cn(G) are finite, then the conjugacy class zeta function of G is
ζGcc(s)=∑n=1∞cn(G)n-s.
The groups considered in the present paper are groups associated to unipotent group schemes, which are obtained from nilpotent Lie lattices; see Section 3.
From here on, let K denote a number field and 𝒪 its ring of integers.
Let 𝐆 be a unipotent group scheme over 𝒪.
The group 𝐆(𝒪) is a finitely generated, torsion-free nilpotent group (𝒯-group for short);
see [20, Section 2.1.1].
We observe that, for a 𝒯-group G, the numbers rn(G) and cn(G) are not all finite.
For this reason, one cannot define representation and conjugacy class zeta functions of G as in Definition 1.1.
In the representation case, many authors have overcome this by considering zeta functions encoding the n-dimensional irreducible complex representations of such groups up to tensoring by one-dimensional representations; see Section 1.3.
Our idea is to investigate zeta functions encoding the relevant data rn(Q) or cn(Q) of principal congruence quotients Q of 𝐆(𝒪).
We define bivariate complex functions: one variable concerns either the dimensions of the representations considered or the cardinalities of the conjugacy classes considered, and the other variable concerns the principal congruence subgroups.
Definition 1.2.
The bivariate representation and the bivariate conjugacy class zeta functions of 𝐆(𝒪) are
𝒵𝐆(𝒪)irr(s1,s2)=∑(0)≠I⊴𝒪ζ𝐆(𝒪/I)irr(s1)|𝒪:I|-s2,
𝒵𝐆(𝒪)cc(s1,s2)=∑(0)≠I⊴𝒪ζ𝐆(𝒪/I)cc(s1)|𝒪:I|-s2,
respectively, where s1 and s2 are complex variables.
These series converge for s1 and s2 with sufficiently large real parts; see Section 2.1.
We remark that the zeta functions defined above depend not only on the group of 𝒪-points 𝐆(𝒪), but implicitly on the ring 𝒪 and the 𝒪-scheme 𝐆, which give rise to the finite quotients 𝐆(𝒪/I), modulo ideals I of 𝒪.
It is convenient and customary to use the short notation 𝐆(𝒪) in the index to indicate this more complex dependency.
In Proposition 2.4, we establish the Euler decompositions
𝒵𝐆(𝒪)∗(s1,s2)=∏𝔭∈Spec(𝒪)∖{(0)}𝒵𝐆(𝒪𝔭)∗(s1,s2),
where ∗∈{irr,cc} and 𝒪𝔭 is the completion of 𝒪 at the nonzero prime ideal 𝔭.
When considering a fixed prime ideal 𝔭, we write simply
𝔬=𝒪𝔭 and 𝐆N:-𝐆(𝔬/𝔭N).
With this notation, the local factor at 𝔭 is given by
(1.1)𝒵𝐆(𝒪𝔭)∗(s1,s2)=𝒵𝐆(𝔬)∗(s1,s2)=∑N=0∞ζ𝐆N∗(s1)|𝔬:𝔭|-Ns2.
Example 1.3.
Let 𝐆(𝒪) be the free Abelian torsion-free group 𝒪m, and let 𝔭 be a nonzero prime ideal of 𝒪 with q=|𝒪:𝔭|. Then, for N∈ℕ0=ℕ∪{0}, we have
rqi(𝐆N)=cqi(𝐆N)={qmNifi=0,0otherwise.
Therefore, for ∗∈{irr,cc},
𝒵𝐆(𝔬)∗(s1,s2)=𝒵𝔬m∗(s1,s2)=∑N=0∞qN(m-s2)=11-qm-s2.
Consequently, 𝒵𝒪m∗(s1,s2)=ζK(s2-m) with ζK(s) denoting the Dedekind zeta function of the number field K.
Moreover, the local factor at 𝔭 satisfies the functional equation
𝒵𝔬m∗(s1,s2)|q→q-1=-qm-s2𝒵𝔬m∗(s1,s2).∎
Certain zeta functions of groups related to representations – or the local factors of such functions – are known to be rational functions satisfying functional equations; for instance, representation zeta functions of certain pro-p groups [1, Theorem A] and local factors of twist representation zeta functions – see Section 1.3 – of groups of the form 𝐆(𝒪) [20, Theorem A], where 𝐆 is a unipotent group scheme obtained from a nilpotent 𝒪-Lie lattice.
As for zeta functions related to conjugacy classes, the so-called class number zeta functions – see Section 1.2 – of certain groups are rational; for instance, the local factors of class number zeta functions of Chevalley groups G(𝔬) [2, Theorem C], where 𝔬 is the valuation ring of a non-Archimedean local field of any (sufficiently large) characteristic, and class number zeta functions of compact p-adic analytic groups [4, Theorem 1.2].
This motivates our main result, which concerns the abovementioned features for the local factors of bivariate representation and conjugacy class zeta functions of groups of the form 𝐆(𝒪) obtained from nilpotent Lie lattices; see Section 3.
Theorem 1.4.
Let O be the ring of integers of a number field K, and let G be a unipotent group scheme obtained from a nilpotent O-Lie lattice Λ.
For each ∗∈{irr,cc}, there exist a positive integer t∗ and a rational function
R∗(X1,…,Xt∗,Y1,Y2) 𝑖𝑛 ℚ(X1,…,Xt∗,Y1,Y2)
such that, for all but finitely many nonzero prime ideals p of O, there exist algebraic integers λ1∗(p),…,λt∗∗(p) for which the following holds.
For any finite extension O of o:-Op with relative degree of inertia f=f(O,o),
𝒵𝐆(𝔒)∗(s1,s2)=R∗(λ1∗(𝔭)f,…,λt∗∗(𝔭)f,q-fs1,q-fs2),
where q=|O:p|.
Moreover, inverting parameters in the rational function R∗ yields that these local factors satisfy the functional equation
𝒵𝐆(𝔒)∗(s1,s2)|q→q-1λj∗(𝔭)→λj∗(𝔭)-1=-qf(h-s2)𝒵𝐆(𝔒)∗(s1,s2),
where h=dimK(Λ⊗K).
The statement of Theorem 1.4 is analogous to [20, Theorem A], and its proof heavily relies on the techniques of [1, 20]; see Section 4.4.
The main tools used in the proof of Theorem 1.4 are the Kirillov orbit method, the Lazard correspondence and 𝔭-adic integration.
Since the rational functions Rirr and Rcc of Theorem 1.4 only depend on the 𝒪-scheme 𝐆, it follows that almost all local factors 𝒵𝐆(𝒪𝔭)∗(s1,s2) only depend on 𝐆 and the chosen prime ideal 𝔭 of 𝒪, not on the group of 𝒪𝔭-points 𝐆(𝒪𝔭).
We keep the notation 𝒵𝐆(𝒪𝔭)∗(s1,s2) to make clear which ring of integers and prime ideal are being considered.
Remark 1.5.
Local multivariate zeta functions counting the number of equivalence classes in some uniformly definable family of equivalence relations are known to be rational functions; see [7, Theorem 1.3].
As an application of this theorem, the authors prove in [7, Section 8] rationality for all local factors of twist representation zeta functions of 𝒯-groups, partially extending [20, Theorem A].
Nevertheless, the techniques of [7] do not assure that these local factors satisfy functional equations, as in [20, Theorem A] or in Theorem 1.4.
We hope that the methods of [7] can be applied to the bivariate zeta functions defined here to show rationality of all of their local factors.
We point out that, if 𝐆(𝒪) is a 𝒯-group as in Theorem 1.4, which has nilpotency class 2, then all local factors of 𝒵𝐆(𝒪)irr(s1,s2) are rational functions; see Remark 4.4 and Proposition 4.8.
Remark 1.6.
The author is not aware whether the results of Theorem 1.4 remain true for groups associated to unipotent group schemes in positive characteristic or for arithmetic groups associated to non-unipotent group schemes since the techniques used here to prove Theorem 1.4 do not apply in such cases.
However, one could obtain a positive answer for the former question if the techniques of [7] do apply for these bivariate zeta functions, in which case [7, Corollary 6.8] would assure that almost all local factors of the bivariate zeta functions of 𝐆(𝔽p[t]) are also rational, where 𝔽p is the field with p elements.
Some applications of bivariate zeta functions are given below in Sections 1.2 and 1.3.
Specifically, we obtain results on previously studied (univariate) zeta functions by specializing the bivariate zeta functions introduced here.
It would be interesting to understand which other kinds of information one can extract from bivariate zeta functions.
