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% Author Package file for use with AMS-LaTeX 1.2
\controldates{25-AUG-1999,25-AUG-1999,25-AUG-1999,25-AUG-1999}
 
\documentclass{era-l} 
\usepackage{amssymb} 

\newtheorem{theorem}{Theorem}[section] 
\newtheorem{corollary}[theorem]{Corollary} 
\newtheorem{conjecture}[theorem]{Conjecture} 
\newtheorem{lemma}[theorem]{Lemma} 
\newtheorem{proposition}[theorem]{Proposition} 

\theoremstyle{remark}
\newtheorem{remark}{Remark}[section] 

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\newtheorem{example}{Example}[section] 
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\newcommand{\func}{\operatorname} 
 
 
\begin{document} 
 
\title{Zeta functions and counting finite $p$-groups} 
\author{Marcus du Sautoy} 
\address{DPMMS, 
16 Mill Lane, 
Cambridge CB2 1SB, UK} 
 
\email{dusautoy@dpmms.cam.ac.uk} 
 
\issueinfo{5}{16}{}{1999}
\dateposted{August 30, 1999}
\pagespan{112}{122}
\PII{S 1079-6762(99)00069-4}
\def\copyrightyear{1999}
\copyrightinfo{1999}{American Mathematical Society} 
 
\subjclass{Primary 20D15, 11M41; Secondary 03C10, 14E15, 11M45} 
 
\date{April 19, 1999} 
 
%\revdate{} 
 
\commby{Efim Zelmanov} 
 
%\keywords{} 
 
%\thanks{} 
 
\begin{abstract} 
We announce proofs of a number of theorems concerning finite $p$-groups 
and 
nilpotent groups.  These include: 
(1) the number of $p$-groups of class $c$ on $d$ generators of order 
$p^n$ 
satisfies a linear recurrence relation in $n$; 
(2) for fixed $n$ the number of $p$-groups of order $p^n$ as one varies 
$p$ 
is given by counting points on certain varieties mod $p$; 
(3) an asymptotic formula for the number of finite nilpotent groups of 
order $n$; 
(4) the periodicity of trees associated to finite $p$-groups of a fixed 
coclass (Conjecture P of Newman and O'Brien). 
The second result offers a new approach to Higman's PORC conjecture. The 
results are established using zeta functions associated to infinite 
groups 
and the concept of definable $p$-adic integrals. 
\end{abstract} 
 
\maketitle 
 
\section{Introduction} 
 
The classification of finite simple groups, the group theorist's Periodic 
table, provides us with the building blocks from which all finite 
groups are 
made. However, even if we take the simplest of the simple groups 
$C_{p}$, a 
cyclic group of order a prime $p$, we are very far from a 
classification of 
the groups that can be constructed from this single building block. 
Indeed 
the groups of order a power of $p$ were generally considered too wild a 
category of groups to hope for any complete classification. 
 
However, there have been two approaches to getting a hold on this wild 
class 
of groups. This paper contributes to the taming of this class of finite 
$p$% 
-groups by announcing smooth behaviours that can be established in both 
these approaches. 
 
The first approach is to ask whether, if we cannot list the $p$-groups of 
order a prime power, we can at least count how many groups of order $% 
p^{n}$ there are. 
 
Define 
\[ 
f(n,p)=\text{the number of groups of order }p^{n} 
\] 
and for $c,d\in {\mathbb N}$ define a more refined counting function $% 
f(n,p,c,d) $ to be the number of groups of order $p^{n},$ of class at 
most $% 
c,$ generated by at most $d$ generators. 
 
Higman and Sims (see \cite{Higman-Enumerating1} and \cite{Sims}) gave an 
asymptotic formula for the behaviour of $f(n,p)$ as $n$ grows: 
\[ 
f(n,p)=p^{\left( 2/27+o(1)\right) n^{3}}\text{ as }n\rightarrow \infty . 
\] 
 
Higman also conjectured in \cite{Higman-Enumerating2} that if you fix 
$n$ 
and vary $p$, then $f(n,p)$ is given by a polynomial in $p$ which depends 
only on the residue class of $p$ modulo some fixed integer $N.$ For 
example,
$f(6,p)$ is described by a quadratic polynomial whose coefficients 
depend on 
the residue of $p$ modulo $60.$ Higman's PORC conjecture, as this has 
become 
known, has withstood any attack since Higman's contribution in \cite 
{Higman-Enumerating2} nearly 40 years ago. 
 
The theme of this paper is to show how zeta functions of groups can be 
used 
to count $p$-groups and in particular offer some new ideas to approach 
Higman's PORC conjecture. 
 
Define 
\begin{equation} 
\zeta _{c,d,p}(s)=\sum_{n=0}^{\infty }f(n,p,c,d)p^{-ns}. 
\label{zeta function p-groups} 
\end{equation} 
We announce the proof of the following smooth behaviour of the counting 
function $f(n,p,c,d)$: 
 
\begin{theorem} 
\label{Rationality}The function $\zeta _{c,d,p}(s)$ is a rational 
function 
in $p^{-s}.$ Hence the function $f(n,p,c,d)$ satisfies a linear 
recurrence relation with constant coefficients when we fix $p,c$ and 
$d$ and vary $n.$ 
\end{theorem} 
 
