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\title[Essential dimension]{Essential dimension}
\author[Zinovy Reichstein]{Zinovy Reichstein}
\thanks{The author is grateful to S.~Cernele, A.~Duncan, 
S.~Garibaldi, R.~L\"otscher,
M.~Macdonald, A.~Merkurjev and A.~Meyer for helpful comments and to 
the National Science and Engineering Council of Canada for
financial support through its Discovery and Accelerator 
Supplement grants}
\address{Department of Mathematics\\
The University of British Columbia\\
1984 Mathematics Road\\
Vancouver, B.C., Canada V6T 1Z2}

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\begin{abstract} 
Informally speaking, the essential dimension of an algebraic object
is the minimal number of independent parameters one needs 
to define it. This notion was initially introduced 
in the context where the objects in question are finite 
field extensions~\cite{bu-r}.  Essential dimension 
has since been investigated in several broader contexts, 
by a range of techniques, and has been found to have 
interesting and surprising connections to many problems in algebra 
and algebraic geometry.

The goal of this paper is to survey some of this research.
I have tried to explain the underlying ideas informally 
through motivational remarks, examples and proof 
outlines (often in special cases, where the argument 
is more transparent), referring an interested
reader to the literature for a more detailed treatment.  
The sections are arranged in rough chronological 
order, from the definition of essential dimension to
open problems.
\end{abstract}

\subjclass[2010]{Primary 14L30, 20G10, 11E72. }

\keywords{Essential dimension, linear algebraic group, Galois cohomology,
cohomological invariant, quadratic form, central simple algebra,
algebraic torus, canonical dimension}

\maketitle

\section{Definition of essential dimension}
\label{sect.def}

Informally speaking, the essential dimension of an algebraic object
is the minimal number of parameters one needs to define it. To motivate 
this notion, let us consider an example, where the object 
in question is a quadratic form.

Let $k$ be a base field, $K/k$ be a field extension and
$q$ be an $n$-dimensional quadratic form over $K$. 
Assume that $\Char(k) \ne 2$ and
denote the symmetric bilinear form associated to $q$ by $b$.
We would now like to see if $q$ can be defined over a smaller
field $k \subset K_0 \subset K$. This means that there is a
$K$-basis $e_1, \dots, e_n$ of $K^n$ such that
$b(e_i, e_j) \in K_0$ for every $i, j = 1,\ldots, n$.
If we can find such a basis, we will say that $q$ {\em descends}
to $K_0$ or that $K_0$ is a {\em field of definition} of $q$.
It is natural to ask if there is a minimal field $K_{min}/k$ (with
respect to inclusion) to which $q$ descends. The answer to this
question is usually ``no". For example, it is not difficult
to see that the ``generic" form 
$q(x_1, \dots, x_n)  = a_1 x_1^2 + \dots + a_n x_n^2$
over the field $K = k(a_1, \dots, a_n)$,
where $a_1, \dots, a_n$ are independent variables,
has no minimal field of definition.
We will thus modify our question: instead of asking
for a minimal field of definition $K_0$ for $q$, we will ask
for the minimal value of the transcendence degree
$\trdeg_k(K_0)$. 
\footnote{One may also 
ask which quadratic forms have a minimal field
of definition. To the best of my knowledge, this is an open question;
see Section~\ref{subsect.incompressible}.}
This number is called the {\em essential dimension}
of $q$ and is denoted by 
$\ed(q)$.

Note that the above definition of $\ed(q)$ 
is in no way particular to quadratic forms.
In a similar manner one can consider fields of definition of
any polynomial in $K[x_1, \dots, x_n]$, any finite-dimensional
$K$-algebra, any algebraic variety defined over $K$, etc.
In each case the minimal transcendence degree 
of a field of definition is an interesting numerical
invariant which gives us some insight into
the ``complexity" of the object in question. 

