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\title[Generalized Igusa functions and ideal growth in nilpotent
     Lie rings]{Generalized Igusa functions and ideal growth in
     nilpotent Lie rings}

      
 \author{Angela Carnevale\thanks{\href{mailto:angela.carnevale@nuigalway.ie}{angela.carnevale@nuigalway.ie}. Partially supported by the  Irish Research Council (GOIPD/2018/319).}\addressmark{1}, Michael M. Schein\thanks{\href{mailto:mschein@math.biu.ac.il}{mschein@math.biu.ac.il}.}\addressmark{2}, and Christopher Voll\thanks{\href{mailto:C.Voll.98@cantab.net}{C.Voll.98@cantab.net}. Partially supported by The
Emmy Noether Minerva Research Institute at Bar-Ilan University
during the preliminary stages of this project}\addressmark{3}}

%% define your authors in the usual way
%% use \addressmark{1}, \addressmark{2} etc for the institutions, and use \thanks{} for contact details
%\author[Optional shorter names]{Longer names me\thanks{\href{mailto:hello@world.c}{hello@world.c}. Longer names me was partially supported by Grant 2017.11.14.$\partial$\;supp.}\addressmark{1}, \and you\addressmark{2}}

%% then use \addressmark to match authors to institutions here
    \address{\addressmark{1} School of Mathematics, Statistics and Applied Mathematics, NUI Galway, Ireland \\
      \addressmark{2} Department of Mathematics, Bar-Ilan University, Ramat Gan, Israel\\
      \addressmark{3} Fakult\"at f\"ur Mathematik,
    Universit\"at Bielefeld, Bielefeld, Germany}

%% put the date of submission here
\received{\today}

%% leave this blank until submitting a revised version

%% put your English abstract here, or comment this out if you don't have one yet
%% please don't use custom commands in your abstract / resume, as these will be displayed online
%% likewise for citations -- please don't use \cite, and instead write out your citation as something like (author year)
\abstract{
     We introduce a new class of combinatorially defined rational
     functions and apply them to deduce explicit formulae for local
     ideal zeta functions associated to the members of a large class
     of nilpotent Lie rings which contains the free class-2-nilpotent
     Lie rings and is stable under direct products. Our results unify
     and generalize a substantial number of previous computations. We
     show that the new rational functions, and thus also the local
     zeta functions under consideration, enjoy a self-reciprocity
     property, expressed in terms of a functional equation upon
     inversion of variables. We establish a conjecture of Grunewald,
     Segal, and Smith on the uniformity of normal zeta functions of
     finitely generated free class-2-nilpotent groups.}

%% put your French abstract here, or comment this out if you don't have one
%\resume{\lipsum[2]}

%% put your keywords here, or comment this out if you don't have them yet
   \keywords{Igusa functions, combinatorial reciprocity theorem, weak
     orders, subgroup growth, normal zeta functions}

%% you can include your bibliography however you want, but using an external .bib file is STRONGLY RECOMMENDED and will make the editor's life much easier
%% regardless of how you do it, please use numerical citations, ie. [xx, yy] in the text

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\addbibresource{biblio.bib}
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\begin{document}

\maketitle
%% note that you DO NOT have to put your abstract here -- it is generated by \maketitle and the \abstract and \resume commands above


This abstract is an exposition of some results from the
preprint~\cite{CSV}.  The objective of our work is twofold.  The first
aim is to introduce a new class of combinatorially defined
multivariate rational functions and to indicate that they satisfy a
self-reciprocity property, expressed in terms of a functional equation
upon inversion of variables. The second is to apply these rational
functions to describe explicitly the local ideal zeta functions
associated to a class of combinatorially defined Lie rings.



\section{Generalized Igusa functions}\label{sec:general.igusa}

In \cref{subsec:gen.igusa.funeq} we introduce generalized Igusa
functions and state the functional equations that they satisfy. In
\cref{subsec:weak.igusa} we record an identity involving weak
order zeta functions, motivated by our applications of Igusa functions
in ideal growth. We collect here some notation and fundamental
notions.


We write $\N=\{1,2,\dots\}$ and, for a subset
  $X\subseteq \N$, set $X_0=X \cup \{0\}$. For $n\in\N_0$ we denote
  $[n]=\{1,\dots,n\}$. %, $[n,m]=\{n,n+1,\dots,m\}$, and
%  $(n,m)=\{n+1,\dots,m-1\}$.  
Given a finite subset $J\subseteq \N_0$,
  we write $J=\{j_1,\dots,j_r\}_<$ to imply that $j_1 < \dots <
  j_r$. %We write $J-n$ for the set $\{j-n \mid j\in\N\}$. 
The power
  set of a set $S$ is denoted $\mcP(S)$.