This is actually a generalization of the following very classical 
formula 
counting finite abelian $p$-groups of rank $d$: 
\begin{equation} 
\zeta _{1,d,p}(s)=\zeta _{p}(s)\zeta _{p}(2s)\cdots \zeta _{p}(ds),
\label{zeta function of finite abelian} 
\end{equation} 
where $\zeta _{p}(s)=(1-p^{-s})^{-1}$, the local Riemann zeta function. The 
idea 
of the proof of Theorem \ref{Rationality}, which uses Denef's work on 
definable integrals, will be explained in section \ref{section-counting 
p-groups}. We shall explain in section \ref{section nilpotent groups} an 
Euler product that can be formed, which allows one to count nilpotent 
groups. 
We can then apply some recent results with Fritz Grunewald \cite 
{duSG-Abscissa} on asymptotics of zeta functions formed as Euler 
products of 
definable integrals to prove the following asymptotic behaviour of the 
function counting finite nilpotent groups: 
 
\begin{theorem} 
\label{asym of counting nilpotent groups}For integers $n,c,d$ define $% 
g(n,c,d)$ to be the number of nilpotent groups of order $n,$ of class at 
most $c,$ generated by at most $d$ elements. Then there exist a rational 
number $\alpha \in {\mathbb Q}$, an integer $\beta \geq 0$ and $\gamma 
\in 
{\mathbb R}$ such that 
\[ 
g(1,c,d)+\cdots +g(n,c,d)\sim \gamma \cdot n^{\alpha }\left( \log 
n\right) 
^{\beta }. 
\] 
\end{theorem} 
 
The analysis of \cite{duSG-Abscissa} also implies the following 
behaviour 
for the original function $f(n,p):$ 
 
\begin{theorem} 
For each $n$ there exist finitely many subvarieties $E_{i}$ $(i\in T)$ 
of a 
variety $Y$ defined over ${\mathbb Q}$ and for each $I\subset T$ a 
polynomial $% 
H_{I}(X)$ such that for almost all primes $p$,
\[ 
f(n,p)=\sum_{I\subset T}c_{p,I}H_{I}(p),
\] 
where 
\[ 
c_{p,I}=\mathrm{ card}\{a\in \overline{Y}({\mathbb F}_{p}):a\in 
\overline{E_{i}}(% 
{\mathbb F}_{p})\text{ if and only if }i\in I\}.
\] 
Here $\overline{Y}$ means reduction of the variety $\operatorname
{mod}p$ which is defined for almost all $p$.
\end{theorem} 
 
So counting $p$-groups is given by counting points on varieties 
$\operatorname{mod}p$.
This reduces Higman's PORC conjecture to showing that the number of 
points 
$\operatorname{mod}p$ on Boolean combinations of the varieties $E_{i}$ varies 
smoothly with $p$. The varieties $E_{i}$ are canonically associated to 
the 
free nilpotent group $F_{n-1,n}$ of class $n-1$ on $n$ generators and 
arise 
from the resolution of singularities of a polynomial defined from the 
structure constants of the associated free nilpotent Lie algebra and its 
associated automorphism group. 
 
The second approach to understanding $p$-groups is more structural and 
involves defining an invariant called the coclass of a $p$-group. A 
group of 
order $p^{n}$ of class $c$ is said to have {\em coclass} $r=n-c.$ The 
concept of coclass was first introduced by Leedham-Green and Newman in 
the 
80's when they made some very insightful conjectures (Conjectures A--E) 
on 
the structure of $p$-groups of a fixed coclass. These conjectures have 
now 
been confirmed and we shall explain the strongest of these in section 
\ref 
{section coclass}. 
 
Since then a new set of conjectures (Conjectures P--S) has been proposed 
by 
Newman and O'Brien \cite{Newman-O'Brien} concerning the structure of a 
directed graph ${\mathcal G}_{r,p}$ that can be associated to 
$p$-groups of 
coclass $r.$ Each such group defines a node in the graph, and directed 
edges 
join two groups $P_{1}\rightarrow P_{2}$ if there exists an exact 
sequence 
\[ 
1\rightarrow C_{p}\rightarrow P_{1}\rightarrow P_{2}\rightarrow 1. 
\] 
 
The graph ${\mathcal G}_{r,p}$ consists of finitely many trees 
${\mathcal F}% 
_{r,p,1},\dots,{\mathcal F}_{r,p,k}$ such that the complement of the 
union of 
these trees is finite. Each tree has a single infinite chain. For each 
$N\in 
{\mathbb N}$ let ${\mathcal F}_{r,p,i}(N)$ denote the tree consisting 
of the 
infinite chain and paths of length at most $N$ from the infinite chain. 
So, 
in horticultural terms, ${\mathcal F}_{r,p,i}(N)$ is the tree 
${\mathcal F}_{r,p,i}$ 
`pruned' so that twigs have length at most $N.$ We wish to announce a 
proof 
of the qualitative part of Conjecture P which concerns the 
periodicity of 
these trees: 
 
\begin{theorem} 
\label{Periodicity}The trees ${\mathcal F}_{r,p,i}(N)$ are ultimately 
periodic. 
\end{theorem} 
 
For $p=2,$ note that there exists some $N$ such that ${\mathcal 
F}_{r,2,i}(N)= $ 
${\mathcal F}_{r,2,i},$ i.e. the twig lengths are bounded. 
 
Essential use is made, as we shall explain, of zeta functions which 
count 
the number of vertices in these trees. We use the flexibility of the 
concept 
of definable $p$-adic integrals as developed by Denef to prove the 
rationality of these zeta functions. 
 
