This paper investigates the average kernel sizes of matrices over certain rings, proves their generating functions are rational, and connects these to counting orbits and conjugacy classes in linear pro-p groups.
Contribution
It introduces explicit formulas for generating functions of kernel sizes over finite quotients and links these to orbit enumeration in pro-p groups using p-adic Lie theory.
Findings
01
Generated functions are rational and have fundamental properties.
02
Explicit formulas are provided for natural families of modules.
03
Connections established between kernel sizes and orbit counts in pro-p groups.
Abstract
Let O be a compact discrete valuation ring of characteristic zero. Given a module M of matrices over O, we study the generating function encoding the average sizes of the kernels of the elements of M over finite quotients of O. We prove rationality and establish fundamental properties of these generating functions and determine them explicitly for various natural families of modules M. Using p-adic Lie theory, we then show that special cases of these generating functions enumerate orbits and conjugacy classes of suitable linear pro-p groups.
Tables2
Table 1. Table 1: Complete list of generic conjugacy class zeta functions of
unipotent algebraic groups of dimension at most 5 5 5
Table 2. Table 2: Examples of generic conjugacy class zeta functions of
unipotent algebraic groups of dimension 6 6 6
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affil0affil0affiliationtext: Department of Mathematics
University of Auckland
Auckland
New Zealand
*Current address:
*School of Mathematics, Statistics and Applied Mathematics
National University of Ireland, Galway
Galway
Ireland
The average size of the kernel of a matrix and orbits of linear groups
Tobias Rossmann
Abstract
Let O be a compact discrete valuation ring
of characteristic zero.
Given a module M of matrices over O,
we study the generating function encoding the average sizes of the kernels of
the elements of M over finite quotients of O.
We prove rationality and establish fundamental properties of these
generating functions and determine them explicitly for various natural
families of modules M.
Using p-adic Lie theory, we then show that special cases of these
generating functions enumerate orbits and conjugacy classes of suitable
linear pro-p groups.
00footnotetext: 2010 Mathematics Subject Classification.
15B33, 05A15, 11M41, 11S80, 20D60, 20D15, 20E45
Keywords.
Average size of a kernel, p-adic integration, orbits of linear groups,
conjugacy classes, finite p-groups, pro-p groups, unipotent groups,
matrix Lie algebras
The author gratefully acknowledges the support of the
Alexander von
Humboldt Foundation in the form of a Feodor Lynen Research Fellowship.
This article is devoted to certain generating functions ZMask(T)
(“ask zeta functions”)
attached to modules M of matrices over compact discrete valuation rings.
The coefficients of ZMask(T) encode the average sizes of the kernels of
the elements of M over finite quotients of the base ring.
Prior to formally defining these functions and stating our main results,
we briefly indicate how our study of the functions ZMask(T) is motivated by
questions from (both finite and infinite) group theory and
probabilistic linear algebra.
Conjugacy classes of finite groups.
Given a finite group G, let k(G) denote the number of its conjugacy
classes.
It is well-known that k(G) coincides with
the number of the (ordinary) irreducible characters of G.
Let Ud⩽GLd be the group scheme of upper unitriangular d×d
matrices.
Raised as a question in [Hig60a], “Higman’s conjecture” asserts that
k(Ud(Fq)) is given by a polynomial in q for each
fixed d⩾1.
Numerous people have contributed to confirming Higman’s conjecture for small d.
In particular,
building on a long series of papers, Vera-López and Arregi [VLA03]
established Higman’s conjecture for d⩽13.
Using a different approach, Pak and Soffer [PS15]
recently provided a confirmation for d⩽16.
While Higman’s conjecture remains open in general and despite
some evidence suggesting that it may fail to hold for large d (see
[PS15]), it nonetheless influenced and inspired numerous results on
related questions; see, in particular, work of
Isaacs [Isa95, Isa07] on character degrees of so-called
algebra groups and work of Goodwin and Röhrle [GR09a, GR09b, GR09c, GR10]
on conjugacy classes of unipotent elements in groups of Lie type.
Orbits of linear groups.
All rings in this article are assumed to be commutative and unital.
Let R be a ring and let V be an R-module with ∣V∣<∞.
Given a linear group G⩽GL(V), it is a classical problem (for
R=Fq) to relate arithmetic properties of the number of orbits of G on
its natural module V to geometric and group-theoretic properties of G; see
e.g. [GLPST16] and the references therein.
This problem is closely related to the enumeration of
irreducible characters (and hence of conjugacy classes).
In particular, if G is a finite p-group of nilpotency class less than p,
then the Kirillov orbit method establishes a bijection between the ordinary
irreducible characters of G and the coadjoint orbits of G on the dual of
its associated Lie ring; cf. [GS09] and see [O'BV15] for
applications of such techniques to the enumeration of characters and
conjugacy classes.
Rank distributions and the average size of a kernel.
In addition to group-theoretic problems such as those indicated above,
this article is also inspired by questions and results from probabilistic linear algebra.
For an elementary example, to the author’s knowledge,
the number
[TABLE]
of d×e matrices of rank r with entries in
a finite field Fq was first recorded by Landsberg [Lan93].
More recently, probabilistic questions surrounding the distribution of ranks
in sets of matrices over finite fields have been studied, see e.g. [Bal68, BKW97, Coo00, DGMS10] and [Kol99, Ch. 3]; for applications,
see [SB10, LMT11].
Let R be a ring, let V and W be R-modules with ∣V∣,∣W∣<∞,
and let M⊂Hom(V,W) be a submodule.
In the following, we are primarily interested in the case that R is finite and M⊂Md×e(R) acts by right-multiplication on V=Rd.
The average size of the kernel of the elements of M
is
[TABLE]
Linial and Weitz [LW00] gave the following formula for
ask(Md×e(Fq)∣);
the same result appeared (with a different proof) in a recent paper by
Fulman and Goldstein [FG15, Lem. 3.2]
which also contains further examples of ask(M∣).
Proposition 1.1**.**
ask(Md×e(Fq)∣)=1+qd−e−q−e.
As we will see later, for a linear p-group G⩽GL(V) with a
sufficiently strong Lie theory, ∣V/G∣ and k(G)
are both instances of ask(g∣) for suitable linear Lie algebras
g.
Orbit-counting and conjugacy class zeta functions.
In the literature, numbers of the form ∣V/G∣, k(G), and ask(M∣V) for R-modules V and W, a linear group G⩽GL(V), and M⊂Hom(V,W) were primarily studied in the case that R=Fq is a finite field.
Instead of individual numbers, we consider families of such numbers obtained
by replacing Fq by the finite quotients of suitable rings.
We will use the following notation throughout this article.
Let K be a non-Archimedean local field and let O be its valuation
ring—equivalently, O is a compact discrete valuation ring with field of
fractions K;
we occasionally write OK instead of O and similarly below.
For example, K could be the field Qp of p-adic numbers (in which case
O=Zp is the ring of p-adic integers) or the field Fq((z))
of formal Laurent series over Fq (in which case O=Fq[[z]]).
Let P denote the maximal ideal of O.
Let K:=O/P be the residue field of K and
let q and p denote the size and characteristic of K, respectively.
We write Pn=P⋯P for the nth ideal power
of P; on the other hand, On=O×⋯×O denotes the
nth Cartesian power of O.
Let On=O/Pn and O∞=O.
Definition 1.2**.**
Let G⩽GLd(O) be a subgroup.
(i)
Let Gn⩽GLd(On) denote the image of G under the natural map
GLd(O)↠GLd(On).
The conjugacy class zeta function of G is
ZGcc(T):=n=0∑∞k(Gn)Tn.
2. (ii)
The orbit-counting zeta function of G is
ZGoc(T):=n=0∑∞∣Ond/G∣Tn.
Referring to these generating functions as “zeta functions” is
justified by various properties recalled or established in the following (e.g. the
existence of meromorphic continuation) for the associated Dirichlet series
ZGcc(q−s) and ZGoc(q−s), at least in
characteristic zero.
Conjugacy class zeta functions were introduced by du Sautoy [dS05] who
established their rationality for O=Zp.
Berman et al. [BDOP13] investigated ZG(O)cc(T) for
Chevalley groups G.
Lins [Lin18] recently determined ZG(O)cc(T) for
certain families of unipotent group schemes G.
Special cases of the functions ZGoc(T) have previously appeared in the literature.
In particular,
Avni et al. [AKOV16b, Thms E, A.5] determined
orbit-counting zeta functions associated with the coadjoint representation
of GLd and group schemes of the form GUd for d=2,3.
Conjugacy class and orbit-counting zeta functions are natural analogues of
the numbers of conjugacy classes and orbits of finite groups from above.
For example, it is a natural generalisation of Higman’s conjecture to ask,
for each fixed d,
whether ZUd(OK)cc(T) is given by a rational function in
qK and T as a function of K.
The definition of ZMask(T).
We now introduce the protagonist of this article.
Let V and W be finitely generated O-modules.
We frequently write Vn=V⊗On and Wn=W⊗On,
where, in the absence of subscripts, tensor products are always taken over O.
Given a submodule M⊂Hom(V,W),
we let Mn denote the image of M under the natural map
Hom(V,W)→Hom(Vn,Wn),a↦a⊗idOn.
Crucially, the module Mn does not merely depend on the abstract
module M but rather on the given embedding of M into Hom(V,W).
In particular, the natural surjection M⊗On↠Mn need not be an isomorphism;
for example, if M=P⊂O=End(O),
then M⊗O1 is isomorphic to O1 but M1=0.
In terms of matrices, for a submodule M⊂Md×e(O),
we obtain Mn⊂Md×e(On) by reducing the entries of all matrices in M modulo Pn.
This article is devoted to generating functions of the following form.
Definition 1.3**.**
Let M⊂Md×e(O) be a submodule and V=Od.
Define the ask zeta function of M to be
[TABLE]
In contrast to the probabilistic flavour of the work on the numbers ask(M∣V)
cited above, our investigations of the functions ZMask(T) draw upon
results and techniques that have been previously applied in asymptotic group
theory and, specifically, the theory of zeta functions
(representation zeta functions, in particular) of groups and other
algebraic structures; for recent surveys of this area, see
[Vol11, Klo13, Vol15].
Conversely, our study of ask zeta functions contributes to asymptotic group
theory: we will see that orbit-counting and conjugacy class zeta functions of
suitable groups are instances of ask zeta functions.
Results I: fundamental properties and examples of ask zeta functions.
Our central structural result on the functions ZMask(T) is the following.
Theorem 1.4**.**
Let O be the valuation ring of a non-Archimedean local field of
characteristic zero.
Let M⊂Md×e(O) be a submodule.
Then ZMask(T) is rational, i.e. ZMask(T)∈Q(T).
For example, Z{0d×e}ask(T)=1/(1−qdT).
At the other extreme, we will obtain the following generalisation of
Proposition 1.1.
Proposition 1.5**.**
Let O be the valuation ring of a non-Archimedean local field of arbitrary
characteristic.
Let q be the residue field size of O.
Then
[TABLE]
Note that since ZMd×e(O)ask(T)=1+(1+qd−e−q−e)T+O(T2),
Proposition 1.5 indeed generalises
Proposition 1.1.
Apart from proving Proposition 1.5, in §5, we will
also determine ZMask(T) for traceless (Corollary 5.10), symmetric
(Proposition 5.13), anti-symmetric (Proposition 5.11),
upper triangular (Proposition 5.15),
and diagonal (Corollary 5.17) matrices.
We will also explain why many of our formulae are of the same shape
as (1.2).
Despite this wealth of explicit examples in arbitrary characteristic,
the author does not know if Theorem 1.4 remains true in
positive characteristic; see §4.3.4.
Our proofs of Theorem 1.4, Proposition 1.5, and various
other results in this article rest upon expressing the functions ZMask(T)
in terms of suitable integrals (Theorem 4.5).
These integrals can then be studied using powerful techniques
developed over the past decades, primarily in the context of
Igusa’s local zeta function
(see [Den91a, Igu00] for introductions).
Our use of these techniques is similar to and inspired by their
applications in the theory of zeta functions of groups
and, in particular, the study of representation growth; see
[GSS88, JZ06, Vol10, AKOV13, SV14].
In particular, Theorem 1.4 follows from rationality
results going back to Igusa and Denef.
Using a theorem of
Voll [Vol10]
we will furthermore see that the identity
[TABLE]
is no coincidence (Theorem 4.18).
Our p-adic formalism is also compatible with our previous
computational work (summarised in [spp1489]) which allows us to
explicitly compute numerous further examples of ZMask(T);
see §9 for some of these.
While “random matrices” over local fields have been studied before
(see e.g. [Eva02]), the author is not aware of previous
applications of the particular techniques employed
(and the point of view taken) here.
Results II: ask zeta functions and asymptotic group theory.
We say that a formal power series F(T)∈Q[[T]] has
bounded denominators if there exists a non-zero a∈Z such that
aF(T)∈Z[[T]].
As usual, for a ring R and R-module V, let gl(V)
denote the Lie algebra associated with the associative algebra
End(V);
that is, gl(V)=End(V) as R-modules
and the Lie bracket of gl(V) is defined in terms of
multiplication in End(V) via [a,b]=ab−ba.
Theorem 1.6**.**
Let O be the valuation ring of a non-Archimedean local field of
characteristic zero.
Let g⊂gld(O) be a Lie subalgebra.
Then Zgask(T) has bounded denominators.
Theorem 1.6
is based on a connection between Zgask(T) and
orbit-counting zeta functions.
For a sketch, let G⩽GLd(O) act on V=Od.
As before, we write On=O/Pn and Vn=V⊗On.
Then G acts on each of the finite sets Vn
and, extending our previous definition of the orbit-counting zeta function
ZGoc(T) (Definition 1.2(ii)), we let
[TABLE]
hence, ZGoc(T)=ZGoc,0(T).
In the setting of Theorem 1.6,
by linearising the orbit-counting lemma using p-adic Lie theory,
we will see that for sufficiently large m,
there exists Gm⩽GLd(O) with
qdmZgask(T)=ZGmoc,m(T).
Theorem 1.6 then follows immediately.