For a variable $Y$ and integers $a,b\in\N_0$ with $a\geq b$, the
associated \emph{Gaussian binomial} is
\begin{equation*}\label{def:gauss}
  \binom{a}{b}_Y = \frac{\prod_{i=a-b+1}^a (1-Y^i)}{\prod_{i=1}^b
    (1-Y^i)}\in \Z[Y].
\end{equation*}

Given $n\in\N$ and a subset $J = \{j_1,\dots,j_r\}_< \subseteq [n-1]$,
the associated \emph{Gaussian multinomial} is defined as
\begin{equation}\label{def:gaussian.multi} \binom{n}{J}_Y = \binom{n}{j_{r}}_Y
  \binom{j_{r}}{j_{r-1}}_Y \cdots \binom{j_2}{j_1}_Y \in \Z [Y].
  \end{equation}

  The following Coxeter group
theoretic interpretation of the Gaussian
multinomials is used in the proof of our reciprocity theorem. Recall that the symmetric group $W=S_n$
of degree $n$ is a Coxeter group, with Coxeter generating system
$S=(s_1,\dots,s_{n-1})$, where $s_i=(i\,i+1)$ denotes the standard
transposition. The \emph{Coxeter length} $\ell(w)$ of an element $w\in
S_n$ is the length of a shortest word for $w$ with elements
from~$S$. We define the (\emph{right}) \emph{descent set} $\Des(w) =
\{ i\in[n-1] \mid \ell(w s_i) < \ell(w) \}$. It is well-known
(\cite[Proposition~1.7.1]{Stanley/12}) that the Gaussian multinomials
\eqref{def:gaussian.multi} satisfy
\begin{equation*}\label{eq:coxlength}
  \binom{n}{J}_Y = \sum_{w\in S_n, \, \Des(w)\subseteq J} Y^{\ell(w)}.
\end{equation*}


\subsection{Generalized Igusa functions and their functional equations}\label{subsec:gen.igusa.funeq}
Let $\underline{n}=(n_1,\dots,n_m)$ be a composition of $N =
\sum_{i=1}^m n_i$ with $m$ parts.  Consider the poset
$C_{\underline{n}}$ of subwords of the word
$v_{\underline{n}}:=a_1^{n_1}a_2^{n_2}\dots a_m^{n_m}$ in ``letters''
$a_1,a_2,\dots,a_m$, each occurring with respective
multiplicity~$n_i$. This poset is naturally isomorphic to the
lattice $$C_{n_1}\times\dots \times C_{n_m},$$ the product of the
chains of lengths $n_i$ with the product order, which we denote
by~``$\leq$''. We write $\widehat{1}=v_{\underline{n}}$ and
$\widehat{0}$ for the empty word.

We denote by $\WO_{\underline{n}}$ the chain (or order) complex of
$C_{\underline{n}}$.  An element $V\in \WO_{\underline{n}}$ is a (possibly empty) chain, or flag, of non-empty subwords of $v_{\underline{n}}$, of the form $V=\{v_1 <
\dots < v_t\}$. On $\WO_{\underline{n}}$ we consider the partial order
defined by refinement of flags, also denoted by~``$\leq$''. Consider
the natural map
\begin{align*}
  \underline\pi:C_{\underline{n}} &\rarr
               [n_1]_0\times\cdots\times[n_m]_0,\\ v =
               a_1^{\alpha_1}\dots a_m^{\alpha_m} &\mapsto
               (\alpha_1,\dots,\alpha_m)=:(\pi_{1}(v),\dots,\pi_{m}(v)).
  \end{align*}

  \begin{dfn}
    We consider the induced morphism of posets
\begin{align*}
 \underline \varphi:\WO_{\underline{n}} &\to \prod_{i=1}^m
  \mathcal P ([n_i-1]),\\ V = \{v_1 < \dots < v_t\} &\mapsto \left(
  \left\{\pi_{i}(v_j) \mid j\in[t] \right\} \cap [n_i-1] \right)_{i=1}^m
  =: \left( \phi_{i}(V)\right)_{i=1}^m.
\end{align*}
\end{dfn}



\begin{figure}
\centering
\begin{tikzpicture}[scale=0.8]
  \node (max) at (-2,6) {$\widehat{1}=a_1^2a_2a_3$};
  \node (h) at (-2,4) {$a_1^2a_3$};
  \node (m) at (0,4) {$ a_1a_2a_3$};
  \node (i) at (-4,2) {$a_1^2$};
  \node (ii) at (-4,4) {$a_1^2a_2$};
  \node (a) at (-2,2) {$a_1a_2$};
  \node (b) at (0,2) {$a_1a_3$};
  \node (c) at (2,2) {$a_2a_3$};
  \node (d) at (-2,0) {$a_1$};
  \node (e) at (0,0) {$a_2$};
  \node (f) at (2,0) {$a_3$};
  \node (min) at (0,-2) {$\widehat{0}=\text{empty word}$};
  \draw (min) -- (d) -- (a) -- (m) -- (b) -- (f) (i) -- (h) -- (b)
  (e) -- (min) -- (f) -- (c) -- (m) (max) -- (ii) -- (a)
  (h) -- (max) -- (m) 
  (ii) -- (i) -- (d) -- (b)  ;
  \draw[preaction={draw=white, -,line width=6pt}] (a) -- (e) -- (c) (m)--(a)--(ii);
\end{tikzpicture}