The proofs of results in sections \ref{section-counting p-groups}--\ref 
{section PORC} can be found in \cite{duS-p-groups}. The proofs of 
results in 
section \ref{section coclass} can be found in \cite{duS-conjectureP}. 
 
\section{Counting $p$-groups\label{section-counting p-groups}} 
 
The key to understanding the zeta function (\ref{zeta function 
p-groups}) 
defined in the introduction is another zeta function first introduced in 
\cite{GSS} to understand another unclassifiable class of groups: 
finitely 
generated nilpotent groups. Let $G$ be any finitely generated group and 
define 
\begin{eqnarray} 
\zeta _{G}^{\triangleleft }(s) &=&\sum_{N\triangleleft G}|G:N|^{-s} 
\label{zeta function of a f.g. group} \\ 
&=&\sum_{n=1}^{\infty }a_{n}^{\triangleleft }(G)n^{-s}.  \nonumber 
\end{eqnarray} 
 
This may be regarded as a noncommutative version of the Dedekind zeta 
function of a number field. The coefficients $a_{n}^{\triangleleft }(G)$ 
denote the number of normal subgroups of index $n$ in $G$ and are finite 
whenever $G$ is finitely generated either as an abstract group or 
topologically. One can also define the zeta function counting all 
subgroups 
of finite index, but it is this normal zeta function which will be 
relevant 
to counting $p$-groups. 
 
When $G$ is a nilpotent group, this zeta function has a natural Euler 
product 
decomposition into a product of local factors (see \cite{GSS}): 
\begin{eqnarray} 
\zeta _{G}^{\triangleleft }(s) &=&\prod_{p\mathrm{ \ prime}}\zeta 
_{G,p}^{\triangleleft }(s),\quad\text{where} 
\label{Euler product for normal subgroups} \\ 
\zeta _{G,p}^{\triangleleft }(s) &=&\sum_{n=0}^{\infty 
}a_{p^{n}}^{\triangleleft }(G)p^{-ns}.  \nonumber 
\end{eqnarray} 
 
Note that $\zeta _{G,p}^{\triangleleft }(s)=\zeta _{\widehat{G}% 
_{p}}^{\triangleleft }(s)$ where $\widehat{G}_{p}$ denotes the pro-$p$ 
completion of $G.$ The first chapter in the theory of zeta functions of 
groups concerned the rationality of these local factors. In \cite{GSS} 
this 
rationality is established for torsion-free nilpotent groups and 
generalized 
in \cite{duS-Annals} to pro-$p$ groups which are $p$-adic analytic. For 
example, when $G={\mathbb Z}_{p}^{d},$ the free abelian pro-$p$ group of 
rank $% 
d, $ the zeta function has the following form: 
\[ 
\zeta _{G}^{\triangleleft }(s)=\zeta _{p}(s)\zeta _{p}(s-1)\cdots \zeta 
_{p}(s-d+1). 
\] 
 
It is instructive to compare this with the formula (\ref{zeta function 
of 
finite abelian}) for the zeta function counting finite abelian 
$p$-groups of 
rank $d.$ There is of course some relation between these two zeta 
functions. 
Each finite abelian $p$-group of rank $d$ is a finite image of the free 
abelian pro-$p$ group ${\mathbb Z}_{p}^{d},$ and vice-versa. The 
crucial point 
though is that $\zeta _{{\mathbb Z}_{p}^{d}}^{\triangleleft }(s)$ 
overcounts 
the finite abelian $p$-groups since each finite abelian $p$-group 
occurs in 
many ways as a quotient of ${\mathbb Z}_{p}^{d}$ by a normal subgroup. 
We shall 
show how we can refine the zeta function (\ref{zeta function of a f.g. 
group}% 
) to take account of this overcounting. 
 
There is nothing special of course about being abelian in this 
connection 
between finite abelian $p$-groups and finite images of a free abelian 
pro-$p$ 
group. Let $F_{c,d}$ be the free nilpotent group of class $c$ on $d$ 
generators. Then the finite $p$-groups of class at most $c$ generated 
by $d$ 
elements are precisely the finite images of the pro-$p$ completion 
$\widehat{% 
(F_{c,d})}_{p}$ of $F_{c,d}.$ 
 
The way to overcome the overcounting is to use the automorphism group $% 
{\mathfrak G}_{p}=\mathrm{ Aut}\widehat{(F_{c,d})}_{p}.$ A pro-$p$ 
group $G$ which 
is free in some variety has the property that an isomorphism of two 
finite 
quotients $G/N_{1}\stackrel{\cong }{\longrightarrow }G/N_{2}$ lifts to 
an 
automorphism of $G.$ Applying this to the free nilpotent pro-$p$ group 
$% 
\widehat{(F_{c,d})}_{p}$ yields a one to one correspondence between 
isomorphism types of finite quotients of order $p^{n}$ and normal 
subgroups 
of index $p^{n}$ equivalent up to the action of the automorphism group 
$% 
{\mathfrak G}_{p}$. We therefore have the following identity of zeta 
functions: 
 
\begin{proposition} 
\[ 
\zeta _{c,d,p}(s)=\sum_{N\triangleleft 
\widehat{(F_{c,d})}_{p}}|\widehat{% 
(F_{c,d})}_{p}:N|^{-s}|{\mathfrak G}_{p}:\mathrm{ Stab}_{{\mathfrak 
G}_{p}}(N)|^{-1}. 
\] 
\end{proposition} 
 