We will now state these observations more formally.
Let $k$ be a base field, $\Fields_k$ be the category of
field extensions $K/k$, $\Sets$ be the category of sets,
and $\mathcal{F} \colon \Fields_k \to \Sets$ be a covariant functor.
In the sequel the word ``functor" will always refer to a functor 
of this type. If $\alpha \in \mathcal{F}(K)$ and $L/K$ is a field
extension, we will denote the image of $\alpha$ in 
$\mathcal{F}(L)$ by $\alpha_L$.

For example, $\mathcal{F}(K)$ could be the set of $K$-isomorphism 
classes of quadratic forms on $K^n$, or of $n$-dimensional $K$-algebras,
for a fixed integer $n$,
or of elliptic curves defined over $K$. In general we think 
of $\mathcal{F}$ as specifying the type of algebraic 
object we want to work with, and elements of $\mathcal{F}(K)$ 
as the of algebraic objects of this type defined over $K$.
         
Given a field extension $K/k$, we will say that  $a\in \mathcal{F}(K)$
  \emph{descends} to an intermediate field $k \subseteq
  K_0 \subseteq K$ if $a$ is in the image of the induced map
  $\mathcal{F}(K_0) \to \mathcal{F}(K)$.
  The \emph{essential dimension} $\ed(a)$ of $a \in \mathcal{F}(K)$
  is the minimum of the transcendence degrees $\trdeg_{k}(K_0)$ taken over
  all fields $k \subseteq K_0 \subseteq K$ such that $a$ descends to $K_0$.
The essential dimension $\ed(\mathcal{F})$ of the functor $\mathcal{F}$
is the supremum of $\ed(a)$ 
taken over all $a\in \mathcal{F}(K)$ with $K$ in $\Fields_k$.
         
These notions are relative to the base field $k$; we will sometimes
write $\ed_k$ in place of $\ed$ to emphasize the dependence on $k$.
If $\mathcal{F} \colon \Fields_k \to \Sets$ be a covariant functor
and $k \subset k'$ is a field extension, we will write
$\ed_{k'}(\mathcal{F})$ for $\ed(\mathcal{F}_{k'})$, where
$\mathcal{F}_{k'}$ denotes the restriction of $\mathcal{F}$ 
to $\Fields_{k'}$.  Is easy to see that in this situation
\begin{equation} \label{e.base-change1}
\ed_k(\mathcal{F}) \ge \ed_{k'}(\mathcal{F}) \, ;
\end{equation}
cf.~\cite[Proposition 1.5]{bf1}.  
In particular, taking $k'$ to be an algebraic closure of $k$, we see
that for the purpose of proving a lower bound of
the form $\ed_k(\mathcal{F}) \ge d$, where $d$ does 
not depend on $k$, we may assume that $k$ is algebraically closed.

Let $\mmu_n$ denote the group of $n$th roots of unity, defined over $k$.
Whenever we consider this group, we will assume that it is smooth, 
i.e., that $\Char(k)$ does not divide $n$.
         
\begin{example} \label{ex.cohomology}
Let $\mathcal{F}(K) := H^r(K, \mmu_n)$ be the Galois cohomology functor.
Assume $k$ is algebraically closed.
If $\alpha \in H^r(K, \mmu_n)$ is non-trivial then
by the Serre vanishing theorem (see, 
e.g., \cite[II.4.2, Prop. 11, p. 83]{serre-gc})
$\ed(\alpha) \ge r$.
\end{example}

\begin{example} \label{ex.functor-of-forms}
Once again, assume that $k$ is algebraically closed.
Let $\Forms_{n, d}(K)$ be the set of homogeneous polynomials
of degree $d$ in $n$ variables. If $\alpha \in \Forms_{n, d}(K)$ is
anisotropic over $K$ then by the Tsen-Lang theorem (see, e.g.,~\cite{pfister}),
$n \le d^{\ed(\alpha)}$ or equivalently, $\ed(\alpha) \ge \log_d(n)$.
\end{example}
         