\caption{The poset $C_{\underline{n}}$ for $\underline{n} = (2,1,1)$.}
\label{fig:plot}
\end{figure}


\begin{dfn} \label{dfn:chain.binomial} Let
  $V=\{v_1  <  \dots <  v_t\}\in \WO_{\underline{n}}$. We define
  $$W_{V}(\bfX) = \prod_{j=1}^t \frac{X_{v_j}}{1-X_{v_j}} \in
  \Q(X_{v_1},\dots,X_{v_t})$$ and
$$\binom{\underline{n}}{V}_\bfY = \prod_{i=1}^m
\binom{n_i}{\varphi_{i}(V)}_{Y_i}\in\Q(Y_1,\dots,Y_m),$$
where $ \underline\varphi(V)=\left(\varphi_{1}(V),\dots,\varphi_{m}(V)\right)$.
\end{dfn}

\begin{exm}
  Let $\underline{n}=(3,2,2)$. For the flag
  $V=\{a_2a_3 < a_1a_2^2 a_3\} \in \WO_{(3,2,2)}$
  we
  find  that
 $$W_V(\bfX)= \frac{X_{a_2 a_3}X_{a_1 a_2^2 a_3}}{(1-X_{a_2
     a_3})(1-X_{a_1 a_2^2 a_3})}$$ and
$$\binom{\underline{n}}{V}_\bfY =\binom{3}{1}_{Y_1} \binom{2}{1}_{Y_2} \binom{2}{1}_{Y_3} = (1 + Y_1 + Y_1^2)(1 + Y_2)(1+Y_3). $$
\end{exm}

The following is the key combinatorial tool of this paper.
\begin{dfn}\label{def:igusa.wo} The \emph{generalized Igusa function
    associated with the composition $\underline{n}$} is
$$I^{\textup{wo}}_{\underline{n}}(\bfY;\bfX) := \sum_{V\in
    \WO_{\underline{n}}} \binom{\underline{n}}{V}_\bfY W_V(\bfX) \in
  \Q(Y_1,\dots,Y_m,(X_r)_{r \leq v_{\underline{n}}}),$$
\end{dfn}

\begin{exm} \label{exm:gen.igusa}
  \ \begin{enumerate}
  \item For $\underline{n}=(N)$, the trivial composition of $N$, we recover
    $I^{\textup{wo}}_{(N)}(\bfY;\bfX) = I_{N}(Y;\bfX)$, the classical
    Igusa zeta function
    \begin{equation*}
I_\wo(Y; \bfX) =
\sum_{I \subseteq [\wo]}\binom{\wo}{I}_{Y} \prod_{i\in
  I}\frac{X_i}{1-X_i} = \frac{\sum_{\sigma \in S_{\wo}} Y^{\ell(\sigma)} \prod_{i \in \mathrm{Des}(\sigma)} X_i}{\prod_{i = 1}^{\wo} (1 - X_i)} \;\in\Q(Y,X_1,\dots,X_\wo).
\end{equation*}
In this case, the functional equation upon inversion of the variables
was found by Igusa;
cf.~\cite[Theorem~4]{Voll/05}. 
  \item For $\underline{n}=(1,\dots,1)$, the all-one composition of
    $N$, we recover $I^{\textup{wo}}_{(1,\dots,1)}(\bfY;\bfX) =
    I^{\textup{wo}}_{N}(\bfX)$, the weak order zeta function introduced in~\cite[Definition~2.9]{SV1/15}. The variables $\bfY$ do not
    appear in this case, as all the polynomials
    $\binom{\underline{n}}{V}_{\bfY}$ are equal to the constant~$1$.  These functions also coincide with certain instances of generating functions
  associated with chain partitions in
  \cite[Section~4.9]{BeckSanyal/18}.
    \item For $\underline{n}=(2,1)$ we obtain
\begin{multline*}
   I^{\textup{wo}}_{(2,1)}(\bfY;\bfX)=\frac{1}{1-X_{a_1^2 a_2}}\left(1+\frac{X_{a_2}}{1-X_{a_2}}+\frac{X_{a_1^2}}{1-X_{a_1^2}}+\right.\\
    (1+Y_1)\left(\frac{X_{a_1}}{1-X_{a_1}}+\frac{ X_{a_1a_2}}{1-X_{a_1a_2}}+\frac{ X_{a_1}}{1-X_{a_1}}\frac{X_{a_1a_2}}{1-X_{a_1a_2}}
    +\right. \\ \left.\left. \frac{X_{a_1}}{1-X_{a_1}}\frac{X_{a_1^2}}{1-X_{a_1^2}}+\frac{X_{a_2}}{1-X_{a_2}}\frac{X_{a_1 a_2}}{1-X_{a_1 a_2}}\right)\right). \end{multline*}      
  \end{enumerate}
\end{exm}

The following ``combinatorial reciprocity theorem'' is the main result
of this section.