$|{\mathfrak G}_{p}:\mathrm{ Stab}_{{\mathfrak G}_{p}}(N)|$ is the size 
of the orbit 
containing $N$ under the action of ${\mathfrak G}_{p}.$ 
 
\begin{theorem} 
\label{rationality section 2}The function $\zeta _{c,d,p}(s)$ is a 
rational 
function in $p^{-s}.$ 
\end{theorem} 
 
\noindent\textit{Idea of proof.} We show how to represent $\zeta 
_{c,d,p}(s)$ by a 
definable $p$-adic integral as considered by Denef in \cite 
{Denef-Rationality}. The group $F_{c,d}$ has an associated ${\mathbb 
Z}$-Lie 
algebra $L$ with the property that for almost all primes $p$, there is a 
one to one index preserving correspondence between normal subgroups in 
$% 
\widehat{(F_{c,d})}_{p}$ and ideals in $L\otimes {\mathbb Z}_{p}$. The 
automorphism group ${\mathfrak G}$ of $L$ has the structure of a 
${\mathbb Q}$% 
-algebraic group with the property that ${\mathfrak G}({\mathbb 
Z}_{p})=\mathrm{ Aut}% 
(L\otimes {\mathbb Z}_{p})\leq \mathrm{ GL}_{n}({\mathbb Z}_{p})$, where 
the underlying 
${\mathbb Z}$-structure of ${\mathfrak G}$ comes from a choice of basis 
$% 
e_{1},\dots,e_{n}$ for $L.$ For almost all primes, ${\mathfrak 
G}({\mathbb Z}_{p})$ 
is isomorphic to ${\mathfrak G}_{p}=\mathrm{ 
Aut}\widehat{(F_{c,d})}_{p}$, and normal 
subgroups equivalent under the action of ${\mathfrak G}_{p}$ correspond 
to 
ideals equivalent under the action of ${\mathfrak G}({\mathbb Z}_{p})$. 
Hence for 
almost all $p$:
\begin{equation} \label{groups=Lie algebras} 
\begin{split}
&\sum_{N\triangleleft\widehat{(F_{c,d})}_{p}}|\widehat{(F_{c,d})}% 
_{p}:N|^{-s}|{\mathfrak G}_{p}:\mathrm{ Stab}_{{\mathfrak 
G}_{p}}(N)|^{-1}\\ 
&\qquad=\sum_{N% 
\triangleleft L\otimes {\mathbb Z}_{p}}|L\otimes {\mathbb 
Z}_{p}:N|^{-s}|{\mathfrak G}(% 
{\mathbb Z}_{p}):\mathrm{ Stab}_{{\mathfrak G}({\mathbb 
Z}_{p})}(N)|^{-1}.
\end{split}
\end{equation} 
This has linearized the problem for almost all primes $p.$ 
 
We now use the concept of $p$-adic integration to express the right hand 
side of (\ref{groups=Lie algebras}). 
 
Let $\mathrm{ Tr}_{n}({\mathbb Z}_{p})$ denote the set of lower 
triangular 
matrices. For each ideal $N\triangleleft L\otimes {\mathbb Z}_{p},$ 
define $% 
M(N)\subset \mathrm{ Tr}_{n}({\mathbb Z}_{p})$ to be the subset 
consisting of all 
matrices $M$ whose rows $\mathbf{ m}_{1},\dots,\mathbf{ m}_{n}$ form a 
basis for $N.$ 
We get a nice description of $\mathrm{ Stab}_{{\mathfrak G}({\mathbb 
Z}_{p})}(N)$ in 
terms of $M\in M(N),$ namely: 
\[ 
\mathrm{ Stab}_{{\mathfrak G}({\mathbb Z}_{p})}(N)={\mathfrak 
G}({\mathbb Z}_{p})\cap M^{-1}% 
\mathrm{ GL}_{n}({\mathbb Z}_{p})M. 
\] 
Let $\mu $ denote the additive normalized Haar measure on$\mathrm{ \ 
Tr}_{n}(% 
{\mathbb Z}_{p})$ and let $\nu $ be the normalized Haar measure on 
${\mathfrak G}({\mathbb % 
Z}_{p}).$ The value of $|{\mathfrak G}({\mathbb Z}_{p}):\operatorname{ 
Stab}_{{\mathfrak G}(% 
{\mathbb Z}_{p})}(N)|^{-1}$ is then just the measure of the set 
${\mathfrak G}(% 
{\mathbb Z}_{p})\cap M^{-1}\mathrm{ GL}_{n}({\mathbb Z}_{p})M.$ Define 
the subset $% 
{\mathcal M}\subset \mathrm{ Tr}_{n}({\mathbb Z}_{p})\times {\mathfrak 
G}({\mathbb Z}_{p})$ by 
\[ 
{\mathcal M}=\left\{ (M,K):M\in \bigcup_{N\triangleleft L\otimes 
{\mathbb Z}% 
_{p}}M(N),K\in {\mathfrak G}({\mathbb Z}_{p})\cap M^{-1}\mathrm{ 
GL}_{n}({\mathbb Z}% 
_{p})M\right\} . 
\] 
Then the right hand side of (\ref{groups=Lie algebras}), and hence 
$\zeta 
_{c,d,p}(s)$ for almost all primes, equals 
\begin{equation} 
(1-p^{-1})^{-n}\int_{{\mathcal M}}|m_{11}|^{s-n}\cdots|m_{nn}|^{s-1}d\mu 
\,d\nu . 
\label{p-adic integral} 
\end{equation} 
 
The task finally is to extend ideas in \cite{GSS} and \cite{duS-Annals} 
to 
prove that this is a definable integral in the language of valued 
fields and 
apply Denef's rationality result for such definable integrals in \cite 
{Denef-Rationality} to prove Theorem \ref{rationality section 2}. The 
finitely many exceptional primes can also be dealt with in a less 
uniform 
manner as compared to the above analysis. 

