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\begin{document}

\title{Essential dimension and algebraic stacks I}

\author[Brosnan]{Patrick Brosnan$^\dagger$}

\author[Reichstein]{Zinovy Reichstein$^\dagger$}

\author[Vistoli]{Angelo Vistoli$^\ddagger$}


\address[Brosnan, Reichstein]{Department of Mathematics\\
The University of British Columbia\\
1984 Mathematics Road\\
Vancouver, B.C., Canada V6T 1Z2}

\address[Vistoli]{Scuola Normale Superiore\\Piazza dei Cavalieri 7\\
56126 Pisa\\Italy}

\email[Brosnan]{brosnan@math.ubc.ca}
\email[Reichstein]{reichst@math.ubc.ca}
\email[Vistoli]{angelo.vistoli@sns.it}

%\date{January 2, 2007}


\begin{abstract} 
  Essential dimension is a numerical invariant of an algebraic group
  $G$ which reflects the complexity of $G$-torsors over a field $K$.  
  In the past 10 years it has been
  studied by many authors in a variety of contexts.  In this
  paper we extend this notion to algebraic stacks.  
  As an application of the resulting theory we obtain new results 
  about the essential dimension of certain algebraic (and, in particular,
  finite) groups which occur as central extensions.
  In particular, we show that the essential dimension of 
  the spinor group $\Spin_n$ grows exponentially with $n$, 
  improving previous lower bounds of Chernousov-Serre 
  and Reichstein-Youssin.  In the last section 
  we apply the result on spinor groups to show that 
  quadratic forms with trivial discriminant and Hasse-Witt 
  invariant are more complex, in high dimensions, than 
  previously expected.
\end{abstract}
\subjclass[2000]{Primary 14A20, 20G15, 11E04}
\thanks{$^\dagger$Supported in part by an NSERC discovery grants and by
PIMS Collaborative Research Group in Algebraic geometry, cohomology 
and representation theory}
\thanks{$^\ddagger$Supported in part by the PRIN Project ``Geometria
sulle variet\`a algebriche'', financed by MIUR}

\maketitle

\tableofcontents

\section{Introduction}
\label{s.intro}

Let $k$ be a field. We will write $\Fields_k$ for the category of
field extensions $K/k$.  Let $F\colon\Fields_k \arr\Sets$ be 
a covariant functor.

\begin{definition} % [A. Merkurjev]
\label{def.merkurjev} Let $a\in F(L)$, where $L$ is an object 
  of $\Fields_k$. We say that $a$ \emph{descends} to 
  an intermediate field $k \subseteq K \subseteq L$ if
  $a$ is in the image of the induced map $F(K) \arr F(L)$.

  The \emph{essential dimension} $\ed(a)$ of $a \in F(L)$ 
  is the minimum of the transcendence degrees $\trdeg_{k}K$ taken over
  all fields $k \subseteq K \subseteq L$ such that $a$ descends to $K$.

  The essential dimension $\ed(a; p)$ of $a$ at a prime integer $p$
  is the minimum of $\ed(a_{L'})$, taken over all finite field 
  extensions $L'/L$ such that the degree $[L':L]$ is prime to $p$.

The essential dimension $\ed F$ of the functor $F$ (respectively, 
the essential dimension $\ed(F; p)$ of $F$ at a prime $p$) is the
supremum of $\ed(a)$ (respectively, of $\ed(a; p)$)
taken over all $a\in F(L)$ with $L$ in $\Fields_k$.  

These notions are relative to the base field $k$. To emphasize this, 
we will sometimes
write $\ed_k(a)$, $\ed_k(a; p)$, $\ed_k F$ or $\ed_k(F; p)$
instead of 
$\ed(a)$, $\ed(a; p)$, $\ed F$ or $\ed(F; p)$, respectively.
\end{definition}

 If the functor $F$ is limit-preserving, a condition that is
 satisfied in all cases that interest us, every element $a \in F(L)$
 has a field of definition $K$ that is finitely generated over $k$,
 so $\ed(a)$ is finite; cf. Remark~\ref{rem.finite}.
 On the other hand, $\ed F$ may be infinite even in cases of interest.
% cf.,~\cite[Theorem 1.8]{brv}.

The following example describes a class of functors which will play 
a key role in the sequel.

\begin{example} \label{ex.edG}
Let $G$ be an algebraic group. Consider the Galois cohomology functor
$F = \H^1(* , G)$ sending $K$ to the set $\H^1(K, G)$ of
isomorphism classes of $G$-torsors over $\Spec(K)$, in the fppf 
topology.  The essential
dimension of this functor is a numerical invariant of $G$, which,
roughly speaking, measures the complexity of $G$-torsors over fields.
This number is usually denoted by $\ed_k G$ or (if 
$k$ is fixed throughout) simply by $\ed G$. Essential dimension 
was originally introduced and has since been extensively studied 
in this context; see~\cite{bur, reichstein, ry, kordonskii0, 
ledet, jly, bf1, lemire, cs, garibaldi}. The more general 
Definition~\ref{def.merkurjev} is due to A.~Merkurjev; see
\cite[Proposition 1.17]{bf1}.
\end{example} 
            
In this paper we define the notion of essential dimension 
for algebraic stacks and apply it in the ``classical" setting
of Example~\ref{ex.edG}.  The stacks that
play a special role in these applications are gerbes 
banded by the multiplicative group $\mu_{n}$; see
\S\S\ref{s.gerbes-and-cd}-\ref{s.gerbes}. Before proceeding
to state out main results, we remark that the notion 
of essential dimension introduced in this paper also leads 
to interesting results for other types of stacks. In particular,
in the forthcoming paper~\cite{brv} we will compute the essential 
dimension of the moduli stack $\cM_{g,n}$ of smooth curves of genus 
$g$ for every $g \ge 0$. We also note that a related (but not 
equivalent) notion of \emph{arithmetic dimension}
has been studied by C.~O'Neil~\cite{ONeil1, ONeil2}.

We will now describe the main results of this paper.
Let \begin{equation}
\label{e.extensions}
1\arr Z \arr G \arr Q\arr 1
\end{equation}
denote an exact sequence of algebraic groups over a field $k$ with $Z$ central
and isomorphic to $\mu_n$ for some integer $n>1$.
For every field extension $K/k$ the 
sequence~\eqref{e.extensions} induces a connecting map
$\partial_K\colon \H^1(K,Q) \arr \H^2(K,Z)$.
We define $\ind(G,Z)$ as the maximal value of 
$\ind\bigl(\partial_K(t)\bigr)$, as
$K$ ranges over all field extensions of $k$ and $t$ ranges over all
torsors in $\H^1(K,Q)$. (Note that $\ind(G,Z)$ does not depend 
on the choice of the isomorphism $Z \simeq \mu_{n}$.)
In~\S\ref{s.extensions} we will prove the following inequality.

\begin{theorem}
\label{thm2}
Let $G$ be an extension as in~\eqref{e.extensions}.  Assume that $n$
a power of a prime integer $p$. Then $\ed(G; p) \geq \ind(G,Z) - \dim G$.
\end{theorem}

In \S\ref{s.cyclic} we will use Theorem~\ref{thm2} to
give an alternative proof of a recent theorem of 
Florence about the essential dimension of a cyclic 
group (in a slightly strengthened form).
We will then use Florence's theorem to settle a particular 
case of a conjecture of Ledet~\cite[Section 3]{ledet},
relating the essential dimensions of the cyclic group $C_n$ and
the dihedral group $D_{n}$ ($n$ odd); see Corollary~\ref{cor.ledet}.

In \S\S\ref{s.ed-pgroups}--\ref{s.ed-pgroups2} we continue to study
essential dimensions of finite $p$-groups.
%  on the assumption
% that the base field $k$ contains a primitive root of unity of
% degree $p^n$, for large $n$. 
Let $G$ be a finite abstract group.  We write $\ed_k G$ (respectively,
$\ed_k(G; p)$) for the essential dimension (respectively, for the essential 
dimension at $p$) of the constant group $G_k$ over the field
$k$.  Let $\exp(G)$ denote the exponent of $G$ and let $\rC(G)$ denote
the center of $G$. 

\begin{theorem}\label{t.p-groups}
Let $G$ be a $p$-group whose commutator $[G, G]$ is central and cyclic.
Then
   \[
   \ed_k(G; p) = \ed_k G = \sqrt{|G/\rC(G)|} + \rank \, \rC(G) - 1 \, . 
   \]
for any base field $k$ of characteristic $\ne p$ which contains 
a primitive root of unity of degree $\exp(G)$.
\end{theorem}

Note that with the above hypotheses, $|G/\rC(G)|$ is a complete
square.  In the case where 
$G$ is abelian we recover the identity $\ed G = \rank(G)$;
see~\cite[Theorem6.1]{bur}. For most finite groups $G$ the best 
previously known lower bounds on $\ed G$ were of the form
\begin{equation} \label{e0.old-bound}
\ed G \ge \rank(A) \, ,
\end{equation}
where $A$ was taken to be an abelian subgroup $A$ of $G$ 
of maximal rank.  Theorem~\ref{t.p-groups} 
represents a substantial improvement over these bounds. 
For example, if $G$ is a non-abelian group of 
order $p^3$ and $k$ contains a primitive root of unity of 
degree $p^2$ then Theorem~\ref{t.p-groups} tells us that $\ed G= p$,
while~\eqref{e0.old-bound} yields only $\ed G \ge 2$. 

In \S\ref{s.ed-pgroups2} we record several interesting corollaries  
of Theorem~\ref{t.p-groups}.  In particular, we show that
$\ed(G; p) \ge p$ for any non-abelian $p$-group $G$;
see Corollary~\ref{cor2.p-groups}. We also 
answer a question of Jensen, Ledet and
Yui~\cite[p.204]{jly} by giving an example of a finite
group $G$ with a normal subgroup $N$ such that
$\ed(G/N) > \ed G$; see Corollary~\ref{cor.jly}.

Another consequence of Theorem~\ref{t.p-groups} is the following 
new bound on $\ed \Spin_n$.  Here by $\Spin_{n}$ we will mean 
the totally split form of the spin group in dimension~$n$ 
over a field $k$.

\begin{theorem} \label{thm.spin} Suppose $k$ is a field of 
characteristic $\neq 2$, and that $\sqrt{-1} \in k$. If 
$n$ is not divisible by $4$ then
\begin{align*}
2^{\lfloor (n -1)/2 \rfloor} - \frac{n(n-1)}{2} \le &
   \ed(\Spin_{n}; 2) \le \ed(\Spin_n) \le  
2^{\lfloor (n - 1)/2 \rfloor} \, . \\
\intertext{If $n$ is divisible by $4$ then}
2^{\lfloor (n-1)/2 \rfloor} - \frac{n(n-1)}{2} + 1 \le &
\ed(\Spin_{n}; 2) \le \ed \Spin_{n} \le  
2^{\lfloor (n - 1)/2 \rfloor} + 1.
\end{align*}
\end{theorem}

Theorem~\ref{thm.spin} is proved in \S\ref{s.spinor}, where
we also prove similar estimates for the essential dimensions 
of pin and half-spin groups.

The lower bound of Theorem~\ref{thm.spin} was surprising to us because 
previously the best known lower bound,
due of V.~Chernousov and J.--P.~Serre~\cite{cs}, was
\begin{equation} \label{e.old-spin}
\ed \Spin_n \geq \begin{cases} \lfloor n/2 \rfloor + 1 &\text{if $n
    \ge 7$
    and $n \equiv 1$, $0$ or $-1 \pmod{8}$} \\
  \lfloor n/2 \rfloor &\text{for all other $n \ge 11$.} \end{cases} 
\end{equation}
(The first line is due to B.~Youssin and the second author 
in the case that $\chr k=0$~\cite{ry}.)  Moreover, in low 
dimensions, M.~Rost~\cite{rost} (cf. also~\cite{garibaldi}) 
computed the following table of exact values:
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\rule{0pt}{14pt}
\begin{tabular}{r@{${}={}$}lr@{${}={}$}lr@{${}={}$}lr@{${}={}$}l}
$\ed \Spin_3$ & $0$ &$ \ed \Spin_4$ & $0$ & $\ed \Spin_5$ & $0$ &
   $\ed \Spin_6$ & $0$ \\
$\ed \Spin_7$ & $4$ & $\ed \Spin_8$ & $5$ & $\ed \Spin_9$ & $5$ &
   $\ed \Spin_{10}$ & $4$\\
$\ed \Spin_{11}$ & $5$ & $\ed \Spin_{12}$ & $6$ & $\ed
\Spin_{13}$ & $6$ &
   $\ed \Spin_{14}$ & $7$.
\end{tabular}
\end{center}
Taken together these results seemed to suggest 
that $\ed\Spin_n$ should be a slowly increasing 
function of $n$ and gave no hint of its exponential growth.

The computation of $\ed\Spin_n$ gives an example of a
split, semisimple, connected linear algebraic group 
whose essential dimension exceeds its
dimension.  (Note that for a semisimple adjoint 
group $G$, $\ed G \le \dim G$; cf. Example~\ref{ex.cgr1}.) 
Since $\ed\SO_n = n - 1$ for every $n \ge 3$
(cf.~\cite[Theorem 10.4]{reichstein}), it also gives 
an example of a split, semisimple, connected linear 
algebraic group $G$ with a central subgroup $Z$ such that 
$\ed G>\ed G/Z$.  

In the last section we follow a suggestion 
of A. Merkurjev and B. Totaro to
apply our results on $\ed\Spin_{n}$ to a problem in the theory 
of quadratic forms. Let $K$ be a field of characteristic 
different from $2$ containing a square root of $-1$, 
$\rW(K)$ be the Witt ring of $K$ and $I(K)$ be 
the ideal of classes of even-dimensional forms 
in $\rW(K)$.  It is well known that if $q$ is 
a non-degenerate $n$-dimensional quadratic form whose 
class $[q]$ in $\rW(K)$ lies in $I^{a}(K)$, then $[q]$ can be 
expressed as the class a sum of $a$-fold Pfister forms. 
It is natural to ask how many Pfister form are needed. 
When $a = 1$ or $2$ it is easy to see that $n$ Pfister forms 
always suffice; see Proposition~\ref{p.PfisterEasy}. 
We prove the following result, which shows that 
the situation is quite different when $a = 3$.

\begin{theorem} \label{t.pfister} 
Let $k$ be a field of characteristic different 
from $2$ and $n$ an even positive integer.
Then there is a field extension $K/k$ and
a class $[q] \in I^3(K)$ represented by an $n$-dimensional
quadratic form $q/K$ such that 
$[q]$ cannot be written as the 
sum of fewer than
%   \[ r \geq \frac{2^{(n+4)/4} -n-2}{7}\,\]
   \[ \frac{2^{(n+4)/4} -n-2}{7}  \]
$3$-fold Pfister forms over $K$.
\end{theorem}

Finally we remark that the stack-theoretic approach to computing 
$\ed G$ developed in this paper has been recently extended and 
refined by Karpenko and Merkurjev~\cite{km2}. Their main
result is a general formula for the essential dimension of any 
finite $p$-group over any field $k$ containing a primitive
$p$th root of unity. This formula may be viewed as a common 
generalization of our Theorems~\ref{t.p-groups} and~\ref{t.cyclic}.

\subsection*{Notation} Throughout this paper, a \emph{variety} 
  over a field $k$ will be a geometrically integral 
  separated scheme of finite type over $k$.  Cohomology groups 
  $\H^i(X, G)$, where $X$ is a scheme and $G$ is an abelian group
  scheme over $X$, will be taken with respect to the fppf 
  topology unless otherwise specified.  We will 
  write $\mu_n$ for the group scheme of $n$-th roots of unity.  
  If $k$ is a field whose characteristic is prime to $n$, 
  $\zeta_n$ will denote a primitive $n$-th root of unity in the algebraic
  closure of $k$.  (Using the axiom of choice, we choose the $\zeta_n$
   once and for all.)  For typographical reasons, we sometimes write $C_n$ for
  the cyclic group $\ZZ/n$.

\subsection*{Acknowledgments}
We would like to thank the Banff International Research Station in
Banff, Alberta (BIRS) for providing the inspiring meeting place where
this work was started.  We are grateful to K.~Behrend, C.-L. Chai,
D.~Edidin, N.~Fakhruddin, M.~Florence, A.~Merkurjev, B.~Noohi, G.~Pappas,
D.~Saltman and B.~Totaro for helpful conversations. 

\section{The essential dimension of a stack: definition and first properties}
\label{s.definition}

We now return to the general setting of Definition~\ref{def.merkurjev}.
As we mentioned in Example~\ref{ex.edG} most of the existing theory
is specific to the Galois cohomology functors
$F = \H^1(*, G)$ for various algebraic $k$-groups $G$. 
On the other hand, many naturally arising functors where the essential
dimension is of interest are not of the form
$F(K) = \H^1(K, G)$ for any algebraic group $G$.  Two such examples are
given below. In this section we identify a class of functors which is
sufficiently broad to include most such examples, yet ``geometric" enough
to allow one to get a handle on their essential dimension. 
These functors are isomorphism classes of objects 
of an algebraic stack; see Definition~\ref{def.ed-stack} below. 
As we shall see in the sequel, these ``new"
functors turns out to be very useful even if one is 
only interested in the ``classical" setting of Example~\ref{ex.edG}.

\begin{example} \label{ex.ed-variety}
Let $X/k$ be a scheme of finite type over a field $k$,
and let $F_X\colon\Fields_k  \arr\Sets$ denote the functor
given by $K\arrto X(K)$.  Then an easy argument due to Merkurjev
shows that $\ed F_X=\dim X$; see~\cite[Proposition 1.17]{bf1}.