\begin{thm}\label{thm:funeq}
  The generalized Igusa function associated with the composition
  $\underline{n}$ of $N = \sum_{i=1}^m n_i$ satisfies the following functional
  equation:
  $$I^{\textup{wo}}_{\underline{n}}(\bfY^{-1};\bfX^{-1}) = (-1)^N
  X_{v_{\underline{n}}}\left(\prod_{i=1}^m Y_i^{-
      \binom{n_i}{2}}\right)
  I^{\textup{wo}}_{\underline{n}}(\bfY;\bfX).$$
\end{thm}

The proof of \cref{thm:funeq} builds on a number of crucial
``inversion properties'' satisfied by the rational functions
$W_V(\bfX)$. The first is rather simple.

\begin{lemma}\label{lem:inv.props}
 For all $V\in \WO_{\underline{n}}$,
$$W_V(\bfX ^{-1})=(-1)^{|V|}  \sum_{Q \leq  V} W_Q(\bfX).$$
\end{lemma}


To formulate the second, more sophisticated inversion property we
denote by $ \WO^{\times}_{\underline{n}}$ the subcomplex of
$ \WO_{\underline{n}}$ of flags of \emph{proper} subwords of
$v_{\underline n}$. Also, for
$I=(I_1,\dots,I_m)\in \prod_{i=1}^m\mathcal P([n_i-1])$ we write
$I^c:=K\setminus I $ for
$ ([n_1-1]\setminus I_1,\dots,[n_m-1]\setminus I_m)$.

\begin{pro}\label{pro:mobius}
For all $I\in \prod_{i=1}^m\mathcal P([n_i-1])$,
\begin{equation*}
\sum_{\substack{V\in \WO^{\times}_{\underline{n}} \\ \underline
    \varphi (V)\supseteq I}} W_V (\bfX ^{-1})
=(-1)^{N-1}\sum_{\substack{V\in \WO^{\times}_{\underline{n}}
    \\ \underline\varphi (V)\supseteq I^c}} W_V (\bfX ).
\end{equation*} 
\end{pro}

\subsection{Weak order zeta functions and generalized Igusa
  functions}\label{subsec:weak.igusa} We record an identity between
instances of weak order zeta functions which is used to show that our computations of the ideal zeta functions of base extensions of the Heisenberg Lie algebra match results obtained earlier by two of the authors~\cite{SV1/15}. 
The identity, which may be of independent interest, compares instances of weak order zeta functions
associated with the all-one-compositions $\underline g$ and
$\underline{2g}$, with $g$ and $2g$ parts, respectively, and holds
when substituting for the variables monomials satisfying certain
relations.

We call a subword of the word $\widehat{1} =
v_{\underline {2g}}:=a_1\cdots a_{2g}$ \emph{radical} if it is of the
form $w=\prod_{i \in \mathcal{J}} a_i a_{i+g}$ for some
$\mathcal{J}\subseteq [g]$.  We
observe that any subword $r \leq v_{\underline {2g}}$ may be written
uniquely in the form $r = \sqrt r \cdot r^\prime r^{\prime \prime}$,
where $\sqrt r = \prod_{i \in \mathcal I} a_i a_{i+g}$ is a radical
word, whereas $r^\prime = \prod_{i \in \mathcal{I}^\prime} a_i$ and
$r^{\prime \prime} = \prod_{i \in {\mathcal{I}^{\prime \prime}}}
a_{i+g}$, and the subsets $\mathcal{I}, \mathcal{I}^\prime,
\mathcal{I}^{\prime \prime} \subseteq [g]$ are disjoint.

In the following result, we omit the non-occurring variable $Y$ from
the generalized Igusa functions $I^{\mathrm{wo}}_{\underline g}$ and
$I^{\mathrm{wo}}_{\underline{2g}}$; cf.\ our remark in
\cref{exm:gen.igusa}~(2).

\begin{pro} \label{pro:igusas.match} Let $g \in \N$. Suppose that
  $\bfy=(y_r \mid r\leq v_{\underline{2g}})$ comprises terms satisfying 
  $y_r = y_{\sqrt r} \cdot \prod_{i \in \mathcal{I}^\prime \cup
    \mathcal{I}^{\prime \prime}} y_{a_i}$.  Then
\begin{equation*}\label{eq:match}
  I^{\mathrm{wo}}_{\underline {2g}} (\mathbf{y}) =\left( \prod_{i =
    1}^g \frac{1 + y_{a_i}}{1 -
    y_{a_i}}\right)I^{\mathrm{wo}}_{\underline {g}} (\mathbf{z}),
\end{equation*} 
where
$\mathbf{z} = (z_{\prod_{i \in \mathcal{I}} a_i} \mid \prod_{i \in
  \mathcal{I}} a_i \leq v_{\underline{g}})$
is given by
$z_{\prod_{i \in \mathcal{I}} a_i} = y_{\prod_{i\in\mathcal{I}}{a_i
    a_{i+g}}}$ for all $\mathcal{I} \subseteq [g]$.
\end{pro}

\section{Applications to ideal growth in Lie rings}
 \subsection{Finite uniformity for ideal zeta functions of nilpotent Lie rings}
  Given an additively finitely generated ring $\mcL$, i.e.\ a finitely
  generated $\Z$-module with some bi-additive, not necessarily
  associative multiplication, the ideal zeta function of $\mcL$ is the
  Dirichlet generating series
  \begin{equation}\label{def:ideal.zeta}
    \zeta^{\ideal}_{\mcL}(s) = \sum_{I \ideal \mcL}|\mcL:I|^{-s},
  \end{equation}
  where $I$ runs over the (two-sided) ideals of $\mcL$ of finite additive
  index in $\mcL$ and $s$ is a complex variable. Prominent examples of
  ideal zeta functions include the Dedekind zeta functions,
  enumerating ideals of rings of integers of algebraic number fields
  and, in particular, Riemann's zeta function $\zeta(s)$.