\section{Counting finite nilpotent groups\label{section nilpotent 
groups}} 
 
Define the following Dirichlet series counting nilpotent groups of 
class at 
most $c$, generated by $d$ elements: 
\[ 
\zeta _{c,d}(s)=\sum_{n=1}^{\infty }g(n,c,d)n^{-s},
\] 
where $g(n,c,d)$ is defined in Theorem \ref{asym of counting nilpotent 
groups}. Analogous to the Euler product for counting normal subgroups of 
finite index (\ref{Euler product for normal subgroups}), we have the 
following: 
 
\begin{proposition} 
\label{Euler product for finite nil} 
\[ 
\zeta _{c,d}(s)=\prod_{p\mathrm{ \ prime}}\zeta _{c,d,p}(s).
\] 
\end{proposition} 
 
This is just an analytic way of saying that a finite nilpotent group is 
a 
direct product of its Sylow $p$-subgroups. 
 
To prove Theorem \ref{asym of counting nilpotent groups} we apply some 
recent work with Grunewald on Euler products of special sorts of 
$p$-adic 
integrals called cone integrals: 
 
\begin{definition} 
Let $\left| dx\right| $ be the normalized additive Haar measure on 
${\mathbb Z}% 
_{p}^{m}.$ We call an integral 
\[ 
Z_{{\mathcal D}}(s,p)=\int_{V_{p}}\left| f_{0}(\mathbf{ x)}\right| 
^{s}\left| g_{0}(% 
\mathbf{ x)}\right| \left| dx\right| 
\] 
{\em a cone integral defined over }${\mathbb Q}$ if 
$f_{0}(\mathbf{ x})$ and $g_{0}(\mathbf{ x})$ are polynomials in the 
variables $\mathbf{ x}=x_{1},\dots,x_{m}$ with coefficients in ${\mathbb 
Q}$ and there exist polynomials $f_{i}(\mathbf{ x})$, $g_{i}(\mathbf{ 
x})$ 
$(i=1,\dots,l)$ over ${\mathbb Q}$ such that 
\[ 
V_{p}=\left\{ \mathbf{ x}\in {\mathbb Z}_{p}^{m}:v(f_{i}(\mathbf{ 
x)})\leq v(g_{i}(\mathbf{ % 
x}))\text{ for }i=1,\dots,l\right\} . 
\] 
The set ${\mathcal D}=\left\{ 
f_{0},g_{0},f_{1},g_{1},\dots,f_{l},g_{l}\right\} $ 
is called the {\em cone integral data}. 
\end{definition} 
 
In \cite{duSG-Abscissa} we prove the following about the Euler product 
of 
cone integrals over ${\mathbb Q}$: 
 
\begin{theorem} 
\label{cone integrals theorem}Let ${\mathcal D}$ be a set of cone 
integral data 
and put 
\[ 
Z_{{\mathcal D}}(s)=\prod_{p\text{ prime }}\left( a_{p,0}^{-1}\cdot 
Z_{{\mathcal D}% 
}(s,p)\right),
\] 
where $a_{p,0}=Z_{{\mathcal D}}(\infty ,p)$ is the constant coefficient 
of $Z_{% 
{\mathcal D}}(s,p),$ i.e. we normalize the local factors to have 
constant 
coefficient $1$. Then the abscissa of convergence $\alpha $ of 
$Z_{{\mathcal D}% 
}(s)$ is a rational number and $Z_{{\mathcal D}}(s)$ has a meromorphic 
continuation to $\Re (s)>\alpha -\delta $ for some $\delta >0.$ 
\end{theorem} 
 
To prove Theorem \ref{asym of counting nilpotent groups} we show that 
for 
almost all primes $p,$ the integrals $(\ref{p-adic integral})$ 
representing $% 
\zeta _{c,d,p}(s)$ can be expressed in terms of cone integrals over 
${\mathbb Q}% 
. $ We can then apply Theorem \ref{cone integrals theorem}, Proposition 
\ref 
{Euler product for finite nil} and a suitable Tauberian theorem to 
deduce 
the asymptotic behaviour of the coefficients $g(n,c,d)$ of the zeta 
function 
$\zeta _{c,d}(s)$ detailed in Theorem \ref{asym of counting nilpotent 
groups}% 
. The finite number of exceptional primes are no worry since we have 
established that all the local factors are rational functions in 
$p^{-s}.$ 
 
\section{PORC and counting points on varieties $\func{mod}p$\label% 
{section PORC}} 
 