In [11], we explicitly compute bivariate zeta functions of three infinite families of nilpotent groups.
As a consequence, we obtain explicit formulae for two (univariate) zeta functions of these groups.
We also provide an application in combinatorics: the formulae for bivariate representation zeta functions of these groups are shown to be related to statistics of certain Weyl groups, leading to formulae for joint distributions of three statistics; see [11, Propositions 5.5 and 5.6].
1.2 Application 1: Class number zeta functions
An advantage of the study of the bivariate zeta functions of Definition 1.2 is that they can be used to investigate (univariate) class number zeta functions, which encode the class numbers of principal congruence quotients of the groups considered.
Recall that the class numberk(G) of a finite group G is the number of its conjugacy classes or, equivalently, the number of its irreducible complex characters.
In particular, k(G)=ζGcc(0)=ζGirr(0).
Definition 1.7.
The class number zeta function of the 𝒯-group 𝐆(𝒪) is
ζ𝐆(𝒪)k(s)=∑(0)≠I⊴𝒪k(𝐆(𝒪/I))|𝒪:I|-s,
where s is a complex variable.
As for the bivariate zeta functions of Definition 1.2, the class number zeta function defined above depend not only on the group of 𝒪-points 𝐆(𝒪), but also on 𝒪 and the 𝒪-scheme 𝐆.
We adopt the notation 𝐆(𝒪) in the index to indicate this more complex dependency.
The term “conjugacy class zeta function” is sometimes used for what we call “class number zeta function”; see, for instance, [2, 17, 18, 4].
Clearly,
(1.2)𝒵𝐆(𝒪)irr(0,s)=𝒵𝐆(𝒪)cc(0,s)=ζ𝐆(𝒪)k(s).
A consequence of Theorem 1.4 is that almost all local factors of the class number zeta function of 𝐆(𝒪) are rational in λi(𝔭), q and q-s and
behave uniformly under base extension.
Moreover, for a finite extension 𝔒 of 𝔬 with relative degree of inertia f=f(𝔒,𝔬), the local factors satisfy
the functional equation
ζ𝐆(𝔒)k(s)|q→q-1λj∗(𝔭)→λj∗(𝔭)-1=-qf(h-s)ζ𝐆(𝔒)k(s).
Rossmann showed independently in [17] that class number zeta functions of certain nilpotent groups G≤GLd(𝒪𝔭) are rational functions and satisfy functional equations.
This is a consequence of [17, Theorems 4.10 and 4.18] together with
the specialization of ask zeta functions to class number zeta functions given in [17, Theorem 1.7].
1.3 Application 2: Twist representation zeta functions
A 𝒯-group of nilpotency class c=2 is called a 𝒯2-group.
The bivariate representation zeta function of a 𝒯2-group 𝐆(𝒪) specializes to its twist representation zeta function, whose definition we now recall.
Nontrivial 𝒯-groups have infinitely many one-dimensional irreducible complex representations.
For this reason, one cannot define the representation zeta function of a 𝒯-group G as in Definition 1.1.
Instead, for a 𝒯-group G, one considers equivalence classes on the set of its irreducible complex representations:
two representations ρ, σ of G are called twist-equivalent if there exists a one-dimensional
representation χ of G such that ρ≅χ⊗σ.
This is an equivalence relation on the set of irreducible complex representations of G, whose equivalence classes are called twist-isoclasses.
Let r~n(G) be the number of twist-isoclasses of n-dimensional irreducible complex representations of G.
If G is a topological group, we only consider continuous representations.
The r~n(G) are all finite; see [8, Theorem 6.6].
Definition 1.8.
The twist representation zeta function of a 𝒯-group G is
ζGirr~(s)=∑n=1∞r~n(G)n-s,
where s is a complex variable.
Twist representation zeta functions of 𝒯-groups have been previously investigated, for instance, in [3, 7, 16, 20, 22].
Explicit examples of (local factors of) twist representation zeta functions of 𝒯-groups can be found in [5, 16, 19, 20, 21].
Let 𝐆(𝒪) be a 𝒯2-group obtained from a unipotent 𝒪-Lie lattice Λ as explained in Section 3.
In Section 4.3, we show that the bivariate representation zeta function of 𝐆(𝒪) specializes to its twist representation zeta function as follows.
For a fixed nonzero prime ideal 𝔭 of 𝒪, let 𝔤=Λ⊗𝒪𝒪𝔭, and let 𝔤′ be the derived Lie sublattice of 𝔤.
Denote by r the torsion-free rank of 𝔤/𝔤′.
Then Proposition 4.11 states
(1.3)∏𝔭∈Spec(𝒪)∖{(0)}((1-qr-s2)𝒵𝐆(𝒪𝔭)irr(s1,s2)|s1→s-2s2→r)=ζ𝐆(𝒪)irr~(s),
provided both the left-hand side and the right-hand side converge.
No specialization of the form (1.3) is expected to exist in case of nilpotency class c>2; see [10, Section 3.3] for details.
In Section 4.3, we exhibit a 𝒯-group of nilpotency class 3, whose bivariate representation zeta function does not specialize to its twist representation zeta function.
We conclude this section with an example which illustrates Theorem 1.4 and specializations (1.2) and (1.3).
Example 1.9.
Let 𝐇(𝒪) denote the Heisenberg group of upper uni-triangular 3×3-matrices over 𝒪.
In Example 4.10, we show that, for a given nonzero prime ideal 𝔭 of 𝒪 with |𝒪:𝔭|=q, the bivariate zeta functions of 𝐇(𝔬) are given by
(1.4)𝒵𝐇(𝔬)irr(s1,s2)=1-q-s1-s2(1-q1-s1-s2)(1-q2-s2),
(1.5)𝒵𝐇(𝔬)cc(s1,s2)=1-q-s1-s2(1-q1-s2)(1-q2-s1-s2).
In particular, these are rational functions in q, q-s1 and q-s2, and
𝒵𝐇(𝔬)∗(s1,s2)|q→q-1=-q3-s2𝒵𝐇(𝔬)∗(s1,s2)
for each ∗∈{irr,cc}.
Specializations (1.2) and (1.3) yield
ζ𝐇(𝔬)k(s)=1-q-s(1-q1-s)(1-q2-s) and ζ𝐇(𝔬)irr~(s)=1-q-s1-q1-s.
The expression of the class number zeta function agrees with the formula given in [17, Section 9.3, Table 1].
This example also occurs in [2, Section 8.2], corrected by a sign mistake.
The expression of the twist representation zeta function accords with [20, Theorem B].
We further note that
ζ𝐇(𝔬)k(s)|q→q-1=-q3-sζ𝐇(𝔬)k(s).∎
3 Group schemes obtained from nilpotent Lie lattices
Here, we explain briefly how the unipotent groups schemes 𝐆 of interest in this work are constructed; for more details see [20, Section 2.1.2].
An 𝒪-Lie lattice is a free and finitely generated 𝒪-module Λ together with an antisymmetric 𝒪-bilinear form [⋅,⋅], which satisfies the Jacobi identity.
Let (Λ,[⋅,⋅]) be a nilpotent 𝒪-Lie lattice of 𝒪-rank h.
Let ℬ=(x1,…,xh) be an 𝒪-basis for Λ.
For each 𝒪-algebra R, set Λ(R)=Λ⊗𝒪R.
Then
ℬR=(x1⊗1R,…,xh⊗1R)
is an R-basis of Λ(R).
If Λ has nilpotency class c and satisfies Λ′⊆c!Λ, where Λ′=[Λ,Λ] is its derived Lie sublattice, we may define a group operation ∗ on Λ(𝒪) by means of the Hausdorff series.
The assumption Λ′⊆c!Λ assures that all denominators in the Hausdorff series will cancel out, so that the coordinates of the operation ∗ in terms of ℬ are given by polynomials f1,…,fh:𝒪2h→𝒪, say, with coefficients in 𝒪. That is, given a=∑i=1haixi and b=∑i=1hbixi∈Λ(𝒪), we have
a∗b=∑i=1hfi(a¯,b¯)xi,
where a¯=(a1,…,ah), b¯=(b1,…,bh)∈𝒪h.
For each 𝒪-algebra R, one may give Λ(R) a group structure by defining the group operation ∗ on Λ(R) as follows:
for alla,b∈Λ(R): a∗b=∑i=1ffi(a¯,b¯)(xi⊗1R),
where
a=∑i=1hai(xi⊗1R),b=∑i=1hbi(xi⊗1R)∈Λ(R),
and a¯=(a1,…,ah), b¯=(b1,…,bh)∈𝒪h.