Of particular interest to us will be the 
functors $\mathcal{F}_G$ given by $K \to H^1(K , G)$, where $G$
is an algebraic group over $k$.  Here,
as usual, $H^1(K, G)$ denotes the set of isomorphism classes of
$G$-torsors over $\Spec(K)$.
%  in the finitely presented faithfully flat (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.
We write $\ed G$ for $\ed \, \mathcal{F}_G$.  Essential 
dimension was originally introduced in this context (and only 
in characteristic $0$);
see~\cite{bu-r, reichstein, ry}.  The above definition of essential
dimension for a general functor $\mathcal{F}$ is due to A.~Merkurjev; see
\cite{bf1}.

In special cases this notion was investigated much earlier.
To the best of my knowledge, the first non-trivial 
result related to essential dimension is due to
F.~Klein~\cite{klein}.  In our terminology,
Klein showed that the essential dimension of the symmetric group
$S_5$ over $k = \CC$, is $2$. (Klein referred to this result as
``Kronecker's theorem", so it may in fact go back even further.)  
The essential dimension of the projective 
linear group $\PGL_n$ first came up in C. Procesi's
pioneering work on universal division algebras 
in the 1960s; see \cite[Section 2]{procesi}.
The problems of computing the essential dimension of the symmetric 
group $S_n$ and the projective linear group $\PGL_n$ remain largely 
open; see Section~\ref{sect.problems}. 

If $k$ is an algebraically closed field 
then groups of essential dimension zero are precisely 
the {\em special groups}, introduced by J.-P.~Serre~\cite{serre}. 
Recall that an algebraic group $G$ over $k$ is called special
if $H^1(K, G) = 0$ for every field extension $K/k$.
Over an algebraically closed field of characteristic zero these
groups were classified by A.~Grothendieck~\cite{grothendieck} in the 1950s.
The problem of computing the essential dimension of an algebraic 
group may be viewed as a natural extension of the problem 
of classifying special groups.
 
\section{First examples}

Recall that an action of an algebraic group $G$ on an algebraic
$k$-variety $X$ is called {\em generically free}
if $X$ has a dense $G$-invariant open subset $U$ 
such that the stabilizer $\Stab_G(x) = \{ 1 \}$
for every $x \in U(\overline{k})$ and {\em primitive} 
if $G$ permutes the irreducible components of $X$.
Here $\overline{k}$ denotes an algebraic closure of $k$.
Equivalently, $X$ is primitive if $k(X)^G$ is a field. 

If $K/k$ is a finitely generated field extension then 
elements of $H^1(K, G)$ can be interpreted as birational 
isomorphism classes of generically free primitive $G$-varieties 
(i.e., $k$-varieties with a generically free primitive $G$-action) 
equipped with a $k$-isomorphism
of fields $k(X)^G \simeq K$;
cf.~\cite[Section 4]{bf1}.
If $X$ is a generically free primitive $G$-variety, and 
$[X]$ is its class in $H^1(K, G)$ then
\begin{equation} \label{e.compression} 
\ed ([X]) = \min \dim(Y) - \dim(G) \, , 
\end{equation}
where the minimum is taken over all dominant
rational $G$-equivariant maps $X \dasharrow Y$ such that
the $G$-action on $Y$ is generically free. 