  It is not hard to verify that, for a general ring $\mcL$, the ideal
  zeta function $\zeta^{\ideal}_{\mcL}(s)$ satisfies an Euler product
  whose factors are indexed by the rational primes:
$$\zeta^{\ideal}_{\mcL}(s) = \prod_{p \textup{
    prime}}\zeta^{\ideal}_{\mcL(\Zp)}(s),$$ where, for a prime $p$,
  $$\zeta^{\ideal}_{\mcL(\Zp)}(s) = \sum_{I \ideal
    \mcL(\Zp)}|\mcL(\Zp): I|^{-s}$$ enumerates the ideals of finite
  index in the completion $\mcL(\Zp) := L\otimes_\Z \Zp$ or,
  equivalently, the ideals of finite $p$-power index in $\mcL$. Here
  $\Zp$ denotes the ring of $p$-adic integers; note that ideals of
  $\mcL(\Zp)$ are, in particular, $\Zp$-submodules of $\mcL(\Zp)$. It
  is, in contrast, a deep result that the Euler factors
  $\zeta^{\ideal}_{\mcL(\Zp)}(s)$ are rational functions in the
  parameter~$p^{-s}$; cf.\ \cite[Theorem~3.5]{GSS/88}.

  Computing these rational functions explicitly for a given ring
  $\mcL$ is, in general, a very hard problem. Solving it is usually
  rewarded by additional insights into combinatorial, arithmetic, or
  asymptotic aspects of ideal growth. It was shown by du Sautoy and Grunewald
  \cite{duSG/00} that the problem, in general, involves the
  determination of the numbers of $\Fp$-rational points of finitely
  many algebraic varieties defined over $\Q$. Only under
  additional assumptions on $\mcL$ may one hope that these numbers are
  given by finitely many polynomial functions in $p$. We say that the
  ideal zeta function of $\mcL$ is \emph{finitely uniform} if there
  are finitely many rational functions
  $W^{\ideal}_1(X,Y),\dots,W^{\ideal}_N(X,Y) \in \Q(X,Y)$ such that
  for any prime $p$ there exists $i\in\{1,\dots,N\}$ such that
  $$\zeta^{\ideal}_{\mcL(\Zp)}(s) = W^{\ideal}_i(p,p^{-s}).$$ If a
  single rational function suffices (i.e.\ $N=1$), we say that the
  ideal zeta function of $\mcL$ is \emph{uniform}. While finite
  uniformity dominates among low-rank examples, including most of
  those included in the book \cite{duSWoodward/08} and those computed
  by Rossmann's computer algebra package
  $\mathsf{Zeta}$~\cite{rossmannzeta, Rossmann/18}, it is not
  ubiquitous: for a non-uniform example in rank~$9$, see
  \cite{duS-ecII/01} and~\cite{Voll/04}.  In general, the ideal zeta
  function of a direct product of rings is not given by a simple
  function of the ideal zeta functions of the factors. It is not even
  clear whether (finite) uniformity of the latter implies (finite)
  uniformity of the former.


\subsection{Main results}
    
We apply the generalized Igusa functions introduced in
\cref{def:igusa.wo} to give constructive proofs of (finite)
uniformity of ideal zeta functions associated to the members of a
large class of nilpotent Lie rings of nilpotency class at most~$2$.

\begin{dfn}\label{def:L}
  Let $\mfL$ denote the class of nilpotent Lie rings of nilpotency
  class at most~$2$ which is closed under direct products and contains
  the following Lie rings:
  \begin{enumerate}
  \item the free class-$2$-nilpotent Lie rings $\mff_{2,d}$ on $d$
    generators, for $d\geq 2$.
   
  \item the free class-$2$-nilpotent products
    $\mfg_{d,d'} = \Z^d * \Z^{d'}$, for $d,d' \geq 0$.  These have
    $\Z$-basis $\{ x_i, y_j, z_{ij} \mid i \in [d], j \in [d'] \}$, with
    the relations $[x_i, y_j] = z_{ij}$; all other pairs of basis
    elements commute.
  \item the higher Heisenberg Lie rings $\mathfrak{h}_d$ for
    $d \geq 1$; these are $d$ copies of the Heisenberg Lie ring
    amalgamated over their centres and have $\Z$-basis
    $\{x_1, \dots, x_d, y_1, \dots, y_d, z \}$, with relation
    $[x_i, y_i] = z$ for all $i \in [d]$.  All other pairs of basis
    elements commute.
  \end{enumerate}
\end{dfn}
Note that $\mfL$ contains the free abelian Lie rings $\Z^d =
\mfg_{d,0}=\mfg_{0,d}$.