This defines a unipotent 𝒪-group scheme 𝐆=𝐆Λ which is isomorphic as a scheme to affine h-space over 𝒪, which represents the group functor R↦(Λ(R),∗).
The 𝒪-group scheme 𝐆 is such that 𝐆(𝒪) is a 𝒯-group of nilpotency class c.
If R is an 𝒪-algebra whose underlying additive group is a finitely generated pro-p group, for instance R=𝒪𝔭, then 𝐆(R) is a finitely generated pro-p group of nilpotency class c.
For Lie lattices Λ of nilpotency class 2, a different construction of such unipotent group schemes is given in [20, Section 2.4.1], in which case the hypothesis Λ′⊆2Λ is not needed.
However, if this condition is satisfied, the unipotent group schemes obtained via such construction coincide with the latter ones.
Remark 3.1.
Definition 1.2 of bivariate zeta functions of groups 𝐆(𝒪) might be extended to 𝒯-groups which are not necessarily of the form 𝐆(𝒪).
As explained in [3, Section 5], every 𝒯-group G is virtually of the form 𝐆(ℤ) for some unipotent ℤ-group scheme 𝐆 obtained from a nilpotent ℤ-Lie lattice.
We might then define 𝒵G∗(s1,s2) as the corresponding bivariate zeta function of a finite index subgroup H of G, which is of the form 𝐆(ℤ).
That is,
𝒵G∗(s1,s2)=𝒵G,H∗(s1,s2):-𝒵𝐆(ℤ)∗(s1,s2),∗∈{irr,cc}.
If G has two subgroups H1=𝐆1(ℤ) and H2=𝐆2(ℤ) of finite index, then H1 and H2 are commensurable, and, therefore, they have the same pro-p completion for all but finitely many prime integers p; see [13, Lemma 1.8].
In particular, 𝒵𝐆1(ℤp)∗(s1,s2)=𝒵𝐆2(ℤp)∗(s1,s2) for all but finitely many primes p; that is, although 𝒵G,H1∗(s1,s2) and 𝒵G,H2∗(s1,s2) may not coincide, they are almost the same in the sense that they coincide except for finitely many local factors.
In this point of view, bivariate zeta functions also yield invariants for 𝒯-groups, namely, domains of convergence and of meromorphy.
More precisely, we prove in [9, Theorem 1] that the maximal domains of convergence of the bivariate zeta functions of groups of the form 𝐆(𝒪) – when finitely many local factors are disregarded – are independent of the ring of integers 𝒪 and admit meromorphic continuations to domains which are also independent of 𝒪.
From now on, we assume that 𝐆 is a unipotent 𝒪-group scheme obtained from a nilpotent 𝒪-Lie lattice Λ.
4 Bivariate zeta functions and 𝔭-adic integrals
Our results rely on the fact that local bivariate representation and local bivariate conjugacy class zeta functions of groups associated to unipotent group schemes can be written in terms of 𝔭-adic integrals.
In Section 4.2, we show how to write most of the local factors of these zeta functions in terms of 𝔭-adic integrals using the methods of [22, Section 2.2].
A consequence is that these local factors are given by rational functions as stated in Theorem 1.4.
In Section 4.4, we use the results of [22, Section 2.1] and [1, Section 4.1] to show that these local factors satisfy functional equations and that they are uniform under base extension, concluding the proof of Theorem 1.4.
The methods of [22, Section 2] were also applied in [17] to show functional equations for the ask zeta functions of modules of matrices over compact discrete valuation rings 𝔒 in characteristic zero.
This result provides functional equations for the class number zeta functions of certain nilpotent groups G≤GLd(𝔒).
Furthermore, these methods were applied to show results similar to [1, Theorem 1.4] for the representation zeta functions of certain p-adic analytic groups, and in [20] for the local factors of twist representation zeta functions of groups of the form 𝐆(𝒪).
We now recall the methods of [22, Section 2.2].
For the rest of this section, let 𝔭 be a fixed nonzero prime ideal of 𝒪 and 𝔬=𝒪𝔭.
Denote by q the cardinality of 𝒪/𝔭 and by p its characteristic.
Recall that, given an element z∈𝔬, the ideal (z)⊲𝒪 has prime factorization (z)=𝔭e𝔭1e1⋯𝔭rer such that 𝔭i≠𝔭 for all i∈[r].
The 𝔭-adic valuation of z is v𝔭(z)=e, and its 𝔭-adic norm is |z|𝔭=q-v𝔭(z).
Equivalently, v𝔭(z)=e and |z|𝔭=q-e if z∈𝔭e∖𝔭e+1.
For each j∈ℕ, denote by ∥⋅∥𝔭 the maximum norm of 𝔬j with respect to |⋅|𝔭; that is, for 𝐳=(z1,…,zj)∈𝔬j, let
∥𝐳∥𝔭=max{|zk|𝔭}k=1j.
For N∈ℕ, we also denote by v𝔭 the function on 𝔬/𝔭N given as follows: let z¯ be the image of z∈𝔬 under 𝔬→𝔬/𝔭N, and assume z∈𝔭e∖𝔭e+1.
Then v𝔭(z¯)=e if 0≤e<N, and v𝔭(z¯)=+∞ otherwise.
We make the following distinction: 𝔭m denotes the m-th ideal power 𝔭⋯𝔭, whilst 𝔭(m) denotes the m-fold Cartesian power 𝔭×…×𝔭.
For k,N∈ℕ, set
Wk(𝔬/𝔭N):-((𝔬/𝔭N)k)*={𝐱∈(𝔬/𝔭N)k∣v𝔭(𝐱)=0},
Wk𝔬:-(𝔬k)*={𝐱∈𝔬k∣v𝔭(𝐱)=0},
and let Wk((0))=(0)k for each k∈ℕ.
Let π∈𝔬 be a uniformizer of 𝔬.
Given a matrix M∈Matm×n(𝔬/𝔭N), we write ν(M)=(m1,…,mϵ) to indicate the elementary divisor type of M, where 0≤ϵ≤min{m,n}.
In the following, denote by Frac(𝔬) the field of fractions of 𝔬.
Let n∈ℕ, and let ℛ(Y¯)=ℛ(Y1,…,Yn) be a matrix of polynomials ℛ(Y¯)ij∈𝔬[Y¯] with
uℛ=max{rkFrac(𝔬)ℛ(𝐳)∣𝐳∈𝔬n}.
For each 𝐦∈ℕ0uℛ, write
𝔑𝐦ℛ(𝔬/𝔭N):-{𝐲∈Wn(𝔬/𝔭N)∣ν(ℛ(𝐲))=𝐦},
𝒩𝐦ℛ(𝔬/𝔭N):-|𝔑𝐦ℛ(𝔬/𝔭N)|.
The number 𝒩𝐦ℛ(𝔬/𝔭N) is zero unless 𝐦=(m1,…,muℛ) satisfies
0=m1≤…≤muℛ≤N.
Let r¯=(r1,…,ruℛ) be a vector of variables.
Consider the Poincaré series
(4.1)𝒫ℛ𝔬(r¯,t)=∑N∈ℕ𝐦∈ℕ0uℛ𝒩𝐦ℛ(𝔬/𝔭N)q-tN-∑i=1uℛrimi.
In [22, Section 2.2], it is shown how to describe the series (4.1) in terms of the 𝔭-adic integral
(4.2)𝒵ℛ𝔬(r¯,t)=11-q-1∫(x,y¯)∈𝔭×Wn𝔬|x|𝔭t×∏k=1uℛ∥Fk(ℛ(y¯))∪xFk-1(ℛ(y¯))∥𝔭rk∥Fk-1(ℛ(y¯))∥𝔭rkdμ,
where μ is the additive Haar measure normalized so that μ(𝔬n+1)=1; Fj(ℛ(y¯)) is the set of nonzero j×j-minors of ℛ(y¯).
More precisely, in [22, Section 2.2] it is shown that (4.1) satisfies
(4.3)𝒫ℛ𝔬(r¯,t)=𝒵ℛ𝔬(r¯,t-n-1).
Suppose now that M∈Matn×n(𝔬/𝔭N) is an antisymmetric matrix.
Then, for some ξ∈[n]0:-{0,1,…,n}, the elementary divisor type of M is of the form
ν(M)=(m1,m1,m2,m2,…,mξ,mξ).
If M is antisymmetric, we write ν~(M)=(m1,m2,…,mξ) for its reduced elementary divisor type, that is, to indicate ν(M)=(m1,m1,m2,m2,…,mξ,mξ).
Assume now that ℛ(Y¯) is antisymmetric, in which case uℛ is even.