In addition to using group theory to deduce properties of ask zeta functions
such as Theorem 1.6,
we will see that, conversely, our methods for studying ask zeta functions
allow us to deduce results on both orbit-counting and conjugacy class zeta
functions.
As we will now sketch,
this direction is particularly fruitful for unipotent groups.
For a Lie algebra g over a ring R, let ad:g→gl(g)
denote its adjoint representation given by ad(a):b↦[b,a] for
a∈g.
Let nd(R)⊂gld(R) denote the Lie algebra of strictly upper triangular d×d matrices.
Theorem 1.7**.**
Let O be the valuation ring of a local field
of characteristic zero and residue characteristic p.
Let g⊂nd(O) be a Lie subalgebra
and let G:=exp(g)⩽Ud(O).
Suppose that p⩾d and that g is an isolated submodule of
nd(O) (i.e. the O-module quotient nd(O)/g is torsion-free).
Then ZGoc(T)=Zgask(T) and ZGcc(T)=Zad(g)ask(T).
We will apply Theorem 1.7 and the methods for
computing ZMask(T) developed below in order to compute “generic”
conjugacy class zeta functions arising from all unipotent algebraic groups of
dimension at most 5 over a number field (see §9.3).
Due to the heavy reliance of the proofs of
Theorems 1.6–1.7 on p-adic Lie theory, it is unclear to the author whether these results have
analogues over local fields of positive characteristic.
Outline.
In §§2–3, we collect
elementary facts on ask(M∣V) and ZMask(T).
We then derive expressions for ZMask(T) in terms of suitable integrals
in §4. In §5, we use these to compute
explicit formulae for ZMask(T) for various modules M.
Next, in §6,
we discuss a relationship between the functions ZMask(T) and
“constant rank spaces” studied extensively in the literature.
A geometric source of interesting examples of ask zeta functions,
determinantal hypersurfaces, is considered in §7.
In §8, we explore the aforementioned connection
between ask, conjugacy class, and orbit-counting zeta functions in
characteristic zero.
In particular, we prove
Theorems 1.6–1.7.
Finally, given that most of the explicit formulae for ZMask(T) obtained in
§§5–6 are quite tame, §9
contains a number of more complicated examples of ZMask(T) and
ZGcc(T).
Acknowledgement
The author is grateful to Angela Carnevale and Christopher Voll for various
discussions, to Eamonn O’Brien for helpful comments, to Christoph Hanselka for
pointing out a flaw in an earlier version, and to the referee for numerous
insightful suggestions.
2 Elementary properties of average sizes of kernels
We collect some elementary observations on average sizes of kernels.
Throughout, unless otherwise stated, let R be a ring, let V and
W be R-modules with ∣V∣,∣W∣<∞, and let M⊂Hom(V,W) be a submodule.
2.1 Rank varieties
In the case of a finite field R=Fq, ask(M∣V) admits a natural geometric interpretation.
Namely, by choosing a basis of M, we may identify M=AFqℓ(Fq),
where ℓ=dimFq(M) and AFqℓ=Spec(Fq[X1,…,Xℓ]).
We may then decompose AFqℓ=i=0∐rVi, where Vi is the
subvariety of maps of rank i.
(Note that if M=Md×e(Fq), then #Vr(Fq) is given by (1.1).)
Then
ask(M∣V)=∑i=0d#Vi(Fq)⋅qd−i−ℓ;
in fact, by replacing q by qf on the right-hand side,
we express ask(M⊗FqFqf∣V⊗FqFqf)
using a formula which is valid for any f⩾1.
However, even for M=Md×e(Fq),
this approach yields a fairly complicated interpretation of
Proposition 1.1.
2.2 Kernels and orbits
One simple yet crucial observation contained in the proof of
[LW00, Thm 1.1] is the following connection between the sizes of the
kernels Ker(a) (a∈M) and those of the “orbits”
xM:={xa:a∈M}≈M/cM(x),
where x∈V and cM(x):={a∈M:xa=0};
note that in contrast to orbits under group actions, the sets xM always overlap.
Lemma 2.1** (Cf. [LW00, Thm 1.1]).**
ask(M∣V)=x∈V∑∣xM∣−1.
We give two proofs of this lemma.
The first is a combinatorial version of a probabilistic argument in
the proof of [LW00, Thm 1.1].
We include it here since our terminology is different
from theirs; similar arguments appear in [Loc07].
By computing #{(x,a)∈V×M:xa=0} in two ways,
we obtain
a∈M∑∣Ker(a)∣=x∈V∑∣cM(x)∣.
Since xM≈M/cM(x) as R-modules, we have
∣cM(x)∣=∣M∣/∣xM∣ and thus
ask(M∣V)=∣M∣−1x∈V∑∣cM(x)∣=x∈V∑∣xM∣−1.
∎
Our second proof of Lemma 2.1 already hints at the
connection between average sizes of kernels and orbits of linear
groups, a subject further explored in §8.
Recall that for a finite group G acting on a finite set X, the
orbit-counting lemma asserts that
∣X/G∣=∣G∣−1g∈G∑∣Fix(g)∣,
where Fix(g)={x∈X:xg=x}.
The rule
a\mapsto a^{*}:=\bigl{[}\begin{smallmatrix}1&a\\
0&1\end{smallmatrix}\bigr{]} yields an isomorphism of (M,+) onto a subgroup M∗
of GL(V⊕W).
We claim that the natural bijection V⊕W→V∐W induces a bijection
(V⊕W)/M∗→x∈V∐W/xM.
Indeed, (x,y)a∗=(x,y+xa) for (x,y)∈V⊕W.
As Fix(a∗)=Ker(a)⊕W,
the orbit-counting lemma yields
[TABLE]
In order to deduce Proposition 1.1
from Lemma 2.1, note that xMd×e(Fq)=Fqe for each non-zero x∈Fqd whence
ask(Md×e(Fq)∣)=1+(qd−1)q−e.
2.3 Interlude: rank distributions and hyperoctahedral groups
We discuss combinatorial consequences of Proposition 1.1.
Reminder: the hyperoctahedral groups Bn.
For background and details on the following, see [Bre94, §3] or [BB05, §8.1].
The hyperoctahedral groupBn={±1}≀Sn is the group of signed
permutations on n letters; we regard Bn as a subgroup of the symmetric
group on {±1,…,±n} and as a Coxeter group in the usual way
(see [BB05, p. 246]).
For σ∈Bn, we write σ=[1σ,…,nσ].
For σ∈Bn, let len(σ) denote the (Coxeter) length of σ
(see [Bre94, Prop. 3.1] for a combinatorial description),
let \operatorname{N}(\sigma):=\#\bigl{\{}i\in\{1,\dotsc,n\}:i^{\sigma}<0\bigr{\}}, and let \operatorname{Des}(\sigma):=\bigl{\{}i\in\{0,\dotsc,n-1\}:i^{\sigma}>(i+1)^{\sigma}\bigr{\}} (where we wrote 0σ=0)
denote the descent set of σ.
For I⊂{0,…,n}, define the quotientBnIc:={σ∈Bn:Des(σ)⊂I}
(see [BB05, §2.4]).
The identity 1∈Bn is the unique element of length [math].
Moreover, since Des(1)=∅, the identity is contained in each
set BnIc.
Let Mdrk=i(Fq):={a∈Md(Fq):rk(a)=i}.
As explained in [CSV17, §3.2], for i=0,…,d,
[TABLE]
whence
[TABLE]
On the other hand, by Proposition 1.1,
ask(Md(Fq)∣)=2−q−d.
The right-hand side of (2.2) is a polynomial in Z[q−1],
the constant term, 2, of which arises from σ=1∈Bn{i}c and i=0,1.
However, the fact that the other terms of (2.2) add up to
−q−d seems much less transparent.
Consider, for example, the case d=2.
For i=0,1, we have qi−i2=1 and the contributions to the right-hand
side of (2.2) are exactly the terms
(−1)N(σ)q−len(σ) indicated in the following tables.
[TABLE]
[TABLE]
For i=2, B2{2}c={1} and the contribution to the right-hand
side of (2.2) is a single summand q−2.
While we can therefore confirm that
[TABLE]
the author is unable to provide a combinatorial explanation of this numerical coincidence.
A further source of such examples is given by
analogues of (2.1) for the numbers of traceless, antisymmetric, and
symmetric d×d matrices over Fq, respectively, due to Carnevale et
al. [CSV17, §3.2]; cf. [SV14].
The average sizes of the kernels of all these spaces of matrices are
known or can be deduced as by-products or our investigations in
§5.3 below.
2.4 Direct sums
Lemma 2.2**.**
Let V′ and W′ be R-modules with ∣V′∣,∣W′∣<∞.
Let M′⊂Hom(V′,W′) be a submodule.
We regard M⊕M′ as a submodule of Hom(V⊕V′,W⊕W′) in the natural way.
Then ask(M⊕M′∣V⊕V′)=ask(M∣V)⋅ask(M′∣V′).
Proof.
[TABLE]
Corollary 2.3**.**
Let R be finite, M⊂Md×e(R) be a
submodule, and M~⊂M(d+1)×e(R) be
obtained from M by adding a zero row to the elements of M in some fixed
position.
Then ask(M~∣)=ask(M∣)⋅∣R∣.
∎
2.5 Matrix transposes
Following Kaplansky [Kap49], we call R an elementary divisor
ring if for each a∈Md×e(R) (and all d,e⩾1), there
exist u∈GLd(R) and v∈GLe(R) such that uav is a diagonal
matrix (padded with zeros according to the shape of a).
For example, any quotient of a principal ideal domain is an elementary divisor ring
(regardless of whether the quotient is an integral domain or not).
Write a⊤ for the transpose of a.
Lemma 2.4**.**
Let R be a finite elementary divisor ring and let M⊂Md×e(R) be a submodule.
Write V=Rd and W=Re.
Then ask(M⊤∣W)=ask(M∣V)⋅∣R∣e−d.
Proof.
Let a=[diag(a1,…,ar)000]∈Md×e(R). Then Ker(a) consists of those
x∈V with aixi=0 for 1⩽i⩽r and Ker(a⊤)
consists of those y∈W with aiyi=0 for 1⩽i⩽r.
∎
2.6 Reduction modulo a and base change R→R/a
Let V and W be finitely generated R-modules, the underlying sets of
which need not be finite. As before, let M⊂Hom(V,W) be a submodule.
Let a◃R with ∣R/a∣<∞.
Define Va:=V⊗RR/a,
Wa:=W⊗RR/a, and let Ma
be the image of the natural map M↪Hom(V,W)→Hom(Va,Wa).
In general, the natural surjection M⊗RR/a↠Ma need not be injective (see the example on p. 1).
However, if M is finitely generated, then M⊗RR/a is
finite and we obtain the following expression for ask(Ma∣Va).
Lemma 2.5**.**
Suppose that M is finitely generated.
Then
[TABLE]
3 Basic algebraic and analytic properties of ZM(T) and ζM(s)
3.1 Average sizes of kernels and Dirichlet series: ζM(s)
While our main focus is on the generating functions ZM(T) from
the introduction, it is natural to also consider a global analogue.
First suppose that R is a ring which contains only finitely many
ideals a of a given finite norm ∣R/a∣.
Given a submodule M⊂Md×e(R) acting on V=Rd
and an ideal a◃R,
let Va=(R/a)d, Wa=(R/a)e,
and let Ma denote the image of the natural map M↪Md×e(R)↠Md×e(R/a) (cf. §2.6).
Definition 3.1**.**
(i)
Define a formal Dirichlet series
[TABLE]
where the sum extends over the ideals of finite norm of R
and s denotes a complex variable.
2. (ii)
Let αM∈[−∞,∞] denote the abscissa of convergence of
ζM(s).
3.2 Abscissae of convergence: local case
Let K be a local field of arbitrary characteristic with valuation
ring O and residue field size q.
Let M⊂Md×e(O) be a submodule acting on V=Od.
Then ζM(s)=ZM(q−s).
Moreover, if O has characteristic zero, then
Theorem 1.4
(proved in §4.3.3)
implies that αM is precisely the
largest real pole of (the meromorphic continuation of) ζM(s).
Recall that, unless otherwise indicated, tensors products are taken over O.
Definition 3.2**.**
The generic orbit rank of M is gor(M):=x∈VmaxdimK(xM⊗K).
Our choice of terminology will be justified by Proposition 4.13.
Proposition 3.3**.**
\max\bigl{(}d-\operatorname{gor}(M),0\bigr{)}\leqslant\alpha_{M}\leqslant d.
Proof.
The upper bound follows since ask(Mn∣Vn)⩽∣Vn∣=qnd and ∑n=0∞qn(d−s) converges for Re(s)>d.
Similarly, the lower bound follows from Lemma 2.1 and
ask(Mn∣Vn)⩾max(∣Vn∣/qngor(M),1).
∎
Let 0⩽r⩽e.
Let M⊂Md×e(O) be obtained from Md×r(O) by inserting e−r zero columns in some fixed
positions.
Then gor(M)=r,
ζM(s)=ζMd×r(O)(s),
and it will follow from Proposition 1.5
that αM=max(d−r,0).
In particular, the bounds in Proposition 3.3 are optimal.
We note that Example 9.1 below illustrates that
the meromorphic continuation of ζM(s) (cf. Theorem 1.4) may have real poles less than d−e.
3.3 Abscissae of convergence in the global case and Euler products
Let k be a number field with ring of integers o.
Let Vk denote the set of non-Archimedean places of k.
For v∈Vk, let kv be the v-adic completion of k and let
ov be its valuation ring.
We let qv denote the size of the residue field Kv of kv.
For an o-module U and v∈Vk, we write Uv:=U⊗oov (regarded as an ov-module).
Let M⊂Md×e(o) be a submodule.
For v∈Vk,
we may identify Mv with the ov-submodule of Md×e(ov)
generated by M.
Proposition 3.4**.**
Let M⊂Md×e(o) be a submodule. Then:
(i)
αM⩽d+1.
2. (ii)
ζM(s)=v∈Vk∏ζMv(s).
Proof.
Let V=od.
The proof of (i) is similar to that of
Proposition 3.3.
