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\begin{document}


\title[Essential dimension]{Some consequences
of the Karpenko-Merkurjev theorem}

\author[Aurel Meyer]{Aurel Meyer$^\dagger$}
\thanks{$^\dagger$ Partially supported by
a University Graduate Fellowship at the University of British Columbia}
\author[Zinovy Reichstein]{Zinovy Reichstein$^{\dagger \dagger}$}
\thanks{$^{\dagger \dagger}$ Partially supported by
NSERC Discovery and Accelerator Supplement grants}
 
\address{Department of Mathematics, University of British Columbia,
 Vancouver, BC V6T 1Z2, Canada}
\subjclass{20D15, 20C15, 20G15} 

\dedicatory{To Andrei Alexandrovich Suslin on the occasion of his 60th birthday}

\keywords{Essential dimension, linear representation, $p$-group,
 algebraic torus}

\begin{abstract} 
 We use a recent theorem of N. A. Karpenko and 
 A. S. Merkurjev to settle several  
 questions in the theory of essential 
 dimension. 
 \end{abstract}

 \maketitle
%\tableofcontents


\section{Introduction}

Let $k$ be a field, $\Fields_k$ be the category of
field extensions $K/k$, $\Sets$ be the category of sets, 
and $F\colon\Fields_k \to \Sets$ be a covariant functor. 
Given a tower of field extensions $k \subset K \subset L$, 
we will denote the image of $a \in F(K)$ under the natural 
map $F(K) \to F(L)$ by $a_L$. Conversely, if $b \in F(L)$ 
lies in the image of this map, we will say that $b$ {\em descends} to $K$.

Given a field extension $K/k$ and $b \in F(L)$, the {\em essential
dimension} $\ed_k(b)$ of $b$ is defined as the minimal transcendence 
degree $\trdeg_k(K)$, as $K$ ranges over all intermediate subfields
$k \subset K \subset L$ such that $b$ descends to $K$. Informally speaking,
this is the minimal number of parameters one needs 
to define $b$.  The essential dimension 
$\ed_k(F)$ of the functor $F$ is the maximal value of $\ed_k(b)$,
as $L$ ranges over all field extensions of $k$
and $b$ ranges over $F(L)$. Informally speaking, this is
the minimal number of parameters required to define any object of $F$.

The {\em essential dimension} $\ed_k(b; p)$ at a prime $p$
is defined as the minimum of $\ed_k(b_{L'})$, taken over all finite field
extensions $L'/L$ such that the degree $[L':L]$ is prime to $p$.
The essential dimension of $\ed_k(F; p)$
of $F$ at a prime $p$ is the supremum of $\ed_k(b; p)$
taken over all $b \in F(L)$ and over all field extensions $L/k$.

An important example where the above notions lead to a rich theory
is the nonabelian cohomology functor
$F_G = H^1(* , G)$, sending a field $K/k$ to the set $H^1(K, G)$ of
isomorphism classes of $G$-torsors over $\Spec(K)$, in the fppf
topology.  Here $G$ is an algebraic group defined over $k$.
The essential dimension of this functor can be thought of as 
a numerical measure of complexity of $G$-torsors over fields
or, alternatively, as the minimal number of parameters required 
to define a versal $G$-torsor. In the case where $G$ is a finite 
(constant) group defined over $k$, which will be the main focus of this paper,
$\ed_k(G)$ is the minimal number of parameters required to describe 
all $G$-Galois extensions.

For details on the notion of essential dimension of a finite group we
refer the reader to~\cite{br}, \cite{reichstein} or \cite[Chapter 8]{jly},
on the notion of essential dimension of a functor to
\cite{bf} or~\cite{brv2} and on essential dimension at a prime $p$
to~\cite{merkurjev}. 

N. Karpenko and A. Merkurjev~\cite{km} recently proved the following 
formula for the essential dimension of a (finite) $p$-group.

 \begin{thm} \label{thm.km} Let $G$ be a $p$-group and
 $k$ be a field of characteristic $\ne p$ containing 
 a primitive $p$th root of unity.
 Then 
\[ \ed_k(G; p) = \ed_k(G) = \min \, \dim(V) \, , \]
 where the minimum is taken over all faithful 
 $k$-representations $G \hookrightarrow \GL(V)$. 
 \end{thm}

The purpose of this paper is to explore some of the consequences 
of this theorem.  The following notation will be used throughout.

We will fix a prime $p$ and a base field $k$ such that
\begin{equation} \label{e.base-field}
\text{$\Char(k) \ne p$ and $k$ contains $\zeta$,}
\end{equation}
where $\zeta$ is a primitive $p$th root of unity if $p\ge 3$ and
a primitive $4$th root of unity if $p = 2$.

For a finite group $H$, we will denote the intersection of 
the kernels of all multiplicative characters $\chi \colon H \to k^*$
by $H'$. 
In particular, if $k$ contains an $e$th root of unity,
where $e$ is the exponent of $H$, then $H' = [H, H]$ is the commutator
subgroup of $H$.

All $p$-groups in this paper will be assumed to be finite.
Given a $p$-group $G$, we set $C(G)$ to be the center of $G$ and
\begin{equation} \label{e.C}
C(G)_p := \{ g \in C(G) \, | \, g^p = 1 \}
\end{equation}
to be the $p$-torsion subgroup of $C(G)$. We will view $C(G)_p$ and its
subgroups as $\mathbb F_p$-vector spaces, and write ``$\dim_{\bbF_p}$"
for their dimensions.  We further set
\begin{equation}\label{e.Ki}
K_i := \bigcap_{[G:H] = p^i}  H' \quad
\text{and} \quad C_i := K_i \cap C(G)_p \, .
\end{equation}
for every $i \ge 0$,
$K_{-1} := G$ and $C_{-1}:= K_{-1} \cap C(G)_p = C(G)_p$.