For each 𝐦∈ℕ0uℛ/2, write
𝔑~𝐦ℛ(𝔬/𝔭N)={𝐲∈Wn(𝔬/𝔭N)∣ν~(ℛ(𝐲))=𝐦},
𝒩𝐦ℛ(𝔬/𝔭N)=|𝔑~𝐦ℛ(𝔬/𝔭N)|.
For ℛ(Y¯) antisymmetric, we assume that the vector of variables r¯ is of the form r¯=(r1,r1,…,ruℛ/2,ruℛ/2) so that
(4.4)𝒫ℛ𝔬(r¯,t)=∑N∈ℕ,𝐦∈ℕ0uℛ/2𝒩𝐦ℛ(𝔬/𝔭N)q-tN-2∑i=1uℛ/2rimi.
Recall the notation [n]={1,…,n} for n∈ℕ.
Given x∈𝔬 with v𝔭(x)=N, 𝐲∈𝔬n and k∈[uℛ], we obtain from [17, Lemma 4.6 (i) and (ii)] the following for the antisymmetric matrix ℛ(𝐲) with ν~(ℛ(𝐲))=(m1,…,muℛ):
∥F2k(ℛ(𝐲))∪xF2k-1(ℛ(𝐲))∥𝔭∥F2k-1(ℛ(𝐲))∥𝔭=∥F2k-1(ℛ(𝐲))∪xF2(k-1)(ℛ(𝐲))∥𝔭∥F2(k-1)(ℛ(𝐲))∥𝔭=q-min(mk,N),
∥F2k(ℛ(𝐲))∪x2F2(k-1)(ℛ(𝐲))∥𝔭∥F2(k-1)(ℛ(𝐲))∥𝔭=q-2min(mk,N).
Therefore, if ℛ(Y¯) is an antisymmetric matrix, the series (4.4) can be described by the 𝔭-adic integral
(4.5)𝒫ℛ𝔬(r¯,t)=𝒵ℛ𝔬(r¯,t-n-1)=11-q-1∫(x,y¯)∈𝔭×Wn𝔬|x|𝔭t-n-1×∏k=1uℛ/2∥F2k(ℛ(y¯))∪x2F2(k-1)(ℛ(y¯))∥𝔭rk∥F2(k-1)(ℛ(y¯))∥𝔭rkdμ.
4.1 The numbers rn(𝐆N) and cn(𝐆N)
Recall the notation 𝐆N=𝐆(𝔬/𝔭N).
We now write the local bivariate zeta functions at 𝔭 in terms of sums encoding the elementary divisor types of certain matrices associated to Λ.
This is done by rewriting the numbers rn(𝐆N) and cn(𝐆N) for n∈ℕ and N∈ℕ0 in terms of numbers 𝒩𝐦ℛ(𝔬/𝔭N) defined at the beginning of Section 4.
In each case, ℛ is one of the two commutator matrices of
Λ, which we now define.
Set 𝔤=Λ(𝔬)=Λ⊗𝒪𝔬.
Let 𝔤′ be the derived Lie sublattice of 𝔤, and let 𝔷 be its center.
Consider the torsion-free 𝒪-ranks
h=rk(𝔤),a=rk(𝔤/𝔷),b=rk(𝔤′),r=rk(𝔤/𝔤′),z=rk(𝔷).
For R either 𝒪 or 𝔬, let M be a finitely generated R-module with a submodule N.
The isolatorι(N) of N in M is the smallest submodule L of M containing N such that M/L is torsion free.
In particular, 𝔷=ι(𝔷); see [20, Lemma 2.5].
Set k=rk(ι(𝔤′)/ι(𝔤′∩𝔷))=rk(ι(𝔤′+𝔷)/𝔷).
The commutator matrices are defined with respect to a fixed 𝔬-basis
ℬ=(e1,…,eh)
of the 𝔬-Lie lattice 𝔤, satisfying the conditions
(ea-k+1,…,ea)is an𝔬-basis forι(𝔤′+𝔷),
(ea+1,…,ea-k+b)is an𝔬-basis forι(𝔤′∩𝔷),
(ea+1,…,eh)is an𝔬-basis for𝔷.
Denote by ⋅¯ the natural surjection 𝔤→𝔤/𝔷.
Let 𝐞=(e1,…,ea).
Then
𝐞¯=(e1¯,…,ea¯)
is an 𝔬-basis of 𝔤/𝔷.
The ei can be chosen so that there are nonnegative integers c1,…,cb with the property that
(πc1ea-k+1¯,…,πckea¯)is an𝔬-basis of𝔤′+𝔷¯,
(πck+1ea+1,…,πcbea-k+b)is an𝔬-basis of𝔤′∩𝔷
by the elementary divisor theorem.
Fix an 𝔬-basis 𝐟=(f1,…,fb) for 𝔤′ satisfying
(f1¯,…,fk¯)=(πc1ea-k+1¯,…,πckea¯)is an𝔬-basis of𝔤′+𝔷¯,
(fk+1,…,fb)=(πck+1ea+1,…,πcbea-k+b)is an𝔬-basis of𝔤′∩𝔷.
For i,j∈[a] and k∈[b], let λijk∈𝔬 be the structure constants satisfying
[ei,ej]=∑k=1bλijkfk.
The following matrices were previously defined in [12, Definition 2.1].
Definition 4.1.
The A-commutator and the B-commutator matrices of 𝔤 with respect to 𝐞 and 𝐟 are
A(X1,…,Xa)=(∑j=1aλijkXj)ik∈Mata×b(𝔬[X¯]),
B(Y1,…,Yb)=(∑k=1bλijkYk)ij∈Mata×a(𝔬[Y¯]),
respectively, where X¯=(X1,…,Xa) and Y¯=(Y1,…,Yb) are independent variables.
For each 𝐲∈𝔬b, the matrix B(𝐲) is antisymmetric.
Fix N∈ℕ.
The congruence quotient 𝐆N is a finite p-group of nilpotency class c.
Set 𝔤N:-Λ⊗𝔬𝔬/𝔭N and 𝔷N=𝔷⊗𝔬𝔬/𝔭N, and let 𝔤N′=𝔤′⊗𝔬𝔬/𝔭N.
Tensoring 𝐞 and 𝐟 with 𝔬/𝔭N yields ordered sets
𝐞N=(e1N,…,eaN) and 𝐟N=(f1N,…,fbN)
such that 𝐞¯=(e1N¯,…,eaN¯) is an 𝔬/𝔭N-basis for 𝔤N/𝔷N and 𝐟N is an 𝔬/𝔭N-basis for 𝔤N′ as 𝔬/𝔭N-modules, where ⋅¯ is the natural surjection 𝔤N→𝔤N/𝔷N.
Given an element ω of 𝔤N^=Hom𝔬(𝔤N,ℂ×), set
BωN:𝔤N×𝔤N→ℂ×,(u,v)↦ω([u,v]).
The radical of BωN is
Rad(BωN)={u∈𝔤N∣for allv∈𝔤N:BωN(u,v)=1}.
Observe that Bω depends only on the restriction of ω to 𝔤N′.
For this reason, given ω~∈𝔤N′^ and any extension ω of ω~ to 𝔤N, we write Bω~ for Bω.
For x∈𝔤N/𝔷N, following [12, Section 3.1], we define
adx:𝔤N/𝔷N→𝔤N′ andadx⋆:𝔤N′^→𝔤N/𝔷N^
y↦[y,x]ω↦ω∘adx.
Observe that, in the definition of the 𝔬/𝔭N-module homomorphism adx, we identify 𝔤N/𝔷N with the 𝔬/𝔭N-submodule of 𝔤N generated by 𝐞N.
The dimensions of the irreducible complex representations and the sizes of conjugacy classes of 𝐆N are powers of p, and, according to [12, Section 3], for c<p, the numbers rpi(𝐆N) and cpi(𝐆N) are given by
(4.6)rpi(𝐆N)=|{ω∈𝔤N′^∣|Rad(BωN):𝔷N|=p-2i|𝔤N/𝔷N|}|×|𝔤N/𝔤N′|p-2i,
(4.7)cpi(𝐆N)=|{x∈𝔤N/𝔷N∣|Ker(adx⋆)|=p-i|𝔤N′^|}||𝔷N|p-i.
The first formula is a consequence of the Kirillov orbit method, which reduces the problem of enumerating the characters of 𝐆N to the problem of determining the indices in 𝔤N of Rad(BωN) for ω∈𝔤N′^; see [12, Theorem 3.1].