Namely, for each a◃o, ask(Ma∣Va)⩽∣o/a∣d and
∑a∣o/a∣d−s=ζk(s−d) converges for
Re(s)>d+1, where ζk(s) is the Dedekind zeta
function of k.
For (ii), it suffices to show that for non-zero coprime ideals
a,b◃o, ask(Mab∣Vab)=ask(Ma∣Va)⋅ask(Mb∣Vb).
To that end, the natural isomorphism o/ab→o/a×o/b yields an (o-module) isomorphism
M⊗oo/ab→(M⊗oo/a)×(M⊗oo/b)
which is compatible with the corresponding isomorphism
Vab→Va×Vb in the evident way.
Hence, for aˉ∈M⊗oo/ab
corresponding to (aˉa,aˉb)∈(M⊗oo/a)×(M⊗oo/b),
we obtain an isomorphism
Ker(aˉ∣Vab)→Ker(aˉa∣Va)×Ker(aˉb∣Vb).
Part (ii) thus follows from Lemma 2.5.
∎
Example 3.5**.**
Let ζk(s) denote the Dedekind zeta function of k.
Then Proposition 1.5 and Proposition 3.4(ii) imply
that
ζMd×e(o)(s)=ζk(s)ζk(s−d+e)/ζk(s+e).
Further analytic properties of ζM(s) in a global setting will be
derived in §4.5.
3.4 Hadamard products
Recall that the Hadamard productF(T)⋆G(T) of formal power
series F(T)=∑n=0∞anTn
and G(T)=∑n=0∞bnTn
with coefficients in some common ring is
[TABLE]
The following is an immediate consequence of Lemma 2.2.
Corollary 3.6**.**
Let K be a local field of arbitrary characteristic with valuation ring O.
Let M=A⊕B⊂Md+e(O)
for submodules A⊂Md(O) and B⊂Me(O).
Then ZM(T)=ZA(T)⋆ZB(T). ∎
We note that Hadamard products of rational generating functions are
well-known to be rational (see [Sta12, Prop. 4.2.5]).
Corollary 3.7**.**
Let M⊂Md×e(O) be a submodule.
Define f=max(d,e).
Let M~⊂Mf(O) be obtained from M by
adding f−d zero rows and f−e zero columns to the elements of M
in some fixed positions.
Then ZM~(T)=ZM(qf−dT). ∎
Thus, various questions on the series ZM(T) are reduced to the case of square matrices.
3.5 Rescaling
Let O be the valuation ring of a non-Archimedean local field K
of arbitrary characteristic.
Let M⊂Md×e(O) be a submodule, V=Od, and
W=Oe.
Definition 3.8**.**
For m⩾0, let
ZMm(T):=n=m∑∞ask(Mn∣Vn)⋅Tn−m∈Q[[T]].
Note that ZM(T)=ZM0(T).
For an O-module U, let Um=PmU
and write Unm for the common value of (Um)n and (Un)m.
Clearly, if n⩽m, then ask(Mnm∣Vn)=ask({0}∣Vn)=∣Vn∣=qnd.
Proposition 3.9**.**
ZMmm(T)=qdm⋅ZM(T).
Proof.
It suffices to show that
ask(Mnm∣Vn)=qdm⋅ask(Mn−m∣Vn−m)
for n⩾m.
Choose π∈P∖P2.
Observe that multiplication by πm induces an O-module
isomorphism Mn−m→Mnm and a monomorphism Vn−m→Vn
with image Vnm.
For a∈M,
[TABLE]
has size ∣Ker(a∣Vn−m)∣⋅qdm.
We conclude that
[TABLE]
4 Rationality of ZM(T) and p-adic integration
Unless otherwise stated, in this section, K is a non-Archimedean
local field of arbitrary characteristic with valuation ring O.
Given a submodule M⊂Md×e(O),
we use the original definition of ask(M∣V) as well as the alternative
formula in Lemma 2.1 to derive two types of expressions for
ZM(T) in terms of p-adic integrals.
In §4.1, we describe a general setting for
rewriting certain generating functions as integrals.
By specialising to the case at hand, we obtain, in
§4.2,
two expressions for ZM(T) (Theorem 4.5)
in terms of functions KM and OM that we introduce.
In §4.3, we derive explicit formulae (in terms
of the absolute value of K and polynomials over O) for these
functions.
These formulae serve two purposes. First, using established
rationality results from p-adic integration, they allow us to deduce
Theorem 1.4.
Secondly, these formulae, in particular the one based on
OM (and hence on Lemma 2.1), lie at the heart of
explicit formulae such as Proposition 1.5 in §5.
4.1 Generating functions and p-adic integrals
Let Z be a free O-module of finite rank d and let U⊂Z be a
submodule.
By the elementary divisor theorem, there exists a unique
sequence
(λ1,…,λd) with 0⩽λ1⩽⋯⩽λd⩽∞ such that Z/U≈⨁i=1dOλi as O-modules;
recall that O∞=O.
We call (λ1,…,λd) the submodule type of U within Z.
Recall that the isolatorisoZ(U) of U in Z is the
preimage of the torsion submodule of Z/U under the natural map Z↠Z/U.
Equivalently, isoZ(U) is the smallest direct summand of Z which
contains U.
Recall that U is isolated in Z if and only if U=isoZ(U);
this is equivalent to each λi from above being either [math] or ∞.
If U is isolated in Z, then we may naturally identify U⊗On
with the image Un of U↪Z↠Zn:=Z⊗On;
to see that this identification may fail if U is not isolated,
consider e.g. U=P⊂O=Z and n=1.
If U is isolated in Z and U has rank ℓ, say, then ∣Un∣=qℓn.
As we will now see, the general case is only slightly more complicated.
We let ν denote the valuation on K with value group Z.
Let ∣⋅∣ be the absolute value on K with ∣π∣=q−1
for π∈P∖P2;
we write ∥A∥=sup(∣a∣:a∈A).
Lemma 4.1**.**
Let U⊂Z be a submodule.
Suppose that U has submodule type (λ1,…,λℓ) within isoZ(U).
Let Un denote the image of the natural map U↪Z↠Zn:=Z⊗On.
Then ∣Un∣=qi=1∑ℓ(n−min(λi,n)).
In particular, for y∈O with ν(y)=n, ∣Un∣=i=1∏ℓ∣y∣∥πλi,y∥.
Proof.
Clearly,
∣Un∣=q∑i=1ℓmin(n−λi,0)
and the first identity follows from min(0,n−a)=n−min(n,a);
the second claim then follows immediately.
∎
The following result is concerned with generating functions
associated with a given family of “weight functions” fn:Un→R⩾0.
Given a free O-module W of finite rank,
let μW denote the Haar measure on W with μW(W)=1.
Lemma 4.2**.**
Let U⊂Z be a submodule.
Suppose that U has submodule type (λ1,…,λℓ) within isoZ(U).
Let Un denote the image of U↪Z↠Zn:=Z⊗On
and let πn:U→Un denote the natural map.
Let N⩾0 and, for n⩾N, let fn:Un→R⩾0 be
given.
Define
[TABLE]
and extend F to a map U×PN→R⩾0 via
F(x,0)≡0.
Let
[TABLE]
where we set g(0,s)≡0.
(Note that g(y,s)=∣y∣s−1⋅∣Uν(y)∣ for
y=0 by Lemma 4.1.)
Suppose that δ⩾0 satisfies
ˉx∈Un∑fn(ˉx)=O(qδn).
Then, writing t=q−s, for all s∈C with Re(s)>δ,
[TABLE]
Proof.
First note that the left-hand side of (4.1)
converges for Re(s)>δ.
Further note that we may ignore the case y=0 on the right-hand
side as it occurs on a set of measure zero.
Let Un=Ker(πn:U↠Un).
Given (x,y)∈U×PN∖{0} with n:=ν(y), the map F is constant
on the open set (x+Un)×(y+Pn+1);
in particular, F is measurable.
Let Rn⊂U be a complete and irredundant set of representatives
for the cosets of Un and let Wn=Pn⋅O×=Pn∖Pn+1.
By Lemma 4.1, μU(Un)=∣Un∣−1=i=1∏ℓ∥πλi,y∥∣y∣ for any y∈Wn; moreover, μO(Wn)=(1−q−1)q−n.
The claim now follows via
[TABLE]
Remark 4.3**.**
The introduction of the additional variable y to express a
generating function as an integral in Lemma 4.2
mimics similar formulae of Jaikin-Zapirain [JZ06, §4] and
Voll [Vol10, §2.2].
4.2 ZM(T) and the functions KM and OM
Let M⊂Md×e(O) be a submodule, V=Od, and
W=Oe.
Definition 4.4**.**
Define
[TABLE]
Note that for y=0, 1⩽KM(a,y)⩽∣y∣−d and 1⩽OM(x,y)⩽∣y∣−gor(M) (see Definition 3.2); these
are the same estimates as in Proposition 3.3.
Using Lemmas 2.1 and 4.2, we obtain
the following formulae for ZM(t) (where t=q−s).
Theorem 4.5**.**
For s∈C with Re(s)>d,
[TABLE]
Proof.
The given formulae for (1−q−1)⋅ZM(q−s)
are based on ask(Mn∣Vn)=∣Mn∣−1⋅∑a∈Mn∣Ker(a)∣ and
ask(Mn∣Vn)=∑ˉx∈Vn∣ˉxMn∣−1 (Lemma 2.1), respectively.
In detail, the first equality follows from Lemma 4.2 with
U=M, Z=Md×e(O),
F(a,y)=∣Mν(y)∣−1KM(a,y)
and g(y,s)=∣y∣s−1⋅∣Mν(y)∣ (for y=0).
For the second equality, use Lemma 4.2 with U=Z=V,
F(x,y)=∣(x+yOd)Mν(y)∣−1=OM(x,y)−1
and
g(y,s)=∣y∣s−d−1 (for y=0).
∎
4.3 Explicit formulae for KM and OM
As before, let M⊂Md×e(O) be a submodule.
In order to use Theorem 4.5 for theoretical
investigations or explicit computations of ZM(T),
we need to produce sufficiently explicit formulae for
KM(a,y) or OM(x,y).
4.3.1 The sizes of kernels and images
Let a∈Md×e(O) have rank r over K.
Fix π∈P∖P2.
By the elementary divisor theorem, there are 0⩽λ1⩽⋯⩽λr, u∈GLd(O), and v∈GLe(O) such that
[TABLE]
We call (λ1,…,λr) the equivalence type of a.
For a set of polynomials f(Y), we write
f(y)={f(y):f∈f}.
Lemma 4.6**.**
Let a∈Md×e(O) have rank r over K and
equivalence type (λ1,…,λr).
Let fi(Y)⊂Z[Y]:=Z[Yij:1⩽i⩽d,1⩽j⩽e] be the set of non-zero i×i minors of the generic
d×e matrix [Yij]∈Md×e(Z[Y]).
(Hence, f0(Y)={1}.)
Let an∈Md×e(On) be the image of the matrix a under the natural
map Md×e(O)↠Md×e(On).
Then:
(i)
∥fi(a)∥=q−λ1−⋯−λi* for 0⩽i⩽r.*
2. (ii)
(Cf. [Vol10, §2.2])*
∥fi(a)∪πnfi−1(a)∥∥fi−1(a)∥=qmin(λi,n) for 1⩽i⩽r.*
3. (iii)
The first part is elementary linear algebra.
Part (ii) then follows from
[TABLE]
For 0⩽i⩽n, the map On→On given by multiplication by πi
has kernel and image size qi and qn−i, respectively.
Parts (iii)–(iv) thus follow from equation (4.2).
∎
4.3.2 A formula for KM
We use Lemma 4.6 in order to derive a formula for KM(a,y).
Definition 4.7**.**
The generic element rank of M is
grk(M):=a∈MmaxrkK(a).
By the following, grk(M) is the rank of a “generic” matrix
in M in any meaningful sense.
Proposition 4.8**.**
Let (a1,…,aℓ) be an O-basis of M and let
λ1,…,λℓ be algebraically independent over K. Then:
(i)
grk(M)=rkK(λ1,…,λℓ)(λ1a1+⋯+λℓaℓ).
2. (ii)
μM({a∈M:rkK(a)<grk(M)})=0.
Proof.
Let r denote the right-hand side in (i).
Then r is the largest number such that some r×r minor,
m(λ1,…,λℓ) say, of λ1a1+⋯+λℓaℓ is non-zero.
In particular, grk(M)⩽r.
Conversely, since m(λ1,…,λℓ)=0, we
find c1,…,cℓ∈O with
m(c1,…,cℓ)=0 whence grk(M)⩾r.
Finally, the well-known fact (provable using induction and Fubini’s
theorem) that the zero locus of a non-zero polynomial over K has
measure zero implies (ii).
∎
Excluding a set of measure zero, we thus obtain the following
formula for KM(a,y).
Corollary 4.9**.**
Let N={a∈M:rkK(a)<grk(M)}
and let fi(Y) be the set of non-zero i×i minors of
the generic d×e matrix.
Then for all a∈M∖N and y∈O∖{0},
As in the proof of Proposition 4.8,
we may replace the fi(Y) in (4.3) by
polynomials in a chosen system of coordinates of M.
We may thus interpret the integral in (4.3)
as being defined in terms of valuations of polynomial expressions in
dimK(M⊗K)+1 variables.
Integrals of this form have been studied extensively.
In particular, using well-known results going back to work of Igusa
and Denef (see [Den91a]), initially for a single polynomial and
later extended to families of polynomials (see, in
particular, work of du Sautoy and Grunewald [dSG00],
Veys and Zúñiga-Galindo [VZG08], and Voll [Vol10]),
we obtain the following two results,
the first of which implies and refines Theorem 1.4.
Theorem 4.10**.**
Let O be the valuation ring of a local field of characteristic
zero and let M⊂Md×e(O) be a submodule.
Then ZM(T)∈Q(T).
More precisely, there exist f(T)∈Z[T], non-zero (a1,b1),…,(ar,br) with (ai,bi)∈Z×N0, and m∈N0 such that
[TABLE]
Moreover, in a global setting, the dependence of Euler factors on
the place is as follows.