In fact, this equality remains true for any algebraic space $X$.
Indeed, an algebraic space $X$ has a stratification by schemes $X_i$.
Any $K$-point $\eta\colon\Spec K \arr X$ must land in one 
of the $X_i$.  Thus $\ed X=\max\ed X_i=\dim X$.
\qed
\end{example}

\begin{example} \label{ex.ed-curves}
Let $\operatorname{Curves}_g$ be the functor that associates to
a field $K/k$
the set of isomorphism classes of smooth algebraic
curves of genus $g$ over $K$. The essential dimension of this  
functor is computed in \cite{brv}.
\end{example}

We are now ready to give the main definition of this section.

\begin{definition} \label{def.ed-stack} Suppose $\cX$ is an algebraic
  stack over $k$.  The \emph{essential dimension} $\ed \cX$ of $\cX$ 
(respectively, the essential dimension $\ed(\cX; p)$ of $\cX$ at 
 a prime integer $p$) is the
 essential dimension (respectively, the essential dimension at $p$)
 of the functor $F_{\cX}\colon\Fields_k \arr\Sets$ which
 sends a field $L/k$ to the set of isomorphism classes of objects in
 $\cX(L)$.\footnote{In the literature the functor
 $F_{\cX}$ is sometimes denoted by $\widehat{\cX}$ or $\overline{\cX}$.}

As in Definition~\ref{def.merkurjev}, we will 
write $\ed(\cX/k)$, or $\ed_k \cX$ when we need 
to be specific about the dependence on the base field $k$. 
Similarly for $\ed(\xi/k)$ or $\ed_k \xi$, where
$\xi$ is an object of $F_{\cX}$, and for $\ed(\cX/k; p)$, 
$\ed_k(\cX; p)$,  $\ed(\xi/k; p)$, $\ed_k(\xi; p)$.
\end{definition}

All of the examples we have considered so far may be viewed as special cases
of Definition~\ref{def.ed-stack}.  If $\cX$ is a scheme of finite type (or an
algebraic space), we recover Example~\ref{ex.ed-variety}. At 
the other extreme, if $\cX = \cB G$ is the classifying stack 
of an agebraic group $G$ defined over $k$ (so that $\cB G(T)$ 
is the category of $G$-torsors on $T$), we recover Example~\ref{ex.edG}. If 
$\cX = \cM_{g}$ is the stack of smooth algebraic curves of genus $g$,
we recover Example~\ref{ex.ed-curves}.

\begin{remark}
\label{r.abuse}
If $G$ is an algebraic group, we will often write $\ed G$ for $\ed\cB G$.
That is, we will write $\ed G$ for the essential dimension of the
stack $\cB G$ and not the essential dimension of the scheme
underlying $G$.  We do this to conform to the, now
standard, notation introduced at the beginning of this paper
(and in Example~\ref{ex.edG}). Of course, by
Example~\ref{ex.ed-variety}, the essential dimension
of the underlying scheme is $\dim G$.   

In fact, the reader may notice that we prefer 
to write $\ed\cB G$ earlier in the paper, where
we are working in a general stack-theoretic setting and $\ed G$
later on, where we are primarily concerned with essential
dimensions of algebraic groups.
\end{remark}

\begin{remark} \label{rem.finite}
$\ed\cX$ takes values in the range
$\{\pm\infty\}\cup\ZZ_{\geq 0}$ (with $-\infty$ occurring 
if and only if $\cX$ is empty). On the other hand, $\ed \xi$ is finite
for every object $\xi \in \cX(K)$ and every field extension $K/k$. 

Indeed, any field $K/k$ can be written as a filtered direct limit
 $K=\colim_I K_i$ of its subfields $K_i$ of finite transcendence degree.
 Since $\cX$ is limit preserving (cf.~\cite[Proposition 4.18]{LMB}),
 $\xi$ lies in the essential image of 
 $\cX(K_i) \arr \cX(K)$ for some $i\in I$ and thus $\ed \xi \le \trdeg_k(K_i) < \infty$.
 \end{remark}
 
We now recall Definitions (3.9) and (3.10) from~\cite{LMB}.
A morphism $f\colon\cX \arr\cY$ of algebraic stacks (over $k$) is said to be
{\em representable} if, for every $k$-morphism $T \to \cY$, where $T$ is 
an affine $k$-scheme, the fiber product $\cX\times_{\cY} T$ is 
representable by a scheme over $T$. A representable morphism
$f\colon\cX \arr\cY$ is said to be 
\emph{locally of finite type and of fiber dimension $\le d$} if the projection
$\cX\times_{\cY} T \to T$ is locally of finite type over $T$ 
and every fiber has dimension $\leq d$.

\begin{proposition}
\label{p.RelDim}
  Let $d$ be an integer $\cX \arr
  \cY$ be a representable $k$-morphism of algebraic stacks
  which is locally of finite type and of fiber dimension at most $d$. 
Let $L/k$ be a field, $\xi \in X(L)$
and $p$ be a prime integer. Then

\smallskip
(a) $\ed_k \xi \leq \ed_k (f(\xi)) + d$,

\smallskip
(b) $\ed_k(\xi; p) \leq \ed_k(f(\xi); p) + d$, 

\smallskip
(c) $\ed_k \cX \leq \ed_k \cY + d$,

\smallskip
(d) $\ed_k(\cX; p) \leq \ed_k(\cY; p) + d$.
\end{proposition}

\begin{proof} 
  (a) By the definition of $\ed_k(f(\xi))$ we can find an intermediate
 field $k \subset K \subset L$ and a morphism
  $\eta \colon\Spec K \arr Y$ such that 
  $\trdeg_k L\leq \ed f(\xi)$ and the following diagram commutes.
\begin{equation*}
\xymatrix{
\Spec L    \ar[r]^-{\xi}\ar[d]  & \cX\ar[d]^{f}\\
\Spec K   \ar[r]^-{\eta}        & \cY      \\
}
\end{equation*}
Let $\cX_K \eqdef \cX\times_{\cY}\Spec K$.  By the hypothesis, $\cX_K$ is an
algebraic space, locally of finite type over $K$ and of relative
dimension at most $d$.  By the commutativity of the above diagram, the
morphism $\xi \colon\Spec L \arr \cX$ factors through $\cX_L$:  
\begin{equation*}
\xymatrix{
\Spec L   \ar@/^/[rrd]^-{\xi} \ar[rd]^-{\xi_0} \ar@/_/[rdd] &  & \\  
 & \cX_K \ar[r] \ar[d] & \cX\ar[d]^{f}\\
 & \Spec K   \ar[r]^-{\eta}        & \cY      \\ }
\end{equation*}
Let $p$ denote
the image of $\xi_0$ in $\cX_K$.  Since $\cX_K$ has dimension at most
$d$, we have $\trdeg_k k(p)\leq d$.  Therefore $\trdeg_k k(p)\leq
\ed(f(\xi)) + d$.  Since $\xi$ factors through $\Spec k(p)$, part 
(a) follows.

\smallskip
(b) Let $L'/L$ be a field extension.
By part (a), $\ed_k(\xi_{L'}) \leq \ed_k(f(\xi_{L'})) + d$.
Taking the minimum over all prime-to $p$ extensions $L'/L$,
we obtain the desired inequality,  
$\ed_k(\xi; p) \leq \ed_k(f(\xi); p) + d$.

\smallskip
(c) follow from (a) and (d) follows from (b) by taking the maximum 
on both sides over all $L/k$ and all $\xi \in \cX(L)$. 
\end{proof}

\begin{corollary}\label{cor.RelDim-quotients}
Let $G$ be an algebraic group defined over $k$ and
$H$ be a closed subgroup of $G$. Then 

\smallskip
(a) $\ed H \le \ed G + \dim G - \dim H$ and

\smallskip
(b) $\ed(H; p) \le \ed(G; p) + \dim G - \dim H$.   

\smallskip
More generally, suppose $G$ is acting on an algebraic 
space $X$ (over $k$). Then

\smallskip
(c) $ \ed{[X/H]} \leq \ed{[X/G]} + \dim G - \dim H$ and

\smallskip
(d) $ \ed([X/H]; p) \leq \ed([X/G]; p) + \dim G - \dim H$.

\smallskip
Here $[X/G]$ and $[X/H]$ denote the quotient stacks for the actions
of $G$ and $H$ on $X$.
\end{corollary}

Recall that the objects in $[X/G](K)$ are, by definition, diagrams of the form
$$
\xymatrix{T \ar[r]^f \ar[d]^\pi &X\\
\Spec(K), & } 
$$ 
where $\pi$ 
% $T \to \Spec(K)$ 
is a $G$-torsor and $f$ is a $G$-equivariant morphism.

\begin{proof}
The natural morphism $[X/H] \arr [X/G]$ of quotient stacks
is easily seen to be representable, of finite type and with 
fibers of dimension $d = \dim G - \dim H$.  Applying 
Proposition~\ref{p.RelDim}(c) and (d) to this morphism,
we obtain the inequalities (c) and (d) respectively. 
Specializing $X$ to a point
(i.e., to $\Spec(k)$) with trivial $G$-action, we obtain (a) and (b).
\end{proof}

In general, the inequality of Proposition~\ref{p.RelDim} only goes in
one direction. However, equality holds in the following 
important special case.  We will say that a morphism $f\colon\cX \arr\cY$
of stacks is \emph{isotropic} if for
  every field extension $K$ of $k$ and every object $\eta$ of $\cY(K)$ there
  exists an object $\xi$ of $\cX(K)$ such that $f(\xi)$ is isomorphic
  to $\eta$.

\begin{proposition}
\label{p.isotropic}
  Let $f\colon\cX \arr\cY$ be an isotropic morphism of stacks fibered
  over $\points_k$.  Then $\ed\cX\geq \ed\cY$ and 
  $\ed(\cX; p) \geq \ed(\cY; p)$ for every prime $p$.
\end{proposition}
\begin{proof} Both inequalities are immediate consequences of 
  the following observation.

  Let $K$ be an extension of $k$ and $\eta$ an object of $\cY(K)$. If
  $\xi$ is an object of $\cX(K)$ such that $f(\xi)$ is isomorphic to
  $\eta$, then clearly a field of definition for $\xi$ is also a field of
  definition for $\eta$. Thus $\ed \xi \ge \ed \eta$. 
\end{proof}

Proposition~\ref{p.isotropic} can be used to deduce the following
well-known inequality. See~\cite{bf1} for another proof.

\begin{theorem} 
\label{t.GenFree}
Let $G$ be a linear algebraic group over a field $k$ admitting a
generically free representation on a vector space $V$.  Then
$$
\ed \cB G\leq \dim V-\dim G.
$$
\end{theorem}
\begin{proof}
  Let $U$ denote a dense $G$-stable Zariski open subscheme of $V$ on
  which $G$ acts freely. Then $[U/G]$ is an algebraic space of
  dimension $\dim V-\dim G$ and the map $[U/G] \arr \cB G$ 
  is representable and isotropic.                       
\end{proof}                            

\section{Preliminaries on gerbes and canonical dimension}
\label{s.gerbes-and-cd}

Let $\cX$ be a gerbe defined over a field $K$ 
\emph{banded} by an abelian $K$-group scheme $G$.
For background material on gerbes we refer the reader to
\cite[p. 144]{milne} and \cite[IV.3.1.1]{G}. 

There is a natural notion of equivalence of gerbes banded by $G$; the
set of equivalence classes is in a natural bijective correspondence with
the group $\H^{2}(K, G)$.  Given a gerbe $\cX$ banded by $G$, we write
$[\cX]$ for its class in $\H^2(K,G)$.  The identity is the class
$[\cB_K G]$ of the neutral gerbe $\cB_{K}G$.

We remark that it makes sense to talk about a gerbe banded by $G$, where
$G$ is not necessarily abelian, but we will not need to work in
this more general setting, which makes the definition 
considerably more involved.

Let $K$ be a field and let $\GG_m$ denote the
multiplicative group scheme over $K$. Recall that the group
$\H^2(K,\GG_m)$ is canonically isomorphic to the Brauer group
$\Br(K)$ of Brauer equivalence classes of central simple algebras
(CSAs) over $K$.  By Wedderburn's structure theorem, any CSA over
$K$ isomorphic to the matrix algebra $\rM_n(D)$ for $D$ a division
algebra over $K$ which is unique up to isomorphism.  Moreover, if
$A$ and $B$ are two Brauer equivalent CSAs, the division algebras
$D$ and $E$ corresponding to $A$ and $B$ respectively are
isomorphic.  For a class $[A]\in\Br(K)$, the \emph{index} of $A$ is
$\sqrt{\dim_K D}$.

Let $\alpha\in \H^2(K,\mu_n)$, where
$n$ is a positive integer prime to $\Char K$.
We define the \emph{index} $\ind \alpha$ to be the index of the image on
$\alpha$ under the composition
$$
\H^2(K,\mu_n) \hookrightarrow \H^2(K,\GG_m)\stackrel{\cong} \larr \Br(K).
$$ 
Note that the index of $\alpha$ is the smallest integer $d$ such that 
$\alpha$ is in the image of the (injective) connecting homomorphism
\begin{equation}
\label{e.partial}
\partial\colon\H^1(K,\PGL_d) \arr \Br(K).
\end{equation}
arising from the short exact sequence
\begin{equation}
\label{e.pgl}
1\arr \GG_m \arr \GL_d \arr \PGL_d \arr 1.
\end{equation}

The \emph{exponent} $\ord([A])$ of a class $[A]\in \Br K$
is defined to be its order
in the Brauer group.  The exponent $\ord([A])$ always
divides the index $\ind([A])$~\cite[Theorem 4.4.5]{herstein}.

In the next section, we address the problem of computing the essential
dimension of a $\mu_d$-gerbe over a field $K$. Our computation will rely, 
in a key way, on the notion of canonical dimension
introduced in~\cite{ber} and~\cite{km}, which we shall now recall.

Let $X$ be a smooth projective variety defined over a field $K$.
We say that $L/K$ is a {\em splitting field} for $X$ if
$X(K) \ne \emptyset$. A splitting field $L/K$ is called {\em generic}
if for every splitting field $L_0/K$ there exists a $K$-place $L \to L_0$.
The canonical dimension of $X$ is defined as the
minimal value of $\trdeg_K(L)$, where $L/K$ ranges over
all generic splitting fields. Note that the function field
$L = K(X)$ is a generic splitting field of $X$; see~\cite[Lemma 4.1]{km}.
In particular, generic splitting fields exist and $\cd(X)$ is finite.

If $X$ is a $K$-scheme and $p$ is a prime integer, we also recall 
the relative variant of this notion from~\cite{km}. 
We say that $L/K$ is a \emph{$p$-generic splitting field} for $X$ if 
for every splitting field $L_0/K$ there is a prime-to-$p$ extension 
$L_1/L$ and a $K$-place $L \to L_1$. The minimal value of $\trdeg_K(L')$, 
as $L'$ ranges over all $p$-generic splitting fields, 
is called the $p$-canonical dimension of $X$ and is denoted 
by $\cd_p(X)$. (Here, as usual, by a prime-to-$p$ extension 
we mean a finite field extension whose degree is prime to $p$.) 

\begin{lemma} \label{lem.cd1}
If $L/K$ is a $p$-generic extension for $X$ then 

\smallskip
(a) any intermediate extension $M/K$, where $K \subset M \subset L$
is also $p$-generic, and

\smallskip
(b) any finite prime-to-$p$ extension $L'/K$ is also $p$-generic.
\end{lemma}

\begin{proof} (a) Let $L_0/K$ be a splitting field for $X$. 
Since $L$ is generic, there is a prime-to-$p$ extension 
$L_1/L_0$ and a $K$-place $L \to L_1$. 
Restricting this place to $M$, we obtain a desired place $M \to L_1$.

Part (b) is an immediate consequence of \cite[Lemma 3.2]{km}.  
\end{proof}

The {\em determination functor} $D_X: \Fields_K\to\Sets$ is defined 
as follows: $D$ associates to a field $L/K$ the empty set, 
if $X(L) = \emptyset$, and a set consisting of one point, which we will
denote by $a(L)$, if  
$X(L) \neq \emptyset$. The natural map $D(L_1) \to D(L_2)$ 
is then uniquely determined for any $K \subset L_1 \subset L_2$.

\begin{lemma} \label{lem.cd2}
Let $X$ be a complete regular $K$-variety then 

\smallskip 
(a) $\cd(X) = \ed(D_X)$.

\smallskip
(b) $\cd_p(X) = \ed(D_X; p)$.
\end{lemma}

\begin{proof} (a) Let $L$ be a generic splitting field for $X$. 
% with $\trdeg_K L =\cd X$.  
By \cite[Lemma 2.1]{km}, any subfield of $L$ containing $F$ is also 
a generic splitting field.  Therefore $\ed D_X \geq \ed a(L) \ge \cd(X)$.  