Our first main result is following theorem. Part (b) may be viewed 
as a variant of Theorem~\ref{thm.km}.

\begin{thm} \label{thm1} Let $G$ be a $p$-group,
$k$ be a base field satisfying~\eqref{e.base-field}
and $\rho \colon G \hookrightarrow \GL(V)$
be a faithful linear $k$-representation of $G$.
Then

\smallskip
(a) $\rho$ has minimal dimension among the faithful linear representations 
of $G$ defined over $k$ if and only if for every $i \ge 0$ 
the irreducible decomposition 
of $\rho$ has exactly \[ \dim_{\bbF_p}\,C_{i-1} - \dim_{\bbF_p} \,C_i \]
irreducible components of dimension $p^i$, each with multiplicity $1$.

\smallskip
(b) $\ed_k(G; p) = \ed_k(G) = \sum_{i = 0}^{\infty} 
(\dim_{\bbF_p} \, C_{i-1} - \dim_{\bbF_p} \, C_i) p^i$. 
\end{thm}

Note that $K_i = C_i = \{ 1 \}$ for large $i$ (say, if $p^i \ge |G|$),
so only finitely many terms in the above infinite sum are non-zero. 
We also remark that the minimal
number of irreducible components in a faithful representations 
of a finite group (but not necessarily a $p$-group) was studied 
in \cite{ta, na}, see also~\cite[Section 4]{lo}. 

We will prove Theorem~\ref{thm1} in section~\ref{sect.thm1};
the rest of the paper will be devoted to its applications.
The main results we will obtain are summarized below.

\subsection*{Classification of $p$-groups of essential dimension $\le p$.}

\begin{thm} \label{thm.ed=p}
Let $p$ be a prime, $k$ be as in~\eqref{e.base-field}
and $G$ be a $p$-group
such that $G' \ne \{ 1 \}$. Then the following conditions are equivalent.

\smallskip
(a) $\ed_k(G) \le p$,

\smallskip
(b) $\ed_k(G) = p$,

\smallskip
(c) The center $C(G)$ is cyclic and $G$ has a subgroup 
$H$ of index $p$ such that $H' = \{ 1 \}$.
% 
% \smallskip
% (d) $G$ is isomorphic to a subgroup of 
% $\bbZ/p^\alpha \wr \bbZ/p=(\bbZ/p^\alpha)^p \rtimes\bbZ/p$, 
% where $\alpha\ge 1$ is an integer such that $k$ contains 
% a primitive $p^\alpha$th root of unity.
\end{thm}

Note that the assumption that $G' \ne \{ 1 \}$ is harmless.
Indeed, if $G' = \{ 1 \}$ then by Theorem~\ref{thm1}(b) 
$\ed_k(G) = \rank(G)$; cf. also~\cite[Theorem 6.1]{br} 
or~\cite[section 3]{bf}. 

\subsection*{Essential dimension of $p$-groups of nilpotency class $2$.}

\begin{thm}\label{thm.p-groups}   
Let $G$ be a $p$-group of exponent $e$ and
$k$ be a field of characteristic $\ne p$ containing
a primitive $e$-th root of unity. Suppose the commutator
subgroup $[G, G]$ is central in $G$. Then

\smallskip
(a) $\ed_k(G; p) = \ed_k(G) \le 
\rank C(G) + \rank [G, G](p^{\lfloor m/2\rfloor}-1)$, 
where $p^m$ is the order of $G/C(G)$.

\smallskip
(b) Moreover, if $[G, G]$ is cyclic then $|G/C(G)|$ is a complete
square and equality holds in (a). That is, in this case
\[ \ed_k(G; p) = \ed_k(G) =\sqrt{|G/C(G)|} + \rank \, C(G) - 1 \, . \]
\end{thm}                                

\subsection*{Essential dimension of a quotient group}

C. U. Jensen, A. Ledet and N. Yui asked if $\ed_k(G) \ge \ed_k(G/N)$ for every
finite group $G$ and normal subgroup $N \triangleleft G$; 
see~\cite[p. 204]{jly}. The following theorem shows that
this inequality is false in general.

\begin{thm} \label{thm.jly}  
Let $p$ be a prime and $k$ be a field of characteristic $\neq p$
containing a primitive $p$th root of unity.
For every real number $\lambda > 0$ there exists 
a $p$-group $G$ and a central subgroup $H$ of $G$ 
such that $\ed_k(G/H) > \lambda \ed_k(G)$.
\end{thm} 

\subsection*{Essential dimension of $\SL_n(\bbZ)$}

G. Favi and M. Florence~\cite{ff} showed that
$\ed_k(\GL_n(\bbZ)) = n$ for every $n \ge 1$ and $\ed_k(\SL_n(\bbZ)) = n-1$ 
for every odd $n$. For details, including the definitions 
of $\ed_k(\GL_n(\bbZ))$ and
$\ed_k(\SL_n(\bbZ))$, see Section~\ref{sect.ff}. For even $n$ Favi and 
Florence showed that $\ed_k(\SL_n(\bbZ)) = n-1$ or $n$ and left the exact
value of $\ed_k(\SL_n(\bbZ))$ as an open question. In this paper
we will answer this question as follows. 

\begin{thm} \label{thm.ff1}
Suppose $k$ is a field of characteristic $\ne 2$. Then
\[ \ed_k (\SL_n(\bbZ);2) = 
\ed_k (\SL_n(\bbZ)) = \begin{cases} \text{$n -1$, if $n$ is odd,} \\
\text{$n$, if $n$ is even} \end{cases}
\]
for any $n \ge 3$.
\end{thm}

\subsection*{Acknowledgement}
Theorems~\ref{thm.p-groups}(b) and~\ref{thm.jly} first 
appeared in the unpublished preprint~\cite{brv1} by P. Brosnan, the second
author and A. Vistoli. We thank P. Brosnan and A. Vistoli for 
allowing us to include them in this paper.  Theorem~\ref{thm.p-groups}(b) 
was, in fact, a precursor to Theorem~\ref{thm.km}; the techniques 
used in~\cite{brv1} were subsequently strengthened and refined 
by Karpenko and Merkurjev~\cite{km} to prove Theorem~\ref{thm.km}. 
The proof of Theorem~\ref{thm.p-groups}(b) in Section~\ref{sect.brv}
may thus be viewed as a result of reverse engineering. We include 
it here because it naturally fits into the framework 
of this paper, because Theorem~\ref{thm.p-groups}(b) is used 
in a crucial way in~\cite{brv2}, and because a proof of this result 
has not previously appeared in print.