An important feature of the functor $H^1(\; \ast \; , G)$ is the
existence of so-called {\em versal objects};
see~\cite[Section I.5]{gms}.  
If $\alpha \in H^1(K, G)$ is a versal torsor then it is easy to see
that $\ed(\alpha) \ge \ed(\beta)$ for any field extension $L/k$
and any $\beta \in H^1(L, G)$.  In other words, $\ed(\alpha) = \ed(G)$.
If $G \to \GL(V)$ is a generically 
free $k$-linear representation of $G$ then the class $[V]$ 
of $V$ in $H^1(k(V)^G, G)$ is versal. 
By \eqref{e.compression}, we see that
\begin{equation} \label{e.compression1} 
\ed (G) = \min \dim(Y) - \dim(G) \, , 
\end{equation} 
where the minimum is taken over all dominant
rational $G$-equivariant maps $V \dasharrow Y$, such that
$G$-action on $Y$ is generically free. 
In particular, \begin{equation} \label{e.linear-rep}
 \ed(G) \le  \dim(V) - \dim(G) \, .
\end{equation}
Moreover, unless $k$ is a finite field and $G$ is special, we only need
to consider closed $G$-invariant subvarieties $Y$ of $V$.
That is,
\begin{equation} \label{e.linear-rep0}
\ed(G) = \min \{ \dim \, \operatorname{Im}(f) \}  - \dim(G) \, , 
\end{equation}
where the minimum is taken over all $G$-equivariant rational
maps $f \colon V \dasharrow V$ such that the $G$-action on 
$\operatorname{Im}(f)$ is generically free;  
see~\cite[Theorem 4.5]{me2}.

\begin{example} \label{ex.adjoint}
Let $G$ be a connected adjoint semisimple group over $k$. Then 
$\ed(G) \le \dim(G)$.
To prove this inequality, apply \eqref{e.linear-rep} to the generically 
free representation $V = \mathcal{G} \times \mathcal{G}$,
where $\mathcal{G}$ is the adjoint representation of $G$ on its Lie algebra.

Note that the inequality $\ed(G) \le \dim(G)$ can fail dramatically
if $G$ is not adjoint; see Corollary~\ref{cor.spin}.
\end{example}

We now turn to lower bounds on $\ed(G)$ for various 
algebraic groups $G$ and more generally, on $\ed(\mathcal{F})$
for various functors $\mathcal{F} \colon \Fields_k \to \Sets$.
The simplest approach to such bounds is to relate the functor 
$H^1(\, \ast \, , G)$ (and more generally, $\mathcal{F}$)
to the functors in Examples~\ref{ex.cohomology} 
or~\ref{ex.functor-of-forms}, using the following lemma,
whose proof is immediate from the definition; cf.~\cite[Lemma 1.9]{bf1}.

\begin{lemma} \label{lem.surjective} 
Suppose a morphism of functors $\phi \colon \mathcal{F} \to 
\mathcal{F}'$ takes $\alpha$ to $\beta$. 
Then $\ed(\alpha) \ge \ed(\beta)$. In particular,
if $\phi$ is surjective then $\ed(\mathcal{F}) \ge \ed(\mathcal{F}')$.
\end{lemma}

A morphism of functors  $\mathcal{F} \to H^d(\, \ast \,, \mmu_n)$ 
is called a {\em cohomological invariant} of degree $d$; it is said to be  
nontrivial if $\mathcal{F}(K)$ contains a non-zero element 
of $H^d(K, \mmu_n)$ for some $K/k$.  
Using Lemma~\ref{lem.surjective} and Example~\ref{ex.cohomology} 
we recover the following observation, due to Serre.

\begin{lemma} \label{lem.coh-inv} Suppose $k$ is algebraically closed.
If there exists a non-trivial cohomological invariant 
$\mathcal{F} \to H^d(\, \ast \,, \mmu_n)$ then $\ed(\mathcal{F}) \ge d$.
\end{lemma}

In the examples below I will, as usual, write 
$\langle a_1, \dots, a_n \rangle$ for the 
quadratic form $(x_1, \dots, x_n)  \mapsto a_1 x_1^2 + \dots + a_n x_n^2$ and
$\ll a_1, \dots, a_r \gg$ for the $r$-fold Pfister form 
$\langle 1, - a_1 \rangle \otimes_K \dots \otimes 
\langle 1, - a_r \rangle$. 

\begin{example} \label{ex.Pf} Suppose $\Char(k) \ne 2$. 
Let $\Pf_r$ be the functor that assigns
to a field $K/k$ the set of $K$-isomorphism classes of
$r$-fold Pfister forms, $q = \ll a_1, \dots, a_r \gg$.
Then $\ed_k(\Pf_r) = r$.