Theorem 4.11**.**
Let k be a number field with ring of integers o.
Recall the notation from §3.3.
Let M⊂Md×e(o) be a submodule.
There are separated o-schemes V1,…,Vr of finite
type and rational functions W1(X,T),…,Wr(X,T)∈Q(X,T)
(which can be written over denominators of the same shape as those in
Theorem 4.10) such that the following holds:
for almost all v∈Vk,
[TABLE]
4.3.4 Local fields of positive characteristic
Suppose that M⊂Md×e(O) is a submodule as before
but that K=K[[z]] is a local field of positive characteristic.
Then the techniques cited above to establish rationality in
Theorem 4.10 do not apply to ZM(T)
and indeed, the author does not know if ZM(T) is necessarily rational
in positive characteristic.
By combining Igusa’s original proof of the rationality of his local zeta function
(see [Den91a, §1.3] for a modern account)
and ideas of du Sautoy and Grunewald [dSG00, §2], we obtain
the following sufficient condition for rationality of ask zeta functions over O:
if every hypersurface embedded inside some affine space over K
admits an embedded resolution of singularities over K, then ZM(T) is
rational for all modules M⊂Md×e(O).
The status of resolution of singularities in positive characteristic is
presently unresolved; see e.g. [Hau10].
In contrast, to such unresolved issues, ask zeta functions in “large”
positive characteristic arising from global models in characteristic
zero are amenable to existing techniques.
Namely, by applying powerful model-theoretic transfer principles such as
[CL10, Thm 9.2.4] to our integrals, we obtain the following.
Theorem 4.12**.**
Let k be a number field with ring of integers o.
Recall the notation from §3.3.
Let M⊂Md×e(o) be a submodule.
Then for almost all v∈Vk,
ZMv(T)=ZM⊗oKv[[z]](T).
In particular, ZM⊗oKv[[z]](T) is rational for
almost all v∈Vk.
∎
We note that Theorems 4.11–4.12 both behave well under local base
extensions; cf. [stability, Thm 2.3] and [Vol17, Rem. 1.6].
4.3.5 A formula for OM
As in the case of KM, we can produce a formula for OM.
Let X=(X1,…,Xd) be algebraically independent over K.
Let C(X)∈Mℓ×e(O[X]) with
O[X]dC(X)=X⋅(M⊗O[X]).
For example,
we may choose generators a1,…,aℓ of M
as an O-module and take
[TABLE]
Let gi(X) be the set of non-zero i×i minors of C(X).
Note that if x∈V, then xM is the row span of C(x) over O
so that, in particular, dimK(xM⊗K)=rkK(C(x)).
Recall the definition of gor(M) in Definition 3.2.
The following is proved in the same way as
Proposition 4.8.
Proposition 4.13**.**
Let X=(X1,…,Xd) be algebraically independent over K.
Then:
(i)
\operatorname{gor}(M)=\dim_{K(\bm{X})}\bigl{(}\bm{X}\,\cdotp(M\otimes K(\bm{X}))\bigr{)}=\operatorname{rk}_{K(\bm{X})}(C(\bm{X})).
2. (ii)
μV({x∈V:dimK(xM⊗K)<gor(M)})=0.
∎
The following analogue of Corollary 4.9 is obtained
using Lemma 4.6(ii),(iv).
Corollary 4.14**.**
Let Z={x∈V:dimK(xM⊗K)<gor(M)}.
Then for all x∈V∖Z and y∈O∖{0},
[TABLE]
Theorem 4.5 thus provides us with the following
counterpart of (4.3):
[TABLE]
Despite the essentially identical shapes of the integrals in
(4.3) and (4.6),
either type might be vastly more useful for explicit computations
of specific examples.
In particular, §5 is concerned with examples of
ZM(T) that can be easily computed using (4.6)
and §6 considers the analogous situations for
(4.3).
A very similar phenomenon was exploited by O’Brien and Voll [O'BV15, §5] in their enumeration of conjugacy classes of certain relatively
free p-groups.
Remark 4.15**.**
We note that the integrals in
(4.3)–(4.6) are almost of the
same shape as those in [AKOV13, Eqn (1.4)].
These similarities can be clarified further by rewriting our integrals
slightly; see the proof of Theorem 4.18 below.
We further that the role of the matrix C(X) here
is similar to that of A(X) in [O'BV15, Def. 2.1].
4.4 Projectivisation
For later applications, in the following, we record “projective”
versions of the integrals in Theorem 4.5 in the spirit of
Voll’s formalism [Vol10, §2.2], as rewritten by Avni et
al. [AKOV13, §3.2].
Let M⊂Md×e(O) be a submodule and
V=Od.
The following lemma is elementary.
Lemma 4.16**.**
Let x∈V, a∈M, y∈O∖{0}, and π∈P∖P2.
Then:
(i)
OM(πx,πy)=OM(x,y).
2. (ii)
KM(πa,πy)=qd⋅KM(a,y).
Proof.
Let W:=Oe.
(i)
For a submodule U⊂W and z∈O,
let Uz denote the image of U in W⊗O/(z).
Since multiplication by π induces an isomorphism
Uz≈(πU)πz, the claim follows by taking U=xM and
z=y.
2. (ii)
Let x∈V.
Then x(πa)≡0(modπyW) if and only if xa≡0(modyW).
Clearly, each residue class modulo yW of such elements has precisely qd lifts
modulo πyW.
∎
Proposition 4.17**.**
Let ℓ:=dimK(M⊗K).
Then:
(i)
(1−q−s)⋅ZM(q−s)=1+(1−q−1)−1(V∖PV)×P∫OM(x,y)∣y∣s−d−1dμV×O(x,y).**
2. (ii)
First, OM(x,y)=1 for x∈V and y∈O×.
In the following,
we write
o as a shorthand for OM(x,y)∣y∣s−d−1dμV×O(x,y).
Using Lemma 4.16 and a change of variables,
we find that PV×P∫o=t⋅V×O∫o.
The claim then follows from Theorem 4.5 and
[TABLE]
2. (ii)
For a∈M and y∈O×,
KM(a,y)=1.
One may then proceed similarly to (i) using
[TABLE]
where κ=∣y∣s−1KM(a,y)dμM×O(a,y).
∎
4.5 Local functional equations and global analytic properties
Functional equations under “inversion of the prime”
are a common (but not universal) phenomenon in the theory of local
zeta functions.
Denef and Meuser [DM91] showed that for a homogeneous polynomial over
a number field, almost all of its associated local Igusa zeta
functions satisfy such a functional equation.
Vastly generalising their result, Voll [Vol10] established
functional equations for numerous types of zeta functions arising in
asymptotic algebra and expressible in terms of p-adic integrals.
For further positive results establishing such functional equations,
see, in particular, work of du Sautoy and Lubotzky [dSL96],
Voll [Vol17], Avni et al. [AKOV13, §4], and
Stasinski and Voll [SV14, Thm A].
Using the formalism developed above, we may deduce the following; recall the
notation from §3.3.
Theorem 4.18**.**
Let k be a number field with ring of integers o.
Let M⊂Md×e(o) be a submodule.
Then for almost all v∈Vk,
[TABLE]
Remark 4.19**.**
(i)
The operation “qv→qv−1” can be unambiguously defined in
terms of an arbitrary formula of the form (4.4);
see [DM91, Vol10] and cf. [stability, Cor. 4.3].
If ZMv(T)=W(qv,T) for almost all
v∈Vk and some W(X,T)∈Q(X,T),
then Theorem 4.18 asserts that
W(X−1,T−1)=(−XdT)⋅W(X,T);
see [stability, §4] and cf. (1.3).
2. (ii)
Using Theorem 4.11–4.12,
Theorem 4.18 also establishes, for
almost all v∈Vk, a functional equation for
ZM⊗oKv[[z]](T);
cf. [Vol17, Cor. 1.3].
We use Voll’s results from [Vol10, §2.1].
Let K=kv for v∈Vk and let O be as before.
Let Hv(s) denote the right-hand side in
Proposition 4.17(i).
Using the surjection GLd(O)→V∖PV which sends a
matrix to its first column, we rewrite Hv(s) in terms of an integral over
GLd(O)×P; cf. [AKOV13, §4].
In the setting of the explicit formula for OMv(x,y)
derived in §4.3.5, we may assume that C(X) is a matrix of
linear forms whence each gi(X) consists of homogeneous
polynomials of degree i.
This allows us to use
[Vol10, Cor. 2.4] which shows that
H_{v}(s)\!\bigm{|}_{q_{v}\to q_{v}^{-1}}=q^{d}H_{v}(s)
for almost all v∈Vk.
∎
Based on work of du Sautoy and Grunewald [dSG00],
Duong and Voll [DV17] studied analytic properties of Euler products of
functions of the same form as the right-hand sides
in Proposition 4.17(i)–(ii).
In particular, their findings allow us to deduce the following.
Theorem 4.20** (Cf. [DV17, Thm A]).**
Let k be a number field with ring of integers o.
Let M⊂Md×e(o) be a submodule and V=od.
(i)
The abscissa of convergence αM of ζM(s) is a rational number.
2. (ii)
There exists δ>0 such that ζM(s) admits
meromorphic continuation to {s∈C:Re(s)>αM−δ}.
This continued function has a pole of order βM, say, at s=αM but no other poles on
the line Re(s)=αM.
3. (iii)
αM* and βM are (and δ can be chosen to be) invariant
under base change of M from o to the ring of integers of an arbitrary
finite extension of k.*
4.6 Reduced and topological ask zeta functions
Let k be a number field with ring of integers o.
Recall the notation from §3.3.
Given a suitable family of local zeta functions indexed by places v∈Vk, associated “reduced” and “topological” zeta functions are
obtained by passing to two different limits “qv→1”.
The original topological zeta functions of Denef and Loeser [DL92] are
singularity invariants attached to polynomials.
Later, du Sautoy and Loeser [dSL04] defined topological subobject zeta
functions of algebraic structures.
Reduced subobject zeta functions were introduced by Evseev [Evs09].
Topological and reduced representation zeta functions of unipotent groups were
studied by the author in [unipotent] and [padzeta, §7], respectively.
The techniques from p-adic integration used above are similar to those
employed in the study of representation zeta functions of unipotent groups.
As a consequence, we immediately obtain adequate notions of reduced
and topological ask zeta functions which we now briefly discuss.
Let M⊂Md×e(o) be a submodule.
Topological ask zeta functions.
Informally, the topological ask zeta function ζMtop(s)∈Q(s)
of M is the constant term of (1−qv−1)ζMv(s) as a series in qv−1;
for a rigorous definition, combine the formalism
developed in [topzeta, §5] (and summarised in [spp1489, §4.2]),
Proposition 4.17, and [unipotent, Pf of Lem. 3.4].
For example, Proposition 1.5 implies that
[TABLE]
We note that, as in the case of subobject [topzeta, Prop. 5.19] and representation zeta
functions [unipotent, Prop. 4.3],
the topological ask zeta function of M only depends on M⊗okˉ, where kˉ is an algebraic closure of k.
Reduced ask zeta functions.
Informally, the reduced ask zeta function ZMred(T)∈Q(T) is obtained from the formal power series ZMv(T) by applying
a limit “qv→1” to each coefficient.
In the present context, this process can be formalised just as in the case of
representation zeta functions of unipotent groups (see [padzeta, §7]).
Moreover, Proposition 4.17 and a variation of [padzeta, Pf of Thm 7.3]
(which relies heavily on arguments due to Duong and
Voll [DV17]) show that in fact ZMred(T)=1/(1−T) for any M.
This is intuitively plausible:
if M~⊂Md×e(On)
is a submodule, then the group (O/P)× acts freely on M~∖{0} and preserves kernels whence
∣M~∣⋅ask(M~∣)≡∣V~∣(mod(q−1)),
where V~=Ond.
In particular, one would expect any reasonable limit of ask(M~∣) as
“q→1” to be 1.
5 Full matrix algebras, classical Lie algebras, and relatives
In this section, let O be the valuation ring of a non-Archimedean local
field of arbitrary characteristic.
Apart from proving Proposition 1.5,
we compute examples of Zg(T),
where g ranges over various infinite families of matrix Lie algebras.
At the heart of these computations lies the notion of “O-maximality”
introduced in §5.1.
5.1 O-maximality
Let M⊂Md×e(O) be a submodule and V=Od.
As we will now see, OM(x,y) is (generically) as large as
possible if and only if ZM(T) coincides with ZMd×gor(M)(T).
Lemma 5.1**.**
Let x∈V and y∈O∖{0}.
Then OM(x,y)⩽∣y∣−gor(M)∥x,y∥gor(M).
Proof.
Let C(X)∈Mℓ×e(O[X]) with
O[X]dC(X)=X⋅(M⊗O[X]).
We may assume that C(x)=0 since otherwise OM(x,y)=1.
As 0Od⋅M={0Oe}, the constant terms of
all non-zero polynomials in C(X) vanish whence
∥C(x)∥⩽∥x∥.
Thus, if C(x) has equivalence type
(λ1,…,λr),
then ∣πλi∣⩽∣πλ1∣=∥C(x)∥⩽∥x∥ for
1⩽i⩽r.
Define m and n via q−m=∥x∥ and n=ν(y).
Then, by Lemma 4.6(iv),
[TABLE]
The above inequality is sharp (cf. the comments after Proposition 3.3):
Lemma 5.2**.**
Let x∈V and y∈O∖{0}.
Then OMd×e(O)(x,y)=∣y∣−e∥x,y∥e.
Proof.
Let (e1,…,ed)⊂Od be the standard basis.
We may assume that x=0.
As xMd×e(O) is generated by {gcd(x1,…,xd)ei:i=1,…,e}, the claim follows from Lemma 4.6(iv).
∎
Lemma 5.3**.**
The following are equivalent:
(i)
OM(x,y)=∣y∣−gor(M)∥x,y∥gor(M)* for all
(x,y)∈V×O outside a set of measure zero.*
2. (ii)
OM(x,y)=∣y∣−gor(M)∥x,y∥gor(M)* for all
x∈V and all y∈O∖{0}.*
Proof.
OM is locally constant on V×(O∖{0}) so
(i) implies (ii); the converse is clear.