The second formula reflects the fact that the Lazard correspondence induces an order-preserving correspondence between subgroups of 𝐆N and sublattices of 𝔤N and maps normal subgroups to ideals.
Moreover, centralizers of elements in 𝐆N correspond to centralizers of elements in 𝔤N under the Lazard correspondence.
The cardinalities of 𝔤N and 𝔤N/𝔷N are powers of q, and hence so are the cardinalities of Rad(BωN)/𝔷 and Ker(adx⋆).
It follows that rn(𝐆N) and cn(𝐆N) can only be nonzero if n is a power of q.
The next step is to relate (4.6) and (4.7) to the commutator matrices of 𝔤N with respect to 𝐞N and 𝐟N.
Using arguments analogous to the ones of [12, Section 2], we define the coordinate systems
ϕN:𝔤N/𝔷N→(𝔬/𝔭N)a,x=∑j=1axjejN↦𝐱=(x1,…,xa),
ψN:𝔤N′^→(𝔬/𝔭N)b,ω=∑j=1byjfjN∨↦𝐲=(y1,…,yb),
where, for N∈ℕ0, 𝐟N∨=(f1N∨,…,fbN∨) is the dual 𝔬/𝔭-basis for
𝔤N′^=Hom𝔬(𝔤N′,ℂ×).
We notice that 𝔤1/𝔷1 and 𝔤1′ are regarded as 𝔬/𝔭-vector spaces in the construction of [12, Section 2].
In the coordinate systems above, we regard 𝔤N/𝔷N and 𝔤N′ as 𝔬/𝔭N-modules for all N∈ℕ.
Lemma 4.2.
Given x∈gN/zN with ϕN(x)=x and ω∈gN′^ with ψN(ω)=y, the following holds:
x∈Rad(BωN)/𝔷Nif and only ifB(𝐲)𝐱tr=0,
ω∈Ker(adx⋆)if and only ifA(𝐱)𝐲tr=0.
Proof.
An element x∈𝔤N/𝔷N belongs to Rad(BωN)/𝔷 exactly when ω[v,x]=1 for all v∈𝔤N/𝔷N, whilst an element ω∈𝔤N′^ belongs to Ker(adx⋆) exactly when ω[v,x]=1 for all v∈𝔤N/𝔷N. Expressing these conditions in coordinates, we see that both expressions hold.
We prove the second claim in detail.
Fix x∈𝔤N/𝔷N with
ϕN(x¯)=𝐱=(x1,…,xa).
It holds that
(4.8)[eiN,x]=∑j=1axj[eiN,ejN]=∑j=1a∑l=1bλijlxjflN.
We want to determine which elements ω∈𝔤N′^ satisfy
ω([v,x])=1 for allv∈𝔤N/𝔷N.
Consider ψN(ω)=𝐲=(y1,…,yb), i.e., ω=∏k=1b(fkN∨)yk.
Because of (4.8), for each i∈[a],
ω([eiN,x])=∏k=1b(fkN∨(∑j=1a∑l=1bλijlxjflN))yk=∏k=1b(fkN∨(fkN))yk∑j=1aλijkxj.
This expression equals 1 exactly when ∑k=1byk∑j=1aλijkxj=0.
Now, by definition, ∑j=1aλijkxj=A(𝐱)ik, where A(𝐱) is the A-commutator matrix of Definition 4.1 evaluated at 𝐱.
Consequently, ω∈Ker(adx⋆) if and only if
∑k=1bA(𝐱)ikyk=0 for alli∈[a],
that is, A(𝐱)𝐲tr=0.
∎
Applying Lemma 4.2 to (4.6), we rewrite the numbers rqi(𝐆N) in terms of solutions of the system
B(𝐲)𝐱tr=0 and, applying Lemma 4.2 to (4.7), we rewrite the numbers cqi(𝐆N) in terms of solutions of the system A(𝐱)𝐲tr=0.
In each case, we consider the elementary divisor type of the corresponding matrix.
Fix an elementary divisor type ν~(B(𝐲))=(m1,…,muB), where
2uB=max{rkFrac(𝔬)B(𝐳)∣𝐳∈𝔬b}.
Since B(𝐲) is similar to the matrix Diag(πm1,πm1,…,πmuB,πmuB,𝟎a-2uB), where 𝟎a-2uB=(0,…,0)∈ℤa-2uB, the system B(𝐲)𝐱tr=0 in 𝔬/𝔭N is
equivalent to
{x1≡x2≡0mod𝔭N-m1,x3≡x4≡0mod𝔭N-m2,x2uB-1≡x2uB≡0mod𝔭N-muB.
For 2uB<a, the elements x2uB+1,…,xa are arbitrary elements of 𝔬/𝔭N, and
|{x∈𝔬/𝔭N∣x≡0mod𝔭N-mj}|=qmj.
Hence the number of solutions of B(𝐲)𝐱tr=0 in 𝔬/𝔭N is
q2(m1+…+muB)+(a-2uB)N.
In other words, ν~(B(𝐲))=(m1,…,muB) implies
|Rad(BωN)/𝔷N|=q2(m1+…+muB)+(a-2uB)N.
Lemma 4.2 then assures that |Rad(BωN)/𝔷N|=q-2i|𝔤N/𝔷N|=qaN-2i whenever B(𝐲) has elementary divisor type (m1,…,muB) satisfying
∑j=1uBmj=uBN-i.
Consequently, for r=rk(𝔤/𝔤′)=h-b, expression (4.6) can be rewritten as
(4.9)rqi(𝐆N)=∑𝐦∈𝒟BN|{𝐲∈(𝔬/𝔭N)b∣ν~(B(𝐲))=𝐦}|qrN-2i,
where
𝒟BN:-{𝐦=(m1,…,muB)∈ℕ0uB|m1≤…≤muB≤N,∑i=1uBmi=uBN-i}.
Analogously, if ν(A(𝐱))=(m1,…,muA), where
uA:-max{rkFrac(𝔬)A(𝐳)∣𝐳∈𝔬a},
the equality A(𝐱)𝐲tr=0 has qm1+m2+…+muA+(b-uA)N solutions in 𝔬/𝔭N. For z=rk(𝔷)=h-a, this yields
(4.10)cqi(𝐆N)=∑𝐦∈𝒟AN|{𝐱∈(𝔬/𝔭N)a∣ν(A(𝐱))=𝐦}|qzN-i,
where
𝒟AN:-{𝐦=(m1,…,muA)∈ℕ0uA|m1≤…≤muA≤N,∑i=1uAmi=uAN-i}.
For a matrix ℛ(Y¯)=ℛ(Y1,…,Yn) of polynomials as the one at the beginning of Section 4 and for
𝐦=(m1,…,muℛ)∈ℕ0uℛ, define
𝔚𝐦ℛ(𝔬/𝔭N):-{𝐲∈(𝔬/𝔭N)n∣ν(ℛ(𝐲))=𝐦}.
Expressions (4.9) and (4.10) are written in terms of cardinalities of such sets, which are related to the numbers 𝒩𝐦ℛ(𝔬/𝔭N) as follows.
Write
𝐦-m=(m1-m,…,muℛ-m) for allm∈ℕ0.
If ℛ(𝐲) is such that v𝔭(𝐲)=v𝔭(ℛ(𝐲)) for all 𝐲∈𝔬n, then
(4.11)|𝔚𝐦ℛ(𝔬/𝔭N)|=𝒩𝐦-m1ℛ(𝔬/𝔭N-m1).
Indeed, the map 𝔑𝐦-m1ℛ(𝔬/𝔭N-m1)→𝔚𝐦ℛ(𝔬/𝔭N) given by 𝐲↦πm1𝐲 is a bijection.
Equality (4.11) provides the following reformulations of (4.9) and (4.10).
Lemma 4.3.
For each i∈N0 and N∈N0,
(4.12)rqi(𝐆N)=∑𝐦∈𝒟BN𝒩𝐦-m1B(𝔬/𝔭N-m1)qrN-2i,
(4.13)cqi(𝐆N)=∑𝐦∈𝒟AN𝒩𝐦-m1A(𝔬/𝔭N-m1)qzN-i.
Remark 4.4.
As explained in Section 3, for 𝒪-Lie lattices Λ of nilpotency class 2, a different construction for 𝐆=𝐆Λ is given in [20, Section 2.4.1], which does not require the assumption Λ′⊆2Λ.
For groups associated to such schemes, a Kirillov orbit method formalism was formulated, which is valid for all primes; see [20, Section 2.4.2].
Consequently, [20, Lemma 2.13] assures that (4.12) holds for all primes 𝔭 if Λ has nilpotency class 2.