We are also grateful to R. L\"otscher for pointing out
and helping us correct an inaccuracy in the proof of
Lemma~\ref{l.Ki}.

\section{Proof of Theorem~\ref{thm1}}
\label{sect.thm1}
Throughout this section we assume $k$ to be as in~\eqref{e.base-field}.
An important role in the proof will be played by the $p$-torsion subgroup
$C(G)_p$ of the center of $G$
and by the descending sequences 
\begin{eqnarray*}
K_{-1} = G \supset K_0 \supset K_1 \supset K_2 \supset \dots & \text{and} \\ 
C_{-1} = C(G)_p \supset C_0 \supset C_1 \supset C_2 \supset \dots & 
\end{eqnarray*}
of characteristic subgroups of $G$ defined in \eqref{e.Ki}.
To simplify the notation, we will write $C$ for $C_{-1} = C(G)_p$ 
for the rest of this section.
We will repeatedly use the well-known fact that
\begin{equation} \label{fact}
\text{A normal subgroup $N$ of $G$ is trivial if and only if $N \cap C$ is
trivial.}
\end{equation}

We begin with three elementary lemmas.

\begin{lem}\label{l.Ki} 
$K_i=\bigcap_{\dim(\rho) \le p^i} \, \ker(\rho)$, where the intersection is
taken over all irreducible representations $\rho$ of $G$ of dimension $\le p^i$.
\end{lem}

\begin{proof} Let $j\le i$.
Recall that every irreducible representation $\rho$ of $G$ of
dimension $p^j$ is induced from a $1$-dimensional representation
$\chi$ of a subgroup $H \subset G$ of index $p^j$; 
see~\cite[(II.4)]{lp} for $p \ge 3$ (cf. also~\cite{volvacev})
and \cite[(IV.2)]{lp} for $p = 2$. (Note that our
assumption~\eqref{e.base-field} on the base field $k$ is
crucial here. In the case where $k = \mathbb{C}$ a more direct
proof can be found in~\cite[Section 8.5]{serre-rep}).

Thus $\ker(\rho)=\ker(\ind_H^G\chi)=\bigcap_{g\in G}g\ker(\chi)g^{-1}$,
and since each $g\ker(\chi)g^{-1}$ contains $(gHg^{-1})^\prime$,
we see that $\ker(\rho)\supset K_j\supset K_i$.
The opposite inclusion is proved in a similar manner.
\end{proof}

\begin{lem}\label{l.rank}
Let $\rho \colon G \to \GL(V)$ an irreducible representation
of a $p$-group $G$. Then 

\smallskip
(a) $\rho(C)$ consists of scalar matrices.
In other words, the restriction of $\rho$ to $C$ decomposes
as $\chi \oplus \ldots \oplus \chi$ ($\dim(V)$ times),
for some multiplicative character $\chi \colon C \to \bbG_m$.
We will refer to $\chi$ as the {\em character associated to} $\rho$.

\smallskip
(b) $C_i=\bigcap_{\dim(\psi) \le p^i} \,  \ker(\chi_{\psi})$, where 
the intersection is taken over all irreducible 
$G$-representations $\psi$ of dimension $\le p^i$ and
$\chi_{\psi} \colon C \to \bbG_m$ denotes the character associated 
to $\psi$. In particular, if $\dim(\rho) \le p^i$ then $\chi_{\rho}$ 
vanishes on $C_i$.
\end{lem}

\begin{proof}
(a) follows from Schur's lemma. 
(b) By Lemma~\ref{l.Ki} 
\[ C_i = C \cap \bigcap_{\dim(\psi) \le p^i} \, \ker(\psi) = 
\bigcap_{\dim(\psi) \le p^i} \, (C \cap \ker(\psi)) = 
\bigcap_{\dim(\psi) \le p^i} \, \ker(\chi_{\psi}) \, . \]
\end{proof}

\begin{lem} \label{lem2.3} Let $G$ be a $p$-group and
$\rho = \rho_1 \oplus \ldots \oplus \rho_m$ be the direct sum of
the irreducible representations $\rho_i \colon G \to \GL(V_i)$.
Let $\chi_i := \chi_{\rho_i} \colon C \to \bbG_m$ be the character
associated to $\rho_i$.

\smallskip
(a) $\rho$ is faithful if and only if $\chi_1, \dots, \chi_m$ span $C^*$
as an $\mathbb F_p$-vector space.

\smallskip
(b) Moreover, if $\rho$ is of minimal dimension among the faithful
representations of $G$ then  $\chi_1, \dots, \chi_m$ form 
an $\mathbb F_p$-basis of $C^*$.
\end{lem}

\begin{proof} (a) By~\eqref{fact},
$\Ker(\rho)$ is trivial if and only if 
$\Ker(\rho) \cap C = \cap_{i = 1}^m \Ker(\chi_i)$ is trivial.
On the other hand, 
$\cap_{i = 1}^m \Ker(\chi_i)$ is trivial if and only if
$\chi_1, \dots, \chi_m$ span $C^*$.

\smallskip
(b) Assume the contrary, say $\chi_m$ is a linear combination of
$\chi_1, \dots, \chi_{m-1}$. Then part (a) tells us that
$\rho_1 \oplus \ldots \oplus\rho_{m-1}$ is a faithful representation of $G$, 
contradicting the minimality of $\dim(\rho)$.
\end{proof}

We are now ready to proceed with the proof of Theorem~\ref{thm1}.
Part (b) is an immediate consequence of part (a) and Theorem~\ref{thm.km}.
We will thus focus on proving part (a). 
In the sequel for each $i \ge 0$ we will set 
\[ \delta_i := \dim_{\bbF_p} \, C_{i-1} - \dim_{\bbF_p} \, C_i \]
and
\[ \Delta_i : = \delta_0 + \delta_1 + \dots + \delta_i = 
\dim_{\bbF_p}\, C - \dim_{\bbF_p}\,C_i \, , \]
where the last equality follows from $C_{-1} = C$.