Indeed, since $q$ is defined over $k(a_1, \dots, a_r)$, 
we have $\ed_k(\Pf_r) \le r$. To prove the opposite
inequality, we may assume that $k$ is algebraically closed.
Let $a_1, \dots, a_r$ be independent variables and
$K = k(a_1, \dots, a_r)$. Then 
the tautological map $\Pf_r(K) \to \Forms_{2^r, 2}(K)$ 
takes $q = \ll a_1, \dots, a_r \gg$ to an anisotropic
form in $2^r$ variables; see, e.g.,~\cite[p.~111]{pfister}.
Combining Lemma~\ref{lem.surjective} 
and Example~\ref{ex.functor-of-forms}
we conclude that $\ed(\Pf_r) \ge r$, as desired. 

Alternatively, the inequality $\ed(\Pf_r) \ge r$ also follows
from Lemma~\ref{lem.coh-inv}, applied to the cohomological 
invariant $\Pf_r \to H^r(\, \ast \, , \mmu_2)$,
which takes $\ll a_1, \dots, a_r \gg$ to 
the cup product $(a_1) \cup \dots \cup (a_r)$.

Since $H^1(\, \ast \,, \G_2)$ is naturally isomorphic to $\Pf_3$,
we conclude that $\ed(\G_2) = 3$. Here $\G_2$ stands 
for the split exceptional 
group of type $\G_2$ over $k$.
\end{example}

\begin{example} \label{ex.O_n} If $\Char(k) \neq 2$ then
$\ed_k(\Orth_n) = n$.  

Indeed, since every quadratic form over $K/k$ 
can be diagonalized, we see that $\ed_k(\Orth_n) \le n$.
To prove the opposite inequality, we may assume that
$k$ is algebraically closed. Define the functor
$\phi \colon H^1(\, \ast \, , \Orth_n) \to \Pf_n$ 
as follows.  Let $b$ be the bilinear form on $V = K^n$, 
associated to $q = \langle a_1, \dots, a_n \rangle \in H^1(K, \Orth_n)$. 
Then $b$ naturally induces a non-degenerate bilinear form on
the $2^n$-dimensional $K$-vector space $\bigwedge(V)$. 
We now set $\phi(q)$ to be the $2^r$-dimensional
quadratic form associated to $\wedge(b)$. 
One easily checks that $\phi(q)$ is the $n$-fold Pfister form
$\phi(q) = \ll a_1, \dots, a_n \gg$.  Since $\phi$ is clearly surjective, 
Lemma~\ref{lem.surjective} and Example~\ref{ex.Pf} 
tell us that $\ed(\Orth_n) \ge \ed(\Pf_n) = n$.

We remark that $\phi(q)$ is closely related to the $n$th Stiefel-Whitney 
class $\sw_n(q)$ (see~\cite[p. 41]{gms}),
and the inequality $\ed(\Orth_n) \ge n$ can also be deduced 
by applying Lemma~\ref{lem.coh-inv} to the cohomological invariant
$\sw_n \colon H^1(K, \Orth_n) \to H^n(K, \mmu_2)$.
\end{example}

\begin{example} \label{ex.abelian}
If $k$ contains a primitive $n$th root of unity 
then $\ed_k(\mmu_n^r) = r$. 

Indeed, the upper bound, 
$\ed(\mmu_n^r) \le r$, follows from~\eqref{e.linear-rep}.
Alternatively, note that
any $(\overline{a_1}, \dots, \overline{a_r}) \in H^1(K, \mmu_n^r)$
is defined over the subfield $k(a_1, \dots, a_r)$ of $K$, of 
transcendence degree $\le r$.