To prove the opposite inequality, it suffices to show that 
$\ed a(F)\leq \trdeg_K L$ for any splitting field $F/K$ and
any generic splitting field $L/K$ of $X$.  In other words, 
given $F$ and $L$ as above, we want 
to construct an intermediate splitting field 
$K \subset F_0 \subset F$ with $\trdeg_K(F_0) \le \trdeg_K(L)$.
By the definition of a generic splitting field, there is a place $L \to F$.  
That is, there is a valuation ring $R$ with quotient field $L$ and a local
homomorphism $f \colon R \to F$. We claim that the residue field
$F_0 = R/m$, where $m$ is the maximal ideal of $R$, has the desired 
properties.  Clearly $K \subset F_0 \subset F$, and
$\trdeg_K(F_0) \le \trdeg_K(R) = \trdeg_K(L)$, so we 
only need to check that $F_0$ is a splitting field for $X$. 

Since $L$ is itself a splitting field, we have a diagram
$$
\xymatrix{\Spec L \ar[r] \ar[d] & X\ar[d]\\
  \Spec R\ar[r] & \Spec K.  }
$$
By the valuative crition, we obtain a map $\Spec R \to X$ and
hence, a map $\Spec F_0 = \Spec R/m \to X$. Thus $F_0$ is 
splitting field for $X$, as claimed.

\smallskip
(b) Let $L$ be a $p$-generic splitting field for $X$. 
Lemma~\ref{lem.cd1} tells us that if $L'/L$
is a prime-to-$p$ extension and $K \subset M \subset L'$ is 
an intermediate splitting field then $M$ is again $p$-generic.
Thus $\ed(a(L); p)$ = minimal value of $\trdeg_K(M) \ge \cd_p(X)$.

To prove the opposite inequality, we need to show that
$\ed(a(F); p) \le \trdeg_K(L)$ for any splitting field $F/K$ and
any $p$-generic splitting field $L/K$. 
Indeed, after replacing $F$ by a prime-to-$p$-extension $F'$, we 
obtain a $K$-place $L \to F'$. Now the same argument as in part (a)
shows that there is an intermediate splitting subfield 
$K \subset F_0 \subset F'$ such that $\trdeg_K(F_0) \le \trdeg_K(L)$.
Thus 
\[ \ed(a(F); p) \le \ed(a(F')) \le \trdeg_K(F_0) \le \trdeg_K(L) \, , \]
as claimed.
\end{proof}

Of particular interest to us will
be the case where $X$ is a Brauer-Severi variety over $K$.
Let $m$ be the index of $X$.  If $m = p^a$ is a prime power then 
\begin{equation} \label{e.cd-primary}
\cd(X) = \cd_p(X) = p^a - 1 \, ; 
\end{equation}
see \cite[Example 3.10]{km} or~\cite[Theorem 11.4]{ber}. 
If the highest power of $p$ dividing $m$ is $p^a$ then
\begin{equation} \label{e.cd_pBS}
\cd_p(X) = p^a - 1 \; ; 
\end{equation}
see \cite[Example 5.10]{km}. On the other hand, if $m$ is divisible by
more than one prime, $\cd(X)$ is not known in general.
Suppose $m = p_{1}^{a_{1}} \dots p_{r}^{a_{r}}$.
Then the class of $P$ in $\Br L$ is the sum of 
classes $\alpha_{1}$, \dots,~$\alpha_{r}$ whose indices 
are $p_{1}^{a_{1}}$, \dots,~$p_{r}^{a_{r}}$.
Denote by $X_{1}$, \dots, $X_{r}$ the Brauer--Severi varieties
with classes $\alpha_{1}$, \dots, $\alpha_{r}$. 
It is easy to see that
$K(X_1 \times \dots \times X_r)$ is a generic splitting field for $X$.
Hence,
\[ \cd(X) \le \dim(X_1 \times \dots \times X_r) =
p_1^{a_1} + \dots + p_r^{a_r} - r \, .  \]
In \cite{ctkm}, Colliot-Th\'el\`ene, Karpenko and Merkurjev conjectured 
that equality holds, i.e.,  
\begin{equation}
\label{e.conjecture} 
\cd(X) = p_1^{a_1} + \dots + p_r^{a_r} - r 
\end{equation}
for all $m \ge 2$. As we mentioned above, this in known 
to be true of $m$ is a prime power (i.e., $r = 1$).
Colliot-Th\'el\`ene, Karpenko and Merkurjev also 
proved~\eqref{e.conjecture} for $m = 6$; see \cite[Theorem 1.3]{ctkm}.

\section{The essential dimension of a $\mu_d$-gerbe}
\label{s.gerbes}

We are now ready to proceed with our main theorem on gerbes.

\begin{theorem}
\label{t.cdP}
Let $d$ be an integer with $d >1$. Let $K$ be a field and $x \in \H^2(K, \mu_d)$, where $\Char K$ 
does not divide $d$. Denote the image of $x$ in $\H^2(K, \GG_m)$ by $y$,
the $\mu_d$-gerbe associated to $x$ by $\cX$, the $\GG_m$-gerbe 
associated to $y$ by $\cY$, and the Brauer--Severi variety 
associated to $y$ by $P$. Then

\smallskip
(a) $\ed(\cY) = \cd P$ and (b) $\ed \cX = \cd P + 1$.

\smallskip
\noindent
Moreover, if the index of $x$ is a prime power $p^r$ then

\smallskip
(c) $\ed(\cY; p) = \ed(\cY) = p^r - 1$ and 
(d) $\ed(\cX; p) = \ed \cX = p^r$.
\end{theorem}

\begin{proof}
  The functor $F_{\cY}:\Fields_K\to\Sets$ sends a field 
  $L/K$ to the empty set, if $P(L) = \emptyset$, and to a set 
  consisting of one point, if $P(L) \neq \emptyset$. In other words,
  $F_Y$ is the determination functor $D_P$ introduced in the previous
  section. Thus $\ed(\cY) = \ed(D_P)$ and 
  $\ed(\cY; p) = \ed(D_P; p)$. On the other hand,
 by Lemma~\ref{lem.cd2}, $\ed(D_P) = \cd(P)$ and $\ed(D_P; p) = \cd_p(P)$.
This, in combination with~\eqref{e.cd-primary}, 
proves parts (a) and (c).

\smallskip (b) First note that the natural map $\cX\to\cY$ is finite
type and representable of relative dimension $1$. By
Proposition~\ref{p.RelDim} this implies that $\ed \cX \leq
\ed(\cY)+1$. By part (a) it remains to prove the opposite inequality,
$\ed \cX \geq \ed(\cY) + 1$. We will do this by constructing an
object $\alpha$ of $\cX$ whose essential dimension is $\ge \ed(\cY) + 1$.

We will view $\cX$ as a torsor for $\cB\mu_d$ in the following
sense:  One has maps 
\begin{align*}
\cX\times\cB\mu_d &\to \cX \\
  \cX\times\cX     &\to \cB\mu_d
\end{align*}
satisfying various compatibilities, where the first map is the action
of $\cB\mu_d$ on $\cX$ and the second map is the ``difference'' of two
objects of $\cX$. For the definition and a discussion
of the properties of these maps, 
see~\cite[Chapter IV, Sections 2.3, 2.4 and 3.3]{G}.  (Note that, in
the notation of Giraud's book, $\cX\wedge\cB\mu_d\cong \cX$ and the
action operation above arises from the map $\cX\times\cB\mu_d\to
\cX\wedge\cB\mu_d$ given in Chapter IV, Proposition 2.4.1.   The
``difference'' operation --- which we will not use here --- arises similarly
from the fact that, in Giraud's notation, $\HOM(\cX,\cX)\cong\cB\mu_d$.)

Let $L = K(P)$ be the function field of $P$. Since $L$ splits $P$, we 
have a natural map $a:\Spec L \to \cY$. Moreover since $L$ 
is a generic splitting field for $P$, 
\begin{align} 
\label{e.cd1}
\ed(a) = \cd(P) = \ed(\cY), \; \; \quad \quad \\
\label{e.cd2}
\ed(a; p) = \cd_p(P) = \ed(\cY; p). % \nonumber
\end{align}
where we view $a$ as an object in $Y$.
Non-canonically lift $a:\Spec L\to\cY$ to a map $\Spec L\to\cX$ using 
the fact that $\cX\to\cY$ is isotropic.  Let $\Spec L(t)\to\cB\mu_d$ 
denote the map classified by 
$(t)\in \H^1(L(t), \mu_d)=L(t)^{\times}/L(t)^{\times d}$.   
Composing these two maps, we obtain an object
$$
\alpha:\Spec L(t)\to \cX\times\cB\mu_d\to\cX.
$$
in $\cX(L(t))$. Our goal is to prove that $\ed(\alpha) \ge \ed(\cY) + 1$.
In other words, given a diagram of the form 
\begin{equation} 
\label{e.oldclaim}
\xymatrix{
\Spec L(t)\ar[r]^{\quad \alpha} \ar[d] &\cX\\
\Spec M\ar[ru]^{\beta}              & \\
} 
\end{equation}
where $K \subset M \subset L$ is an intermediate field, we
want to show that $\trdeg_K(M) \ge \ed(\cY) + 1$. Assume the contrary:
there is a diagram as above with $\trdeg_K(M) \le   \ed(\cY)$.
Let $\nu \colon L(t)^* \to \ZZ$ be the usual discrete valuation 
corresponding to $t$ and consider two cases.

\smallskip
{\bf Case 1.} Suppose
the restriction $\nu_{| M}$ of $\nu$ to $M$ is non-trivial.
Let $M_0$ denote the residue field of $\nu$ and $M_{\geq 0}$ denote 
the valuation ring.  Since $\Spec M\to \cX\to\cY$, 
there exists an $M$-point of $P$.   Then by the
valuative criterion of properness for $P$, there exists an 
$M_{\geq 0}$-point and thus an $M_{0}$-point of $P$.  
Passing to residue fields, we obtain the diagram
\[ \xymatrix{
\Spec L\ar[r]^{a}\ar[d] &\cY\\
\Spec M_0\ar[ru]              & .\\
}  
\]
which shows that 
$\ed(a) \le \trdeg_K M_0 = \trdeg_K M-1\leq\ed(\cY) -1$,
contradicting~\eqref{e.cd1}. 

\smallskip
{\bf Case 2.} Now suppose the restriction of $\nu$ to $M$ is trivial.
The map $\Spec L\to \cX$ sets up an
isomorphism $\cX_L\cong\cB_L\mu_d$.  The map $\Spec L(t)\to\cX$ factors
through $\cX_L$ and thus induces a class in 
$\cB \mu_d(L(t)) = \H^1(L(t),\mu_d)$.  This class 
is $(t)$. Tensoring the diagram~\eqref{e.oldclaim}
with $L$ over $K$, we obtain
$$
\xymatrix{
\Spec L(t)\otimes L\ar[r]^{\alpha}\ar[d] & \cX_L\cong\cB_L \mu_d\\
\Spec M\otimes L\ar[ru]^{\beta}             & \\
} 
$$ 
Recall that $L = K(P)$ is the function field of $P$. 
Since $P$ is absolutely irreducible, the tensor products
$L(t)\otimes L$ and $M \otimes L$ are fields.
The map $\Spec M\otimes L\to \cB_L\mu_d$ is classified by some 
$m\in (M\otimes L)^{\times}/(M\otimes L)^{\times d}=\H^1(M\otimes L,\mu_d)$.   
The image of $m$ in $L(t) \otimes L$ would have to be equal to $t$ modulo
$d$-th powers. We will now derive a contradiction by comparing
the valuations of $m$ and $t$.

To apply the valuation to $m$, we lift $\nu$ from $L(t)$ 
to $L(t) \otimes L$. That is, we define $\nu_L$ as the valuation on
$L(t) \otimes L = (L \otimes L)(t)$ 
corresponding to $t$. Since
$\nu_L(t) = \nu(t) = 1$, we conclude that
$\nu_L (m)\equiv 1\pmod d$. This shows that
$\nu_L$ is not trivial on $M \otimes L$ and thus
$\nu$ is not trivial on $M$, contradicting our assumption. 
This contradiction completes the proof of part (b).

\bigskip
The proof of part (d) proceeds along similar lines, except for a small
complication in prime characteristic, which is resolved 
by Lemma~\ref{lem.inseparable} below. For the sake of completeness 
we outline the argument.

Once again, applying Proposition~\ref{p.RelDim} to the representable
projection map $\cX \to \cY$ of relative dimension $1$, we obtain 
the inequality $\ed(\cX; p) \le \ed(\cY; p) + 1$. By part (c) 
it remains to show that $\ed(\cX; p) \ge \ed(\cY; p) + 1$. We 
will do this by showing that 
\begin{equation} \label{e.partd}
\ed(\alpha; p) \ge \ed(\cY; p)+ 1 \, ,
\end{equation}
where $\alpha \colon \Spec L(t) \to \cX$ is the same object in $\cX(L(t))$ 
we constructed in the proof of part (b). Here we continue to denote
the function field of $P$ by $L = K(P)$.  Once again, assume 
the contrary: there exists a finite field extension $L'/L(t)$ 
of degree prime to $p$, such that $\ed(\alpha_{L'}) \le \ed(\cY; p)$. 
Let $L'_s$ be the separable closure of $L(t)$ in $L'$. 
By Lemma~\ref{lem.inseparable}
$\ed(\alpha_{L'_s}) = \ed(\alpha_{L'})$. Hence, after replacing
$L'$ by $L'_s$, we may assume that $L'$ is a finite separable extension
of $L(t)$ of degree prime to $p$. The assumption that 
$\ed(\alpha_{L'}) \le \ed(\cY; p)$ means, by definition,
that there exists a commutative diagram 
\begin{equation} \label{e.claim(d)} \xymatrix{
  \Spec(L') \ar[rd] \ar[dd]_{\text{\tiny prime-to-$p$}} & & \\
   & \Spec M \ar[rd]^{\beta} &  \\
   \Spec L(t) \ar[rr]^{\alpha}  & & \cX  } 
\end{equation}
where $K \subset M \subset L'$ is an intermediate field and
$\trdeg_K(M) \le \ed(\cY; p)$. 

Let $\nu \colon L(t)^* \to \ZZ$ be the discrete valuation 
corresponding to $t$ and $\nu_1', \dots, \nu_r'$ be the liftings 
of $\nu$ to $L'$.  Since $L'/L$ is separable,
\cite[Proposition XII.18]{lang} tells us that 
\[ e_1f_1 + \dots + e_r f_r = [L':L] \, , \]
where $e_i$ and $f_i$ are, respectively, the ramification index and
the residue class degree of $\nu_i'$. Since $[L':L]$ is prime to $p$,
at least one of the valuations $\nu_1', \dots, \nu_r'$, say 
$\nu' \colon (L')^* \to \frac{1}{e} \ZZ$, has the property that
its ramification index $e$ and residue class degree $f$ are both
prime to $p$. 

Once again we consider two cases.
If the restriction of $\nu'$ to $M$ is non-trivial then
after passing to residue fields and arguing as in Case 1 
of the proof of part (b), we arrive at the diagram
$$ \xymatrix{
  \Spec(L'_0) \ar[rd] \ar[dd]_{\text{\tiny degree $f$}} & & \\
   & \Spec M_0 \ar[rd]  &  \\
   \Spec L \ar[rr]  &  & \cY  } 
$$ 
which shows that 
$\ed(a; p) \le \trdeg_K M_0 = \trdeg_K M-1 \leq\ed(\cY; p) -1$,
contradicting~\eqref{e.cd2}.

If the restriction of $\nu'$ to $M$ is trivial then the induced valuation 
on $M \otimes_K L'$ is trivial, and we argue as in Case 2 in the proof of
part (b).  Tensoring the diagram~\ref{e.claim(d)} with $L$ over $K$ and
identifying $X_L$ with $\cB_L\mu_d$ via our chosen map 
$\phi \colon \Spec(L) \to \cX$, we obtain the diagram
\[ \xymatrix{
  \Spec(L' \otimes_K L) \ar[rd] \ar[dd]_{\text{\tiny prime-to-$p$}} & & \\
   & \Spec M \otimes_K L \ar[rd]^{\beta \otimes_K \phi} &  \\
   \Spec L(t) \otimes_K L \ar[rr]^{\alpha \otimes_K \phi} & & \cB_L \mu_d.} 
\]
Since $L$ is absolutely irreducible, 
$L' \otimes_K L$, $M \otimes_K L$ and $L(t) \otimes_K L$ are fields.
The map $\Spec M\otimes_K L\to \cB_L\mu_d$ gives rise to an
$m\in (M\otimes L)^{\times}/(M\otimes L)^{\times d}=\H^1(M\otimes L,\mu_d)$   
whose image in $L'\otimes_K L$ is equal to $t$, modulo $d$-th powers. 

Lifting $\mu$ from $L'$ to $L' \otimes_K L$
(so that $\nu'_L$ is trivial on $1 \otimes_K L$), we see that
\[ \nu'_L(m) - \nu'_L(t) = \nu'_L(m) - 1 \in d \frac{1}{e} \ZZ \, . \]
Since $d$ and $e$ are relatively prime, this shows that
$\nu'_L(m) \ne 0$. We conclude that $\nu'_L$ is non-trivial on
$M \otimes_K L$. Consequently, 
$\nu'$ is non-trivial on $M$, contradicting our assumption.
This contradiction completes the proof of part (d) (modulo
Lemma~\ref{lem.inseparable} below).
\end{proof}

\begin{lemma} \label{lem.inseparable} Let $F$ be a functor from 
the category of field extensions of $K$ to the category of sets.