Our proof will proceed in two steps. In Step 1 we will construct
a faithful representation $\mu$ of $G$ such that for every $i \ge 0$
exactly $\delta_i$ irreducible components of $\mu$ have dimension $p^i$.
In Step 2 we will show that $\dim(\rho) \ge \dim(\mu)$ 
for any other faithful representation $\rho$ of $G$, and moreover 
equality holds if and only if $\rho$ has exactly $\delta_i$ 
irreducible components of dimension $p^i$, for every $i \ge 0$. 

\smallskip
{\bf Step 1:}
We begin by constructing $\mu$.  By definition, 
\[ C = C_{-1} \supset C_{0} \supset C_1 \supset \dots  \, ,  \] 
where the inclusions are not necessarily strict.
Dualizing this flag of $\mathbb F_p$-vector spaces, we 
obtain a flag 
\[ (0) = (C^*)_{-1} \subset (C^*)_0 \subset (C^*)_1 \subset \dots
\] 
of $\mathbb F_p$-subspaces of $C^*$, where
\[ \text{$(C^*)_i := \{ \chi \in C^* \, | \, \chi$ is trivial 
on $C_i \} \simeq (C/C_i)^*$.} \]
Let $\Ass(C) \subset C^*$ be the set of characters of $C$ associated 
to irreducible representations of $G$, and let $\Ass_i(C)$ be the set
of characters associated to irreducible representations of 
dimension $p^i$. Lemma~\ref{l.rank}(b) tells us that
\[ \text{$\Ass_0(C) \cup \Ass_1(C) \cup \dots \cup \Ass_i(C)$ spans $(C^*)_i$} \]
for every $i \ge 0$.  Hence, we can choose a basis 
$\chi_1, \dots, \chi_{\Delta_0}$ of $(C^*)_0$ from $\Ass_0(C)$, then
complete it to a basis 
$\chi_1, \dots, \chi_{\Delta_1}$ of $(C^*)_1$ by choosing
the last $\Delta_1 - \Delta_0$ characters from $\Ass_1(C)$, 
then complete this basis of $(C^*)_1$ to a basis of $(C^*)_2$ by choosing
$\Delta_2 - \Delta_1$ additional characters from $\Ass_2(C)$, 
etc. 
We stop when $C_i = (0)$, i.e., $\Delta_i = \dim_{\bbF_p}\,C$.  

By the definition of $\Ass_i(C)$, each $\chi_j$ is the associated character 
of some irreducible representation $\mu_j$ of $G$. By our construction
\[ \mu = \mu_1 \oplus \dots \oplus \mu_{\dim_{\bbF_p}(C)} \]
has the desired properties. 
Indeed, since 
$\chi_1, \dots, \chi_{\dim_{\bbF_p}(C)}$ form a basis of $C^*$, Lemma~\ref{lem2.3}
tells us that $\mu$ is faithful. 
On the other hand, by 
our construction exactly
\[ \delta_i - \delta_{i-1} = \dim_{\bbF_p}\,C_i^* - \dim_{\bbF_p}\,C_{i-1}^* =
\dim_{\bbF_p}\,C_{i-1} - \dim_{\bbF_p}\,C_i \] 
of the characters $\chi_1, \dots, \chi_c$ come from $\Ass_i(C)$. 
Equivalently, exactly
\[ \dim_{\bbF_p}\,C_{i-1} - \dim_{\bbF_p}\,C \]
of the irreducible representations
$\mu_1, \dots, \mu_c$ are of dimension $p^i$. 

\medskip
{\bf Step 2:} Let $\rho \colon G \to \GL(V)$ be a faithful linear 
representation of $G$ of the smallest possible dimension, 
\[ \rho = \rho_1 \oplus \ldots \oplus \rho_c \] 
be its irreducible decomposition, and $\chi_i \colon C \to \bbG_m$ 
be the character associated to $\rho_i$. By Lemma~\ref{lem2.3}(b), 
$\chi_1, \dots, \chi_c$ form a basis of $C^*$. 
In particular, $c = \dim_{\bbF_p}\,C$ 
and at most $\dim_{\bbF_p}\,C - \dim_{\bbF_p}\,C_i$ of the characters 
$\chi_1, \dots, \chi_c$
can vanish on $C_i$. On the other hand, by
Lemma~\ref{l.rank}(b) every representation of dimension
$\le p^i$ vanishes on $C_i$.  Thus if 
exactly $d_i$ of the irreducible representations 
$\rho_1, \dots, \rho_c$ have dimension $p^i$ then
\[ d_0 + d_1 + d_2 + \ldots + d_i \le \dim_{\bbF_p}\,C - \dim_{\bbF_p}\,C_i 
\]
for every $i \ge 0$. For $i \ge 0$, set
$D_i := d_0 + \dots + d_i$ = number of representations of dimension $\le p^i$
among $\rho_1, \dots, \rho_c$. 
We can now write the above inequality as
\begin{equation} \label{e.inequality}
\text{$D_i \le \Delta_i$ for every $i \ge 0$.} 
\end{equation}
Our goal is to show that
$\dim(\rho) \ge \dim(\mu)$ and that equality holds if and only if
exactly $\delta_i$ of the irreducible representations 
$\rho_1, \dots, \rho_{\dim_{\bbF_p}(C)}$ have dimension $p^i$.
The last condition translates into $d_i = \delta_i$ 
for every $i \ge 0$, which is, in turn equivalent to
$D_i = \Delta_i$ for every $i \ge 0$. 