To prove the opposite inequality
we may assume that $k$ is algebraically closed. 
Now apply Lemma~\ref{lem.coh-inv} to the cohomological invariant
\[ H^1(K, \mmu_n^r) = K^*/(K^*)^n \times \dots \times K^*/(K^*)^n
\to H^r(K, \mmu_n) \]
given by $(\overline{a_1}, \dots, \overline{a_r}) 
\to (a_1) \cup \dots \cup (a_r)$.  
Here $\overline{a}$ denotes the class of $a \in K^*$ in $K^*/(K^*)^n$.  
\end{example}
    
\begin{remark} \label{rem.subgroup}
Suppose $H$ is a closed subgroup of $G$ and $G \to \GL(V)$ is
a generically free linear representation. Since every 
rational $G$-equivariant map $V \dasharrow Y$ is also $H$-equivariant, 
\eqref{e.compression1} tells us that
\begin{equation} \label{e.subgroup}
\ed(G) \ge \ed(H) + \dim(H) - \dim(G) \, .
\end{equation}
In particular, if a finite group $G$ contains 
a subgroup $H \simeq (\ZZ/p \ZZ)^r$ for some prime $p$
and if $\Char(k) \neq p$ (so that we can identify
$(\ZZ/p\ZZ)^r$ with $\mmu_p^r$ over $\overline{k}$)
then 
\[ \ed_k(G) \ge \ed_{\overline{k}}(G) \ge \ed_{\overline{k}}(H) = r \, . \]
In the case where $G$ is the symmetric group $\Sym_n$ and
$H \simeq (\ZZ/2\ZZ)^{[n/2]}$ is the subgroup generated
by the commuting 2-cycles $(12)$, $(34)$, $(56)$, etc., this yields
$\ed(S_n) \ge [n/2]$; cf.~\cite{bu-r}.
\end{remark}
    
\begin{example} \label{ex.PGLn}
Recall that elements of $H^1(K, \PGL_n)$ are in a natural 
bijective correspondence with isomorphism classes
of central simple algebras of degree $n$ 
over $K$. Suppose $n = p^s$ is a prime power, and $k$ contains
a primitive $p$th root of unity.  Consider the morphism of functors
$\phi \colon H^1(K, \PGL_n) \to \Forms_{n^2, p}$
given by sending a central simple $K$-algebra $A$ to the degree $p$
trace form $x \to \Tr_{A/K}(x^p)$. 

If $a_1, \dots, a_{2s}$ are independent 
variables over $k$, $K = k(a_1, \dots, a_{2s})$, and 
\[ A = (a_1, a_2)_p \otimes_K \otimes \dots \otimes (a_{2s-1}, a_{2s})_p \]
is a tensor product of $s$ symbol algebras of degree $p$ then
one can write out $\phi(A)$ explicitly and show that it is anisotropic 
over $K$; see~\cite{hermite-joubert}. Lemma~\ref{lem.surjective}
and Example~\ref{ex.functor-of-forms} now tell us that
$\ed_k(A) \ge \ed_{\overline{k}}(A) \ge 2s$. Since $\trdeg_k(K) = 2s$,
we conclude that, in fact 
\begin{equation} \label{e.pgln-old}
\text{$\ed_k(A) = 2s$ and consequently, $\ed_k(\PGL_{p^s}) \ge 2s$.}
\end{equation}

The following alternative approach to proving~\eqref{e.pgln-old}
was brought to my attention by P.~Brosnan.
Consider the cohomological invariant
given by the composition of the natural 
map $H^1(K, \PGL_n) \to H^2(K, \mmu_n)$, 
which sends a central simple algebra to its Brauer class, and
the divided power map $H^2(\, \ast \,, \mmu_n) \to H^{2s}(\, \ast \,, \mmu_n)$; 
see~\cite[Appendix]{kahn}.  
The image of $A$ under the resulting cohomological invariant
\[ H^1(\, \ast \, , \PGL_n) \to H^{2s}(\, \ast \, ,  \mmu_n)  \]
is $(a_1) \cup (a_2) \cup \dots \cup (a_{2s}) \neq 0$ in
$H^{2s}(K, \mmu_n)$.  Lemma~\ref{lem.coh-inv} now tells us that
$\ed_k(A) \ge 2s$, and~\eqref{e.pgln-old} follows.
The advantage of this approach is that it shows that the essential 
dimension of the Brauer class of $A$ is also $2s$. 
\end{example}