4.2 𝔭-adic integrals
We now write the local factors of the bivariate zeta functions of 𝐆(𝒪) in terms of Poincaré series such as (4.1).
Recall from Section 4.1 that the dimensions of irreducible complex representations as well as the sizes of the conjugacy classes of 𝐆(𝔬) are powers of q, allowing us to write the representation and the conjugacy class zeta functions of the congruence quotient 𝐆N=𝐆(𝔬/𝔭N) as
ζ𝐆Nirr(s)=∑i=0∞rqi(𝐆N)q-is and ζ𝐆Ncc(s)=∑i=0∞cqi(𝐆N)q-is.
These sums are finite since 𝐆N is a finite group.
Applying this to (1.1), the definition of the local factors of the bivariate zeta functions, one obtains
𝒵𝐆(𝔬)irr(s1,s2)=∑N=0∞∑i=0∞rqi(𝐆N)q-is1-Ns2,
𝒵𝐆(𝔬)cc(s1,s2)=∑N=0∞∑i=0∞cqi(𝐆N)q-is1-Ns2.
Recall that z=rk(𝔷)=h-a and r=rk(𝔤/𝔤′)=h-b. If c=2 or p>c>2, then (4.12) yields
(4.14)𝒵𝐆(𝔬)irr(s1,s2)=∑N=0∞∑i=0∞∑𝐦∈𝒟BN𝒩𝐦-m1B(𝔬/𝔭N-m1)q-(s2-r)N-(2+s1)i.
If p>c, then (4.13) yields
(4.15)𝒵𝐆(𝔬)cc(s1,s2)=∑N=0∞∑i=0∞∑𝐦∈𝒟AN𝒩𝐦-m1A(𝔬/𝔭N-m1)q-(s2-z)N-(1+s1)i.
We now show how to rewrite these sums as Poincaré series of the form (4.1).
In preparation for this, we need two lemmata.
Lemma 4.5.
Let s be a complex variable, (am)m∈N0 a sequence of real numbers and q∈Z≥2.
Provided both series converge, the following holds:
∑N=1∞∑m=0N-1amq-sN=q-s1-q-s(∑N=0∞aNq-sN).
Proof.
This is due to a short manipulation of geometric series [10, Lemma 3.2.9].
∎
Lemma 4.6.
Let s and t be complex variables.
Let R(Y¯)=R(Y1,…,Yn) be a matrix of polynomials R(Y¯)ij∈o[Y¯].
If R is not antisymmetric, set
u=max{rkFrac(𝔬)ℛ(𝐳)∣𝐳∈𝔬n}.
Otherwise, set u=12max{rkFrac(o)R(z)∣z∈on}. Moreover, let q=|o/p|.
Provided both series converge, the following holds:
(4.16)∑N=0∞∑i=0∞∑𝐦∈𝒟N(uN-i)𝒩𝐦-m1ℛ(𝔬/𝔭N-m1)q-sN-ti=11-q-s(1+∑N=1∞∑𝐦∈ℕ0u𝒩𝐦ℛ(𝔬/𝔭N)q-(s+ut)N+t∑j=1umj),
where, for each c∈N0,
𝒟N(c):-{𝐦=(m1,…,mu)∈ℕ0u|m1≤…≤mu≤N,∑i=1umi=c}.
Proof.
Set 𝐦=(m1,…,mu) and recall the notation
𝐦-m=(m1-m,…,mu-m) form∈ℕ0.
As 𝒩𝐦ℛ(𝔬/𝔭N)=0 unless 0=m1≤m2≤…≤mu≤N, in which case
0≤∑j=1umj≤uN,
the condition ∑j=1umj=uN-i implies that the only values of i which are relevant for the sum (4.16) are 0≤i≤uN.
Hence, the expression on the left-hand side of (4.16) can be rewritten as
(4.17)1+∑N=1∞∑i=0uN∑𝐦∈𝒟N(uN-i)𝒩𝐦-m1ℛ(𝔬/𝔭N-m1)q-sN-ti.
In the following, we make use of the notation
𝒟≤N(c):-{𝐦=(m1,…,mu)∈ℕ0u|m1≤…≤mu≤N,∑i=1umi≤c}.
Restricting the summation in (4.17) to m1=0 yields
∑N=1∞∑𝐦∈𝒟≤N((u-1)N)m1=0𝒩𝐦ℛ(𝔬/𝔭N)q-sN-t(uN-∑j=2umj).
Since 𝒩𝐦ℛ(𝔬/𝔭N)=0 unless 0=m1≤m2≤…≤mu≤N, we may rewrite this sum as
∑N=1∞∑𝐦∈ℕ0u𝒩𝐦ℛ(𝔬/𝔭N)q-(s+ut)N+t∑j=1umj-:𝒮(s,t).
Our goal now is to write the part of the summation in (4.17) with m1>0 in terms of 𝒮(s,t).
Restricting the summation in (4.17) to m1>0 yields
(4.18)∑N=1∞∑i=0uN∑𝐦∈𝒟N(uN-i)m1>0𝒩𝐦-m1ℛ(𝔬/𝔭N-m1)q-sN-ti=∑N=1∞∑m=1N∑𝐦∈𝒟≤N(uN)m1=m𝒩𝐦-m1ℛ(𝔬/𝔭N-m1)q-sN-t(uN-∑j=1umj).
By writing mj′=mj-m1, we obtain ∑j=1umj=um1+∑j=2umj′.
Moreover, for each m∈[N],
{𝐦-m∣𝐦∈𝒟≤N(uN),m1=m}={𝐦∈𝒟≤N-m(u(N-m))∣m1=0}.
Then we may rewrite (4.18) as
(4.19)∑N=1∞∑m=1N∑𝐦∈𝒟≤N-m(u(N-m))m1=0𝒩𝐦ℛ(𝔬/𝔭N-m)q-sN-t(u(N-m)-∑j=2umj)=∑N=1∞q-sN∑m=0N-1∑𝐦∈𝒟≤m(um)m1=0𝒩𝐦ℛ(𝔬/𝔭m)qt∑j=2umj-tum=∑N=1∞q-sN∑m=0N-1∑𝐦∈ℕ0u𝒩𝐦ℛ(𝔬/𝔭m)qt∑j=1umj-tum.
Apply Lemma 4.5 to (4.19) by setting
am:-∑𝐦∈ℕ0u𝒩𝐦ℛ(𝔬/𝔭m)qt∑j=1umj-tum.
This gives that (4.19) equals
q-s1-q-s(1+∑N=1∞∑𝐦∈ℕ0u𝒩𝐦ℛ(𝔬/𝔭N)q-(s+ut)N+t∑j=1umj)=q-s1-q-s(1+𝒮(s,t)).
Combining the expressions for the parts of the sum with m1=0 and m1>0 yields
∑N=0∞∑i=0∞∑𝐦∈𝒟N(uN-i)𝒩𝐦ℛ(𝔬/𝔭N)q-sN-ti=1+𝒮(s,t)+q-s1-q-s(1+𝒮(s,t))=11-q-s(1+𝒮(s,t)).∎
Proposition 4.7.
If either c=2 or p>c>2, then
(4.20)𝒵𝐆(𝔬)irr(s1,s2)=11-qr-s2(1+∑N=1∞∑𝐦∈ℕ0uB𝒩𝐦B(𝔬/𝔭N)×q-N(uBs1+s2+2uB-r)-2∑j=1uBmj(-s1-2)2).
Moreover, if p>c, then
(4.21)𝒵𝐆(𝔬)cc(s1,s2)=11-qz-s2(1+∑N=1∞∑𝐦∈ℕ0uA𝒩𝐦A(𝔬/𝔭N)×q-N(uAs1+s2+uA-z)-∑j=1uAmj(-s1-1)).
Proof.
By setting s=s2-r and t=2+s1 and considering ℛ to be the B-commutator matrix of Λ on the left-hand side of (4.16), we obtain (4.14).
Under these substitutions, Lemma 4.6 shows (4.20).
Analogously, by setting s=s2-z and t=1+s1 and considering ℛ to be the A-commutator matrix of Λ, the left-hand side of (4.16) equals (4.15), so that, under these substitutions, Lemma 4.6 shows (4.21).
∎
Expression (4.20) is of the form (4.4) with
t=uBs1+s2+2uB-r and rk=-s1-22 for eachk∈[uB],
whilst (4.21) is (4.1) with
t=uAs1+s2+uA-z and rk=-s1-1 for eachk∈[uA].
Therefore, these choices of t and r¯ applied to (4.5) and to (4.3) yield the following.