Indeed, setting $D_{- 1} := 0$ and $\Delta_{-1} := 0$, we have,
\begin{eqnarray*}
\dim(\rho)  - \dim(\mu) = \sum_{i = 0}^{\infty} (d_i - \delta_i) p^i =  
\sum_{i = 0}^{\infty} (D_i - \Delta_i)p^i - 
\sum_{i = 0}^{\infty} (D_{i-1} - \Delta_{i-1})p^i \\ 
= \sum_{i = 0}^{\infty} (D_i - \Delta_i)(p^i - p^{i + 1}) \ge 0 \, ,    
\end{eqnarray*}
where the last inequality follows from~\eqref{e.inequality}. Moreover,
equality holds if and only if $D_i = \Delta_i$ for every $i \ge 0$, as claimed.
This completes the proof of Step 2 and thus of Theorem~\ref{thm1}.
\qed

\section{Proof of Theorem~\ref{thm.ed=p}}

Since $K_0 = G'$ is a non-trivial normal subgroup of $G$, we see
that $K_0 \cap C(G)$ and thus $C_0 = K_0 \cap C(G)_p$ is non-trivial. 
This means that in the summation formula of Theorem~\ref{thm1}(b) at
least one of the terms 
\[ (\dim_{\bbF_p}\,C_{i-1} - \dim_{\bbF_p}\,C_i) p^i \]
with $i \ge 1$ will be non-zero. Hence,
$\ed_k(G) \ge p$; this shows that (a) and (b) are equivalent.
Moreover, equality holds 
if and only if (i) $\dim_{\bbF_p}\,C_{-1} = 1$, (ii) $\dim_{\bbF_p}\,C_0 = 1$ 
and (iii) $C_1$ is trivial. 
Since we are assuming $K_0 = G' \ne \{ 1 \}$ and hence,
$C_0 = K_0 \cap C(G)_p \ne \{ 1 \}$ by~\eqref{fact},
(ii) follows from (i) and thus can be dropped.

It now suffices to prove that (i) and (iii) are equivalent to condition (c)
of the theorem.
Since $C_{-1} = C(G)_p$, (i) is equivalent to $C(G)$ being cyclic. 
On the other hand, (iii) means that
\begin{equation} \label{e.K_1}
K_1 = \bigcap_{[G:H] = p} \, H'
\end{equation}
intersects $C(G)_p$ trivially.  Since $K_1$ is a normal subgroup 
of $G$, \eqref{fact} tells us that (iii) holds if and only 
if $K_1 = \{ 1 \}$. 

It remains to show that $K_1 = \{ 1 \}$ if and only
if $H' = \{ 1 \}$ for some subgroup $H$ of $G$ of index $p$.
One direction is obvious: if $H' = \{ 1 \}$ for some $H$ of index $p$
then the intersection~\eqref{e.K_1} is trivial.
To prove the converse, assume the contrary: the intersection~\eqref{e.K_1} 
is trivial but $H' \ne \{ 1 \}$ for every subgroup $H$ of index $p$.
Since every such $H$ is normal in $G$ (and so is $H'$),~\eqref{fact}
tells us that that $H' \ne \{ 1 \}$ if and only if 
$H' \cap C(G) \ne \{ 1 \}$. Since $C(G)$ is cyclic, the latter 
condition is equivalent to $C(G)_p \subset H'$. Thus
\[ C(G)_p \subset K_1 = \bigcap_{[G:H] = p} \,  H' \, , \]
contradicting our assumption that $K_1 = \{ 1 \}$.

To sum up, we have shown that (c) is equivalent 
to conditions (i) and (iii) above, and that these conditions are 
in turn, equivalent to (a) (or to (b)). This completes 
the proof of Theorem~\ref{thm.ed=p}.

\begin{remark} \label{rem.ed=p}
$p$-groups that have a faithful representation 
of degree $p$ over a field $k$, satisfying~\eqref{e.base-field}
are described in \cite[II.4, III.4, IV.2]{lp}; see also~\cite{volvacev}.  
% They are exactly the subgroups of $\bbZ/p^\alpha\wr\bbZ/p$, where
% $k$ contains a primitive root of unity of degree $p^{\alpha}$.
% Thus yields the following variant of Theorem~\ref{thm.ed<=p}:
Combining this description with Theorem~\ref{thm.km} 
yields the following variant of Theorem~\ref{thm.ed=p}.

{\em Let $k$ be a field satisfying~\eqref{e.base-field} and
$G$ be a $p$-group such that $G' \ne \{ 1 \}$.
Then the following conditions are equivalent:

\smallskip
(a) $\ed_k(G) \le p$, 

\smallskip
(b) $\ed_k(G) = p$,

\smallskip
(c) $G$ is isomorphic to a subgroup of 
$\bbZ/p^\alpha \wr \bbZ/p=(\bbZ/p^\alpha)^p \rtimes\bbZ/p$, 
for some $\alpha\ge 1$ such that $k$ contains 
a primitive root of unity of degree $p^\alpha$.}
\qed
\end{remark}

\section{Proof of Theorems~\ref{thm.p-groups} and ~\ref{thm.jly}}
\label{sect.brv}
\begin{proof}[Proof of Theorem~\ref{thm.p-groups}]
Since the commutator $K_0=[G,G]$ is central, $C_0=K_0\cap C(G)_p$ 
is of dimension $\rank [G,G]$ and the $p^0$ term in 
the formula of Theorem~\ref{thm1} is $(\rank C(G)-\rank [G,G])$.