∎
Definition 5.4**.**
We say that M is O-maximal if it satisfies one of the two
equivalent conditions in the preceding lemma.
Proposition 1.5 serves as a blueprint for ZM(T)
whenever M is O-maximal:
Corollary 5.5**.**
M* is O-maximal if and only if
ZM(T)=ZMd×gor(M)(O)(T).*
Proof.
The “only if” part follows by combining (4.6) and
Lemma 5.2.
Conversely, suppose that OM(x,y)<∣y∣−gor(M)∥x,y∥gor(M)
for some x∈V and y∈O∖{0}.
Using the fact that both sides of this inequality are locally constant
functions of (x,y),
Lemmas 5.1–5.2,
and Theorem 4.5,
we conclude that for sufficiently large s∈R,
ζM(s)>ζMd×gor(M)(O)(s).
In particular, ZM(T)=ZMd×gor(M)(O)(T).
∎
The following is a “projective” characterisation of O-maximality.
Lemma 5.6**.**
M* is O-maximal if and only if
OM(x,y)=∣y∣−gor(M)
for all (x,y)∈(V∖PV)×P outside a set of
measure zero.*
Proof.
Necessity of the given condition being clear,
suppose that
OM(x,y)=∣y∣−gor(M)
for all (x,y)∈((V∖PV)×P)∖Z, where
Z⊂V×O has measure zero.
Choose π∈P∖P2.
For each n⩾0, we recursively define a set Z(n)⊂V×O
of measure zero such that OM(πnx,y)=∣y∣−gor(M)∥πn,y∥gor(M)
for all (x,y)∈((V∖PV)×O)∖Z(n).
By assumption and since OM(x,y)=1 for all x∈V and y∈O×,
we may take Z(0):=Z.
Suppose that Z(n) has been defined with the aforementioned properties and let
Z(n+1):={(x,y)∈V×O:(x,π−1y)∈Z(n)};
note that Z(n+1) has measure zero.
Let (x,y)∈((V∖PV)×O)∖Z(n+1).
We may assume that y∈P, say y=πz.
Then, since (x,z)∈Z(n), using
Lemma 4.16(i), we obtain
[TABLE]
Since {(πnx,y):n⩾0,(x,y)∈Z(n)} has measure zero,
the claim follows.
∎
We will repeatedly use the following lemma to prove O-maximality.
Lemma 5.7**.**
Let X=(X1,…,Xd). Let C(X)∈Mℓ×e(O[X]) with O[X]dC(X)=X⋅(M⊗O[X]).
Suppose that there exists an N⩾0 such that
for i=1,…,gor(M), the ideal of O[X] generated
by the i×i minors of C(X) contains each of
X1N,…,XdN.
Then M is O-maximal.
Proof.
Let Z={x∈V:dimK(xM⊗K)<gor(M)} as in
Corollary 4.14.
Let x∈Od∖(Pd∪Z) and let y∈O∖{0}.
As in §4.3.5, let
gi(X) be the set of non-zero i×i minors of
C(X).
Then, by assumption and since ∥x∥=1,
we have ∥gi(x)∥=1 for i=0,…,gor(M) whence OM(x,y)=∣y∣−gor(M) by
Corollary 4.14.
Thus, M is O-maximal by Lemma 5.6.
∎
For a geometric interpretation of Lemma 5.7 in a global setting, see
Proposition 6.9.
Our proof of Proposition 1.5 and other computations
in §5.3 rely on the following.
Lemma 5.8**.**
Let a0,…,ar∈C and write
σj=a0+⋯+aj.
Suppose that the integral
[TABLE]
is absolutely convergent.
Then
[TABLE]
In particular, in the special case a1=⋯=ar−1=0, we obtain
[TABLE]
Proof.
Both claims follow by induction
from the identities
(a) F0(a0)=1−q−1−a01−q−1 and
(b) Fr(a0,…,ar)=Fr−1(a0+a1+1,a2,…,ar)⋅1−q−a0−11−q−a0−2
for r⩾1.
The formula for F0(a0) in (a) is well-known and easily proved.
By performing a change of variables according to whether ∣x1∣⩽∣y∣ or ∣x1∣>∣y∣, we find that
F_{r}(a_{0},\dotsc,a_{r})=F_{r-1}(a_{0}+a_{1}+1,a_{2},\dotsc,a_{r})\,\cdotp\bigl{(}1+\int_{\mathfrak{P}}\lvert y\rvert^{a_{0}}\operatorname{d\!}\mu_{\mathfrak{O}}(y)\bigr{)}
whence (b) follows readily.
∎
Let M⊂Md×e(O) be any submodule.
By combining Lemma 5.1 and (5.1),
we thus obtain another interpretation of the lower bound in
Proposition 3.3 in the form
\alpha_{M}\geqslant\alpha_{\operatorname{M}_{d\times\operatorname{gor}(M)}(\mathfrak{O})}=\max\bigl{(}d-\operatorname{gor}(M),0\bigr{)}.
2. (ii)
We note that (1.2) could also be derived in an
elementary fashion (without using p-adic integration)
using Lemma 2.1 and ad hoc computations with generating
functions.
Such an approach quickly becomes cumbersome for more complicated
examples such as most of those in §9.
The author is, moreover, unaware of elementary proofs of general
results such as Theorems 1.4 and 4.18.
5.3 Classical Lie algebras and relatives
Reminder.
Let R be a ring.
Recall the definitions of the special linear,
orthogonal, and symplectic Lie algebras
[TABLE]
These are Lie subalgebras of gld(R) and gl2d(R), respectively.
Finally, we let trd(R) and nd(R) denote the Lie subalgebras of
gld(R) consisting of upper triangular matrices and strictly upper
triangular matrices, respectively.
We now determine Zg(T), where g is one of the Lie algebras from
above.
Of course, the case g=gld(O) is covered by Proposition 1.5.
Next, clearly, Zsl1(O)(T)=1/(1−qT).
The general case of sld(O) offers nothing new.
Corollary 5.10**.**
Let d>1. Then Zsld(O)(T)=Zgld(O)(T)=(1−T)21−q−dT.
Proof.
It suffices to show that x⋅sld(O)⊃x⋅gld(O)
for all x∈Od∖{0}.
Let xℓ have minimal valuation among the entries of x.
Let ej∈Od be the jth unit vector.
By our proof of Lemma 5.2, x⋅gld(O) is generated by {xℓej:1⩽j⩽d}.
Note that x⋅sld(O) is spanned by all xiej and xiei−xjej for 1⩽i,j⩽d with i=j.
It thus only remains to show that xℓeℓ∈x⋅sld(O).
Since d>1, we may choose j=ℓ.
Since ∣xj∣⩽∣xℓ∣,
xℓeℓ=(xℓeℓ−xjej)+xℓxjxℓej∈x⋅sld(O) whence the claim follows.
∎
Proposition 5.11**.**
Let char(K)=2.
Then Zsod(O)(T)=ZMd×(d−1)(O)(T)=(1−T)(1−qT)1−q1−dT.
Remark 5.12**.**
(i)
It is instructive to first determine ask(sod(Fq)∣) for odd q.
If F is any field with char(F)=2, then it is easy to
see that x⋅sod(F)=x⊥ for all x∈Fd∖{0}, where the orthogonal complement is taken with
respect to the standard inner product.
In particular, if x=0, then dimF(x⋅sod(F))=d−1.
Using Lemma 2.1,
we conclude that ask(sod(Fq)∣)=1+qd−1qd−1=1−q1−d+q for odd q;
this identity
was first proved probabilistically by Fulman and Goldstein [FG15, Lem. 5.3].
We note that for char(K)=2 and x∈Od, while we
still have an inclusion
x⋅sod(O)⊂x⊥, equality does not, in general, hold;
indeed, x⊥ is always an isolated submodule of Od.
2. (ii)
While we assumed that char(K)=2 in Proposition 5.11,
we do allow char(O/P)=2.
Note, however, that in this case, sod(O)n(=sod(O)⊗On) is properly
contained in the set of all skew-symmetric d×d-matrices over On.
Part (i) of the preceding remark implies that gor(sod(O))=d−1.
Given elements z1,…,zℓ of some ring, we recursively
define an (2ℓ)×ℓ matrix
[TABLE]
for instance, m(z1) is the 0×1-matrix and m(z1,z2)=[−z2z1].
Let eij∈Md(O) be the elementary matrix with 1 in
position (i,j) and zeros elsewhere.
Then the eij−eji for 1⩽i<j⩽d generate sod(O) as
an O-module whence
the rows of m(X1,…,Xd) span X⋅sod(O[X]).
In other words, the matrix m(X1,…,Xd) plays the role of
C(X) in §4.3.5 for M=sod(O).
By induction, we may assume that ±X2i,…,±Xdi are
i×i minors of m(X2,…,Xd) for all 1⩽i⩽d−2
so that ±X1j,…,±Xdj are j×j
minors of m(X1,…,Xd) for 1⩽j⩽d−1.
Thus, sod(O) is O-maximal by Lemma 5.7
and the claim follows from Corollary 5.5.
∎
For a ring R, let Symd(R)={a∈Md(R):a⊤=a}.
Proposition 5.13**.**
ZSymd(O)(T)=Zgld(O)(T)=(1−T)21−q−dT.
Proof.
This proof is similar to that of Proposition 5.11
and we use the same notation.
By considering the images of the first unit vector in Od under the
matrices e11 and
e1j+ej1 (2⩽j⩽d), we find that gor(Symd(O))=d.
Given z1,…,zℓ, recursively define an (2ℓ+1)×ℓ matrix
[TABLE]
An induction similar to the one in the proof of Proposition 5.11
shows that X1i,…,Xdi are i×i
minors of m′(X1,…,Xd) for 1⩽i⩽d.
As X⋅Symd(O[X]) is the row span of
m′(X1,…,Xd) over O[X],
the claim follows from Lemma 5.7 and Corollary 5.5.
∎
Proposition 5.14**.**
Zsp2d(O)(T)=Zgl2d(O)(T)=(1−T)21−q−2dT.
Proof.
We proceed along the same lines as the preceding two proofs.
Let eij denote the usual elementary matrix, now of size 2d×2d.
Using these matrices, it is easy to see that (1,0,…,0)⋅sp2d(O)=O2d whence gor(sp2d(O))=2d.
As an O-module, sp2d(O) is generated by the following
matrices:
(i) eij−ed+j,d+i (1⩽i,j⩽d),
(ii) ei,d+i,ed+i,i (1⩽i⩽d), and
(iii) ei,d+j+ej,d+i,ed+i,j+ed+j,i (1⩽i<j⩽d).
Write X=(X1,…,Xd) and X′=(X1′,…,Xd′).
Define m′(z1,…,zℓ) as in the proof of Proposition 5.13.
Then (X,X′)⋅sp2d(O[X,X′]) is generated by
the rows of
[TABLE]
Using what we have shown about the minors of m′(X)
in the proof of Proposition 5.13, we conclude that
Xij and (Xi′)j (i=1,…,d;j=1,…,2d) arise as
j×j minors of m~(X,X′).
Again, the claim thus follows from
Lemma 5.7 and Corollary 5.5.
∎
In contrast to the above examples,
neither nd(O) nor trd(O) (for d>1) is O-maximal.
Proposition 5.15**.**
(i)
Znd(O)(T)=(1−qT)d(1−T)d−1.
2. (ii)
Ztrd(O)(T)=Znd+1(O)(q−1T)=(1−T)d+1(1−q−1T)d.
Proof.
Since nd+1(O) is obtained from trd(O) by adding a zero row
and a zero column, by Corollary 3.6, it
suffices to prove (i).
Let x∈Od∖{0}.
Then xnd(O) is generated by
[TABLE]
in particular, gor(nd(O))=d−1.
Moreover, by (4.6) and Lemma 5.2,
[TABLE]
Hence, using (4.6) and Lemma 5.8,
(1−q−1)Znd(O)(q−s)=Fd−1(s−2,−1,…,−1).
∎
5.4 Diagonal matrices
Let dd(O)⊂gld(O) be the subalgebra
of diagonal matrices.
Clearly d1(O)=gl1(O) so that Zd1(O)(T)=(1−T)21−q−1T.
It turns out that the functions Zdd(O)(T)
have essentially been computed by Brenti [Bre94, Thm 3.4] in a different
context;
the author is grateful to Angela Carnevale for pointing this out.
First recall the definitions of Bn, Des(σ), and
N(σ) from §2.3.
For σ∈Bn, let dB(σ):=#Des(σ);
the function dB is known as the “descent statistic”.
Define a polynomial
hn(X,Y):=σ∈Bn∑XN(σ)YdB(σ).
Theorem 5.16** ([Bre94, Thm 3.4(ii)]).**
i=0∑∞(i(X+1)+1)nYi=(1−Y)n+1hn(X,Y)* for n⩾1.*
The following marks a departure from the simplicity of previous
examples
of ZM(T).
Corollary 5.17**.**
Zdd(O)(T)=(1−T)d+1hd(−q−1,T).
Proof.
By Corollary 3.6,
Zdd(O)(T) is the dth Hadamard power
of Zd1(O)(T).
Since
Zd1(O)(T)=(1−T)21−q−1T=∑i=0∞(1+i−iq−1)Ti,
the claim follows from Theorem 5.16.
∎
We note that permutation statistics have previously featured in explicit
formulae for representation zeta functions [SV14, BC17];
see also [CV17, CSV17].
6 Constant rank spaces
By a constant rank space over a field F,
we mean a subspace M⊂Md×e(F) such that
all non-zero elements of M have the same rank, say r;
we then say that M has constant rank r.
Such spaces have been studied extensively in the literature
(see e.g. [Bea81, Syl86, Wes87, IL99, BFM13]), often in the context of
vector bundles on projective space.
A problem of particular interest is
to find, for given d and r, the largest possible dimension of a subspace
of Md(C) of constant rank r.
Apart from trivial examples such as band matrices (see Example 6.6
below),
the construction of constant rank spaces
(in particular those of large dimension)
seems to be challenging.