The proof of Theorem \ref{cone integrals theorem} relies on proving an 
explicit formula for cone integrals in terms of a resolution of 
singularities $h:Y\rightarrow {\mathbb A}^{m}$ of the polynomial $% 
F=\prod_{i=0}^{l}f_{i}g_{i}$, where ${\mathcal D}=\left\{ 
f_{0},g_{0},f_{1},g_{1},\dots,f_{l},g_{l}\right\} $ is the associated cone 
integral data. Let $E_{i},i\in T,$ be the irreducible components defined 
over ${\mathbb Q}$ of the reduced scheme $\left( h^{-1}(D)\right) 
_{\mathrm{ red}}$,
where $D=${$\mathrm{ Spec}$}$\left( \frac{{\mathbb Q}[\mathbf{ 
x]}}{(F)}\right) .$ Then 
there exist rational functions $P_{I}(x,y)\in {\mathbb Q}(x,y)$ for 
each $% 
I\subset T$ with the property that for almost all primes $p$,
\begin{equation} 
Z_{{\mathcal D}}(s,p)=\sum_{I\subset T}c_{p,I}P_{I}(p,p^{-s}), 
\label{formula as points times uniform zeta function}
\end{equation} 
where 
\[ 
c_{p,I}=\mathrm{ card}\{a\in \overline{Y}({\mathbb F}_{p}):a\in 
\overline{E_{i}}(% 
{\mathbb F}_{p})\text{ if and only if }i\in I\}\text{ } 
\] 
and $\overline{Y}$ means the reduction $\func{mod}p$ of the scheme $Y.$ 
Since $\zeta _{c,d,p}(s)$ are given by cone integrals for almost all 
primes $% 
p,$ we get a corresponding uniform description for $\zeta _{c,d,p}(s)$ 
as in 
(\ref{formula as points times uniform zeta function}). 
 
Let $Y(n)$ and $E_{i}(n)$ ($i\in T(n))$ be the varieties arising from a 
resolution of singularities of the polynomial $F_{n}=% 
\prod_{i=0}^{l}f_{i}g_{i}$ associated to the cone integrals 
representing $% 
\zeta _{n-1,n,p}(s).$ Since $f(n,p,n-1,n)=f(n,p),$ the explicit 
formulas in 
\cite{duSG-Abscissa} imply the following uniformity for $f(n,p):$ 
 
\begin{theorem} 
For each $I\subset T(n)$ there exists a polynomial $H_{I}(X)\in 
{\mathbb Q}[X]$ 
such that 
\[ 
f(n,p)=\sum_{I\subset T(n)}c_{p,I}H_{I}(p),
\] 
where 
\[ 
c_{p,I}=\mathrm{ card}\{a\in \overline{Y(n)}({\mathbb F}_{p}):a\in 
\overline{% 
E_{i}(n)}({\mathbb F}_{p})\text{ if and only if }i\in I\}\text{ .} 
\] 
\end{theorem} 
 
In \cite{duS-p-groups}, the polynomial $F_{n}$ associated to $\zeta 
_{n-1,n,p}(s)$ is defined explicitly in terms of the structure 
constants of 
the free nilpotent Lie algebra of class $n-1$ on $n$ generators and its 
associated automorphism group. 
 
By using the recent language of motivic integration as developed by 
Denef 
and Loeser \cite{Denef-Loeser-Arcs} and \cite{Denef-Loeser-motivic Igusa}, 
it can be shown that the varieties $E_{i}(n)$ sitting inside a suitable 
completion of the Grothendieck ring of varieties, are in fact not 
accidental 
but are canonically associated to the cone integrals expressing $\zeta 
_{n-1,n,p}(s)$. One approach to Higman's PORC conjecture therefore is to 
understand the varieties $E_{i}(n)$ ($i\in T(n))$ arising from the 
resolution of $F_{n}.$ 
 
We conjecture a stronger uniformity for the behaviour of the rational 
functions $\zeta _{c,d,p}(s)$ as we vary $p$ than provided by 
$(\ref{formula 
as points times uniform zeta function})$: 
 
\begin{conjecture} 
\label{Uniformity}Fix integers $c$ and $d$. There exist finitely many rational 
functions $W_{i}(X,Y)$ $\in {\mathbb Q}(X,Y)$ $(i=1,\dots,N)$ such that 
if $p\equiv i% 
\operatorname{mod}N$, then 
\[ 
\zeta _{c,d,p}(s)=W_{i}(p,p^{-s}). 
\] 
The rational functions $W_{i}(X,Y)$ have the form 
\[ 
W_{i}(X,Y)=\frac{P_{i}(X,Y)}{% 
(1-X^{a_{i1}}Y^{b_{i1}})\cdots(1-X^{a_{id_{i}}}Y^{b_{id_{i}}})}. 
\] 
\end{conjecture} 
 
Note in particular that this conjecture implies the following: 
 
\begin{corollary} 
\label{Uniformity implies PORC}Suppose Conjecture \ref{Uniformity} is 
true. 
Then for $n\in {\mathbb N}$ and $i=1,\dots,N$ there exist polynomials $% 
r_{n,i}(X)\in {\mathbb Q}[X]$ such that if $p\equiv i\operatorname{mod}N$, then 
\[ 
f(n,p,c,d)=r_{n,i}(p),
\] 
i.e. the function $f(n,p,c,d)$ is PORC in $p.$ 
\end{corollary} 
 
Since $f(n,p,n-1,n)=f(n,p)$, this includes Higman's PORC conjecture as a 
special case. We can actually deduce a stronger corollary from 
Conjecture 
\ref{Uniformity}, which gives some relationship between the polynomials 
$% 
r_{n,i}(X)$ as we vary $n.$ 
 
Since $\zeta _{c,d,p}(s)$ can be expressed as a cone integral for 
almost all 
$p$, the explicit expression in (\ref{formula as points times uniform 
zeta 
function}), valid for almost all primes, implies that the degrees of the 
numerators and denominators of the rational functions $\zeta 
_{c,d,p}(s)$ 
can be bounded independently of $p.$ This bound on the degrees combined 
with 
the conjectured PORC behaviour of $f(n,p,c,d)$ would in fact imply 
Conjecture \ref{Uniformity}. 
 