Let $Q=G/C(G)$ which is abelian by assumption. 
Let $h_1,...,h_s$ be generators of $[G,G]$, where 
$s=\rank [G,G]$, so that 
\[ [G,G]=\bbZ/p^{e_1}h_1\oplus
\dots \oplus \bbZ/p^{e_1}h_1 \, , \]
written additively.  For $g_1,g_2\in G$ the commutator can then 
be expressed as \[ [g_1,g_2]=\beta_1(g_1, g_2) h_1+ 
\ldots +\beta_s(g_1, g_2) h_s \, . \]
Note that each $\beta_i(g_1, g_2)$ depends on $g_1,g_2$ only
modulo the center $C(G)$. Thus each $\beta_i$ descends to a
skew-symmetric bilinear form
\[
Q\times Q\rightarrow \bbZ/p^{e_i} 
\]
which, by a slight abuse of notation, we will continue to denote 
by $\beta_i$. 
Let $p^m$ be the order of $Q$. For each form $\beta_i$
there is an isotropic subgroup $Q_i$ of $Q$ of order 
at least $p^{\lfloor (m+1)/2 \rfloor}$ (or equivalently, of
index at most $p^{\lfloor m/2 \rfloor}$ in $Q$); see \cite[Corollary 3]{at}.
Pulling these isotropic subgroups back to $G$, we obtain 
subgroups $G_1, \dots, G_s$ of $G$ of index $\le p^{\lfloor m/2 \rfloor}$ 
with the property that $G_i' = [G_i, G_i]$ lies in the subgroup of 
$C(G)$ generated by $h_1, \dots, h_{i-1}, h_{i+1}, \dots, h_s$.
In particular, $G_1' \cap \dots \cap G_s' = \{ 1 \}$.
Thus, all $K_i$ (and hence, all $C_i$) in \eqref{e.Ki} are trivial 
for $i\ge \lfloor m/2 \rfloor$, and Theorem~\ref{thm1} tells us that
\begin{eqnarray*} \ed_k(G) =  \dim_{\bbF_p}\, C_{-1} - \dim_{\bbF_p}\, C_0 +
\sum_{j = 1}^{\lfloor m/2 \rfloor} 
(\dim_{\bbF_p} \, C_{j-1} - \dim_{\bbF_p} \, C_j)  p^j
\le  \\
 \dim_{\bbF_p}  \, C_{-1} - \dim_{\bbF_p} \, C_0 +
\sum_{j = 1}^{\lfloor m/2 \rfloor} 
(\dim_{\bbF_p} \, C_{j-1} - \dim_{\bbF_p} \, C_j)  \cdot p^{\lfloor m/2 \rfloor} = \\ 
\rank \, C(G)  +  \rank \, [G, G] (p^{\lfloor m/2 \rfloor} - 1) \, . 
\end{eqnarray*}

(b) In general, the skew-symmetric bilinear forms $\beta_i$ 
may be degenerate. However, 
if $[G,G]$ is cyclic, i.e., $s = 1$, then we have only one form, 
$\beta_1$, which is easily seen to be non-degenerate.  
For notational simplicity, we will write $\beta$ instead of $\beta_1$. 
To see that $\beta$ is non-degenerate,
suppose $\overline{g} := g$ (modulo $C(G)$) 
lies in the kernel of $\beta$ for some $g \in G$. Then by definition 
\[ \beta(g, g_1) = g g_1 g^{-1} g_1^{-1} = 1 \]
for every $g_1 \in G$. Hence, $g$ is central in $G$,
i.e., $\overline{g} = 1$ in $Q = G/C(G)$, as claimed.
     
We conclude that the order of $Q=G/C(G)$ is a perfect square, say $p^{2i}$, 
and $Q$ contains a maximal isotropic subgroup $I \subset Q$ of order 
$p^i = \sqrt{|G/C(G)|}$; see \cite[Corollary 4]{at}. 
The preimage of $I$ in $G$ is a maximal abelian subgroup of 
index $p^i$.  Consequently, $K_0= [G,G], K_1 , \ldots, 
K_{i-1}$ are all of rank $1$ and $K_i$ is trivial, 
where $p^i=\sqrt{|G/C(G)|}$.
Moreover, since all of these groups lie 
in $[G, G]$ and hence, are central, we have $C_i = (K_i)_p$ and thus
\[ \text{$\dim_{\mathbb F_p}(C_0) = \dim_{\mathbb F_p}(C_1) = 
\ldots = \dim_{\mathbb F_p}(C_{i-1}) = 1$ and
$\dim_{\mathbb F_p}(C_i) = 0$} \, . \] 
Specializing the formula of Theorem~\ref{thm.p-groups} to this
situation, we obtain part (b).
\end{proof}

% \begin{example} \label{ex.extraspecial}
% Recall that a $p$-group $G$ is called \emph{extra-special} if its
% center $C$ is cyclic of order $p$, and the quotient $G/C$ is elementary
% abelian. The order of an extra special $p$-group $G$ is an odd power
% of $p$; the exponent of $G$ is either $p$ or $p^{2}$;
% cf.~\cite[III. 13]{huppert}.  Note that every non-abelian
% group of order $p^{3}$ is extra-special. For extra-special
% $p$-groups Theorem~\ref{thm.p-groups}(b) reduces to the following.
% 
% {\em Let $G$ be an extra-special $p$-group of order $p^{2m+1}$.
% Assume that the characteristic of $k$ is different from $p$,
% that $\zeta_{p} \in k$, and $\zeta_{p^{2}} \in k$ if the exponent
% of $G$ is $p^{2}$. Then $\ed_k(G) = p^{m}$.}   
% \end{example}

\begin{proof}[Proof of Theorem~\ref{thm.jly}]
Let $\Gamma$ be the non-abelian group of order $p^3$ given by
generators $x, y, z$ and relations $x^p = y^p = z^p = [x,z]=[y,z]=1$, $[x, y] = z$.
Choose a multiplicative character $\chi \colon H \to k^*$
of the subgroup $A = \langle x, z \rangle \simeq (\mathbb Z/p \mathbb Z)^2$
which is non-trivial on the center $\langle z \rangle$ of $\Gamma$
and consider the $p$-dimensional induced representation $\Ind_A^{\Gamma}(\chi)$.
Since the center $\langle z \rangle$ of $\Gamma$ does not lie in
the kernel of $\Ind_A^{\Gamma}(\chi)$, we conclude 
that $\Ind_A^{\Gamma}(\chi)$ is faithful. Thus we have constructed
a faithful $p$-dimensional representation of $\Gamma$ defined over $k$. 
Consequently
\begin{equation} \label{e.upper1}
\ed_k(\Gamma) \le p \, .
\end{equation}
Taking the direct sum of $n$ copies of this representation, we
obtain a faithful representation of $\Gamma^n$ of dimension $np$.
Thus for any $n \ge 1$ we have
   \begin{equation} \label{e.upper2}
   \ed_k\Gamma^n \le np \, .
   \end{equation}
(We remark that both \eqref{e.upper1} and~\eqref{e.upper2} are in fact
equalities.  Indeed, if $\zeta_{p^2}$ is a primitive root of unity 
of degree $p^2$ then
\[ \ed_k(\Gamma) \ge \ed_{k(\zeta_{p^2})} (\Gamma) = 
\sqrt{p^2} + 1 - 1 = p \, , \]
where the middle equality follows from Theorem~\ref{thm.p-groups}(b). 
Hence, we have $\ed_k(\Gamma) = p$. Moreover, 
by~\cite[Theorem 5.1]{km}, $\ed_k\Gamma^n = n \cdot \ed_k(\Gamma) = np$. 
However, we will only need the upper bound~\eqref{e.upper2} in the sequel.)