Our main ``global'' result produces explicit formulae for almost all
Euler factors of the ideal zeta functions associated to Lie rings
obtained from the members of $\mfL$ by base extension with general
rings of integers of number fields. In particular, we show that these
zeta functions are finitely uniform and, more precisely, that the
variation of the Euler factors is uniform among
unramified primes with the same decomposition behaviour in the
relevant number field.
\begin{thm}\label{thm:main.global}
  Let $\mcL$ be an element of $\mfL$, let $g\in\N$, and
  $\bff = (f_1,\dots,f_g) \in\N^g$. There exists an explicitly
  described rational function $W^{\ideal}_{\mcL,\bff} \in \Q(X,Y)$
  such that the following holds:

 Let $\mcO$ be the ring of integers of a number field and set
 $\mcL(\mcO) = \mcL \otimes \mcO$.  If a rational prime $p$ factorizes
 in $\mcO$ as $p \mcO =\mfp_1 \mfp_2 \cdots\mfp_g$, for pairwise
 distinct prime ideals $\mfp_i$ in $\mcO$ of inertia degrees
 $(f_1,\dots,f_g)$, then
 $$\zeta^{\ideal}_{\mcL(\mcO),p}(s) =
 W^{\ideal}_{\mcL,\bff}(p,p^{-s}).$$ In particular,
 $\zeta^{\ideal}_{\mcL(\mcO)}(s)$ is finitely uniform and
 $\zeta^{\ideal}_{\mcL}(s)=\zeta^{\ideal}_{\mcL(\Z)}(s)$ is uniform.
\end{thm}


A special case of \cref{thm:main.global} establishes part of a
conjecture of Grunewald, Segal, and Smith on the normal subgroup
growth of free nilpotent groups under extension of
scalars. In~\cite{GSS/88}, they introduced the concept of the
\emph{normal zeta function}
$$\zeta_G^{\ideal}(s) = \sum_{H\ideal G}|G:H|^{-s}$$ of a torsion-free
finitely generated nilpotent group $G$, enumerating the normal
subgroups of $G$ of finite index in $G$. As $G$ is nilpotent, it also
satisfies an Euler product decomposition
$$\zeta_G^{\ideal}(s) = \prod_{p \textup{
    prime}}\zeta_{G,p}^{\ideal}(s),$$ whose factors enumerate the
normal subgroups of $G$ of $p$-power index. If $G$ has nilpotency
class two, then its normal zeta function coincides with the ideal zeta
function of the associated Lie ring $\mcL_G := G/Z(G) \oplus Z(G)$;
see \cite[Remark on p.~206]{GSS/88}.  Thus, $\zeta^{\ideal}_G(s) =
\zeta^{\ideal}_{\mcL_G}(s)$. Moreover, every class-$2$-nilpotent Lie
ring $\mcL$ arises in this way and gives rise to a torsion-free
finitely generated nilpotent group $G(\mcL)$; see
\cite[Section~1.2]{Voll/17} for details.
\cref{thm:main.global} thus has a direct corollary pertaining
to the normal zeta functions of the finitely generated
class-$2$-nilpotent groups corresponding to the Lie rings in $\mfL$.
Since the groups associated to the free class-$2$-nilpotent Lie rings
$\mff_{2,d}$ are the finitely generated free class-$2$-nilpotent
groups~$F_{2,d} = G(\mff_{2,d})$, \cref{thm:main.global}
implies the Conjecture on p.~188 of \cite{GSS/88} for the case $*
=\ideal$ and $c=2$. The conjecture for normal zeta functions had
previously been established only for $d=2$ (\cite[Theorem~3]{GSS/88};
see also \cref{subsubsec:previous}). We are not aware of any
other case for which the conjecture has been proven or refuted.

\cref{thm:main.global} is a direct consequence of the following
uniform ``local'' result. In the following, $\lri$ will denote a
compact discrete valuation ring of arbitrary characteristic and
residue field of characteristic $p$ and cardinality $q$. Thus, $\lri$
may, for instance, be a finite extension of the ring $\Zp$ of $p$-adic
integers (of characteristic zero) or a ring of formal power series of
the form $\Fq\llbracket T\rrbracket$ (of positive characteristic).
The \emph{$\lri$-ideal zeta function}
  $$\zeta^{\ideal\,\lri}_L (s)= \sum_{I \triangleleft L}|L:I|^{-s}$$
of an $\lri$-algebra $L$ of finite $\lri$-rank is defined as in
\eqref{def:ideal.zeta}, with $I$ ranging over the $\lri$-ideals of
$L$, viz.\ $(\ad L)$-invariant $\lri$-submodules of~$L$. Note that
every element $\mcL$ of $\mfL$ may, after tensoring over $\Z$ with
$\lri$, be considered a free and finitely generated $\lri$-Lie
algebra. Given an $\lri$-module~$R$, we write~$L(R)=L\otimes_\lri R$.