Note that if M⊂Md×e(Fq) has constant rank r and
dimension ℓ,
then
[TABLE]
In §6.1, we consider a natural analogue,
K-minimality, of the concept of O-maximality
studied in §5.1.
We then derive interpretations of these notions in a global
setting in §6.2—in particular, we will see
that K-minimality is related to constant rank spaces.
6.1 K-minimality
Let O be the valuation ring of a non-Archimedean local field K of
arbitrary characteristic.
Let M⊂Md×e(O) be a submodule.
Recall the definition of KM from Definition 4.4.
Lemma 6.1**.**
Let F:Oℓ→M be an O-module isomorphism,
w∈Oℓ, and y∈O∖{0}.
Then KM(F(w),y)⩾∣y∣grk(M)−d∥w,y∥−grk(M).
Proof.
We may assume that w=0.
Let a:=F(w) have equivalence type (λ1,…,λr) (see
§4.3.1); of course,
r⩽grk(M).
Let m:=min(ν(w1),…,ν(wℓ)), n=ν(y), and a:=F(w).
Then m⩽min(ν(aij):1⩽i,j⩽d)=λ1⩽λ2⩽⋯⩽λr
and Lemma 4.6(iii) shows that
For an O-module isomorphism F:Oℓ→M, the following are equivalent:
(i)
KM(F(w),y)=∣y∣grk(M)−d∥w,y∥−grk(M)* for all (w,y)∈Oℓ×O
outside a set of measure zero.*
2. (ii)
KM(F(w),y)=∣y∣grk(M)−d∥w,y∥−grk(M)* for all
w∈Oℓ and y∈O∖{0}.
∎*
Definition 6.3**.**
We say that M is K-minimal if there exists an O-module
isomorphism F:Oℓ→M which satisfies one of the equivalent
conditions from the preceding lemma.
Clearly, if x∈On and a∈GLn(O), then ∥xa∥=∥x∥.
We conclude that
if the condition in the preceding definition is satisfied for some
isomorphism F:Oℓ→M, then it holds for all of them.
Lemma 5.8 and arguments as in the proof of Corollary 5.5
now imply the following.
Proposition 6.4**.**
Let r=grk(M) and ℓ=dimK(M⊗K).
Then M is K-minimal if and only if
[TABLE]
The following sufficient condition for K-minimality is proved
similarly to Lemma 5.7.
Lemma 6.5**.**
Let (a1,…,aℓ) be an O-basis of M.
Suppose that there exists N⩾0 such that
for 1⩽i⩽grk(M),
the ideal generated by the i×i minors of
X1a1+⋯+Xℓaℓ∈Md×e(O[X1,…,Xℓ])
contains X1N,…,XℓN.
Then M is K-minimal. ∎
Example 6.6** (Band matrices).**
Let r⩾1 and define
[TABLE]
By Lemma 6.5 and
Proposition 6.4 (with d=2r−1 and ℓ=r),
ZBr(T)=(1−qr−1T)21−q−1T.
6.2 A global interpretation
Henceforth, let k be a number field with ring of integers o;
recall the notation from §3.3.
Let kˉ be an algebraic closure of k.
Let M⊂Md×e(o) be a submodule.
The following can be proved similarly to Proposition 4.8 (and Proposition 4.13).
Lemma 6.7**.**
(i)
a∈Mmaxrkk(a)=aˉ∈M⊗okˉmaxrkkˉ(aˉ)=grk(Mv)* for all v∈Vk.*
2. (ii)
x∈odmaxdimk(x⋅(M⊗ok))=ˉx∈kˉdmaxdimkˉ(ˉx⋅(M⊗okˉ))=gor(Mv)* for all v∈Vk. ∎*
Extending Definitions 3.2 and 4.7,
we let grk(M) and gor(M) be the common number in (i) and
(ii), respectively.
We will now see that K-minimality is closely related to constant rank
spaces.
Proposition 6.8**.**
(i)
Let (a1,…,aℓ)⊂M be a k-basis of M⊗ok.
Let Ii be the ideal of k[X1,…,Xℓ] generated
by the i×i minors of X1a1+⋯+Xℓaℓ.
Then M⊗okˉ is a constant rank space
if and only if there exists N⩾0 such that X1N,…,XℓN∈Ii for i=1,…,grk(M).
2. (ii)
If M⊗okˉ is a constant rank space, then Mv is
K-minimal for almost all v∈Vk.
Proof.
(i)
Let V(I)⊂kˉℓ be the algebraic set corresponding to
I◃k[X]:=k[X1,…,Xℓ] and
let r=grk(M).
Then M⊗okˉ is a constant rank space if and only if
V(Ii)⊂{0} for i=1,…,r
which, by Hilbert’s Nullstellensatz,
is equivalent to Ii⊃⟨X1,…,Xr⟩ for
i=1,…,r.
2. (ii)
This follows from (i) and Lemma 6.5;
note that if f∈o[X] belongs to the k[X]-ideal generated by
g1,…,gr∈o[X], then f also belongs to the
ov[X]-ideal generated by the gi, at least for almost all v∈Vk.
∎
In view of Lemma 2.1, we say that a subspace M′⊂Md×e(F) (where F is a field) has
constant orbit dimension if all F-spaces xM′ for x∈Fd∖{0} have the same dimension.
The following counterpart of Proposition 6.8
is then proved in the same way.
Proposition 6.9**.**
(i)
Let (a1,…,aℓ)⊂M be a k-basis of M⊗ok.
Let Ji be the ideal of k[X]:=k[X1,…,Xd] generated
by the i×i minors of
[TABLE]
Then
M⊗okˉ has constant orbit dimension.
if and only if there exists N⩾0 such that X1N,…,XdN∈Ji
for i=1,…,gor(M).
2. (ii)
If M⊗okˉ has constant orbit dimension, then Mv is
O-maximal for almost all v∈Vk. ∎
7 Smooth determinantal hypersurfaces
In a series of papers, Voll [Vol04, Vol05, Vol09]
developed geometric techniques for studying the normal subgroup growth of
finitely generated torsion-free nilpotent groups of class 2.
Under suitable genericity assumptions on the Pfaffian hypersurface
attached to such a group, he produced an explicit formula [Vol05, Thm 3]
for almost all of its local normal subgroup zeta functions in terms of
numbers of rational points of the aforementioned hypersurface.
The following is an analogue of Voll’s result for ask zeta functions.
Here, the role of the Pfaffian hypersurface of a group is
played by the determinantal hypersurface associated with a matrix of linear
forms, a classical topic in algebraic geometry (see Remark 7.9).
Throughout this section, K is a non-Archimedean local field of arbitrary
characteristic with valuation ring O.
Recall that K=O/P denotes the residue field of K.
Theorem 7.1**.**
Let a1,…,aℓ∈Md(O), where ℓ⩾1.
Let X=(X1,…,Xℓ) consist of algebraically independent
variables over O.
Write a(X)=X1a1+⋯+Xℓaℓ∈Md(O[X]) and
let M={a(x):x∈Oℓ}⊂Md(O).
Let F(X):=det(a(X)).
Suppose that the following smoothness condition is satisfied:
[TABLE]
Let H:=Proj(O[X]/(F(X)))⊂POℓ−1.
Then:
[TABLE]
A proof of Theorem 7.1 will be given below.
We henceforth use the notation of Theorem 7.1 and assume that
condition (SM) is satisfied.
Note that the latter assumption is certainly satisfied if H is smooth as a scheme over O.
Let Ii(X) be the ideal of O[X] generated by the i×i minors of
a(X).
For an ideal J(X)◃O[X] and an element b of an
(associative, commutative, unital) O-algebra B, we write
J(b)={f(b):f∈J(X)}◃B.
Lemma 7.2** (Cf. [CT79, Pf of Thm 2.2]).**
(i)
∂Xi∂F(X)∈Id−1(X)* for i=1,…,ℓ.*
2. (ii)
Let ˉx∈Kℓ∖{0}.
Then rkK(a(ˉx))∈{d−1,d}.
Proof.
See [CT79, p. 426] for (i).
For (ii),
if rkK(a(ˉx))<d−1 for ˉx∈Kℓ,
then F(ˉx)=0 and Id−1(ˉx)={0}.
Part (i) and (SM) then imply that
ˉx=0.
∎
Corollary 7.3**.**
If d⩾2, then
Oℓ→M,x↦a(x)
is an isomorphism of O-modules.
Proof.
Let x∈Oℓ with a(x)=0 and
suppose that x=0.
Choose π∈P∖P2.
Then x=πmy for some m⩾0 and an element y∈Oℓ
whose image in Kℓ is non-zero.
Then a(y)=0 but also rkK(a(y)⊗K)⩾d−1>0 by Lemma 7.2(ii), a contradiction.
∎
First suppose that d=1.
Then a(X)=F(X) is linear.
Moreover, condition (SM) implies that M=M1(O)
and #H(K)=(qℓ−1−1)/(q−1).
The claim is now easily verified using a direct computation and
Proposition 1.5.
Let d⩾2.
Write U=Oℓ and
fix π∈P∖P2.
Let x∈U∖PU and y∈P∖{0}.
By Lemma 7.2(ii),
for each i=1,…,d−1,
some i×i minor of a(x) belongs to O×.
Hence, I0(x)=…=Id−1(x)=O.
As F(X)=0,
grk(M)=d.
Using Corollaries 4.9 and 7.3,
where we wrote ω as a shorthand for ∣y∣s−1∥F(x),y∥−1dμU×O(x,y).
In order to evaluate the integral in (7.1),
we decompose the domain of integration into sets
of the form (x0+PU)×P for x0∈U∖PU.
If F(x0)≡0(modP),
then clearly
[TABLE]
where A(q,t):=q−ℓt/(1−t).
Suppose that F(x0)≡0(modP).
Using (SM) and Hensel’s lemma [Bou89, Ch. III, §4.5, Cor. 2],
a measure-preserving change of coordinates transforms the
map induced by F(X) on x0+PU into the map induced by
X1, say,
on PU. Thus,
[TABLE]
By an elementary calculation,
[TABLE]
and thus
(x0+PU)×P∫ω=(1−q−1)B(q,t),
where B(q,t)=q1−ℓt⋅(1−q−1t)/(1−t)2.
The condition F(x0)≡0(modP) is equivalent to
the image of x0 in Pℓ−1(K) being contained in H(K).
Therefore,
[TABLE]
and the claim follows from
1+(1−q−ℓ)1−tt=1−t1−q−ℓt
and B(q,t)−A(q,t)=(q−1)(1−t)2q−ℓt.
∎
Remark 7.4**.**
By the Chevalley-Warning Theorem, #H(K)=0 implies that
ℓ⩽d; see e.g. [Ser73, Ch. 1, §2, Cor. 1].
Suppose that #H(K)=0.
Then M is readily seen to be K-minimal (cf. the proof of
Theorem 7.1) and M⊗K has constant rank d.
The formula ZM(T)=(1−T)(1−qd−ℓT)1−q−ℓT given by
Theorem 7.1 in this case agrees with
Proposition 6.4.
Example 7.5** (Diagonal 2×2 matrices).**
Let a(X)=diag(X1,…,Xℓ).
Then condition (SM) is satisfied if and only if ℓ⩽2.
Using Theorem 7.1 with ℓ=2 and #H(K)=2, we
recover the special case d=2 of Corollary 5.17.
Example 7.6** (Univariate polynomials).**
(i)
(Local case.)
Let d⩾1 and
f(X)=Xd+cd−1Xd−1+⋯+c1X+c0∈O[X].
Let
[TABLE]
be the companion matrix of f(X).
Let
af(X,Y)=X⋅1d−Y⋅bf∈Md(O[X,Y]),
where 1d denotes the d×d identity matrix.
Then det(af(X,Y))=Yd⋅f(Y−1X) is the
homogenisation of f(X).
Let Mf(O)={af(x,y):x,y∈O}⊂Md(O);
note that Mf(O) has O-rank 2 and that grk(Mf(O))=d.
We now assume that the image of f(X) in K[X] has no repeated roots in K.
Then condition (SM) is satisfied for a(X,Y)=af(X,Y) and
F(X,Y)=det(af(X,Y)).
We may therefore apply Theorem 7.1 (with ℓ=2) to obtain
[TABLE]
where nf(K)=#{x∈K:f(x)=0}∈{0,1,…,d}.
If the image of f(X) in K[X] is irreducible, then we are in the special
case nf(K)=#H(K)=0
discussed in Remark 7.4.
At the other extreme, if f(X) splits into d
linear factors
over K (which is possible if and only if q⩽d), then nf(K)=d.
2. (ii)
(Global case.)
Let k be a number field.
Let f(X)=Xd+cd−1Xd−1+⋯+c0∈Z[X] be the
minimal polynomial of an integral primitive element of k/Q.
For a (rational) prime p, the reduction fp(X)∈Fp[X] of f(X) modulo p is
separable if and only if p does not divide the discriminant disc(f(X)) of f(X).
Let Mf⊂Md(Z) be the module of matrices generated by the identity
matrix 1d and the companion matrix of f(X).
Then, using the notation from (i), we may identify
Mf⊗ZZp=Mf(Zp).
In particular, if p∤disc(f(X)), then we obtain a
formula (7.2) for ZMf⊗ZZp(T)
which depends on the number nf(Fp) of roots of f(X) in Fp.
Remark 7.7**.**
For sufficiently large primes p,
Voll [Vol04, Prop. 3] gave an explicit formula for almost all local
normal subgroup zeta functions associated with an indecomposable D∗-group attached to a power of an irreducible
polynomial f(X) over Q; for background on D∗-groups, see [GS84].
Voll’s formula depends on the number of roots of f(X) modulo p and has
essentially the same shape as (7.2);
we note that the matrix B in [Vol04, Eqn (16)] (and in
[GS84, §6]) plays the same role as af(X,Y) above.
Example 7.8** (Y2=X3−X).**
Let E⊂PZ2 be defined by the
homogenisation of Y2=X3−X.
Then E⊗ZF is an elliptic curve
for every field F of characteristic distinct from 2.
The curve E⊗ZQ has been previously used to
show that various group-theoretic counting problems exhibit
arithmetically “wild” behaviour.