\smallskip
(a) Suppose that the natural map $F(A) \to F(B)$ is bijective 
for every algebraic inseparable field extension $B/A$. Then  
$\ed(a) = \ed(a_B)$ for every algebraic inseparable field 
extension $B/A$ and every $a \in F(A)$.

\smallskip 
(b) Let $\cX$ be a gerbe over a field $K$ banded by $\mu_d$, 
$A$ a field over $K$, and $B/A$ an algebraic inseparable 
field extension. Suppose $d$ is not divisible by $\Char K$.
Then $\ed(a) = \ed(a_B)$ for every $a \in \cX(A)$.
\end{lemma}

\begin{proof} (a) The inequality $\ed(a) \ge \ed(a_B)$ follows 
immediately from Definition~\ref{def.merkurjev}. To prove 
the opposite inequality it suffices to show that if
$a_B$ descends to an intermediate field $M$, where
$K \subset M \subset B$, then $a$ descends to an
intermediate field $M_0$, where $K \subset M_0 \subset A$ and
$\trdeg_K(M_0) \le \trdeg_K(M)$. To prove this assertion, set
$M_0 = M \cap A$ and $p = \Char K$. Then for every
$m \in M$ there is a power of $p$, say $e = p^r$ such that
$m^e$ lies in both $A$ and $M$ (and hence, in $M_0$). 
This shows that $M$ is algebraic and inseparable over $M_0$;
cf.~\cite[VII.7]{lang}. Since $M_0$ is a subfield of $M$,
the inequality $\trdeg_K(M_0) \le \trdeg_K(M)$ is obvious.
(In fact, equality holds, since $M$ is algebraic over $M_0$.)
The commutative diagram 
\[ \xymatrix{
F(M) \ar[r] & F(B) \ni a_B                                    \\
F(M_0) \ar[u]^{\simeq} \ar[r] & F(A) \in a \ar[u]^{\simeq} }  
\]
shows that $a$ descends to $M_0$.

\smallskip
(b) By part (a) it suffices to show that the natural map $\cX(A) \to \cX(B)$
is an isomorphism for every algebraic inseparable extension $B/A$. If $A$
doesn't split the class $[\cX] \in \H^1(K, \mu_d)$ then neither does
$B$, so, $\cX(A) = \cX(B) = \emptyset$, and there is nothing to prove.
If $A$ splits $[\cX]$, i.e., there is a map $\Spec(A) \to \cX$ 
then we can use this map to trivialize $\cX$. The map
$\cX(A) \to \cX(B)$ can then be identified with the natural
map $\H^1(A, \mu_d) \to \H^1(B, \mu_d)$ or equivalently, 
\[ A^*/(A^*)^d \to B^*/(B^*)^d \, . \]
We want to show that this map is an isomorphism. To do this, we recall that
for every $y \in B^*$ there exists an $r \ge 0$ such that $y^{p^r}$ 
lies in $A$. Since $p$ and $d$ are relatively prime, both injectivity 
and surjectivity of the above map follow.
\end{proof}

In view of Theorem~\ref{t.cdP}(a), Conjecture~\eqref{e.conjecture} 
of Colliot-Th\'el\`ene, Karpenko and Merkurjev be restated
in the language of essential dimension of gerbes as follows. 

 \begin{conjecture}\label{conj:ed-product}
 If $\cX$ is a gerbe banded by $\mu_{n}$ over a field $K$, 
 let $p_{1}^{a_{1}} \dots p_{r}^{a_{r}}$ be the decomposition 
 into prime factors of the index of the class of $\cX$ in 
 the Brauer group of $K$. Then
    \[
    \ed\cX = p_{1}^{a_{1}} + \dots + p_{r}^{a_{r}} - r + 1.
    \]
 \end{conjecture}
When the index is $6$ this follows from \cite[Theorem 1.3]{ctkm}.
 
In view of the fact that the conjecture holds for $r = 1$, it
can also be rephrased as follows: if $m$ and $n$ are relatively prime 
positive integers, $\cX$ and $\cY$ are gerbes banded by 
$\mu_{m}$ and $\mu_{n}$, then
    \[
    \ed(\cX \times \cY) = \ed\cX + \ed\cY - 1.
    \]

Back to the language of canonical dimension, one 
could ask the following more general question. 
Let $X$ and $Y$ be smooth projective varieties over 
a field $K$. Assume that there are no rational functions 
$X \darr Y$ or $Y \darr X$. Then is it true 
that $\cd(X \times Y) = \cd(X) + \cd(Y)$? 
A positive answer to this question would imply 
Conjecture~\ref{conj:ed-product}.

\section{Proof of Theorem~\ref{thm2}}
\label{s.extensions}

In this section $k$ is a field and $p$ is a prime number 
not equal to $\chr k$.  We begin with some preliminary facts.  

\begin{proposition}
\label{p.observation}
  Let $U$ be an integral algebraic space locally of finite type over
  $k$ with function field $K\eqdef k(U)$, and let 
$f\colon\cX \arr U$ 
be a stack over $U$. Let $\cX_K$ denote the pullback of $\cX$ 
to $\Spec K$.  Then,  
\begin{align*}
\ed \cX &\geq \ed (\cX_K/K) + \dim U;\\
\ed (\cX;p) &\geq \ed (\cX_K/K; p) + \dim U.
\end{align*}
\end{proposition}
\begin{proof}
  Let $j:\cX_K\to\cX$ denote the inclusion.  If $\xi:\Spec L\to\cX_K$
  be a morphism, then it is easy to see that
  $\ed(\xi/K)=\ed(j\circ\xi /k)+\dim U$.  This implies the first
  inequality of the proposition directly.  The second inequality then
  follows by taking the infinum of $\ed(\xi_M/K)$ 
for all prime-to-$p$ extensions $M/L$. 
\end{proof}

Let $\cX$ be a locally noetherian stack over a field $k$ with
presentation $P\colon X \arr\cX$.  Recall that the \emph{dimension
of $\cX$ at a point $\xi\colon\Spec K \arr \cX$} is given by
$\dim_x(X)-\dim_x P$ where $x$ is an arbitrary point of $X$ lying over
$\xi$~\cite[(11.14)]{LMB}. Let $\cY$ be stack-theoretic closure of the
image of $\xi$; that is, the intersection of all the closed substacks
$\cY_{i}$ such that $\xi^{-1}(\cY_{i}) = \spec K$. The morphism $\xi$
factors uniquely through $\cY \subseteq \cX$. We defined \emph{the
  dimension of the point $\xi$} to be the dimension of the stack $\cY$
at the point $\spec K \arr \cY$.

\begin{proposition}
\label{p.RelDim2}
  Let $\cX \arr \cY$ be a morphism of algebraic stacks over a field $k$.  Let 
$K/k$ be a field extension and let $y\colon\Spec K \arr \cY$ be a
point of 
dimension $d\in\ZZ$.  Let 
$\cX_K\eqdef \cX\times_{\cY}\Spec K$.  Then
\begin{align*}
  \ed (\cX_K/K)&\leq \ed (\cX/k) - d;\\
  \ed (\cX_K/K;p)&\leq \ed(\cX/k;p) - d.
\end{align*}
\end{proposition}


\begin{proof}
By~\cite[Theorem 11.5]{LMB}, $\cY$ is the disjoint union of a finite
family of locally closed, reduced substacks $\cY_i$ such that each
$\cY_i$ is an fppf gerbe over an algebraic space $Y_i$ with structural
morphism $A_i\colon\cY_i \arr Y_i$.  We can therefore replace $\cY$ by one of
the $\cY_i$ and assume that $\cY$ is an fppf gerbe over an algebraic
space $Y$.  Without loss of generality, we can assume that $Y$ is an
integral affine scheme of finite type over $k$.  

Let $p$ be the image of $y$ in $Y$.  
Since $\cY$ is limit-preserving, we can find an integral affine scheme
$U$ equipped with a morphism $i\colon U \arr \cY$ and a dominant morphism
$j\colon\Spec K \arr U$ such that $y$ is equivalent to $i\circ j$. 
We can also assume that the composition $U \arr
\cY \arr Y$ is dominant. 

Since $\cY$ is a gerbe over $Y$, it follows that $U \arr \cY$ is
representable of fiber dimension at most $\dim U-d$.  Now, form the
following diagram with Cartesian squares.
$$
\xymatrix{
  \cX_K\ar[r]\ar[d] & \Spec K\ar[d]\\
  \cX_U\ar[r]\ar[d] & \ar[d]U\\
  \cX\ar[r]         & \cY
}
$$
Since the vertical maps in the lower square are finite type and representable of fiber
dimension at most $\dim U-d$, it follows from
Propositions~\ref{p.RelDim} and~\ref{p.observation} that
\begin{align*}
\ed (X_K/K) &\leq \ed (\cX_{k(U)}/k(U))\\
            &\leq \ed\cX_U -\dim U\\
            &\leq \ed \cX  +\dim U -d + \dim U\\
            &\leq \ed\cX -d.\qedhere
\end{align*}

Similarly, $\ed(\cX_K/K; p)\leq \ed(\cX;p)-d$,
\end{proof}


 We now proceed with the proof of Theorem~\ref{thm2}, 
 which we restate for the convenience of the reader.  Let
 \begin{equation}
 \label{e.extensionsb}
 1\arr Z \arr G \arr Q\arr 1
 \end{equation}
 denote an extension of groups over a field $k$, with $Z$ central 
 and isomorphic to $\mu_n$ for some integer $n>1$.  Recall from the 
 introduction, that we defined $\ind(G,Z)$ as the maximal value of the index 
 $\ind\bigl(\partial_K(t)\bigr)$, as $K$ ranges over all field extensions 
 of $k$ and $t$ ranges over all torsors in $\H^1(K,Q)$. 
 
 \begin{theorem}
 \label{thm2b} 
 Let $G$ be an extension as in~\eqref{e.extensionsb}.  Assume that
 $n=p^r$ for some non-negative integer $r$. Then
 $\ed_k(G; p) \geq \ind(G,Z) - \dim G$.
 \end{theorem}

\begin{proof}
Let $K/k$ be a field extension and let $t:\Spec K\to\cB Q$ be a
$Q$-torsor over $\Spec K$. The dimension of $\cB Q$ at the
point $t$ is $-\dim Q = - \dim G$.   Let $\cX$ denote the pull-back in the
following diagram.
$$\xymatrix{ 
\cX\ar[r]\ar[d] & \Spec K\ar[d]^{t}\\
\cB G\ar[r]     & \cB Q        \\
}
$$
By Proposition~\ref{p.RelDim2}, $\ed(\cX/K;p)\leq \ed(\cB G/k;p) + \dim G$.
On the other hand, since $\cB G$ is a gerbe banded by $Z$ over $\cB
Q$, $\cX$ is a gerbe banded by $Z$ over $\Spec K$.   Therefore, by
Theorem~\ref{t.cdP}(d), $\ed (\cX/K; p)=\ind\partial_K(t)$.  
By substitution, $\ind\partial_K(t)-\dim G \leq \ed_k(\cB G; p)$.   Since this
inequality holds for all field extensions $K/k$ and all $Q$-torsors
$t$ over $K$, the theorem follows.
\end{proof}

\begin{remark} Suppose $G$ is a simple algebraic group whose center $Z$ is
cyclic.  It is tempting to apply Theorem~\ref{thm2} to the natural
sequence
\[ 1 \arr Z \arr G \arr G^{\mathrm{ad}} \arr 1 \] where the adjoint group
$G^{\mathrm{ad}}$ is $G/Z$. Given a torsor $t \in \H^1(K, G^{\mathrm{ad}})$, 
the central simple algebra representing $\partial_K(t) \in \H^2(K, Z)$ 
is called {\em the Tits algebra} of $t$. The values of the index of the
Tits algebra were studied in~\cite{tits}, 
where this index is denoted by $b(X)$ (for a group of type $X$) 
and its possible values are listed on p. 1133. A quick look at this 
table reveals that for most types these indices are
smaller than $\dim G$, so that the bound of Theorem~\ref{thm2}
becomes vacuous. The only exception are groups of types $B$ and $D$,
in which case Theorem~\ref{thm2} does indeed, give interesting
bounds; cf. Remark~\ref{rem.clifford}.
\end{remark}

\begin{remark} \label{rem.ctkm}
If $n \ge 2$ is not necessarily a prime power, 
then
% suppose Conjecture~\ref{conj:ed-product} holds. 
the above argument shows that
\[ \ed_k G \ge \cd(\partial_K(t))-\dim G \, .\]
Here by $\cd(\partial_K(t))$ we mean the canonical dimension
of the Brauer-Severi variety representing the class of 
$\partial_K(t)$ in $\H^2(K, \mu_n)$.  Assuming that 
Conjecture~\ref{conj:ed-product} holds, we obtain 
the following (conjectural) inequality: if 
$\ind(G,Z)= p_1^{a_1} \dots p_r^{a_r}$ is
the prime factorization of $\ind(G,Z)$ then 
\[
\ed_k G \ge p_1^{a_i} + \dots + p_r^{a_r} - r + 1 - \dim G \, . 
\]
As we remarked at the end of \S\ref{s.gerbes}, 
Conjecture~\ref{conj:ed-product} (and thus the above inequality)
is known to be true if $\ind(G, Z)$ is a prime power (i.e., $r = 1$)
or $\ind(G, Z) = 6$. 
\end{remark}

\section{Florence's theorem}
\label{s.cyclic}

In this section we give an alternative proof of 
a recent theorem of Florence (in a slightly strengthened form) 
by combining the lower bound of Theorem~\ref{thm2} 
with the Brauer--Rowen Theorem. The idea to use the Brauer--Rowen 
theorem in this context is due to Florence. Thus while our proof is
different from the one in~\cite{florence}, it is not entirely 
independent.

\begin{theorem}[M.~Florence {\cite{florence}}]
\label{t.cyclic}
Let $p$ be a prime, $k$ a field of characteristic $\neq p$.
Suppose $\zeta_{p^n} \in k$ but $\zeta_{p^{n+1}}  \notin k$ for some 
integer $n \ge 1$.  Moreover, if $p = 2$ and $n = 1$, assume also
that $k(\zeta_4) \neq k(\zeta_8)$. Then
\[ \ed_k (C_{p^m}; p) = \ed_k C_{p^m} =
\begin{cases} p^{m-n} &  \text{if } n < m ,\\
1   &  \text{if }n\geq m. 
\end{cases} \]
\end{theorem}

Before proceeding with the proof, we remark that, in general,
the value of $\ed_k C_{p^m}$ is not known if $\zeta_p \not \in k$. 
On the other hand, for the sake of computing $\ed_k(C_{p^m}; p)$, 
we may replace $k$ by $k(\zeta_p)$ and then apply
Theorem~\ref{t.cyclic}. Indeed, since $[k(\zeta_p) : k]$ 
is prime to $p$, $\ed_k(C_{p^m}; p) = \ed_{k(\zeta_p)}(C_{p^m}; p)$.

\begin{proof} If $m\leq n$, then $C_{p^m}=\mu_{p^m}$.  Therefore $\ed
  C_{p^m} =1$ (see~\cite[Example 2.3]{bf1}) and hence, $\ed(C_{p^m}; p) = 1$ 
as well.  We can therefore restrict our attention to the case where $n<m$.

  We first show that $\ed C_{p^m} \leq p^{m-n}$.  To do this, pick a
  faithful character $\chi\colon  C_{p^n} \arr\GG_m$ defined over $K$ and set
  $V\eqdef \ind_{C_{p^n}}^{C_{p^m}} \chi$.  A simple calculation shows
  that $V$ is faithful, thus, $V$ is generically free since $C_{p^m}$
  is finite.  By Theorem~\ref{t.GenFree}, it follows that $\ed
  C_{p^m} \leq \dim V=p^{m-n}$.

  It remains to show that $\ed(C_{p^m}; p) \ge p^{m-n}$. 
By Theorem~\ref{thm2}
it suffices to show that $\ind(C_{p^m}, C_{p^n}) \ge p^{m-n}$.  
To establish this inequality, we will view the representation $V$
as a homomorphism $\rho\colon C_{p^m} \arr \GL(V)$ of algebraic groups.  Let
$\pi:\GL(V) \arr \PGL(V)$ denote the obvious projection and note that
the kernel of $\pi\circ\rho$ is exactly $C_{p^n}$. It follows that we
have a commutative diagram
$$
\xymatrix{ 0\ar[r] & C_{p^n}\ar[r]\ar[d] & C_{p^m}\ar[r]^{\rho}\ar[d]
& C_{p^{m-n}}\ar[r]\ar[d]^{\iota} & 1\\ 1\ar[r] & \GG_m\ar[r]
&\GL(V)\ar[r]^{\pi} &\PGL(V)\ar[r] & 1 \\ }
$$
where the rows are exact and the columns are injective.

Let $K/k$ be a field extension and let $t\in \H^1(K,C_{p^m})$ be a
torsor.  Let 
$\iota_*\colon \H^1(K,C_{p^{m-n}}) \arr \H^1\bigl(K, \PGL(V)\bigr)$
denote the map induced by $\iota$.  Then, from the commutativity of
the above diagram (and the injectivity of the columns), it follows
that $\ind_K(t)$ is the index of the CSA $\iota_*(t)$.