To prove Conjecture \ref{Uniformity} is likely to depend on 
understanding 
the two pieces of the integral (\ref{p-adic integral}): 
 
(1) one coming from the algebraic group ${\mathfrak G}$ and the 
behaviour of the 
measure ${\mathfrak G}({\mathbb Z}_{p})\cap M^{-1}\mathrm{ 
GL}_{n}({\mathbb Z}_{p})M;$ and 
 
(2) the other coming from understanding the uniformity of the zeta 
function 
counting normal subgroups in the free nilpotent groups. 
 
The uniformity of these normal zeta functions in (2) was first raised in 
\cite{GSS}, where one rational function $W(X,Y)$ is conjectured to 
describe $% 
\zeta _{F_{c,d},p}^{\triangleleft }(s)$ for almost all $p$. However, the 
relationship with Higman's PORC conjecture implies that the conjecture made 
there 
is much more significant than was first realized. 
 
In \cite{GSS} the uniformity of $\zeta _{F_{c,d},p}^{\triangleleft 
}(s)$ is 
proved for $c=2$ and $d$ arbitrary by use of Hall polynomials. Recently 
in 
joint work with Fritz Grunewald \cite{duSG-2gen}, we have also proved 
the 
uniformity for $d=2$ and arbitrary class $c.$ 
 
It remains to analyze the integral coming from the algebraic group in 
these 
cases. Note that the true PORC behaviour (i.e. dependence on residue 
classes) of $f(n,p)$ should be a result of the analysis of the algebraic 
group, since the conjectured uniformity of $\zeta 
_{F_{c,d},p}^{\triangleleft 
}(s)$ in \cite{GSS} does not depend on residue classes of $p.$ 
 
It should be pointed out that the uniformity of $\zeta 
_{F_{c,d},p}^{\triangleleft }(s)$ is something special about free 
nilpotent 
groups. It was asked in \cite{GSS} whether such uniformity might hold 
for a 
general nilpotent group. However, I have produced an example of class 2, 
Hirsch length 9 nilpotent group $G$ whose local factors $\zeta 
_{G,p}^{\triangleleft }(s)$ depend on the behaviour of the number of 
points $% 
\func{mod}p$ on the elliptic curve $y^{2}=x^{3}-x,$ a behaviour which 
is far 
from the uniformity previously expected. This example is explained in 
\cite 
{duSG-class2}.
 
\section{Coclass and conjecture P\label{section coclass}} 
 
\begin{definition} 
A $p$-group of order $p^{n}$ and class $c$ is said to have {\em coclass 
}$% 
r=n-c.$ 
\end{definition} 
 
We begin with explaining the idea of the proof of the following: 
 
\begin{theorem} 
\label{rationality of coclass}Let $c(r,n,p)$ denote the number of 
$p$-groups 
of order $p^{n}$ and coclass $r$ and define the Poincar\'{e} series of 
$p$% 
-groups of coclass $r$ to be 
\[ 
Z_{r,p}(X)=\sum_{n=0}^{\infty }c(r,n,p)X^{n}. 
\] 
For each $p$ and $r,$ $Z_{r,p}(X)$ is a rational function. 
\end{theorem} 
 
We use the same idea as explained in section \ref{section-counting 
p-groups} 
to count finite groups by counting normal subgroups in some suitable 
infinite group. However, unlike counting $p$-groups of class $c$ on $d$ 
generators where there is a suitable free object whose finite images are 
precisely the groups we want to count, coclass is not a good variety of 
groups and no such free object exists. However, all is not lost. We can 
find 
a group whose finite images include all $p$-groups of coclass $r$ which 
is 
not so big as the free group itself. This depends on the remarkable fact 
(conjectured by Leedham-Green and Newman and confirmed by Leedham-Green 
\cite 
{Leedham-GreenJLMS} and Shalev \cite{Shalev-Invent}) that $p$-groups of 
coclass $r$ are almost of class two in the following sense: 
 
\begin{theorem}[Conjecture A] 
\label{conjecture A}There exists a positive integer $h=h(p,r)$ such that 
every $p$-group of coclass $r$ has a normal subgroup of class at most $2$
$(1$ 
if $p=2)$ and index dividing $p^{h}.$ 
\end{theorem} 
 
A $p$-group of coclass $r$ is generated by at most $r+1$ elements. 
 
\begin{definition} 
Define the group ${\mathcal G}_{r}$ by 
\[ 
{\mathcal G}_{r}=F/\gamma _{3}\left( \gamma _{h}(F)\cdot 
F^{p^{h}}\right),
\] 
where $F$ is the free $(r+1)$-generator pro-$p$ group. 
\end{definition} 
 
The following is then a Corollary of Theorem \ref{conjecture A}: 
 
\begin{corollary} 
Every finite $p$-group of coclass $r$ is an image of ${\mathcal G}_{r}.$ 
\end{corollary} 
 