The center of $\Gamma$ is $\langle z \rangle$; denote it by $C$. 
The center of $\Gamma^n$ is then isomorphic to $C^n$. Let $H_n$ be the subgroup 
of $C^n$ consisting of $n$-tuples $(c_1, \dots, c_n)$ such that
$c_1 \cdots c_n = 1$.  The center $C(\Gamma^n/H_n)$ of 
$\Gamma^n/H_{n}$ is clearly cyclic of order $p$
(it is generated by the class
of the element $(z, 1, \dots, 1)$ modulo $H_n$), and
the commutator 
$[\Gamma^n/H_{n}, \Gamma^n/H_n]$ is central. Hence,
\begin{equation} \label{e.lower}
\ed_k(\Gamma^n/H_{n}) \ge
\ed_{k(\zeta^2)} (\Gamma^n/H_{n}) = 
\sqrt{p^{2n}} + 1 - 1 = p^n \, \, ,
\end{equation}
where the middle equality follows from Theorem~\ref{thm.p-groups}(b). 
Setting $G = \Gamma^n$ and $H = H_n$, and comparing~\eqref{e.upper2} 
with~\eqref{e.lower},
we see that the desired inequality $\ed_k(G/H) > \lambda \ed_k(G)$ 
holds for suitably large $n$.
\end{proof}

\section{Proof of Theorem~\ref{thm.ff1}}  
\label{sect.ff}

Recall that the essential dimension of the group $\GL_n(\bbZ)$ 
over a field $k$, or $\ed_k(\GL_n(\bbZ))$ for short, is defined 
as the essential dimension of this functor
\[ H^1( \ast , \GL_n(\bbZ))\colon
 K \to \text{\{$K$-isomorphism classes of $n$-dimensional $K$-tori\}} \, , \]
where $K/k$ is a field extension.  Similarly
$\ed_k(\SL_n(\bbZ))$ is defined as the essential dimension 
of the functor
\[ \begin{array}{r} H^1( \ast , \SL_n(\bbZ)) \colon
K \to \text{\{$K$-isomorphism classes of $n$-dimensional $K$-tori} \\
\text{with $\phi_T \subset \SL_n(\bbZ)$ \}} \, , \end{array} \]
where $\phi_T \colon \Gal(K) \to \GL_n(\bbZ)$ is the natural representation
of the Galois group of $K$ on the character lattice of $T$.
The essential dimensions $\ed_k(\GL_n(\bbZ); p)$ and
$\ed_k(\SL_n(\bbZ); p)$ are respectively the essential dimensions 
of the above functors at a prime $p$.

G. Favi and M. Florence~\cite{ff} showed that for $\Gamma = \GL_n(\bbZ)$ or
$\SL_n(\bbZ)$, 
\begin{equation}\label{e.edL}
\ed_k(\Gamma)=\max \{\ed_k(F)| F\mbox{ finite subgroup of }\Gamma\}.
\end{equation} 
 From this they deduced that
\[ \ed_k(\GL_n(\bbZ)) = n, \quad \text{and} \quad
\ed_k(\SL_n(\bbZ)) = \begin{cases} \text{$n -1$, if $n$ is odd,} \\
\text{$n -1$ or $n$, if $n$ is even.} \end{cases} \]
For details, see \cite[Theorem 5.4]{ff}.

Favi and Florence also proved that $\ed_k(\SL_2(\bbZ)) = 1$
if $k$ contains a primitive 12th root of unity and asked whether
$\ed_k(\SL_n(\bbZ)) = n-1$ or $n$ in the case where $n \ge 4$ is
even; see~\cite[Remark 5.5]{ff}.  In this section we will prove
Theorem~\ref{thm.ff1} which shows that the answer is always $n$.

A minor modification of the arguments in \cite{ff} shows 
that~\eqref{e.edL} holds also for essential dimension at a prime $p$:
\begin{equation}\label{e.edLp}
\ed_k(\Gamma;p)=\max \{\ed_k(F;p)| F\mbox{ a finite subgroup of }\Gamma\},
\end{equation} 
where $\Gamma = \GL_n(\bbZ)$ or $\SL_n(\bbZ)$.
The finite groups $F$ that Florence and Favi used to find 
the essential dimension of $\GL_n(\bbZ)$ and $\SL_n(\bbZ)$ ($n$ odd) 
are $(\bbZ/2 \bbZ)^n$ and  $(\bbZ/2 \bbZ)^{n-1}$ respectively.
Thus $\ed_k(\GL_n(\bbZ); 2) = \ed_k(\GL_n(\bbZ)) = n$ for every $n \ge 1$ 
and $\ed_k(\SL_n(\bbZ); 2) = \ed_k(\SL_n(\bbZ)) = n-1$ if $n$ is odd. 

Our proof of Theorem~\ref{thm.ff1} will rely on part (b) of
the following easy corollary of Theorem~\ref{thm1}.

\begin{cor} \label{cor.km3} 
Let $G$ be a $p$-group, and $k$ be as in~\eqref{e.base-field}.