We claim that there is a field extension $K/k$ and
a $t\in \H^1(K,C_{p^m})$ 
such that $\iota_*(t)$ is a division algebra. From this it will 
easily follow that $\ind_K(t)=\dim V = p^{m-n}$.
In fact, we will take $t\in \H^1(K,C_{p^m})$ to be a versal 
element, constructed as follows.
Let $L=K(x_1,\ldots, x_{p^{m-n}})$ denote the field obtained
by adjoining $p^{m-n}$ independent variables to $K$, and let
$C_{p^{m-n}}$ act on $L$ by cyclically permuting the variables, i.e., 
$a \cdot x_i=x_{i+a} \pmod{p^{m-n}}$ for any $a \in C_{p^{m-n}}$.
Let $K =L^{C_{p^{m-n}}}$.  Then $L/K$ defines a $C_{p^{m-n}}$-torsor $t$ 
over $K$.  

The CSA corresponding to $\iota_*(t) \in \H^1(K, \PGL_n)$ is 
the Brauer-Rowen algebra $R_{p^n, p^m, p^m}$; cf.~\cite[\S 7.3]{rowen2}.  
The fact that this algebra has index $n$ is a variant of 
a theorem of Brauer; for a proof see \cite[Theorem 2.17]{florence} or
(for $k = \QQ(\zeta_p)$ only)~\cite[Theorem~7.3.8]{rowen2}. 
\end{proof}

\smallskip
Let $D_n$ be the dihedral group of order $2n$.
Ledet~\cite[Section 3]{ledet} conjectured that if $n$ is odd
then $\ed_k C_n = \ed_k D_n$ over any field $k$ of characteristic zero. 
As an application of Theorem~\ref{t.cyclic}
we will now prove Ledet's conjecture in the case where $n = p^r$ 
is a prime power and $k$ contains a primitive $p$th root of unity.

\begin{corollary} \label{cor.ledet}
Let $p$ be an odd prime, $m \ge 1$ be an integer, and
$k$ be a field containing a primitive $p$th root of unity.
Then $\ed_k D_{p^m} = \ed_k C_{p^m}$.
\end{corollary}

\begin{proof}  If $\zeta_{p^m} \in k$ then we know that
\[ \ed_k C_{p^m} = \ed_k D_{p^m} = 1 \, ; \]
see the proof of~\cite[Theorem 6.2]{bur}.  Thus we may 
assume $\zeta_{p^m} \not \in k$.
Let $n$ be the largest integer such that
$\zeta_{p^n} \in k$. By our assumption $1 \le n \le m-1$.
% By Theorem~\ref{t.cyclic} $\ed_k C_{p^m} = p^{m - n}$.
Since $C_{p^m} \subset D_{p^m}$, we
have $\ed_k D_{p^m} \ge \ed_k C_{p^m}$.

To prove the opposite inequality, note that
$D_{p^m} \simeq C_{p^m} \rtimes C_2$
has a subgroup isomorphic to $D_{p^n} = C_{p^n} \rtimes C_2$
of index $p^{m-n}$. As we pointed out above,
$\zeta_{p^n} \in k$ implies $\ed_k D_{p^n} = 1$.  Thus
\[ \ed_k D_{p^m} \le (\ed_k D_{p^n}) \cdot [D_{p^m} : D_{p^n}]
 = 1 \cdot p^{m - n} = \ed C_{p^m} \, .  \]
where the inequality is given by \cite[Section 3]{ledet} and
the last equality by Theorem~\ref{t.cyclic}.
\end{proof}

% We conclude this section with an observation about 
% the essential dimension of the cyclic group $C_{n^2}$, 
% where $n$ is not a prime power. 
% 
% \begin{remark} \label{rem.cyclic}  
% Let $n = p_{1}^{a_{1}} \dots p_{r}^{a_{r}}$ be the prime decomposition 
% of an integer $n \ge 2$ and let $k$ be a field. Suppose
% $\zeta_n \in k$. Under this assumption, 
% $[k(\zeta_{n^2}) :k]$ always divides $n$; assume 
% $[k(\zeta_{n^2}): k] = n$ (this occurs, for example, if $k = \QQ(\zeta_n)$). 
% 
% If Conjecture~\ref{conj:ed-product} holds then
% \[ \ed_k(C_{n^2}) \ge p_1^{a_1} + \dots + p_r^{a_r} - r + 1 \, . \] 
% \end{remark}
% 
% In the case $n = 6$, where Conjecture~\ref{conj:ed-product} 
% is known to hold, this yields $\ed_k(C_{36}) \ge 4$.
% 
% \begin{proof}
% By the formula of Remark~\ref{rem.ctkm}, applied to the exact sequence
% \[ 1 \to C_n \to C_{n^2} \to C_n \to 1 \]
% of cyclic groups, it suffices to show that $\ind(C_{n^2}, C_n) = n$.
%  The boundary map 
%   \[
%    \partial_K \colon K^*/(K^*)^n  = \H^{1}(K, C_n) \arr \H^{2}(K, C_n)
%    \]
% sends $(t)$ to the class of the symbol algebra $(t, \zeta_n)_n$; 
% see, e.g.,~\cite[Theorem 7.1]{vela}. The index 
% of this symbol algebra clearly divides $n$. Taking $K = k(t)$, 
% where $t$ is an independent variable over $k$, and applying
% Wedderburn's criterion (cf. e.g., \cite[Corollary 15.1d]{pierce}), 
% one easily checks that in this case the cyclic algebra $(t, \zeta_n)_n$ has
% exponent (and hence, index) $n$. Thus $\ind(C_{n^2}, C_n) = n$, as claimed.
% \end{proof}

\section{A theorem about the essential dimension of a finite group}
\label{s.ed-pgroups}

% In the next three sections will be concerned with essential 
% dimensions of finite groups (mostly of $p$-groups). 
The following proposition, extending~\cite[Theorem 5.3]{bur} 
and \cite[Theorem 4.5]{kang}, will be used in the proof 
of Theorem~\ref{t.p-groups} in the next section.

\begin{theorem} \label{thm5.3} 
Let $G$ be a finite group, $H$ be a central cyclic subgroup 
of $G$, and $\chi \colon G \to k^*$ be a character of $G$ 
whose restriction to $H$ is faithful. Assume $\Char k$ does 
not divide $|G|$. If $H$ is maximal 
among central cyclic subgroups of $G$ then 

\smallskip
(a) $\ed_k G = \ed_k(G/H) + 1$.

\smallskip
(b) Moreover, if $G$ is a $p$-group then $\ed_k(G; p) = \ed_k(G/H; p) + 1$.
\end{theorem}

\begin{proof} 
Let $\phi \colon G \to G/H \hookrightarrow \GL(V)$
be a faithful representation of $G/H$. Then
$\phi \oplus \chi \colon G \to \GL(V \times \AA^1)$ is
a faithful representation of $G$.  

(a) To prove the inequality $\ed G \le \ed(G/H) + 1$, 
recall that $\ed(G/H)$ is the minimal dimension 
of a faithful $G/H$-variety $X$ which admits 
a dominant $G/H$-equivariant rational map $f \colon V \darr X$; see
\cite{bur}. Then 
\[ f \times \id \colon V \times \AA^1 \darr X \times \AA^1 \]  
is a dominant rational map of $G$-varieties. Thus
\[ \ed G \le \dim(X \times \AA^1) = \ed(G/H) + 1 \, . \]

We will now prove the opposite inequality, $\ed G \ge \ed(G/H) + 1$.
Choose a dominant rational map  
\[ V \times \AA^1 \darr Y \]  
of faithful $G$-varieties such that $\dim Y = \ed G$.
Equivalently, if $K = k(V)$, we choose $G$-invariant subfield $F \subset K(t)$ 
such that the $G$-action on $F$ is faithful and $\trdeg_k(F) = \ed G$.
(Here, of course, $F = k(Y)$.) 

Let $\nu \colon K(t) \to \ZZ$ be the natural valuation 
associated to $t$ (and trivial on $K$). Restricting $\nu$ 
to $F$ we obtain a discrete valuation $\nu_{\, | F}$ of $F$.
Denote the residue field for this valuation by $F_0$.

\smallskip
{\bf Claim:} (i) $\nu_{\, | F}$ is non-trivial, and 
(ii) the inertia group of $\nu_{\, | F}$ (i.e., the subgroup of
$G$ that acts trivially on $F_0$) is precisely $H$.

\smallskip
To prove (i), assume the contrary.
Then there is a natural $G$-equivariant 
embedding of $F$ in the residue field $K(t)_0 = K$. Since $H$ 
acts faithfully on $F$ and trivially on $K$, this is impossible.

To prove (ii), let $H_F \subset G$ be the inertia group
for $\nu_{\, | F}$. By our construction $H$ is the inertia group 
for the valuation $\nu$ on $K(t)$. Hence, $H \subset H_F$. 
On the other hand, since $\Char k$ is prime to $|G|$, $H_F$ is cyclic 
(see~\cite[Corollaries 2 and 3 to Proposition IV.7]{serre-lf})
and central (see~\cite[Propositioni IV.10]{serre-lf}).
By our assumption $H$ is maximal among central cyclic subgroups 
of $G$; thus $H = H_F$, proving (ii).

\smallskip
We are now ready to complete the proof of the inequality 
$\ed G \ge \ed(G/H) + 1$ (and thus of part (a)).  By (i) 
the $G$-equivariant inclusion $F \hookrightarrow K(t)$ induces 
a $G/H$-equivariant
inclusion $F_0 \hookrightarrow K(t)_0 = K$ of residue fields. In other
words, we have a rational dominant $G/H$-equivariant map  $V \darr Y_0$
where $k(Y_0) = F_0$. Moreover, by (ii) $G/H$ acts faithfully on $F_0$
(or equivalently, on $Y_0$).  Since $V$ is a faithful linear representation
of $G/H$, we have $\ed(G/H) \le \dim Y_0$. On the other hand,
\[ \dim Y_0 = \trdeg_k(F_0) = \trdeg_k(F) - 1 = \dim Y - 1 \le \ed G - 1 
\, . \]
and the desired inequality $\ed(G/H) \le \ed G - 1$ follows.

\smallskip
(b) To prove the inequality $\ed(G; p) \le \ed(G/H; p) + 1$, 
recall that $\ed(G/H)$ is the minimal dimension 
of a faithful $G/H$-variety $X$ which admits 
a diagram 
\[ \xymatrix{ V' \ar@{-->}[d]_{\text{prime-to-$p$}}
\ar@{-->}[dr] &   \cr
V & X \, , } \]
of dominant rational $G/H$-equivariant maps,  where $X$ is 
a generically free $G$-variety, $\dim V' = \dim V$
and $[k(V'): k(V)]$ is prime to $p$. (Note that, 
in the definition of $\ed(G; p)$ we require that $G$ should 
transitively permute the connected components of $V'$ (and of $X$). However, 
if $G$ is a $p$-group and the degree of the cover $V' \to V$
is prime to $p$, a simple counting argument shows that $G$ has to fix 
an irreducible component of $V'$. Consequently,
in this situation $V'$ and $X$ are necessarily irreducible.) 
Multiplying every variety 
in the above diagram by $\AA^1$ (on which $G$ acts via the character 
$\chi$), we obtain the diagram
\[ \xymatrix{ V' \times \AA^1 \ar@{-->}[d]_{\text{prime-to-$p$}}
\ar@{-->}[dr] &   \cr
V \times \AA^1 & X \times \AA^1 \, , } \]
of dominant rational map of faithful $G$-varieties, which shows that
\[ \ed(G; p) \le \dim(X \times \AA^1) = \ed(G/H; p) + 1 \, . \]

To prove the opposite inequality, $\ed(G; p) \le \ed(G/H; p) + 1$,
choose a diagram
\[ \xymatrix{ W \ar@{-->}[d]_{\text{prime-to-$p$}} \ar@{-->}[dr] &   \cr
V \times \AA^1 & Y \, , } \]
of rational maps of faithful $G$-varieties, where $[k(W): k(V \times \AA^1)]$
is finite and prime to $p$ and $\dim Y = \ed_k(G; p)$.
Equivalently, we have a diagram of field extensions
\[ \xymatrix{ L \ar@{-}[d]_{\text{prime-to-$p$}} \ar@{-}[dr] &   \cr
K(t) & F \, , } \]
where $K$, $K(t)$, $L$ and $F$ are the function fields of
$V$, $V \times \AA^1$, $W$ and $Y$, respectively. 

First observe that after replacing $L$ by the separable closure $L_s$
of $K(t)$ in $L$ and $F$ by $F \cap L_s$, we may assume without loss 
of generality that $L$ is separable over $K(t)$. Indeed, the $G$-action 
on $L$ and $F$ descends to (and remains faithful on) $L_s$ and $F \cap L_s$.
Moreover, some power of every $x \in F$ lies in $F \cap L_s$; hence,
$F$ is algebraic over $F \cap L_s$, i.e., the two have 
the same transcendence degree over $k$.  

 From now on we will assume that $L$ is separable over $K(t)$ 
(or equivalently, $W$ is separable over $V \times \AA^1$). Let $\nu$
be the natural discrete valuation on $K(t)$, by the degree in $t$ 
(the same valuation we considered in part (a)).  We claim that
there exists a $G$-invariant lifting $\eta$ of $\nu$ to $L$
whose residue degree (i.e., the degree of the induced extension
$L_0/K$ of the residue fields) in prime to $p$.

Clearly the inertia group $H_L \subset G$ of $L$ (with respect to $\eta$)
is contained in the inertia group of $K(t)$, which is equal to $H$
by our construction. On the other hand, the cyclic $p$-group $H/H_L$ 
faithfully acts on
the extension $L_0/K$ whose degree is prime to $p$. This is only
possible if $H/H_L = \{ 1 \}$, i.e., $H = H_L$. Thus if the above claim 
is established, we can use the same argument as in part (a)
(with $K(t)$, $\nu$ replaced by $L$, $\eta$) to complete the proof 
of the inequality $\ed(G; p) \le \ed(G/H; p) + 1$.

To prove the claim, denote the liftings of $\nu$ on $K(t)$ to $L$
by $\eta_1, \dots, \eta_r$. As usual, we will denote the ramification index 
of $\eta_i$ by $e_i$ and the residue degree by $f_i$.
By \cite[Proposition XII.18]{lang} 
\[ e_1 f_1 + \dots + e_r f_r = [L:K(t)] \,  \]
or, equivalently,
\[ \sum_{e, f \ge 1} |V_{e, f}| e f = [L: K(t)] \, \]
where $V_{e, f}$ be the subset of $\{ \eta_1, \dots, \eta_n \}$
consisting of those valuations $\eta_i$ with $e_i = e$ and $f_i = f$.
(Note that $V_{e, f} = \emptyset$ for all but finitely many pairs $(e, f)$.)
Thus there exist integers $e, f \ge 1$ such that $e$, $f$, and 
$|V_{e, f}|$ are all prime to $p$.
The $p$-group $G$ permutes the elements of $V_{e, f}$; since
$|V_{e, f}|$ is prime to $p$, $G$ must fixed a valuation
$\eta \in V_{e, f}$.  This is the valuation we were looking for.
(Note that the residue degree $f$ of $\eta$ is prime 
to $p$ by our construction.) This completes the proof 
of the claim and thus of part (b).
\end{proof}

\begin{corollary} \label{cor5.3}
Let $G$ be a finite $p$-group and $S$ be a direct
summand of $\rC(G)$ such that $[G, G] \cap S = \{ 1 \}$.
If $\zeta_e \in k$, where $e$ is the exponent of $S$ then 
$\ed_k G = \ed_k(G/S) + \rank(S)$.
\end{corollary}

\begin{remark} \label{rem5.2} 
The following elementary observation will be useful in
the sequel.