Recall that
a+z=rk(𝔤/𝔷)+rk(𝔷)=rk(𝔤)=h and b+r=rk(𝔤′)+rk(𝔤/𝔤′)=h.
For k∈ℕ, write 𝟏k=(1,…,1)∈ℤk.
Proposition 4.8.
If either c=2 or p>c>2, then
(4.22)𝒵𝐆(𝔬)irr(s1,s2)=11-qr-s2(1+𝒵B𝔬((-s1-22)𝟏uB,uBs1+s2+2uB-h-1)).
Moreover, if p>c, then
(4.23)𝒵𝐆(𝔬)cc(s1,s2)=11-qz-s2(1+𝒵A𝔬((-s1-1)𝟏uA,uAs1+s2+uA-h-1)).
Specialization (1.2) applied to (4.22) and to (4.23) yields
(4.24)ζ𝐆(𝔬)k(s)=11-qz-s(1+𝒵A𝔬(-𝟏uA,s+uA-h-1)),=11-qr-s(1+𝒵B𝔬(-𝟏uB,s+2uB-h-1)).
Remark 4.9.
Formula (4.24) coincides with the 𝔭-adic integral obtained from the 𝔭-adic integral [17, formula (4.3)] together with the specialization given in [17, Theorem 1.7].
In fact, for each x∈𝔤, let adx:𝔤→𝔤′ be the adjoint homomorphism
adx(z)=[z,x] for allz∈𝔤.
As in Section 4.1, let ℬ=(e1,…,eh) be a basis of 𝔤 with the properties described there; we use the notation that was set up in this context.
For each x∈𝔤, we can write x=∑i=1hxiei for some xi∈𝔬.
Let 𝐱=(x1,…,xh)∈𝔬h.
The b×h-matrix representing the linear transformation adx is such that its submatrix composed of its first a columns is the transpose A(𝐱)tr of the A-commutator matrix of Λ, and the remaining columns have only zero entries.
We observe that the abovementioned integrals of [17] are taken over 𝔬×𝔬a instead of 𝔭×Wa𝔬 as in (4.24). Formula (4.24) coincides with the abovementioned 𝔭-adic integral due to [10, Lemma 2.2.4].
Example 4.10.
Let 𝐇(𝒪) be the Heisenberg group over 𝒪 considered in Example 1.9.
The unipotent group scheme 𝐇 is obtained from the ℤ-Lie lattice
Λ=〈x1,x2,y∣[x1,x2]-y〉.
The commutator matrices of 𝔤=Λ(𝔬) with respect to the ordered sets 𝐞=(x1,x2) and 𝐟=(y) are
A(X1,X2)=[X2-X1] and B(Y)=[0Y-Y0].
The A-commutator matrix has rank 1, and the B-commutator matrix has rank 2 over the respective fields of rational functions, that is, uA=uB=1.
Moreover, h=rk(𝔤)=3, and
F1(A(X1,X2))={-X1,X2},F2(B(Y))={Y2}.
In particular, if (x1,x2)∈W2𝔬, i.e., v𝔭(x1,x2)=0, then ∥F1(A(x1,x2))∥𝔭=1.
Also, if y∈W1𝔬, then, in particular, v𝔭(y2)=0, which gives ∥F2(B(y))∥𝔭=1.
It follows from Proposition 4.8 that
𝒵𝐇(𝔬)irr(s1,s2)=11-q2-s2(1+(1-q-1)-1∫(w,y)∈𝔭×W1𝔬|w|𝔭s1+s2-2dμ),
𝒵𝐇(𝔬)cc(s1,s2)=11-q1-s2(1+(1-q-1)-1∫(w,x1,x2)∈𝔭×W2𝔬|w|𝔭s1+s2-3dμ).
Expressions (1.4) and (1.5) for 𝒵𝐇(𝔬)irr(s1,s2) and 𝒵𝐇(𝔬)cc(s1,s2) given in Example 1.9 are then consequence of the following well-known fact: for k∈ℕ and t∈ℂ,
(4.25)∫w∈𝔭k|w|𝔭tdμ=q-k(t+1)(1-q-1)1-q-k(t+1),
provided the 𝔭-adic integral on the left-hand side converges.
∎
4.3 Twist representation zeta functions
In this section, we assume that 𝐆 is the unipotent group scheme associated to a nilpotent 𝒪-Lie lattice Λ of nilpotency class 2 without the assumption Λ′⊆2Λ, constructed as in [20, Section 2.4.1].
We provide a univariate specialization of the bivariate representation zeta function of 𝐆(𝔬), which results in the twist representation zeta function of this group.
According to [20, Corollary 2.11], the twist representation zeta function of 𝐆(𝔬) is given by
ζ𝐆(𝔬)irr~(s)=1+𝒵B𝔬(-s2𝟏uB,uBs-b-1),
where b=rk(𝔤′), 2uB=max{rkFrac(𝔬)B(𝐳)∣𝐳∈𝔬b}, 𝟏uB=(1,…,1)∈ℤuB and 𝒵B𝔬(r¯,t) is the integral
𝒵ℛ𝔬(r¯,t) given in (4.5) with ℛ(Y¯) being regarded as the B-commutator matrix B(Y¯) of 𝔤.
Recall that r=rk(𝔤/𝔤′)=h-b.
Proposition 4.8 states
(1-qr-s2)𝒵𝐆(𝔬)irr(s1,s2)=1+𝒵B𝔬(-2-s12𝟏uB,uBs1+s2+2uB-h-1).
Comparing the expressions for ζ𝐆(𝔬)irr~(s) and (1-qr-s2)𝒵𝐆(𝔬)irr(s1,s2), we obtain the desired specialization.
Proposition 4.11.
If G(o) has nilpotency class 2, then
(1-qr-s2)𝒵𝐆(𝔬)irr(s1,s2)|s1→s-2s2→r=ζ𝐆(𝔬)irr~(s),
provided both the left-hand side and the right-hand side converge.
In the following example, we exhibit a 𝒯-group of nilpotency class 3, whose bivariate representation zeta function does not specialize to its twist representation zeta function.
Example 4.12.
Consider the following free nilpotent ℤ-Lie lattice on 2 generators of class 3:
𝔣3,2=〈x1,x2,y,z1,z2∣i,j∈{1,2}:[x1,x2]-y,[y,xi]-zi,[zi,xj],[zi,y],[z1,z2]〉.
Let 𝔉3,2 denote the unipotent group scheme obtained from 𝔣3,2, and denote by 𝔷3,2 and by 𝔣3,2′ the center and the derived Lie lattice of 𝔣3,2, respectively.
The B-commutator matrix of 𝔣3,2 with respect to 𝐞=(y,x1,x2) and 𝐟=(z1,z2,y) is
B(Y1,Y2,Y3)=[0Y1Y2-Y10Y3-Y2-Y30].
Thus uB=1, F0(B(Y¯))={1} and F2(B(Y¯))⊇{Y12,Y22,Y32}.
It follows from Proposition 4.8 and (4.25) that
(4.26)𝒵𝔉2,3(𝔬)irr(s1,s2)=11-q2-s2(1+(1-q-1)-1∫(w,y1,y2,y3)∈𝔭×W3𝔬|w|𝔭s1+s2-4dμ)=1-q-s1-s2(1-q2-s2)(1-q3-s1-s2).
By implementing his methods in Zeta [15], Rossmann provides in [16, Table 1] the following formula for the twist representation zeta function of 𝔣3,2 – denoted by L5,9 in [16] –, provided q is sufficiently large:
(4.27)ζ𝔉3,2(𝔬)irr~(s)=(1-q-s)2(1-q1-s)(1-q2-s).
Comparing (4.26) and (4.27), we see that there is no specialization of the form (1.3) for the bivariate representation zeta function of 𝔉3,2(𝔬) that leads to its twist representation zeta function.
For the sake of completeness, we now calculate the bivariate conjugacy class and the class number zeta functions of 𝔉3,2(𝔬).
The A-commutator matrix of 𝔣3,2 with respect to 𝐞 and 𝐟 is
A(X1,X2,X3)=[X2X30-X10X30-X1-X2].
Thus uA=2,
F0(A(X¯))={1},F1(A(X¯))={-X1,±X2,X3},F2(A(X¯))⊇{X12,-X22,X32}.
Hence
𝒵𝔉2,3(𝔬)cc(s1,s2)=11-q2-s2(1+(1-q-1)-1∫(w,x1,x2,x3)∈𝔭×W3𝔬|w|𝔭2s1+s2-4dμ)=1-q-2s1-s2(1-q2-s2)(1-q3-2s1-s2).