In particular, using determinantal representation in the sense of the
present section, du Sautoy [dS01] constructed a nilpotent group of
Hirsch length 9 whose local normal subgroup zeta function at a
prime p depends on the number #E(Fp) of Fp-rational points of E.
The precise shapes of these local zeta functions were first determined by Voll [Vol04].
Due to the “wild” behaviour of #E(Fp) as a function of p (see
[dS01, Pf of Thm 2.1]),
du Sautoy’s result disproved earlier predictions on the growth of (normal)
subgroups of nilpotent groups.
His construction has since been used to demonstrate that other
group-theoretic counting problems can be “wild”
(e.g. the enumeration of representations [Vol11, Ex. 2.4]
or of “descendants” [dSVL12]). We will now see that the
present setting is no exception.
Let
[TABLE]
and M={a(x,y,z):x,y,z∈Z}⊂M3(Z).
Then E is defined by det(a(X,Y,Z))=0.
Suppose that q is odd
so that F(X,Y,Z):=det(a(X,Y,Z)) satisfies
condition (SM).
By applying Theorem 7.1,
we thus obtain
[TABLE]
In particular,
Theorem 4.11 accurately reflects the
general dependence of ZMv(T) on a place v∈Vk
for a module of matrices M over the ring of integers o of a number
field k.
However, just as in the study of zeta functions of groups, it
is presently unclear if anything meaningful can be said about the varieties
Vi⊗ok “required” to produce formulae (4.4)
as M varies over all modules of matrices over o.
Remark 7.9**.**
Determinantal representations of projective hypersurfaces
(i.e. representations of defining polynomials as determinants of matrices of linear forms)
over the complex numbers (or over algebraically closed fields) have been
studied extensively; see e.g. [Dol12, §§4.1, 9.3], [Bea00] and [KV12].
In particular, in the smooth case, only curves and cubic surfaces over C generically
admit determinantal representations.
Ishitsuka [Ish17, Cor. 8.3] showed
that over a local field (of arbitrary characteristic), every smooth plane
cubic admits a determinantal representation.
He further showed [Ish17, Thm 1.1(i)] that
the same is true of a positive proportion (measured by height) of smooth plane cubics
over Q.
Remark 7.10**.**
Let k be a number field with ring of integers o;
recall the notation from §3.3.
Let a(X)=a1X1+⋯+aℓXℓ∈Md(o[X]) for
ℓ⩾1 and a1,…,aℓ∈Md(o).
Let F(X):=det(a(X)) and M:={a(x):x∈oℓ}.
Define H:=Proj(o[X]/(F(X))) and suppose that H⊗ok is smooth over k.
For almost all v∈Vk, Theorem 7.1 then provides
a formula for ZMv(T) in terms of #H(Kv).
In this special case,
for almost all v∈Vk,
the functional equation in Theorem 4.18 follows immediately from
the identity
[TABLE]
a consequence of the Weil conjectures applied to
the smooth projective variety
H⊗oKv;
cf. [DM91, Pf of Thm 4].
8 Orbits and conjugacy classes of linear groups
In this section, we use p-adic Lie theory to relate ask, orbit-counting, and
conjugacy class zeta functions.
In §8.1, we recall properties
of saturable pro-p groups and Lie algebras.
In §8.2, we prove that orbit-counting zeta functions over
Zp are rational.
In §8.3 we compare group stabilisers and Lie
centralisers under suitable hypotheses and this allows us to deduce
Theorem 1.6 in §8.4.
Finally, in §8.5, we prove Theorem 1.7.
8.1 Reminder: saturable pro-p groups and Lie algebras
We briefly recall Lazard’s [Laz65] notion of (p-)saturability
of groups and Lie algebras using González-Sánchez’s [Gon07]
equivalent formulation.
Let g be a Lie Zp-algebra whose underlying Zp-module is free of
finite rank.
A potent filtration of g is a central series
g=g1⊃g2⊃⋯ of ideals (i.e. [gi,g]⊂gi+1 for all i) with
i=1⋂∞gi=0 and such
that
[gi,p−1,g]:=[gi,p−1g,…,g]:=[…[[gi,g],g],…,g]⊂pgi+1
for all i⩾1.
We say that g is saturable if it admits a potent filtration.
Proposition 8.1**.**
([AKOV13, Prop. 2.3])
Let O be the valuation ring of a non-Archimedean local field
K⊃Qp.
Let e(K/Qp) denote the ramification index of K/Qp.
Let g be a Lie O-algebra whose underlying O-module is free of
finite rank.
Let m>p−1e(K/Qp).
Then gm (=Pmg) is saturable as a Zp-algebra.
We note that, as before, subalgebras of an O-algebra are
understood to be O-subalgebras; whenever we consider
Zp-subalgebras, we will explicitly state as much.
Similarly to the case of Lie algebras, a torsion-free finitely generated
pro-p group G which admits a central series G=G1⩾G2⩾⋯ of closed subgroups with i=1⋂∞Gi=1
and [Gi,p−1G]⩽Gi+1p is saturable.
If g is a saturable Lie Zp-algebra, then the underlying topological
space of g can be endowed with the structure of a saturable pro-p group
using the Hausdorff series. Conversely, every saturable pro-p group gives
rise to a saturable Lie Zp-algebra and these two functorial operations
furnish mutually quasi-inverse equivalences between the categories of
saturable Lie Zp-algebras and saturable pro-p groups
(defined as full subcategories of all Lie Zp-algebras and pro-p
groups, respectively); see [Gon07, §4] for an overview and
[Laz65, Ch. 4] for details.
While the general interplay between subalgebras and subgroups
is subtle, we note the following fact.
Lemma 8.2**.**
Let g be a saturable Lie Zp-algebra and let h be a
saturable subalgebra of finite additive index.
Let G and H be the saturable pro-p groups associated with g and
h via the Hausdorff series (so that H⩽G).
Then ∣G:H∣=∣g:h∣.
Proof.
This can be proved in the same way as [GS09, Lem. 3.2(4)].
∎
8.2 Orbits of p-adic linear groups
Let O be the valuation ring of a non-Archimedean local field K.
Recall that P denotes the maximal ideal of O
and that q and p denote the size and characteristic of the residue
field of K, respectively.
Further recall the definition of ZGoc(T)
(Definition 1.2(ii)).
Although we will not need it in the sequel, since it might be of
independent interest, we note the following rationality statement for
ZGoc(T).
Theorem 8.3**.**
Let G⩽GLd(Zp).
Then ZGoc(T)∈Q(T).
More precisely, there are a1,…,ar∈Z and b1,…,br∈N such that
∏i=1r(1−paiTbi)ZGoc(T)∈Q[T].
Remark 8.4**.**
(i)
The author does not know if the conclusion of Theorem 8.3
remains valid if G is allowed to be a subgroup of GLd(Fq[[z]]).
The proof of Theorem 8.3 below combines
basic p-adic Lie theory and a powerful model-theoretic result due to
Cluckers [HMRC17, App. A].
Both of these ingredients are only available in characteristic zero.
In the latter case, this reflects the mysterious nature of the model theory of
local fields of (small) positive characteristic.
2. (ii)
Avni et al. [AKOV16b, Thms E, A.5] gave
formulae for the “similarity class zeta functions”
associated with the groups GLd(O) and GUd(O) for d=2,3.
In addition to being consistent with Theorem 8.3,
their formulae are valid for all local fields K subject to the sole
assumption that q be odd in case of GUd(O).
As explained in (i),
the techniques employed here seem incapable of establishing rationality
results in such great generality.
Lemma 8.5**.**
Let Gˉ be the closure of G⩽GLd(O) in GLd(O).
Then ZGˉoc(T)=ZGoc(T).
Proof.
The open subgroups Γi:={a∈GLd(O):a≡1(modPi)}
form a fundamental system of neighbourhoods of the identity
in GLd(O).
The claim follows since GΓi=GˉΓi for all i⩾0.
∎
Our theorem will follow immediately from [HMRC17, Thm A.2]
once we have established that ∼n is definable (definably in
n) in the subanalytic language used in [HMRC17, App. A].
By the preceding lemma, we may assume that G=Gˉ.
It then follows from the well-known structure theory of
p-adic analytic groups (see [Laz65, DdSMS99])
that there exists an open saturable (or, more restrictively, uniform) subgroup H⩽G of the form H=exp(p2h), where
h⊂gld(Zp) is a suitable saturable (or uniform)
Zp-subalgebra.
Let T be a transversal for the right cosets of H in G.
Then, for x,y∈Zpd, x∼ny if and only if
[TABLE]
The claim thus follows since the d×d-matrix exponential
exp(p2X) is given by d2 power series in d2 variables which
all converge on gld(Zp).
∎
Remark 8.6**.**
Let K/Qp be a finite extension.
Then we may regard G⩽GLd(O) as
a Zp-linear group of degree d∣K:Qp∣ via the regular
representation of O over Zp.
In particular,
Theorem 8.3
implies that ZGoc(T) is rational
provided that K/Qp is unramified.
The following questions are inspired by Theorems 4.11–4.12 and [BDOP13, Thm C].
Question 8.7**.**
Let k be a number field with ring of integers o.
Let G⩽GLd⊗k be an algebraic group over k.
Let G denote the schematic closure of G in GLd⊗o.
(i)
Do there exist V1,…,Vr and W1,…,Wr∈Q(X,T) as
in Theorem 4.11 such that
[TABLE]
for almost all places v∈Vk?
2. (ii)
Do we have
ZG(ov)oc(T)=ZG(Kv[[z]])oc(T) for almost all v∈Vk?
By combining Theorem 4.11 and
Corollary 8.19 below, we see that
Question 8.7(i) has a positive answer if G is unipotent.
We conclude this subsection with some elementary examples of ZGoc(T).
Example 8.8**.**
(i)
It is easy to see that the rule
x↦min(ν(x1),…,ν(xd),n) induces a bijection Ond/GLd(O)→{0,…,n}
whence
[TABLE]
2. (ii)
Let p=2.
The number of orbits of ⟨−1⟩⩽GL1(O) on On is 1+(qn−1)/2
so that
[TABLE]
3. (iii)
Let G=⟨[0110]⟩⩽GL2(O).
It is easy to see that
[TABLE]
We note that the fact that the preceding examples as well as
the formula in [AKOV16b, (1.12)] all satisfy functional equations under
the operation “(q,T)→(q−1,T−1)” does not seem to be explained by any of the results in the
present article (e.g. Theorem 4.18).
8.3 Lie centralisers and group stabilisers
Let O be the valuation ring of a local field K⊃Qp.
Let e(K/Qp) denote the ramification index of K/Qp.
As expected, for suitable matrix algebras and groups,
the equivalence between saturable pro-p groups
and Lie Zp-algebras recalled in §8.1
can be made explicit using exponentials and logarithms.
In line with our previous notation (see §3.5), we write
gldm(O):={a∈gld(O):a≡0(modPm)}.
Moreover, we write GLdm(O):={a∈GLd(O):a≡1(modPm)}.
Proposition 8.9** ([Klo05, Lem. B.1]).**
Let m>p−1e(K/Qp).
The formal exponential and logarithm series
converge on gldm(O) and GLdm(O), respectively, and
define mutually inverse bijections
exp:gldm(O)→GLdm(O) and log:GLdm(O)→gldm(O).
Hence, if m>p−1e(K/Qp) and g⊂gldm(O) is a saturable
subalgebra, then we may identify the saturable pro-p group associated with
g in the sense of §8.1 with exp(g)⩽GLdm(O).
For x∈Od and n⩾0, we write xmodPn for the
image of x in Ond.
Lemma 8.10**.**
Let g⊂glde(K/Qp)(O) (=pgld(O)) be a saturable
subalgebra.
Then
[TABLE]
is a saturable subalgebra of g
for all x∈Od.
Proof.
Write e=e(K/Qp) and
cn:=cg(xmodPn);
obviously, cn is a subalgebra of g.
Let g=γ1(g)⊃γ2(g)⊃⋯ be the lower central
series of g.
Then γp(cn)⊂cn+e.
Indeed, each element, a say, of γp(cn) is a sum
of matrix products c(ph) for suitable c∈cn and h∈gld(O) and clearly, xc(ph)≡0(modPn+e).
Let g=g1⊃g2⊃⋯ be a potent filtration of g.
Then cn=cn∩g1⊃cn∩g2⊃⋯ is
a central series of cn with
i=1⋂∞(cn∩gi)=0.
It is in fact a potent filtration
for pg∩cn+e=pcn
whence
[(cn∩gi),p−1cn]⊂pgi+1∩γp(cn)⊂pg∩cn+e=pcn.
∎
Lemma 8.11**.**
Let m>p−1e(K/Qp) and a∈gldm(O).
Then there exists u∈GLd1(O) with exp(a)=1+au.
Proof.
Let g(X):=∑i=0∞(i+1)!1Xi so that exp(X)=1+Xg(x) in Q[[X]].
Let a∈gldm(O) be non-zero.
Let b be an entry of a of minimal valuation.
Then νK(bi/(i+1)!) is a lower bound for the valuation
of each entry of (i+1)!1ai for i⩾0.
It is well-known that νp((i+1)!)⩽i/(p−1)
(see e.g. [Coh07, Lem. 4.2.8(1)]) so that νK((i+1)!)⩽e(K/Qp)i/(p−1).
Therefore, νK(bi/(i+1)!)⩾i(νK(b)−e(K/Qp)/(p−1))=:f(i).
Clearly, f(i)→∞ as i→∞ and f(i)>0 for i>0.
Hence, the series g(a) converges in Md(O) to an element
of GLd1(O).
∎
By combining the preceding two lemmas, we obtain the following.
Corollary 8.12**.**
Let m>max(e(K/Qp)−1,p−1e(K/Qp)),
let g⊂gldm(O) be a saturable subalgebra,
and let x∈Od.
Then exp(cg(xmodPn))=Stexp(g)(xmodPn).
Proof.
Let a∈g and
write exp(a)=1+au for u∈GLd1(O).
Then
Let O be the valuation ring of a local field
K⊃Qp. Recall that e(K/Qp) denotes the ramification index of K/Qp.