\begin{thm}\label{thm:main.local}
 Let $\boldsymbol{{\mcL}} = (\mcL_1,\dots,\mcL_g)$ be a family of
 elements of $\mfL$ and $\boldsymbol{f}=
 (f_1,\dots,f_g)\in\N^g$. There exists an explicit rational function
 $W^{\ideal}_{\boldsymbol{\mcL},\boldsymbol{f}}\in\Q(X,Y)$ such that
 the following holds:

 Let $\lri$ be a compact discrete valuation ring and
 $(\Lri_1,\dots,\Lri_g)$ be a family of finite unramified extensions
 of $\lri$ with inertia degrees~$(f_1,\dots,f_g)$. Consider the
 $\lri$-Lie algebra
 $$L = \mcL_1(\Lri_1)\times\dots \times \mcL_g(\Lri_g).$$ For every
 finite extension $\Lri$ of~$\lri$, of inertia degree $f$ over $\lri$,
 say, the $\Lri$-ideal zeta function of $L(\Lri)$ satisfies
 $$\zeta^{\ideal\,\Lri}_{L(\Lri)}(s) =
 W^{\ideal}_{\boldsymbol{\mcL},\boldsymbol{f}}(q^f,q^{-fs}).$$ The
 rational function $W^{\ideal}_{\boldsymbol{\mcL},\boldsymbol{f}}$
 satisfies the functional equation
\begin{equation}\label{equ:funeq.local.newer}
  W^{\ideal}_{\boldsymbol{\mcL},\boldsymbol{f}}(X^{-1},Y^{-1}) =
  (-1)^{N_0} X^{\binom{N_0}{2}} Y^{N_0+N_1}
  W^{\ideal}_{\boldsymbol{\mcL},\boldsymbol{f}}(X,Y),
 \end{equation}
 where
 \begin{equation*}
   N_0 = \mathrm{rk}_{\mfo} L = \sum_{i=1}^g{f_i} \rk_\Z(\mcL_i) \quad \textup{
     and } \quad N_1 = \mathrm{rk}_{\mfo} (L/Z(L)) = \sum_{i=1}^g{f_i} \rk_\Z(\mcL_i/Z(\mcL_i)).
 \end{equation*}
\end{thm}


\begin{rem}
 In~\cite[Conjecture~1.4]{SV1/15} it was suggested that a functional
 equation should hold for {\emph{all}} local factors
 $\zeta^{\ideal}_{\mathfrak{f}_{2,2}(\mcO),p}(s)$, where
 $\mathfrak{f}_{2,2}$ is the Heisenberg Lie ring and $\mcO$ is a
 number ring; if $p$ ramifies in $\mcO$, then the symmetry factor must
 be modified from that of~\eqref{equ:funeq.local.newer}.  Some cases
 of the conjecture were proved in~\cite[Corollary~3.13]{SV2/16}.
 There is computational evidence, due to T.~Bauer, that other Lie
 rings in the class $\mfL$ also exhibit the remarkable property of the
 local factors $\zeta^{\ideal}_{\mcL(\mcO),p}(s)$ at ramified primes
 $p$ being described by rational functions satisfying functional
 equations.  Those computations, together with the results of this
 paper, suggest the following natural question: how do the local
 factors $\zeta^{\ideal}_{\mcL(\mcO),p}(s)$ behave at ramified primes,
 and how does the structure of $\mcL$ govern their behaviour?  A further generalization of our generalized Igusa functions will be necessary to provide explicit formulae in ramified cases.
\end{rem}
  
\subsection{Previous and related work}\label{subsubsec:previous}
\cref{thm:main.global,thm:main.local} generalize and
unify several previously known results.
  \begin{enumerate} 
  \item The most classical may be the formula for the $\lri$-ideal
    zeta function
     \begin{equation*}\label{equ:abelian}
       \zeta_{\lri^n}(s) := \zeta^{\ideal\,\lri}_{\lri^n}(s) =
       \prod_{i=1}^n\frac{1}{1-q^{-s+i-1}}
     \end{equation*}  of the (abelian Lie) ring $\lri^n = \mfg_{0,n}(\lri) =
       \mfg_{n,0}(\lri)$; cf.\ \cite[Proposition~1.1]{GSS/88}.
     \item The ideal zeta functions of the \emph{Grenham Lie rings}
       $\mfg_{1,d}$ were given in \cite[Theorem~5]{Voll/05}.
     \item Formulae for the ideal zeta functions of the free
       class-$2$-nilpotent Lie rings $\mff_{2,d}$ on $d$ generators
       are the main result of \cite{Voll/05a}.
   \item The paper \cite{SV1/15} contains formulae for all local
     factors of the ideal zeta functions of the Lie rings
     $\mff_{2,2}(\mcO) = \mfg_{1,1}(\mcO) = \mathfrak{h}_1(\mcO)$,
     i.e.\ the \emph{Heisenberg Lie ring} over an arbitrary number
     ring~$\mcO$, which are indexed by primes unramified in
     $\mcO$. The uniform nature of these functions had already been
     established in \cite[Theorem~3]{GSS/88}. Formulae for factors
     indexed by non-split primes are given in \cite{SV2/16}.
     \item 
     The ideal zeta functions of the Lie rings $\mathfrak{h}_d \times
     \mfo^r$ were computed in~\cite[Proposition~8.4]{GSS/88}, whereas
     for the direct products $\mathfrak{h}_d \times \cdots \times
     \mathfrak{h}_d$ they were computed in~\cite{Bauer/13}.
   \item The ideal zeta function of the Lie ring $\mfg_{2,2}$ was
     computed in \cite[Theorem~11.1]{Paajanen/08}.
 \end{enumerate}