Specialization (1.2) yields
ζ𝔉2,3(𝔬)k(s)=1-q-s(1-q2-s)(1-q3-s).
This formula agrees with the one given in [17, Section 9.3, Table 1].
∎
4.4 Local functional equations – Proof of Theorem 1.4
Proposition 4.8 assures that, for each ∗∈{irr,cc}, almost all local factors of 𝒵𝐆(𝒪)∗ are given by a rational function R∗ in certain parameters, as stated in Theorem 1.4.
In this section, we conclude the proof of Theorem 1.4 by showing that the integrals given in Proposition 4.8 behave uniformly under base extension and that they satisfy functional equations.
Fix a nonzero prime ideal 𝔭 satisfying the conditions of Proposition 4.8.
Let L be a finite extension of K=Frac(𝒪) with ring of integers 𝒪L.
For a fixed prime ideal 𝔓 of 𝒪L dividing 𝔭, write 𝔒 for the localization 𝒪L,𝔓.
Denote the relative degree of inertia by f=f(𝔒,𝔬), and hence |𝔒/𝔓|=qf.
Set 𝔤L=Λ(𝔒), and let 𝔷L and 𝔤L′ be the center and the derived Lie sublattice of 𝔤L, respectively.
Since 𝒪L is a ring of integers of a number field L, we can choose ordered sets 𝐞 and 𝐟 as the ones of Section 4.1 such that 𝐞¯ and 𝐟 are bases of 𝔤L/𝔷L and 𝔤L′,
respectively.
Let A(X¯) and B(Y¯) be the commutator matrices of 𝔤L with respect to 𝐞 and 𝐟; see Definition 4.1.
Consider the functions
𝒵𝐆(𝔒)irr~(s1,s2):-1+𝒵B𝔒((-s1-2)/2,uBs1+s2+2uB-h-1),
𝒵𝐆(𝔒)cc~(s1,s2):-1+𝒵A𝔒(-s1-1,uAs1+s2+uA-h-1),
where 𝒵B𝔒(r¯,t) and 𝒵A𝔒(r¯,t) are the integrals given in (4.5) and (4.2), respectively.
We have shown in Proposition 4.8
𝒵𝐆(𝔒)irr(s1,s2)=11-qf(r-s2)𝒵𝐆(𝔒)irr~(s1,s2),
𝒵𝐆(𝔒)cc(s1,s2)=11-qf(z-s2)𝒵𝐆(𝔒)cc~(s1,s2).
It is clear that the terms (1-qf(r-s2))-1 and (1-qf(r-s2))-1 are given by rational functions in qf and q-fs2, and that they satisfy functional equations under inversion of qf. Therefore, to prove Theorem 1.4, it suffices to show that 𝒵𝐆(𝔒)irr~(s1,s2) and 𝒵𝐆(𝔒)cc~(s1,s2) behave uniformly under base extension and satisfy functional equations.
We first show that the integrands of 𝒵𝐆(𝔒)irr~(s1,s2) and 𝒵𝐆(𝔒)cc~(s1,s2) are defined over 𝒪, that is, that only their domains of integration vary with the ring 𝔒.
The 𝔒-bases 𝐞¯ and 𝐟 are only defined locally, and hence so are the matrices A(X¯) and B(Y¯).
We must assure that there exist 𝔒-bases 𝐞¯ and 𝐟 as the ones of Section 4.1 such that the commutator matrices A(X¯) and B(Y¯), defined with respect to 𝐞 and 𝐟, are defined over 𝒪, and hence so are the sets of polynomials Fj(A(X¯)) and F2j(B(Y¯)).
Since the matrix B(Y¯) is the same as the one appearing in the integrands of [20, formula (2.8)] and A(X¯) is obtained in an analogous way, the argument of [20, Section 2.3] also holds in this case.
Namely, we choose an 𝒪-basis 𝐟 for a free finite-index 𝒪-submodule of the isolator i(Λ′) of the derived 𝒪-Lie sublattice of Λ; see Section 4.1.
By [20, Lemma 2.5], 𝐟 can be extended to an 𝒪-basis 𝐞 for a free 𝒪-submodule M of finite index of Λ.
If the residue characteristic p of 𝔭 does not divide |Λ:M| or |i(Λ′):Λ′|, this basis 𝐞 may be used to obtain an 𝔒-basis for Λ(𝔒) by tensoring the elements of 𝐞 with 𝔒.
Remark 4.13.
The condition “p does not divide |i(Λ′):Λ′|” is missing in [20], but this omission does not affect the proof of [20, Theorem A] since this condition only excludes a finite number of prime ideals 𝔭.
This was first pointed out in [3, Section 3.3].
We now recall the general integrals given in [22, Section 2.1] and show that the integrals 𝒵𝐆(𝔒)irr~(s1,s2) and 𝒵𝐆(𝔒)cc~(s1,s2) are special cases of such integrals, so that the arguments given in [1, Section 4] assure that they satisfy functional equations.
Recall that [u]={1,…,u} for u∈ℕ.
Fix l,m,n∈ℕ.
For each k∈[l], let Jk be a finite index set.
Fix I⊆[n-1].
Further, fix nonnegative integers eikj and finite sets Fkj(Y¯) of polynomials over 𝔬 for k∈[l], j∈Jk and i∈I.
Also, let 𝒲(𝔬)⊆𝔬m be a union of cosets modulo 𝔭(m).
Define
(4.28)𝒵𝒲(𝔬),I(s¯)=∫𝔭(|I|)×𝒲(𝔬)∏k=1l∥⋃j∈Jk(∏i∈Ixieikj)Fkj(y¯)∥𝔭skdμ,
where s¯=(s1,…,sl) is a vector of complex variables and
x¯=(xi)i∈I and y¯=(y1,…,ym)
are independent integration variables.
In [22, Corollary 2.4], by studying the transformation of the integral (4.28) under a principalization (Y,h) of the ideal ∏k,j(Fkj(Y¯)), Voll proved a functional equation for (4.28) under inversion of the parameter q under certain invariance and regularity conditions.
In particular, it is required that the principalization (Y,h) has good reduction modulo 𝔭, a condition that is satisfied for almost all prime ideals 𝔭.
We now relate the integrals of Proposition 4.8 with the general integral (4.28).
Set I={1}, and write x1=x.
Set n=b, m=b2, l=2uB+1, Jk={1,2} if k∈[uB] and Jk={1} if uB<k≤2uB+1.
We also set 𝒲(𝔒)=GLb(𝔒) and
kjFkje1kj≤uB1F2k(B(y¯))0uB<k≤2uB1F2(k-1-uB)(B(y¯))02uB+11{1}1≤uB2F2(k-1)(B(y¯))2
We see that, with this set-up, the integral (4.28) is equal to
(4.29)𝒵GLb(𝔒),{1}(s¯)=∫𝔓×GLb(𝔒)∥x∥𝔓s2uB+1×∏k=1uB∥F2k(B(Y¯))∪x2F2(k-1)(B(Y¯))∥𝔓sk×∏k=uB+12uB∥F2(k-1-uB)(B(Y¯))∥𝔓skdμ.
Set
𝐚1irr=(-12𝟏uB,12𝟏uB,uB),𝐚2irr=(𝟎uB,𝟎uB,1),𝐛irr=(-𝟏uB,𝟏uB,2uB-h-1),
where 𝟏uB=(1,…,1)∈ℤuB and 𝟎uB=(0,…,0)∈ℤuB.
Although the domain of integration of the integral (4.29) involves GLb(𝔒), the integrand only depends on the entries of the first column, say, as explained in [1, Section 4.1.3].
It follows that
𝒵𝐆(𝔒)irr~(s1,s2)=1+11-q-f(∏k=1b-1(1-q-fk))-1×𝒵GLb(𝔒),{1}(𝐚1irrs1+𝐚2irrs2+𝐛irr).
Analogously, for n=a, m=a2, one can find appropriate data l∈ℕ, Jk, e1jk, and Fkj(X¯) such that
𝒵𝐆(𝔒)cc~(s1,s2)=1+11-q-f(∏k=1a-1(1-q-fk))-1×𝒵GLa(𝔒),{1}(𝐚1ccs1+𝐚2ccs2+𝐛cc)
for 𝐚1cc=(-𝟏uA,𝟏uA,uA), 𝐚2cc=(𝟎uA,𝟎uA,1),
𝐛cc=(-𝟏uA,𝟏uA,uA-h-1).
Theorem 1.4 then follows by the arguments given in [1, Section 4].