Since ${\mathcal G}_{r}$ is free in some variety, it has the property 
that 
isomorphisms between finite quotients lift to automorphisms of 
${\mathcal G}% 
_{r}. $ The group ${\mathcal G}_{r}$ and its automorphism group are 
compact $p $% 
-adic analytic groups. We can use the ideas in section 
\ref{section-counting 
p-groups} and \cite{duS-Annals} to express the zeta function counting 
normal 
subgroups of ${\mathcal G}_{r}$ up to equivalence under the action of 
the 
automorphism of ${\mathcal G}_{r}$ as a definable integral, now in the 
analytic 
language for the $p$-adic integers as developed in \cite{Denef-van denDries}.
 However, although every finite $p$-group of coclass $r$ occurs as an 
image 
of ${\mathcal G}_{r},$ it also has other quotients that we do  not want 
to count. 
So we need to add a sentence in the definition of our definable 
integral to 
make sure we only count normal subgroups corresponding to quotients of 
coclass $r.$ 
 
But at first sight, coclass looks a very undefinable condition. As the 
index 
of the normal subgroup increases, for the quotient to have coclass $r,$ 
we 
need to check that the class is also increasing. How can we capture this 
statement in a definable sentence, which necessarily bounds the length 
of 
commutators that we could check? 
 
Here a corollary to the proof of Conjecture A comes to the rescue. Each 
nontrivial layer of the lower central series $\gamma _{n}(G)/\gamma 
_{n+1}(G)$ contributes 
\begin{equation*}
\log _{p}|\gamma _{n}(G)/\gamma _{n+1}(G)|-1\end{equation*}
to the 
coclass. The following corollary of the proof of Conjecture A says that 
coclass is realized in a bounded image of the group. 
 
\begin{theorem} [Proposition 4.5 of 
\cite{Shalev-Invent}]\label{corollary to conjecture A}
There exists an integer $g=g(p,r)$ such that for every $p$-group of 
coclass $% 
r,$ if $n>g$, then 
\[ 
|\gamma _{n}(G)/\gamma _{n+1}(G)|\leq p.
\] 
\end{theorem} 
 
Therefore, to check whether a quotient has coclass $r$ can be done by 
analyzing only the size of the first $g$ layers of the lower central 
series. 
It is therefore possible to express $Z_{r,p}(p^{-s})$ as a $p$-adic 
integral 
definable in the analytic language of the $p$-adic numbers, and 
therefore get 
a proof of Theorem \ref{rationality of coclass} by appealing to the 
proof of 
the rationality of such integrals established in \cite{Denef-van denDries}. 
 
Theorem \ref{rationality of coclass} is not strong enough, however, to 
establish the periodicity of the associated trees ${\mathcal 
F}_{r,p,i}(N)$ 
defined in the introduction. Certainly the periodicity of these trees 
implies a recurrence relation for the number of vertices at each layer. 
But 
such a recurrence relation says nothing about the shape of the tree 
being 
periodic. 
 
We have to squeeze the flexibility of the concept of definable to its 
limits 
to prove this periodicity. It can be proved that there is a bound on the 
valency of the tree ${\mathcal F}={\mathcal F}_{r,p,i}$ which we shall 
call $v.$ Let $P_{n}$ 
($n\in {\mathbb N}$) denote the $p$-groups which are the vertices of the 
infinite chain in ${\mathcal F}$. From each group $P_{n}$ define the 
twig $% 
{\mathfrak t}_{n}(N)$ to be the directed graph with unique root $P_{n}$ 
together 
with all nonmainline groups of length at most $N$ from $P_{n}.$ Let 
${\mathfrak % 
T}(N,v)$ be the set of finite (directed) graphs of length bounded by $N$ 
with a unique root and the valency of each node bounded by $v.$ Then for 
each ${\mathfrak t}\in {\mathfrak T}(N,v)$ define a zeta function 
\begin{eqnarray*} 
\zeta _{{\mathcal F}(N),{\mathfrak t}}(s) &=&\sum_{{\mathfrak 
t}_{n}(N)={\mathfrak t}% 
}|P_{n}|^{-s} \\ 
&=&|P_{1}|^{-s}\sum_{{\mathfrak t}_{n}(N)={\mathfrak t}}p^{-(n-1)s}.
\end{eqnarray*} 
 
\begin{theorem} 
${\mathcal F}(N)$ is ultimately periodic if and only if for all 
${\mathfrak t}\in 
{\mathfrak T}(N,v),$ $\zeta _{{\mathcal F}(N),{\mathfrak t}}(s)$ is a 
rational function 
in $p^{-s}.$ 
\end{theorem} 
 
The coefficients of $\zeta _{{\mathcal F}(N),{\mathfrak t}}(s)$ are 
just $1$ or $0$ 
according to whether the twig of $P_{n}$ of length $N$ is isomorphic to 
$% 
{\mathfrak t}$ or not. Therefore, rationality implies periodicity for 
occurrence 
of $1$'s in the series. 
 
Therefore, Theorem \ref{Periodicity} is a corollary of the following: 
 
\begin{theorem} 
\label{rationality of twigs}$\zeta _{{\mathcal F}(N),{\mathfrak t}}(s)$ 
is a 
rational function in $p^{-s}.$ 
\end{theorem} 
 
The strategy for the proof of Theorem \ref{rationality of twigs} is to 
take 
the definable integral describing the number of all $p$-groups of 
coclass $r$ 
and to add more definable conditions so that we are only counting 
$p$-groups 
which are on the infinite chain of the tree ${\mathcal F}$ with twigs 
of a 
certain shape ${\mathfrak t}$. 
 
The details with examples of some of the rational functions for small 
$r$ 
and $p$ are contained in \cite{duS-conjectureP}. 
 
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\end{document}