\subsection{Informal overview of the proofs of \cref{thm:main.local,thm:main.global}}\label{subsec:overview} 
Here we summarize the principal ideas behind our approach, which
greatly generalize those of~\cite{SV1/15}.  Let $L$ be an $\lri$-Lie
algebra with derived subalgebra $L^\prime = [L,L]$.  If $L$ is class-$2$-nilpotent,
then an $\lri$-sublattice $\Lambda \leq L$ is an $\lri$-ideal if
$[\overline{\Lambda}, L] \leq \Lambda \cap L^\prime$, where
$\overline{\Lambda} = (\Lambda + L^\prime)/L^\prime$.  For simplicity
of exposition we will assume that $L^\prime =
Z(L)$, i.e.~that $L$ has no abelian direct summands.  By an argument
going back to~\cite[Lemma~6.1]{GSS/88}, the computation of $\zidealo_L
(s)$ is reduced to a summation over pairs $(\overline{\Lambda},
\mathrm{M})$, where $\overline{\Lambda} \leq L / L^\prime$ and
$\mathrm{M} \leq L^\prime$ are $\lri$-sublattices such that
$[\overline{\Lambda}, L] \leq \mathrm{M}$.  Recall that the
$\Lri$-elementary divisor type of a finite-index $\Lri$-sublattice
$\Lambda \leq \Lri^n$, where $\Lri$ is a compact discrete valuation
ring with maximal ideal $\mathfrak{M}$, is the partition $(\lambda_1,
\dots, \lambda_n)$ such that
$$ \Lri^n / \Lambda \simeq \Lri / \mathfrak{M}^{\lambda_1} \times
\cdots \times \Lri / \mathfrak{M}^{\lambda_n}.$$ Given the
$\lri$-elementary divisor type $\lambda(\overline{\Lambda})$ of
$[\overline{\Lambda},L]$, the lattices $\mathrm{M}$ satisfying this
condition are enumerated by a formula going back to work of Birkhoff~\cite{Birkhoff/35}
(see~\cite[Lemma~1.4.1]{Butler/94} and also \cite{Dyubyuk/48,
  Delsarte/48, Yeh/48}).

An essential ingredient of our method, therefore, is an effective
description of the $\lri$-elemen\-ta\-ry divisor type
$\lambda(\overline{\Lambda})$ in terms of the structure of
$\overline{\Lambda}$.  For the $\lri$-Lie algebras considered in this
paper, this is accomplished as follows.  The quotient $L/ L^\prime$
decomposes, as an $\lri$-module, into a direct sum of $m$ components,
which are viewed as free modules over finite extensions $\Lri_1,
\dots, \Lri_m$ of $\lri$.  For each component, we consider the
$\Lri_i$-elementary divisor type $\nu^{(i)}$ of the $\Lri_i$-lattice
generated by the projection of $\overline{\Lambda}$ onto that
component.  
The Lie rings in the class $\mfL$ of \cref{def:L} satisfy the combinatorial condition, crucial to our method, that 
the parts of the partition
$\lambda(\overline{\Lambda})$ are given by the minima of term-by-term
comparisons among the elementary divisor types $\nu^{(1)}, \dots, \nu^{(m)}$.  Assuming this, we deduce a
purely combinatorial expression for $\zidealo_L (s)$.

Analogously to the argument of~\cite{SV1/15}, we break up the ideal zeta function into a sum of finitely many pieces on which
the Gaussian multinomial coefficients and the dual partitions occurring in Birkhoff's formula are constant.  The sum
over each piece yields a product of Gaussian multinomials and
geometric progressions; these, in turn, are assembled into generalized
Igusa functions introduced in \cref{sec:general.igusa}. The combinatorial machinery of the weak orders of \cref{subsec:gen.igusa.funeq} is
required to keep track of the relative sizes of the parts of the
different partitions $\nu^{(i)}$; this is necessary in order to specify domains of
summation over which the dual partition
$\lambda(\overline{\Lambda})^\prime$ is constant. As in
\cite{SV1/15}, Dyck words of fixed length turn out to be suitable
indexing objects for the finitely many pieces.
An intrinsic advantage
 of this combinatorial point of view over the general (and typically
 immensely more powerful) algebro-geometric approach is that,
 structurally, $\Zp$ only enters as a compact discrete valuation
 ring. The effect of passage to various other such local rings,
 including those of positive characteristic, is therefore easy to
 control. 

\acknowledgements{The research of all three authors was supported by a grant from the GIF, the German--Israeli Foundation for Scientific Research and Development  (1246/2014)}

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\end{document}