\smallskip
(a) If $C(G)_p \subset K_i$ then $\ed_k(G)$ is divisible by $p^{i+1}$.

\smallskip
(b) If $C(G)_p \subset G'$ then $\ed_k(G)$ is divisible by $p$.  

\smallskip
(c) If $C(G)_p \subset G^{(i)}$, where $G^{(i)}$ denotes the $i$th 
derived subgroup of $G$, then $\ed_k(G)$ is divisible by $p^{i}$.
\end{cor}

\begin{proof} 
(a) $C(G)_p \subset K_i$ implies $C_{-1} = C_{0} = \dots = C_{i}$.
Hence, in the formula of Theorem~\ref{thm1}(b)
the $p^0, p^1, \dots, p^{i}$ terms appear with 
coefficient $0$. All other terms are divisible by $p^{i+1}$,
and part (a) follows.

(b) is an immediate consequence of (a), since $K_0 = G'$.

(c) By ~\cite[Theorem V.18.6]{huppert} $G^{(i)}$ is contained
in the kernel of every $p^{i-1}$-dimensional representation
of $G$. Lemma~\ref{l.Ki} now tells us that $G^{(i)} \subset K_{i-1}$
and part (c) follows from part (a).
\end{proof}

\begin{proof}[Proof of Theorem~\ref{thm.ff1}]
We assume that $n = 2d \ge 4$ is even. 
To prove Theorem~\ref{thm.ff1} it suffices 
to find a 2-subgroup $F$ of $\SL_n(\bbZ)$ of essential 
dimension $n$.

Diagonal matrices and permutation matrices generate a subgroup
of $\GL_n(\bbZ)$ isomorphic to $\mu_2^n \rtimes \Symn$. 
The determinant function restricts to a homomorphism  
\[ \det \colon \mu_2^n \rtimes \Symn \to \mu_2 \]
sending $((\epsilon_1, \dots, \epsilon_n), \tau)) \in \mu_2^n \rtimes \Symn$
to the product 
$\epsilon_1 \epsilon_2 \cdots \epsilon_n \cdot \sign(\tau)$. Let $P_n$ be 
a Sylow $2$-subgroup of $\Symn$ and $F_n$ be the kernel of
$\det \colon \mu_2^n \rtimes P_n \to \mu_2$.
By construction $F_n$ is a finite $2$-group contained in $\SL_n(\bbZ)$.
Theorem~\ref{thm.ff1} is now a consequence of the following proposition.

\begin{prop} \label{prop.ff2} If $\Char(k) \ne 2$ then 
$\ed_k(F_{2d}) = 2d$ for any $d \ge 2$.
\end{prop}

To prove the proposition,
let \[ D_{2d} = \{ \diag(\epsilon_1, \dots, \epsilon_{2d}) \, | \, 
\text{each $\epsilon_i = \pm 1$ and
$\epsilon_1 \epsilon_2 \cdots \epsilon_{2d} = 1$} \} \]
be the subgroup of ``diagonal" matrices contained in $F_{2d}$.

Since $D_{2d} \simeq \mu_2^{2d-1}$ has essential dimension $2d-1$, 
we see that $\ed_k(F_{2d}) \ge \ed_k(D_{2d}) = 2d - 1$.  On the other 
hand the inclusion $F_{2d} \subset \SL_{2d}(\bbZ)$ gives rise
to a $2d$-dimensional representation of $F_{2d}$, which remains faithful
over any field $k$ of characteristic $\neq 2$. 
Hence, $\ed_k(F_{2d}) \le 2d$. We thus conclude that 
\begin{equation} \label{e.ff2}
\text{$\ed_k(F_{2d}) = 2d - 1$ or $2d$.}
\end{equation} 
Using elementary group theory, one easily 
checks that \begin{equation} \label{e.check}
C(F_{2d}) \subset [F_{2d}, F_{2d}] \subset F_{2d}' \, . 
\end{equation}
Thus, if $k^\prime\supset k$ is a field as in~\eqref{e.base-field}, $\ed_{k^\prime}(F_{2d})$ is even by Corollary~\ref{cor.km3}; since $\ed_k(F_{2d})\ge \ed_{k^\prime}(F_{2d})$,
\eqref{e.ff2} now tells us that $\ed_k(F_{2d}) = 2d$.
This completes the proof of Proposition~\ref{prop.ff2} and thus 
of Theorem~\ref{thm.ff1}.
\end{proof}

\begin{remark} \label{rem.d=1}
The assumption that $d \ge 2$ is essential in the proof 
of the inclusion~\eqref{e.check}. In fact, $F_2 \simeq \bbZ/4 \bbZ$,
so~\eqref{e.check} fails for $d = 1$.
\end{remark}

\begin{remark}  
Note that for any integers $m, n \ge 2$, $F_{m + n}$ contains 
the direct product $F_m \times F_n$.  Thus 
\[ \ed_k(F_{m + n}) \ge  \ed_k(F_m \times F_n) = \ed_k(F_m) + \ed_k(F_n) \, ,  \]
where the last equality follows from~\cite[Theorem 5.1]{km}. 
Thus Proposition~\ref{prop.ff2} only needs to be proved for $d = 2$ 
and $3$ (or equivalently, $n = 4$ and $6$); 
all other cases are easily deduced from these by applying
the above inequality recursively, with $m = 4$.  In particular,
the group-theoretic inclusion~\eqref{e.check} only needs to be 
checked for $d = 2$ and $3$. Somewhat to our surprise, this 
reduction does not appear to simplify the proof of 
Proposition~\ref{prop.ff2} presented above to any 
significant degree.
\end{remark}

\begin{remark}
It is interesting to note that while 
the value of $\ed_k(\SL_2(\bbZ))$ depends on the 
base field $k$ (see~\cite[Remark 5.5]{ff}),
for $n \ge 3$, the value of $\ed_k(\SL_n(\bbZ))$ does not
(as long as $\Char(k) \ne 2$).
\end{remark}

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\end{document}