\smallskip
{\em Let $G$ be a finite group and $S$ be
a normal subgroup such that $[G, G] \cap S = \{ 1 \}$. 
Then $\rC(G/S) = \rC(G)/S$.}

\smallskip
To prove this assertion, suppose $g \in G$ projects 
to a central element of $G/S$.  Then for every $x \in G$ the commutator
$c = g x g^{-1} x^{-1}$ lies in both $[G, G]$ and $S$.
Hence, $c = 1$, and thus $g \in \rC(G)$, as desired.  \qed
\end{remark}

\begin{proof}[Proof of Corollary~\ref{cor5.3}]
Suppose $\rC(G)$ as $S \oplus T$. Write
$S = H_1 \oplus \dots \oplus H_r$
as a direct sum of cyclic groups, where $r = \rank(S)$. 
We argue by induction on $r$. When $r = 0$, i.e., $S = \{ 1 \}$,
there is nothing to prove.  For the induction 
step, note that by Remark~\ref{rem5.2},
\[ S/H_r \simeq H_1 \oplus \dots \oplus H_{r-1} \]
is a direct summand of $\rC(G/H_r) = S/H_r \oplus T$. Thus by
the induction assumption $\ed(G/H_r) = \ed(G/S) + r -1$.
On the other hand, by Theorem~\ref{thm5.3}
$\ed G = \ed(G/H_r) + 1$, and the corollary follows. 
\end{proof}

\section{Proof of Theorem~\ref{t.p-groups}}

Let $Z$ be a maximal cyclic subgroup of $\rC(G)$ containing $[G, G]$.
Then $\rC(G) = Z \oplus S$ for some central subgroup $W$ of $G$. 
By Corollary~\ref{cor5.3}, 
\[ \ed G = \ed(G/S) + \rank(S) = \ed(G/S) + \rank \, \rC(G)  - 1 \, . \] 
By Remark~\ref{rem5.2}, $|\rC(G/S)| = |\rC(G)|/|S|$. Thus the quantity
$|G/\rC(G)|$ does not change when we replace $G$ by $G/S$.
We conclude that it suffices to prove Theorem~\ref{t.p-groups}
for $G/S$. In other words, we may assume without loss of 
generality that the center $Z = \rC(G)$ of $G$ is cyclic. 
In this case the theorem reduces to the identity
\[ \ed(G; p) = \ed G = \sqrt{|G/Z|} \, . \]
To prove this identity it suffices to establish 
Lemma~\ref{lem.p-groups2} below.
Indeed, part (a) of this lemma tells us that
$\ed(G; p) \geq \sqrt{|G/Z|}$ (see Theorem~\ref{thm2}), and 
part (b) implies the inequality,
$\ed G \leq \sqrt{|G/Z|}$ (see Theorem~\ref{t.GenFree}), and
Theorem~\ref{t.p-groups} follows. 

\begin{lemma} \label{lem.p-groups2}
Let $G$ be a $p$-group such that the center $Z = \rC(G)$ is cyclic
and the quotient group $A \eqdef G/Z$ is abelian. 
(or equivalently, $[G, G] \subset Z$).  Then

\smallskip
(a) $\ind(G,Z) \geq \sqrt{|A|}$, and

\smallskip
(b) $G$ has a faithful linear representation of dimension $\sqrt{|A|}$.
\end{lemma}

\begin{proof} We will use additive notation for the groups $Z$ and $A$, 
multiplicative for $G$. In this situation we can define a skew-symmetric
bilinear form $\omega \colon A \times A  \arr Z$ by
   \[
   \omega(a_1, a_2) = g_1g_2 g_1^{-1} g_2^{-1} \, ,
   \]
where $a_i = g_i$, modulo $Z$, for $i = 1, 2$. (Note that
$\omega(a_1, a_2)$ is independent of the choice of $g_1$ and $g_2$.)
Clearly $g$ lies in $\rC(G)$ if and only if its image $a$ lies in
the kernel of $\omega$; i.e., $\omega(a,b) = 0$ for every $b \in A$. 
Since we are assuming that $\rC(G) = Z$, we conclude that the kernel of
$\omega$ is trivial; i.e., $\omega$ is a {\em symplectic form} on $A$.
It is well known (see for example 
\cite[\S 3.1]{tignol-amitsur}) that the order of $A$, 
which equals the order of $G/\rC(G)$, is a complete square.

Fix a generator $z$ of $Z$. We recall the basic 
result on the structure of a symplectic form $\omega$ on
a finite abelian group $A$ (the proof is 
easy; it can be found, e.g., in~\cite[\S 3.1]{wall}
or~\cite[\S 7.1]{tignol-amitsur}). 
There exist elements $a_1$, \dots,~$a_{2r}$ in $A$ 
and positive integers $d_{1}$, \dots,~$d_{r}$ with 
the following properties.

\begin{enumeratea}
\item $d_{i}$ divides $d_{i-1}$ for each $i = 2$, \dots,~$r$, 
and $d_{r} > 1$.

\item Let $i$ be an integer between $1$ and $r$. 
If $A_{i}$ denotes the subgroup of $A$ generated by $a_i$ and $a_{r+i}$, 
then there exists an isomorphism $A_{i} \simeq 
(\ZZ/d_{i}\ZZ)^{2}$ such that 
$a_{i}$ corresponds to (1,0) and $a_{r+i}$ to $(0,1)$.

\item The subgroups $A_{i}$ are pairwise orthogonal 
with respect to $\omega$.

\item $\omega(a_{i}, a_{r+i}) = z^{n/d_{i}}\in Z$.

\item $A = A_{1} \oplus \dots \oplus A_{r}$.
\end{enumeratea}


Then the order of $A$ is $d_{1}^{2} \dots d_{r}^{2}$, 
hence $\sqrt{|A|} = d_{1} \dots d_{r}$.

Let $G_{i}$ be the inverse image of $A_{i}$ in $G$; 
note that $G_{i}$ commutes with $G_{j}$ for any $i \ne j$.

Let $u_{1}$, \dots,~$u_{2r}$ be indeterminates, and 
set $K \eqdef k(u_{1}, \dots, u_{2r})$. 
Identify $Z$ with $\mu_{n}$ by sending $z$ into $\zeta_{n}$. 
Consider the boundary map
   \[
   \partial_{i}\colon \H^{1}(K, A_{i}) \arr \H^{2}(K, Z)
   \]
obtained from the exact sequence
   \[
   1 \arr Z \arr G_{i} \arr A_{i} \arr 1.
   \]

\begin{claim}
There exists a class $\xi_{i} \in \H^{1}(K,A_{i})$ 
such that $\partial_{i}\xi_{i}$ is the class of the cyclic 
algebra $(u_{i}, u_{r+i})_{d_{i}}$ in $\Br K$.
\end{claim}

To see that part (a) follows from the claim,
consider the commutative diagram
   \[
   \xymatrix{
   1 \ar[r] & Z ^{r} \ar[r]\ar[d]^{m}& {}\prod_{i} G_{i}\ar[r] \ar[d]
     & \prod_{i}A_{i} \ar[r]\ar[d] & 1\\
   1 \ar[r] & Z \ar[r] & G \ar[r] & A \ar[r] & 1
   }
   \]
in which $m$ is defined by the formula 
$m(z_{1}, \dots, z_{r}) = z_{1} \dots z_{r}$, and 
the homomorphism $\prod_{i} G_{i} \arr G$ is induced
by the inclusions $G_{i} \subseteq G$. This yields 
a commutative diagram
   \[
   \xymatrix{
   {}\prod_{i}\H^{1}(K,A_{i}) \ar[r]^{\prod_{i}\partial_{i}} \ar[d]\ar[r]&
   {}\H^{2}(K, Z)^{r}  \ar[d]^{m_{*}}\\
   {}\H^{1}(K,A) \ar[r]^{\partial} &{}\H^{2}(K,Z)
   }
   \]
in which the map $m_{*}$ is given by
$m_{*}(\alpha_{1}, \dots, \alpha_{r}) = \alpha_{1} \dots \alpha_{r}$. 
So, if $\xi \in \H^{1}(K,A)$ is the image of 
$(\xi_{1}, \dots , \xi_{r})$ then $\partial\xi$ 
is the class of the product
   \[
   (u_{1}, u_{r+1})_{d_{1}} \otimes_{K} (u_{2}, u_{r+2})_{d_{2}}
      \otimes_{K} \dots \otimes_{K} (u_{r}, u_{2r})_{d_{r}},
   \]
whose index is $d_{1} \dots d_{r}$. Hence 
$\ind(G,Z) \geq d_{1} \dots d_{r} = \sqrt{|A|}$, as needed.

We now proceed with the proof of the claim. Choose a power of $p$, 
call it $d$, that is divisible by the order of $Z$ 
and by the order of each $a_{i}$. 
Consider the group $\Lambda(d)$ defined by the presentation
   \[
   \langle x_1, x_2, y \mid x_1^d = x_2^d = y^d = 1 , \; 
   x_1 x_2 = y x_2 x_1, \; x_1 y = yx_1, \; x_2 y = y x_2 \rangle.
   \]
Call $\rho_{i}\colon \Lambda(d) \arr G_{i}$ the homomorphism obtained 
by sending $x_{1}$ to $a_{i}$, $x_{2}$ to 
$a_{r+i}$, and $y$ to $z^{n/d_{i}} = \omega(a_{i}, a_{r+i})$.

Let $\zeta_{d}$ be a primitive $d$-th root of $1$ 
in $k$ such that $\zeta_{n} = \zeta_{d}^{n/d}$. 
The subgroup $\generate{y}$ in $\Lambda(d)$ is 
cyclic of order $d$; we fix the isomorphism 
$\generate{y} \simeq \mu_{d}$ so that $y$ corresponds 
to $\zeta_{d}$. The restriction of $\rho_{i}$ 
to $\generate{y} \arr Z$ corresponds to 
the homomorphism $\mu_{d} \arr \mu_{n}$ defined 
by $\alpha \arrto \alpha^{d/d_{i}}$. We have a commutative diagram
   \[
   \xymatrix{
   1 \ar[r]
 &\ar[r]\ar[d]_{\substack{\alpha\\\downarrow\\\alpha^{d/d_{i}}}{}} \mu_{d} & 
     {}\Lambda(d)\ar[r]\ar[d]^{\rho_{i}}
     & {}(\ZZ/d\ZZ)^{2} \ar[d]\ar[r] & 1\\
   1 \ar[r] & \mu_{n} \ar[r] & G_{i} \ar[r] & A_{i} \ar[r] & 1\hsmash{.}
   }
   \]
We have $\H^{1}\bigl(K, (\ZZ/d\ZZ)^{2}\bigr) = (K^{*}/{K^{*}}^{d})^{2}$. According to \cite[Example 7.2]{vela}, the image of the element 
$(u_{i}, u_{r+i})\in \H^{1}\bigl(K, (\ZZ/d\ZZ)^{2}\bigr)$ 
is the cyclic algebra $(u_{i}, u_{r+i})_{d}$; 
hence, if $\xi_{i}$ is the image in $\H^{1}(K, A_{i})$ 
of $(u_{i}, u_{r+i})$, the image of $\xi_{i}$ in 
$\H^{2}(K, \mu_{d})$ is the algebra 
$(u_{i}, u_{r+i})_{d}^{\otimes d/d_{i}}$, which is equivalent 
to $(u_{i}, u_{r+i})_{d_{i}}$. This concludes the proof of 
Lemma~\ref{lem.p-groups2}(a).

We now turn to the proof of Lemma~\ref{lem.p-groups2}(b). 
Suppose $|\rC(G)| = p^{h}$ and $|A| = p^{2m}$; we want to construct 
a faithful representation of $G$ of dimension~$p^{m}$. 
By \cite[\S 3.1]{tignol-amitsur} $A$ contains a Lagrangian 
subgroup $L$ of order $p^{m}$. 
Denote by $H$ the inverse image of $L$ in $G$; 
then $H$ is an abelian subgroup of $G$ of order $p^{h+m}$. 
Since $\zeta_{p^{e}} \in k$ we can embed $Z$ in $k^{*}$ 
and extend this embedding to a homomorphism 
$\chi\colon H \arr k^{*}$. We claim that 
the representation $\rho\colon G \arr \GL_{p^{m}}$ 
induced by $\chi$ is faithful.

It is enough to show that $\rho(g) \neq \id$ for any $g \in G$ 
of order~$p$, or, equivalently, that 
$\rho\mid_{\generate{g}}$ is non-trivial for any such $g$. 
If $s \in G$ consider the subgroup $H_{s} 
\eqdef s\generate{g}s^{-1} \cap H$ of $H$, which is embedded 
in $\generate{g}$ via the homomorphism 
$x \arrto s^{-1}xs$. By Mackey's formula 
(\cite[\S 7.3]{serre}), $\rho\mid_{\generate{g}}$  
contains all the representations of $\generate{g}$ 
induced by the restrictions $\chi\mid_{H_{s}}$ via the embedding above.

If $g \notin H$ then 
$H_{1} = \generate{g} \cap H = \{1\}$: 
we take $s = 1$, and we see that $\rho\mid_{\generate{g}}$ 
contains a copy of the regular representation 
of $\generate{g}$, which is obviously non-trivial.

Assume $g \in H$. Then $H_{s} = \generate{sgs^{-1}}$ for any $s \in G$; it is enough to prove that $\chi(sgs^{-1}) \neq 1$ for some $s \in G$. If $\chi(g) \neq 1$ then we take $s = 1$. Otherwise $\chi(g) = 1$; in this case $g \notin \rC(G)$, because $\chi\mid_{\rC(G)}\colon  \rC(G) \arr k^{*}$ is injective. Hence the image $\overline{g}$ of $g$ in $A$ is different from $0$, and we can find $s \in G$ such that $\omega(\overline{s}, \overline{g}) \neq 1$. Then 
\[   \chi(sgs^{-1}) = \chi\bigl(\omega(\overline{s}, \overline{g})g\bigr)
   = \chi\bigl(\omega(\overline{s}, \overline{g})\bigr) \chi(g)
   = \chi\bigl(\omega(\overline{s}, \overline{g})\bigr)
   \neq 1. \]
This concludes the of Lemma~\ref{lem.p-groups2} 
and thus of Theorem~\ref{t.p-groups}.
\end{proof}

\section{More on $p$-groups}
\label{s.ed-pgroups2}

In this section we will discuss some
consequences of Theorem~\ref{t.p-groups}.

\begin{example} \label{ex.extraspecial}
Recall that a $p$-group $G$ is called \emph{extra-special} if its 
center $Z$ is cyclic of order $p$, and the quotient $G/Z$ 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[pp. 145--146]{robinson}.  Note that every non-abelian 
group of order $p^{3}$ is extra-special. For extra-special
$p$-groups Theorem~\ref{t.p-groups} 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 $\ed(G; p) = \ed G = p^{m}$.}
\end{example}

\begin{example} \label{ex3.p-groups}
Let $p$ be an odd prime and $G = C_{p^r} \ltimes C_{p^s}$ be the natural
semidirect product of cyclic groups of order $p^r$ and $p^s$
(in other words, $C_{p^s}$ is identified with the unique subgroup
of $C_{p^r}^*$ of order $p^s$).  If $s \le r/2$ then
\[ \ed_k(C_{p^r} \ltimes C_{p^s}; p) =
\ed_k(C_{p^r} \ltimes C_{p^s}) = p^s \, , \] 
for any field $k$ containing a primitive $p$th root of unity $\zeta_p$.
\end{example}    

\begin{proof} The center $\rC(G)$ of $G$ is the (unique) subgroup 
of $C_{p^r}$ of order
$p^s$. Since $s \le r/2$, this subgroup is central. Thus, 
if $\zeta_{p^r} \in k$, the equality $\ed_k G = p^s$
is an immediate consequence of Theorem~\ref{t.p-groups}. But
we are only assuming that $\zeta_p \in k$, so
Theorem~\ref{t.p-groups} only tells us that $\ed_k G \ge p^s$.  
To prove the opposite inequality, we argue as follows.
Let $F$ be the prime subfield of $k$.
By~\cite[Corollary to Proposition 2]{ledet},
$\ed_{F(\zeta_p)}(G) \le p^s$.
Since we are assuming that $F(\zeta_p) \subset k$, 
we conclude that $\ed_k G \le p^s$ as well.
\end{proof}

\begin{corollary} \label{cor2.p-groups}
Suppose $k$ is a base field of characteristic $\neq p$.
If $G$ is a non-abelian finite $p$-group then $\ed(G; p) \ge p$.
\end{corollary}

\begin{proof}
Assume the contrary: let $G$ be a non-abelian $p$-group 
of smallest possible order such that $\ed G < p$. 
Since $G$ has a non-trivial center,  
there exists a cyclic central subgroup $Z \subset G$.
The short exact sequence 
\begin{equation} \label{e.exact-sequence}
1  \arr Z  \arr G  \arr G/Z  \arr 1
\end{equation}
gives rise to the exact sequence of pointed sets
   \[
   \H^1(K, G) \arr \H^1(K, G/Z) \stackrel{\partial_K}\larr \H^2(K, Z)
   \]
for any field extension $K$ of our base field $k$.
We will now consider two cases.

{\bf Case 1.} Suppose the map $\H^1(K, G) \arr \H^1(K,
G/Z)$ is not surjective for some $K/k$. Then $\partial_K$ is
non-trivial, and Theorem~\ref{thm2} tells us that $\ed(G; p) \ge p$, a
contradiction.

{\bf Case 2.} Suppose the map $\H^1(K, G)  \arr \H^1(K, G/Z)$ is 
surjective for every $K/k$. Then
the morphism $\cB G \arr \cB(G/Z)$ is isotropic, and 
Proposition~\ref{p.isotropic} implies that
$p > \ed(G; p) \ge \ed(G/Z; p)$. By the minimality of $G$, the group 
$G/Z$ has to be abelian. Consequently, $[G, G] \subset Z$ is cyclic 
and central in $G$. Since $G$ is non-abelian, $|G/\rC(G)| \ge p^2$. 
Theorem~\ref{t.p-groups} now tells us that 
\[ \ed(G; p) = \sqrt{|G|/|\rC(G)|} + \rank \, \rC(G) - 1 \ge p \, , \] 
a contradiction.
\end{proof}

We will conclude this section by answering the following question
of Jensen, Ledet and Yui~\cite[p. 204]{jly}.

\begin{question}
\label{q:ljy} Let $G$ be a finite group and $N$ be a normal subgroup.
Is it true that $\ed G \ge \ed(G/N)$?
\end{question}

The inequality $\ed G \ge \ed(G/N)$ is known to hold in many 
cases; cf., e.g., Theorem~\ref{thm5.3}. 
We will now show that it does not hold in general, even if $H$ 
is assumed to be central.

\begin{corollary} \label{cor.jly}
For every real number $\lambda > 0$ there exists a finite $p$-group $G$, 
with a central subgroup $H \subset G$ such that $\ed(G/H) > \lambda \ed G$.
\end{corollary}

\begin{proof} Let $\Gamma$ be a non-abelian group of order $p^3$.
The center of $\Gamma$ has order $p$; denote it by $C$. The
center of $\Gamma^n = \Gamma \times \dots \times \Gamma$ 
($n$ times) is then $C^n$. Let $H_n$ be the subgroup of $C^n$ 
consisting of $n$-tuples $(c_1, \dots, c_n)$ such that 
$c_1 \dots c_n = 1$.  Clearly 
   \[
   \ed\Gamma^n \le n \cdot \ed\Gamma = np \, ;
   \]
see Example~\ref{ex.extraspecial}.