Proposition 8.13**.**
Let m>max(e(K/Qp)−1,p−1e(K/Qp)),
let g⊂gldm(O) be a saturable subalgebra,
and let G=exp(g).
Then Zg(T)=ZGoc(T).
Proof.
Write V=Od and Vn=V⊗On.
Recall that gn denotes the image of g under the natural map
gld(O)→gld(On).
For n⩾0,
by combining Lemma 2.1, Lemma 8.2, and
Corollary 8.12, we obtain
Let m>max(e(K/Qp)−1,p−1e(K/Qp)).
Propositions 3.9 and 8.13 show that
qdmZg(T)=Zgmm(T)=Zexp(gm)oc,m(T)∈Z[[T]].
∎
8.5 Orbits and conjugacy classes of unipotent groups
Let O be the valuation ring of a local field K⊃Qp.
Recall that nd(O)⊂gld(O) is the Lie algebra of all
strictly upper triangular matrices and
that Ud denotes the group scheme of upper unitriangular d×d
matrices.
The following is well-known.
Proposition 8.14**.**
Let p⩾d.
(i)
All Zp-subalgebras of nd(O) and closed subgroups of
Ud(O) are saturable.
2. (ii)
exp* and log define polynomial bijections
between nd(O) and Ud(O).*
Proof.
Noting that subalgebras of nd(O) and closed subgroups of
Ud(O) are nilpotent of class at most d−1<p,
their lower central series constitute potent filtrations.
This proves (i). Part (ii) follows since ad=0 for a∈nd(O).
∎
A simple variation of Proposition 8.14 yields the following.
Corollary 8.15**.**
Let g⊂gld(O) be a subalgebra.
Suppose that g is nilpotent of class at most c and that
ac+1=0 (in Md(O)) for all a∈g.
Further suppose that p>c.
(i)
g* is saturable.*
2. (ii)
G:=exp(g)* is a saturable subgroup of GLd(O).*
3. (iii)
exp* and log define mutually inverse polynomial bijections
between g and G.
∎*
By going through the proof of Theorem 1.6,
we now easily obtain the following.
Corollary 8.16**.**
Let the notation be as in Corollary 8.15.
Then Zgask(T)=Zexp(g)oc(T) and
thus, in particular, Zgask(T)∈Z[[T]]. ∎
This proves the first part of Theorem 1.7.
We note that Corollary 5.17 shows that we may not, in
general, relax the assumptions in Corollary 8.16
and merely assume that g⊂gld(O) is a nilpotent subalgebra.
The following completes the proof of Theorem 1.7.
Proposition 8.17**.**
Let g⊂gld(O) be an isolated subalgebra
consisting of nilpotent matrices.
Suppose that p⩾d.
Then Zexp(g)cc(T)=Zad(g)ask(T).
Proof.
By Engel’s theorem, g is GLd(K)-conjugate to a subalgebra of
nd(K) and hence nilpotent of class at most d−1;
in particular, ad(a)d=0 for all a∈g.
By Corollary 8.15,
G:=exp(g) is a saturable subgroup of GLd(O).
Let Ad:G→GL(g)
denote the adjoint representation of G;
hence, Ad(g):g→g,a↦log(exp(a)g)=ag
for g∈G.
Recall that Gn denotes the image of G in GLd(On)
and gn that of g in gld(On).
Clearly, a≡0(modPn) if and only if exp(a)≡1(modPn) for a∈g.
We may thus identify conjugacy classes of Gn with Ad(G)-orbits on gn.
As g is isolated within gld(O),
we may identify gn=g⊗On
and obtain ZGcc(T)=ZAd(G)↷goc(T).
By Corollary 8.15,
ad(g) is a saturable subalgebra of gl(g).
The Hausdorff series shows that
log(exp(b)exp(a))=∑i=0∞i!1[b,ia]
for a,b∈g (see [GS09, Eqn (3)])
whence
Ad(exp(a))=exp(ad(a)) for all a∈g.
Thus, Ad(G)=exp(ad(g)) and
Corollary 8.16 shows that
ZAd(G)↷goc(T)=Zlog(Ad(G))↷gask(T)=Zad(g)↷gask(T).
∎
Remark 8.18**.**
The conclusion of Proposition 8.17 does not generally hold if
g is not isolated.
For a simple example, take g=P⋅n2(O).
Then ad(g)={0}⊂gl(g)≈gl1(O) and thus
Zad(g)ask(T)=1/(1−qT)=1+qT+O(T2).
On the other hand, the reduction of exp(g) modulo P is trivial
whence Zexp(g)cc(T)=1+T+O(T2).
Using the well-known equivalence between unipotent algebraic groups
and nilpotent finite-dimensional Lie algebras over a field of
characteristic zero (see [DG70, Ch. IV]),
Corollary 8.16 and Proposition 8.17
now imply the following global result.
Corollary 8.19**.**
Let k be a number field with ring of integers o.
Let G⩽Ud⊗Zk be a unipotent algebraic group
over k and let G⩽Ud⊗Zo be the
associated o-form of G (i.e. the schematic
closure of G).
Let g⊂nd(k) be the Lie algebra of
G and g=g∩nd(o).
Then for almost all v∈Vk,
ZG(ov)oc(T)=Zgvask(T) and
ZG(ov)cc(T)=Zad(gv)ask(T).
∎
Using Theorem 4.18, we further establish the following functional
equations for orbit-counting and conjugacy class zeta functions arising from
unipotent algebraic groups.
Corollary 8.20**.**
Let the notation be as in Corollary 8.19.
Then for almost all v∈Vk,
[TABLE]
and
[TABLE]
Corollary 8.21**.**
Let k be a number field with ring of integers o.
Let G⩽Ud⊗Zk
and H⩽Ue⊗Zk
be unipotent algebraic groups over k with
o-forms G and H as above.
Suppose that for almost all v∈Vk,
ZG(ov)cc(T)=ZH(ov)cc(T).
Then dimk(G)=dimk(H). ∎
Proof.
Corollary 8.20 and [stability, §4] allow us to
recover dimk(G) from \bigl{(}\mathsf{Z}^{\mathsf{cc}}_{\mathsf{G}(\mathfrak{o}_{v})}(T)\bigr{)}_{v\in\mathcal{V}_{k}\setminus S}
for any finite set S⊂Vk.
∎
We note that there are examples of non-isomorphic groups
G and H (of the same dimension) which satisfy
ZG(ov)cc(T)=ZH(ov)cc(T) for almost all v∈Vk;
see Table 2 in §9.3.
9 Further examples
9.1 Computer calculations: Zeta
The author’s software package Zeta [Zeta]
for Sage [Sage] can compute numerous types of “generic local”
zeta functions in fortunate (“non-degenerate”) cases.
The techniques used by Zeta were developed over the course of
several papers; see [padzeta], in particular, and [spp1489] for an
overview and references to other pieces of software that Zeta relies upon.
When performing computations, Zeta proceeds by attempting to explicitly
compute certain types of p-adic integrals.
Fortunately, the integrals in (4.3) and
(4.6) can both be encoded in terms of the
“representation data” introduced in [unipotent, §5] whence
the author’s computational techniques apply verbatim to the functions
ZM⊗ZO(T), where M is Z-defined.
In detail, given a submodule M⊂Md×e(Z),
Zeta can be used to attempt to construct a rational
function W(X,T)∈Q(X,T) with the following property:
for almost all primes p and all finite extensions K/Qp,
ZM⊗ZOK(T)=W(qK,T);
we note that for various reasons, Zeta may well fail to
construct W(X,T) even if it exists.
Given M⊂Md×e(Z), Zeta can
also be used to attempt to construct a formula as in Theorem 4.11.
We note that while the techniques used by Zeta can, at least in
principle, be used to construct an explicit number CM such that all
primes p which needed to be excluded above satisfy p<CM, such a
number CM is not presently determined.
The remainder of this section is devoted to a number of examples
of functions ZM(T) and ZGcc(T) (via
Theorem 1.7) computed with the help of Zeta.
Throughout, O denotes the valuation ring of a non-Archimedean local field
K⊃Qp with residue field size q.
9.2 Examples of ask zeta functions
Example 9.1** (Small poles and unbounded denominators).**
Let
[TABLE]
Then for sufficiently large p,
[TABLE]
Hence, the real poles of ζM(s) are −1 and [math]; it is easy
to see that gor(M)=3 (see Definition 3.2).
This example illustrates that,
in contrast to the case of Md×e(O),
d−gor(M) is generally not a lower bound for the real poles
of ζM(s).
Note that ZM(T) has unbounded denominators—the author has
found comparatively few modules of square matrices with this
property (and initially suspected they did not exist).
Example 9.2**.**
Suppose that p=2 and let
[TABLE]
For sufficiently large p,
[TABLE]
Since ZM(T)=1+(2q2+4q+4q−1−q−2−8)T+O(T2), we see that,
in contrast to O-maximal or K-minimal cases (see §5–6),
the complexity of ask(M1∣V1) is in general a poor indicator of that of ZM(T).
We note that by Corollary 8.16 and since
ZM(T)∈Z[[T]], the module M cannot be a Lie subalgebra of n6(O).
Indeed, this is readily verified directly even though M is
listed among Lie algebras in [Roj16, Table 5].
Rojas’s article [Roj16] provides numerous examples of Lie subalgebras g⊂nd(Z), say, for d⩽6.
For many of these, we may use Zeta to compute
Zg⊗ZZp(T).
Here, we only include one example.
Example 9.3**.**
Let p=2 and let
[TABLE]
Then g is a Lie subalgebra of n5(O) of nilpotency class 4, listed
as L5,6 (de Graaf’s [dG07] notation) in
[Roj16, Table 3].
For sufficiently large p,
[TABLE]
Numerous examples (including the case of Md×e(O)) show
that ζM(s) may have a pole at zero and
Example 9.1 shows that negative poles can arise
even for modules of square matrices.
In contrast,
all of the author’s computations are consistent with the following
question having a positive answer.
Question 9.4**.**
Let k be a number field with ring of integers o.
Let g⊂nd(o) be a Lie subalgebra.
Is it the case that for almost all v∈Vk, every real pole
of ζgv(s) is positive?
Supposing that Question 9.4 indeed has a positive answer,
if G⩽Ud⊗Zk is an algebraic group over k with
associated o-form G⩽Ud⊗Zo (see
Corollary 8.19),
then we may evaluate the meromorphic continuation of
ZG(ov)oc(qv−s) at s=0 for almost all v∈Vk.
Inspired by similar questions regarding the behaviour at zero of
local subalgebra [topzeta, Conj. IV], submodule [cyclic, Conj. E],
and representation [padzeta, Qu. 8.5] zeta functions,
it would then be interesting to see if one can interpret the resulting
rational numbers, say in terms of properties of the orbit space ovd/G(ov).
9.3 Examples of conjugacy class zeta functions
Let k be a number field with ring of integers o.
Morozov [Mor58] classified nilpotent Lie algebras of dimension at
most 6 over an arbitrary field of characteristic
zero—equivalently, he classified unipotent algebraic groups of
dimension at most 6 over these fields.
A recent computer-assisted version of this classification (valid for
fields of characteristic =2) is due to de Graaf [dG07].
We use his notation and let Ld,i (or Ld,i(a)) denote the
ith Lie k-algebra (with parameter a) given in [dG07, §4].
Table 1 provides a complete list of “generic conjugacy
class zeta function” associated with nilpotent Lie k-algebras of
dimension at most 5 in the following sense: for each such
algebra g, let G be its associated unipotent algebraic
group over k. After choosing an embedding G⩽Ud⊗Zk, we obtain an o-form G of G as in
Corollary 8.19.
Then for almost all v∈Vk and all finite extensions
K/kv, ZG(O)cc(T) is given in Table 1.
In contrast to dimension at most 5, Zeta is unable to
compute generic conjugacy class zeta functions associated with every
nilpotent Lie k-algebra of dimension 6.
Nevertheless, Table 2 contains numerous examples of such functions;
we only included examples corresponding to ⊕-indecomposable algebras.
Clearly, generic conjugacy class zeta functions of direct products of
algebraic groups are Hadamard products of the zeta functions corresponding to the factors.
We note that L3,2≈n3(K) and L6,19(−1)≈n4(K).
A formula for ZU3(O)cc(T) was previously given
in [BDOP13, §8.2].
This formula is incorrect due to a sign mistake.
More substantially, the computation in [BDOP13, §8.2] relies on
[BDOP13, Prop. 6.2] and what seems to be a variation of the integral
formalism developed in [BDOP13] for unipotent groups; this is however not
explained.
Said integral formalism in [BDOP13] appears to be essentially different
from the methods developed and applied here.
Possible further directions.
A refinement of Higman’s conjecture (see §1) predicts that
k(Ud(Fq)) is a polynomial in q−1 with non-negative
coefficients.
In recent years, the same question has been studied for groups of
Fq-rational points of unipotent radicals of Borel subgroups of more
general algebraic groups such as Chevalley groups of small rank; see, in particular,
work of Goodwin et al. [Goo06, GR09a, GMR14, GMR16].
In this spirit, an elementary calculation using the formulae in
Table 1–2 shows that the coefficients of the
generic conjugacy class zeta functions associated with n3(K) and n4(K) are
polynomials with non-negative coefficients in q−1, generalising the known
cases of the corresponding coefficients of T.
The same is true of the generic conjugacy class zeta functions associated with
L4,3; the latter algebra is isomorphic to the nilradical of a Borel
subalgebra of sp4(K).
It would be interesting to further explore to what extent such
non-negativity properties are satisfied by the coefficients of ask zeta
functions in the setting of Goodwin et al.
For another intriguing problem, let fc,d be the free nilpotent
Lie ring of class c on d generators and write fc,d(R):=fc,d⊗ZR.
O’Brien and Voll [O'BV15, §5] gave a combinatorial description of
k(exp(fc,d(Fq))) under mild assumptions on q.
The generic conjugacy class zeta functions associated with f3,2
and f2,3 can be found in Tables 1
and 2, respectively.
Lins computed the conjugacy class zeta functions associated with f2,d for all d; see [Lin18, Cor. 1.11].
It seems challenging to determine the conjugacy class zeta functions
fc,d in general.