On the other hand, $\Gamma^n/H_{n}$, is easily seen to be extra-special 
of order $p^{2n+1}$, so $\ed(\Gamma^n/H_{n}) = p^{n}$, again by 
Example~\ref{ex.extraspecial}. Setting $G = \Gamma^n$ and
$H = H_n$, we see that the desired inequality
$\ed(G/H) > \lambda \ed G$, holds for suitably large $n$.
\end{proof}

\section{Spinor groups}
\label{s.spinor}

In this section we will prove Theorem~\ref{thm.spin} stated in the
introduction. 
% For an introduction to the structure of this group
% and the theory of Clifford algebras and spin modules, we refer the reader
% to \cite[\S 20.2]{fulton-harris} and \cite{chevalley}.
% 
% Recall that by $\Spin_{n}$ we mean the totally split form of the spin
% group in dimension~$n$ over a field of characteristic not equal to
% $2$.  Let us be explicitly about this.

As usual, we will write 
$\langle a_1,\ldots, a_n \rangle$ for the rank $n$-quadratic form $q$
given by $q(x_1,\ldots, x_n)=\sum_{i=1}^n a_i x_i^2.$   Set 
$h$ to be the standard hyperbolic quadratic form given by
$h(x,y)=xy$.  (Thus $h\cong \langle 1, -1 \rangle$).  For each
$n\ge 0$ define
\begin{equation}
h_n =
\begin{cases}
  h_n^{\oplus n/2}, & \text{if $n$ is even,}\\
  h_n^{\oplus (n-1/2)} \oplus \langle 1 \rangle, & \text{if $n$ is odd.}
\end{cases}
\end{equation}
Set $\Spin_n=\Spin(h_n)$; this is the totally split spin group
which appears in the statement of Theorem~\ref{thm.spin}.
We also denote the totally split orthogonal and special orthogonal groups by
$\Orth_n\eqdef\Orth(h_n)$ and $\SO_n\eqdef\SO(h_n)$.

Now, one of the hypotheses of Theorem~\ref{thm.spin} is that
$\zeta_{4} \in k$.  Therefore we can write $\Spin_n$ 
as $\Spin(q)$, where
   \[
   q(x_{1}, \dots, x_{n}) = -(x_{1}^{2} + \dots + x_{n}^{2}).
   \]

Consider the subgroup $\Gamma_{n} \subseteq \SO_{n}$ consisting 
of diagonal matrices, which is isomorphic to 
$\mu_{2}^{n-1}$. Call $G_{n}$ the inverse image of 
$\Gamma_{n}$ in $\Spin_{n}$. It is a constant group scheme over $k$.  
Denote by $\mu_{2}$ the kernel 
of the homomorphism $\Spin_{n} \arr \SO_{n}$.

\begin{lemma}
Every $\Spin_{n}$-torsor over an extension $K$ of $k$ admits reduction 
of structure to $G_{n}$; i.e., the natural map
$\H^1(K, G_n) \to \H^1(K, \Spin_n)$ is surjective for 
any field extension $K/k$.
\end{lemma}

\begin{proof}
Let $P \arr \Spec K$ be a $\Spin_{n}$-torsor: we are claiming that 
the $K$-scheme $P/G_{n}$ has a rational point. We have 
$P/G_{n} = (P/\mu_{2})/\Gamma_{n}$. However 
$P/\mu_{2} \arr \Spec K$ is the $\SO_{n}$ torsor associated 
with $P \arr \Spec K$, and every $\SO_{n}$-torsor admits
reduction of structure group to $\Gamma_{n}$.
\end{proof}

This means that the natural morphism $\cB G_{n} \arr \cB\Spin_{n}$ 
is isotropic; so from Corollary~\ref{cor.RelDim-quotients} 
and Proposition~\ref{p.isotropic} we get the bounds
   \begin{equation}\label{eq:bound-spin}
   \ed G_{n} - \dim\Spin_{n} \leq \ed\Spin_{n} \leq \ed G_{n};
   \end{equation}
cf.~also \cite[Lemma 1.9]{bf1}.
Of course $\dim \Spin_{n} = n(n-1)/2$; we need to compute 
$\ed G_{n}$. The structure of $G_{n}$ is well understood; 
in particular, it is very clearly described in \cite{wood}. 
Recall that $\Spin_{n}$ is a subgroup of the group 
of units in the Clifford algebra $A_{n}$ of the quadratic 
form $-(x_{1}^{2} + \dots + x_{n}^{2})$. The algebra $A_{n}$ 
is generated by elements $e_{1}$, \dots,~$e_{n}$, 
with relations $e_{i}^{2} = -1$ and 
$e_{i}e_{j} + e_{j}e_{i} = 0$ for all $i \neq j$. 
The element $e_{i}$ is in $\Pin_{n}$, and image 
of $e_{i}$ in $\rO_{n}$ is the diagonal matrix with 
$-1$ as the $i$-th diagonal entry, and $1$ as all the other 
diagonal entries. The kernel of the homomorphism 
$\Pin_{n} \arr \rO_{n}$ is $\{\pm 1\}$.
(For background material on the theory of Clifford algebras 
and spin modules, we refer the reader to~\cite{chevalley}, 
\cite[\S 20.2]{fulton-harris} or~\cite{adams}).

For any $I \subseteq \{1, \dots, n\}$ 
write $I = \{i_{1}, \dots, i_{r}\}$ with 
$i_{1} < i_{2} < \dots < i_{r}$ and set 
$e_{I} \eqdef e_{i_{1}} \dots e_{i_{r}}$. 
The group $G_{n}$ consists of the elements of 
$A_{n}$ of the form $\pm e_{I}$, 
where $I \subseteq \{1, \dots, n\}$ has an even number 
of elements. The element $-1$ is central, and 
the commutator $[e_I, e_J]$ is given by
   \[
   [e_{I}, e_{J}] = (-1)^{|I \cap J|} .
   \]
It is clear from this description that $G_n$ is a $2$-group
of order $2^n$, the commutator $[G_n, G_n] = \{ \pm 1 \}$ is cyclic,
and the center $\rC(G)$ is given by 
\[ \rC(G_n) = \begin{cases} 
\text{$\{ \pm 1 \} \simeq \ZZ/2 \ZZ$, if $n$ is odd,} \\
\text{$\{ \pm 1, \pm e_{\{ 1, \dots, n \}} \} \simeq \ZZ/4 \ZZ$, 
if $n \equiv 2$ (mod $4$),} \\
\text{$\{ \pm 1, \pm e_{ \{ 1, \dots, n \} } \} \simeq \ZZ/2 \ZZ \times \ZZ/2 \ZZ$, 
if $n$ is divisible by $4$.} \end{cases} \]
Theorem~\ref{t.p-groups} now tells us that
\[ \ed(G_n) = \begin{cases} \text{$2^{(n-1)/2}$, if $n$ is odd,} \\ 
 \text{$2^{(n-2)/2}$, if $n \equiv 2$ (mod $4$),} \\
 \text{$2^{(n-2)/2} + 1$, if $n$ is divisible by $4$.} \end{cases} \]
Substituting this into~\eqref{eq:bound-spin}, we obtain the bounds of
Theorem~\ref{thm.spin}.
\qed

\begin{remark} \label{rem.pin} The same argument can be applied to
$\Pin$ groups. (For the definition of $\Pin$ groups, see, e.g., \cite{adams}.)
Here we replace $G_n$ the inverse image $G_n'$ 
of the diagonal subgroup of $O_n$ in $\Pin_n$. One easily checks that
\[ \rC(G_n') \simeq \begin{cases} 
\text{$\ZZ/2 \ZZ$, if $n$ is even,} \\
\text{$\ZZ/4 \ZZ$, if $n \equiv 1$ (mod $4$),} \\
\text{$\ZZ/2 \ZZ \times \ZZ/2 \ZZ$, 
if $n \equiv 3$ (mod $4$).} \end{cases}  \]
This yields the following bounds on the essential dimensions of $\Pin_n$:
\begin{eqnarray*} 
2^{\lfloor n/2 \rfloor} - \frac{n(n-1)}{2} \le
   \ed_k \Pin_{n} \le  2^{\lfloor n/2 \rfloor}, & 
\text{if $n \not\equiv 1 \pmod 4$,} \\
2^{\lfloor n/2 \rfloor} - \frac{n(n-1)}{2} + 1 \le
   \ed_k \Pin_{n} \le  2^{\lfloor n/2 \rfloor} + 1, &
\text{if $n \equiv 1 \pmod 4$.} \end{eqnarray*}
for any field $k$ of characteristic $\ne 2$
containing $\zeta_4$. 
\end{remark}

\begin{remark}
When $n \leq 14$ the lower bound of Theorem~\ref{thm.spin} is 
negative and the upper bound is much larger than the true value 
of $\ed\Spin_{n}$. For $n = 15$ and $16$ our inequalities yield
   \[
   23 \leq \ed \Spin_{15} \leq 128
   \]
and
   \[
   9 \leq \ed \Spin_{16} \leq 129.
   \]
When $n = 16$ our lower bound coincides with the lower 
bound~\eqref{e.old-spin} of Reichstein--Youssin and 
Chernousov--Serre, while for $n = 15$ it is substantially larger. 
When $n \geq 17$ the exponential part of the lower bound takes 
over, the growth becomes fast and the gap between the lower 
bound and the upper bound proportionally small. For values 
of $n$ close to $15$ our estimates are quite imprecise; 
it would be interesting to improve them.
\end{remark}

\begin{remark} 
The lower bounds in Theorem~\ref{thm.spin}
hold over any field of characteristic 
different from $2$ (and for any form of the Spin group).

On the other hand, if we do not assume that $\zeta_{4} \in k$,
we have the following slightly weaker upper bound
   \[
   \ed\Spin_{n} \leq 2^{\lfloor(n-1)/2\rfloor} + n - 1
   \]
for the totally split form of the spin group in dimension~$n$, 
over $k$.  To prove this inequality, we observe that
a generically free representation of $\Spin_{n}$ 
can be constructed 
by taking a spin, or half-spin, representation $V$ of 
$\Spin_{n}$ of dimension $2^{\lfloor(n-1)/2\rfloor}$, 
and adding a generically free representation $W$ of $\SO_{n}$. 
Since the essential dimension of $\SO_{n-1}$ 
is $n-1$ over any field of characteristic different from~$2$,
there is an $\SO_n$-equivariant dominant rational map 
$f \colon W \darr X$, where $\dim X = 
\dim\SO_n + n-1$. Now
$\id \times f \colon V \times W \darr V \times X$ is 
a $\Spin_n$-equivariant dominant rational map 
$V \times W$.  Consequently, 
\begin{align*}
 \ed \Spin_n & \ge \dim(V \times X) - \dim\Spin_n \\
 & =  2^{\lfloor(n-1)/2\rfloor} + \dim\SO_n + n - 1 - \dim\Spin_n \\
 & = 2^{\lfloor(n-1)/2\rfloor} + n - 1 \, , \end{align*}
as claimed.
\end{remark}

\begin{remark} \label{rem.clifford}
It is natural to ask whether the inequality
\[ \ed \Spin_n  \ge 2^{\lfloor (n -1)/2 \rfloor} - \frac{n(n-1)}{2} \]
can be proved by a direct application of Theorem~\ref{thm.spin}
to the exact sequence 
\begin{equation}
\label{e.SpinExact}
 1 \arr \mu_2 \arr \Spin_n \arr \SO_n \arr 1 \,  
\end{equation}
without considering the finite subgroup $G_n$ of $\Spin_n$.
The answer is ``yes.'' 
%%%%%%%%%%%%%%%
% Note, however, that we see no way of obtaining
% the slightly stronger lower bound which we obtain when $n$ is
% divisible by $4$.
%%%%%%%%%%%%%%%

Indeed, consider the associated coboundary map
 \[ \xymatrix{ \H^1(K, \SO_{m}) \ar@{->}[r]^{\partial_K} & \H^2(K, \mu_2).} \]
A class in $\H^1(K, \SO_m)$ is represented by a $m$-dimensional
quadratic form $q$ of discriminant $1$ defined over $K$. The
class of $\partial_K(q) \in \H^2(K, \mu_2)$ is then
the Hasse-Witt invariant of $q$; following Lam~\cite{lam},
we will denote it by $c(q)$. (Note that since we are assuming that
$-1$ is a square in $k$, the Hasse invariant and the Witt
invariant coincide; see \cite[Proposition V.3.20]{lam}.)
Our goal is thus to show that for every $n \ge 1$ there exists 
a quadratic form $q_n$ of dimension $n$ and discriminant $1$ such 
that $c(q_n)$ has index $2^{\lfloor (n -1)/2 \rfloor}$.

If $n$ is even this is proved in~\cite[Lemma 5]{m2}.
(Note that in this case $c(q) \in \H^2(K, \mu_2)$ is the class of               
the Clifford algebra of $q$.) If $n = 2r + 1$ is odd, set 
$K = k(a_1, b_1, a_2, b_2, \dots)$, where 
$a_1, b_1, a_2, b_2, \dots$ are independent variables,
and define $q_{2r +1}$ recursively by                      
\[ \text{$q_1 = \langle 1 \rangle$ and 
$q_{2r + 1} = \langle a_r, b_r \rangle \oplus
a_r b_r \rangle \otimes q_{2r -1} \rangle$.} \]   
One easily sees by induction on $r$ that every $q_{2r + 1}$
has discriminant $1$. Moreover, a direct computation 
using basic properties of the Hasse-Witt invariant 
(see, e.g., \cite[V.3.15 and 3.16]{lam}) shows that 
$c(q_{2r + 1})$ is the class of the tensor product 
$(a_1, b_1)_2 \otimes_K \dots \otimes_K (a_r, b_r)_2$ 
of quaternion algebras.  This class has index $2^r$, as claimed.
\qed

In summary, this approach recovers the lower bound of Theorem~\ref{thm.spin} 
in the case where $n$ is not divisible by $4$. In the case where
$n$ is divisible by $4$ Theorem~\ref{thm.spin} gives a slightly 
stronger lower bound. 
% it cannot be used to do so if $n$ is divisible by $4$ (in the latter 
% Any lower bound on $\ed \Spin_n$ obtained in this way
% (i.e., directly from Theorem~\ref{thm2b}) will necessarily 
% be of the form $\ed \Spin_n  \ge 2^m - \frac{n(n-1)}{2}$ for some 
% integer $m \ge 0$.  
% Thus, while this approach recovers the lower 
% bound of Theorem~\ref{thm.spin} if $n$ is not divisible by $4$,
% it cannot be used to do so if $n$ is divisible by $4$ (in the latter 
% case the lower bound of Theorem~\ref{thm.spin} is slightly stronger). 
\end{remark}

To conclude this section, we will now prove similar bounds on the essential 
dimensions on half-spin groups. We begin with the following
simple corollary of~\cite[Theorem 1.1]{cgr}, 
which appears to have been previously overlooked.

\begin{lemma} \label{lem.cgr}
Let $G$ be a closed (but not necessarily connected)
subgroup of $\GL_{n}$ defined over a field $k$. Assume 
that $\Char k = 0$ and either $k$ is algebraically closed or
$G$ is connected.  Then $\ed G \leq n$.
\end{lemma}

\begin{proof}
By~\cite[Theorem~1.1]{cgr}, there exists a finite 
$k$-subgroup $S \subseteq G$ such that every $G$-torsor over $\spec K$, 
admits reduction of structure to $S$, for any field extension $K/k$.
This means that the morphism $\cB S \arr \cB G$ is isotropic 
and thus \[ \ed G \leq \ed S \, ; \]
cf. also~\cite[Lemma 1.9]{bf1}. 
The restriction of the representation 
$G \subseteq \GL_{n}$ to $S$ is faithful; since $S$ is a finite group 
over a field of characteristic zero, any faithful representation of $S$ 
is necessarily generically free. By Theorem~\ref{t.GenFree},
$\ed S \leq n$, and hence, $\ed G \leq n$, as claimed.
\end{proof}

\begin{example} \label{ex.cgr1}
Suppose $G/k$ satisfies one of the conditions 
of Lemma~\ref{lem.cgr} and the centralizer $C_G(G^0)$
of the connected component of $G$ is trivial. Then 
the adjoint representation of $G$ is faithful and
Lemma~\ref{lem.cgr} tells us that $\ed G \le \dim G$.
In particular, this inequality is valid for every connected semisimple 
adjoint group $G$. (In the case of simple adjoint groups, a stronger 
bound is given by \cite[Theorem 1.3]{lemire}.)
\end{example}

We are now ready to proceed with our bounds on the essential 
dimension of half-spin groups. Recall that 
the \emph{half-spin group} $\HSpin_{n}$ is defined, for every $n$ 
divisible by $4$, as $\Spin_{n}/\generate{\eta}$, where $\eta$ 
is an element of the center of $\Spin_{n}$ different from $-1$. 
(There are two such elements, but
the resulting quotients are isomorphic.)

\begin{theorem} \label{thm.half-spin}
(a) Suppose $k$ is a field of characteristic $0$ and $\zeta_{4} \in k$. Then
   \[
   2^{(n-2)/2} - \frac{n(n-1)}{2} \leq \ed\HSpin_{n} \leq 2^{(n-2)/2}
   \]
for any positive integer $n$ divisible by $4$. 
\end{theorem}

The conditions that $\Char k = 0$ and $\zeta_4 \in k$ are used 
only in the proof of 
the upper bound.  The lower bound of Theorem~\ref{thm.half-spin}
remains valid for any base field $k$ of characteristic $\ne 2$.

\begin{proof}
The group $\HSpin_{n}$ contains $G_{n}/\generate{\eta} \simeq G_{n-1}$, 
which is an extra-special group of order $2^{n-1}$. By 
Example~\ref{ex.extraspecial}
$\ed(G_{n}/\generate{\eta}) = 2^{(n-2)/2}$ and thus
%   \begin{align*}
\[   \ed \HSpin_n \ge 
\ed(G_{n}/\generate{\eta}) - \dim \HSpin_n = 2^{(n-2)/2} - \frac{n(n-1)}{2} 
\, , \]
%    \end{align*}
as in the proof of Theorem~\ref{thm.spin}.

For the upper bound notice that one of the two half-spin 
representations of $\Spin_{n}$ descends to $\HSpin_{n}$, 
and is a faithful representation of $\HSpin_{n}$ of dimension
$2^{(n-2)/2}$. The upper bound now follows from Lemma~\ref{lem.cgr}
\end{proof}

\section{Pfister numbers}
\label{s.arason}

Let $k$ be a field of characteristic not equal to $2$ and write
$\rW(k)$ for the Witt ring of $k$; see~\cite[Chapter 2]{lam}.  Let
$I=I(k)$ denote the ideal of all even dimensional forms in the Witt
ring.  Then, for any integer $a>0$, $I^a$ is generated as an abelian
group by the $a$-fold Pfister forms~\cite[Proposition 1.2]{lam}.

Let $q$ be a quadratic form of rank $n>0$ whose class $[q]$ in $\rW(k)$
lies in $I^a$ for $a>0$.  Define the \emph{$a$-Pfister number} of $q$ to be
the minimum number $r$ appearing in a representation 
$$
q=\sum_{i=1}^r \pm p_i
$$
with the $p_i$ being $a$-fold Pfister forms.  The
\emph{$(a,n)$-Pfister number} $\Pf_k(a, n)$ is the supremum of the
$a$-Pfister number of $q$ taken over all field extensions $K/k$ and 
all $n$-dimensional forms $q$ such that $[q] \in I^a(K)$.   

We have the following easy (and probably well-known) result.

\begin{proposition}
\label{p.PfisterEasy}  
Let $k$ be a field of characteristic not equal to
  $2$ and let $n$ be a positive even integer. 
  \begin{enumeratea}
  \item % The $(1,n)$-Pfister number of $k$ is $\leq n$.
  $\Pf_k(1,n) \leq n$.
  \item % The $(2,n)$-Pfister number of $k$ is $\leq n-2$.
  $\Pf_k(2,n) \leq n - 2$.
  \end{enumeratea}
\end{proposition}
  
\begin{proof}
  (a) Immediate from the identity 
\[ \langle a_1, a_ 2 \rangle = 
\langle 1, a_1 \rangle - \langle 1, - a_2 \rangle = \da{-a_1} - \da{a_2} \]
in the Witt ring.

  (b) Let $q=\langle a_1,\ldots, a_n\rangle$ be an $n$-dimensional
quadratic form over $K$. Recall that 
  $q \in I^2(K)$ iff $n$ is even and $d_{\pm}(q)=1$, modulo 
$(K^*)^2$~\cite[Corollary II.2.2]{lam}. Here $d_{\pm}(q)$ 
is the \emph{signed determinant}
  given by $(-1)^{n(n-1)/2}d(q)$ where $d(q)=\prod_{i=1}^n a_n$ is the
  determinant~\cite[p. 38]{lam}.  

To explain how to write $q$ as a sum of $n-2$ Pfister forms,
we will temporarily assume that $\zeta_4 \in K$. In this case 
we may assume that $a_1 \dots a_n = 1$. Since $\langle a, a \rangle$
is hyperbolic for every $a \in K^*$, we see that
$q = \langle a_1, \dots, a_n \rangle$ is Witt equivalent to
\[ \ll a_2, a_1 \gg \oplus 
\ll a_3, a_1a_2 \gg \oplus \cdots \oplus \ll a_{n-1}, a_1 \dots a_{n-2} \gg \, . \]
By inserting appropriate powers of $-1$, we can modify this formula so that
it remains valid even if we do not assume that $\zeta_4 \in K$, as follows:
\[ q = \langle a_1, \dots, a_n \rangle \simeq
\sum_{i=2}^n (-1)^i \da{(-1)^{i+1} a_i, (-1)^{i(i-1)/2 +1} a_1 \dots a_{i-1}} 
\qedhere \]
\end{proof}


We do not have an explicit upper bound on $\Pf_k(3, n)$; 
however, we do know that $\Pf_k(3, n)$ is finite for any $k$ and any $n$.
To explain this, let us recall that $I^3(K)$ is the set of all classes
$[q]\in\rW(K)$ such that $q$ has even dimension, trivial signed determinant
and trivial Hasse-Witt invariant~\cite{invol}.  

Let $n$ be a positive integer. Let $q$ be a
non-degenerate $n$-dimensional quadratic form over $K$ whose whose signed
determinant is $1$. The class of $q$ in $\H^{1}(K, \Orth_{n})$ lies in
$\H^{1}(K, \SO_{n})$. 
%%%%%% This was commented out previously
We say that $q$ \emph{admits a spin structure}
if its class is in the image of $\H^{1}(K, \Spin_{n})$ into $\H^{1}(K,
\SO_{n})$. As pointed out in Remark~\ref{rem.clifford}, the
obstruction to admitting a spin structure is the Hasse-Witt invariant
$c(q)$.  Thus, the forms in $I^3$ are exactly the even dimensional
forms admitting a spin structure. 
%%%%%%%% Previously commmented
The following result was suggested to us by Merkurjev and Totaro.

\begin{proposition}
\label{p.PfisterEasy2} Let $k$ be a field of characteristic different from~$2$.
  Then $\Pf_k(3,n)$ is finite.
\end{proposition}
\begin{proof}[Sketch of proof]
  Let $E$ be a versal torsor for $\Spin_n$ over a field extension
  $L/k$; cf. \cite[Section I.V]{gms}.
  Let $q_L$ be the quadratic form over $L$ corresponding to
  $E$ under the map $\H^1(L,\Spin_{n})\to \H^1(L,\Orth_{n})$.  The
  $3$-Pfister number of $q_L$ is then an upper bound for the
  $3$-Pfister number of any $n$-dimensional form in $I^3$ 
over any field extension $K/k$.
\end{proof}

\begin{remark}
  For $a > 3$ the finiteness of $\Pf_k(a,n)$ is an open problem.
\end{remark}

The goal of this section is to prove Theorem~\ref{t.pfister} 
stated in the introduction, which says that
   \[ \Pf_k(3, n) \ge \frac{2^{(n+4)/4} -n-2}{7}  \]
for any field $k$ be a field of characteristic different from $2$
and any positive even integer $n$. 
Since this is a lower bound on $\Pf_k(3, n)$, we may assume, 
without loss of generality that $k$ contains $\zeta_4$ in the
proof.  To simplify matters, this assumption will be in force for
the remainder of this section.

For each extension $K$ of $k$, denote by $\rT_{n}(K)$ the image of
$\H^{1}(K, \Spin_{n})$ in $\H^{1}(K, \SO_{n})$. We will view
$\rT_{n}$ as a functor $\Fields_{k} \arr \Sets$. The essential dimension of
this functor is closely related to the essential dimension of
$\Spin_{n}$.

\begin{lemma} \label{lem.T-Spin} 
% We have
%    \[
$\ed\Spin_{n} - 1 \leq \ed\rT_{n} \leq \ed\Spin_{n}$.
%    \]
\end{lemma}
\begin{proof}
  In the language of~\cite[Definition 1.12]{bf1}, we 
have a fibration of functors
\[
\H^1(\_\,, \mu_{2})\leadsto \H^1(\_\,,\Spin_n)\arr \rT_{n}(\_\,).
\]
The first inequality then follows from~\cite[Proposition 1.13]{bf1}
and the second follows from Proposition~\ref{p.isotropic} 
(or \cite[Lemma 1.9]{bf1}).
\end{proof}

\begin{lemma}\label{lem:sum-spin}
  Let $q$ and $q'$ be non-degenerate even-dimensional quadratic forms 
  over $K$. Suppose
  that $q$ admits a spin structure. Then $q \oplus q'$ admits a spin
  structure if and only if $q'$ admits a spin structure.
\end{lemma}
\begin{proof}
Immediate from the 
fact that $I^3(K)$ is an ideal of $W(K)$. 
%identity $c(q\oplus q')=c(q)+c(q')$~\cite[V.3.15]{lam}.
\end{proof}

Let $h_{K}$ be the standard $2$-dimensional hyperbolic form
$h_{K}(x,y) = xy$ over an extension $K$ of $k$.  For each
$n$-dimensional quadratic form $q \in I^3(K)$, 
let $\ed_{n}(q)$ denote the essential dimension of the class of $q$
in $\rT_{n}(K)$.

\begin{lemma}\label{lem:decrease-ed}
Let $q$ be an $n$-dimensional quadratic form over $K$ whose
class in $W(K)$ lies in $I^3(K)$. 
Then for any positive integer $s$
   \[
   \ed_{n+2s}(h_{K}^{\oplus s} \oplus q)\geq \ed_{n}(q)
      - \frac{s(s + 2n -1)}{2}.
   \]
\end{lemma}

\begin{proof}
  Set $m \eqdef \ed_{n+2s}(h_{K}^{\oplus s} \oplus q)$. Let $F$ be a
  field of definition of $h^{\oplus s}_{K} \oplus q$ of transcendence
  degree~$m$, and let $\widetilde{q}$ be an $(n + 2s)$-dimensional
  quadratic form with a spin structure over $F$
  such that $\widetilde{q}_K$ is $K$-isomorphic
  to $h_{K}^{\oplus s} \oplus q$. Let $X$ be the Grassmannian of
  $s$-dimensional subspaces of $F^{n+2s}$ which are totally isotropic
  with respect to $\widetilde{q}$; the dimension of $X$ is precisely
  $s(s + 2n -1)/2$.

  The variety $X$ has a rational point over $K$; hence there exists an
  intermediate extension $F \subseteq E \subseteq K$ such that
  $\trdeg_{F}E \leq s(s + 2n -1)/2$, with the property that
  $\widetilde{q}_{E}$ has a totally isotropic subspace of
  dimension~$s$. Then $\widetilde{q}_{E}$ splits as $h_{E}^{s} \oplus
  q'$. By Witt's Cancellation Theorem, $q'_K$ is $K$-isomorphic to $q$; 
  hence $\ed_{n}(q) \leq m + s(s + 2n -1)/2$, as claimed.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{t.pfister}]
If $n \leq 10$ then the statement 
is vacuous, because then $2^{(n+4)/4}-n-2 \leq 0$, 
so we assume that $n \geq 12$. We may also assume 
without loss of generality that
$\zeta_{4} \in k$. In this case $\rW(K)$ is a
$\ZZ/2$-vector space; it follows that the $3$-Pfister number 
of a form $q$ is the smallest $r$ appearing in an expression
\[
q=\sum_{i=1}^r \da{a_i,b_i,c_i}.
\]  
in $\rW(K)$.  Choose an $n$-dimensional form $q$ 
such that $[q] \in I^3(K)$ and $\ed_{n}(q) = \ed\rT_{n}$. 
(Here we view $q$ as an object in $T_n(K)$.) Suppose 
that $q$ is equivalent in the Witt ring to 
$\sum_{1 = 1}^{r}\da{a_{i}, b_{i}, c_{i}}$.

Let us write a Pfister form $\da{a,b,c}$ as
   \[
   \da{a,b,c} = \generate{1} \oplus \da{a,b,c}_{0},
   \]
where
   \[
   \da{a,b,c}_{0} \eqdef
   \langle a_i, b_i, c_i, a_ib_i, a_i c_i, b_i c_i, a_i b_i c_i \rangle.
   \]
Set
   \[
   \phi \eqdef \sum_{1 = 1}^{r}\da{a_{i}, b_{i}, c_{i}}_{0}
   \]
if $r$ is even, and
   \[
   \phi \eqdef \generate{1} \oplus \sum_{1 = 1}^{r}\da{a_{i}, b_{i}, c_{i}}_{0}
   \]
if $r$ is odd. Then $q$ is equivalent to 
$\phi$ in the Witt ring, and hence, $\phi \in I^3(K)$. The dimension 
of $\phi$ is $7r$ or $7r+1$, depending on the parity of $r$. 

We claim that $n < 7r$. Indeed, assume the contrary. Then 
the dimension of $q$ is less than of equal to the dimension of $\phi$, so 
$q$ is isomorphic to a form of type $h_{K}^{s} \oplus \phi$. Thus 
\[  \frac{3n}{7} \geq 3r \geq\ed_{n}(q) = \ed\rT_{n} 
\stackrel{\text{\tiny by Lemma~\ref{lem.T-Spin}}}{\ge} 
 \ed \Spin_{n} - 1 
\, . \]
% where the last inequality is given by Lemma~\ref{lem.T-Spin}.
The resulting inequality fails for every even $n \ge 12$ because, for 
such $n$, $\ed \Spin_n \ge n/2$; see~\eqref{e.old-spin}. 

So we may assume that $7r \geq n$; then there is an isomorphism between 
the quadratic forms $\phi$ 
and a form of the type $h_{K}^{\oplus s}\oplus q$. 
By comparing dimensions we get the equality $7r = n + 2s$ when $r$ is even, and $7r+1 = n + 2s$ when $r$ is odd.
The essential dimension of the form 
$\phi$ as an element 
of $\rT_{7r}(K)$ or $\rT_{7r+1}(K)$ is at most $3r$, 
while Lemma~\ref{lem:decrease-ed} tells us that this essential 
dimension is at least $\ed(q) - s(s + 2n -1)/2$. 
 From this, Lemma~\ref{lem.T-Spin} 
and Theorem~\ref{thm.spin} we obtain the chain of inequalities
   \begin{align} 
   3r &\geq \ed_{n}(q) - \frac{s(s + 2n -1)}{2} \nonumber \\
   &= \ed\rT_{n} - \frac{s(s + 2n -1)}{2} \label{e.quadratic} \\
   &\geq \ed\Spin_{n} - 1 - \frac{s(s + 2n -1)}{2} \nonumber \\
   &\geq 2^{(n-2)/2} - \frac{n(n-1)}{2} - 1 - \frac{s(s + 2n -1)}{2}. \nonumber
   \end{align}

Now suppose $r$ is even. Substituting 
$s = (7r - n)/2$ into the inequality~\eqref{e.quadratic}, we obtain
   \[
   \frac{49r^{2} + (14n+10)r - 2^{(n+4)/2} - n^{2} + 2n - 8}{8} \geq 0.
   \]
We interpret the left hand side as a quadratic polynomial in $r$. 
The constant term of this polynomial is negative
for all $n \geq 8$; hence this polynomial has one positive real 
root and one negative real root. Denote the positive root by $r_+$.
The above inquality is then equivalent to $r \geq r_+$. 
By the quadratic formula 
   \[ r_+ = \frac{\sqrt{49 \cdot 2^{(n+4)/2} + 168n - 367} -(7n + 5)}{49}
\ge
      \frac{2^{(n+4)/4} -n-2}{7}\, . \]
   This completes the proof of Theorem~\ref{t.pfister} when $r$ is
   even. If $r$ is odd then substituting $s = (7r+1-n)/2$ 
   into~\eqref{e.quadratic},
   we obtain an analogous quadratic inequality, whose positive root is
   \[ r_+ = \frac{\sqrt{49 \cdot 2^{(n+4)/2} + 168n - 199} -(7n + 12)}{49}\\
      \geq  \frac{2^{(n+4)/4} -n-2}{7}\, , \] 
and Theorem~\ref{t.pfister} follows.
\end{proof}

\bibliographystyle{amsalpha}
%\bibliography{largs}
%%%  Bib file for Ed.  To contain appropriate BibItems
b\def\cprime{$'$}
\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}
\providecommand{\MR}{\relax\ifhmode\unskip\space\fi MR }
% \MRhref is called by the amsart/book/proc definition of \MR.
\providecommand{\MRhref}[2]{%
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}
\providecommand{\href}[2]{#2}
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\end{thebibliography}

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