\section{The fixed point method}

The following lower bound on $\ed(G)$ was conjectured by Serre and 
proved in~\cite{gr}.
Earlier versions of this theorem have appeared in~\cite{ry} 
and~\cite{cs}.
 
\begin{theorem} \label{thm.gille-r}
If $G$ is connected, $A$ is a finite abelian subgroup of $G$ and
$\Char(k)$ does not divide $|A|$, then
$\ed_k(G) \ge \rank(A) - \rank \, C^0_G(A)$.
\end{theorem}
 
Here $\rank(A)$ stands for the minimal number of generators of $A$
and $\rank \, C^0_G(A)$ for the dimension of the maximal 
torus of the connected group $C^0_G(A)$.
Note that if $A$ is contained in a torus $T \subset G$ then $\rank(C^0_G(A))
\ge \rank(T) \ge \rank(A)$, 
and the inequality of Theorem~\ref{thm.gille-r} 
becomes vacuous. Thus we are primarily interested in non-toral
finite abelian subgroups $A$ of $G$.  These subgroups 
have come up in many different contexts, starting with the work 
of Borel in the 1950s. For details and further references, 
see~\cite{ry}.

The proof of Theorem~\ref{thm.gille-r} relies on the following
two simple results.
 
\begin{theorem}[Going Down Theorem] \label{going-down}
Suppose $k$ is an algebraically closed base field and
$A$ is an abelian group such that $\Char(k)$ does not divide $|A|$.
Suppose $A$ acts on $k$-varieties $X$ and $Y$ and
$f \colon X \dasharrow Y$ is an $A$-equivariant rational map.
If $X$ has a smooth $A$-fixed point and $Y$ is complete 
then $Y$ has an $A$-fixed point.
\end{theorem}

A short proof of Theorem~\ref{going-down}, 
due to J. Koll\'ar and E. Szab\'o, can be found  
in~\cite[Appendix]{ry}.

\begin{lemma} \label{lem.fixed-pt} 
Let $A$ be a finite abelian subgroup, acting faithfully on an irreducible
$k$-variety $X$. 
Suppose $\Char(k)$ does not divide $|A|$. If $X$ has a smooth
$A$-fixed point then $\dim(X) \ge \rank(A)$.
\end{lemma}

The lemma follows from the fact that the $A$-action on the tangent space
$T_x(X)$ at the fixed point $x$ has to be faithful; 
see~\cite[Lemma 4.1]{gr}.

For the purpose of proving Theorem~\ref{thm.gille-r} we may
assume that $k$ is algebraically closed.  To convey the flavor of the proof 
I will make the following additional assumptions: 
(i) $C_G(A)$ is finite and (ii) $\Char(k) = 0$.
The conclusion then reduces to 
\begin{equation} \label{e.ry}
\ed(G) \ge \rank(A) \, .  
\end{equation}
This special case of Theorem~\ref{thm.gille-r} is proved
in~\cite{ry} but I will give a much simplified argument here,
based on~\cite[Section 4]{gr}. 

Let $G \to \GL(V)$ be a generically free representation. 
By~\eqref{e.compression1} we need to show that 
if $V \dasharrow Y$ is a $G$-equivariant
dominant rational map and the $G$-action on $Y$ is 
generically free, then 
\begin{equation} \label{e.Y}
\dim(Y) - \dim(G) \ge \rank(A) \, .
\end{equation} 
To see how to proceed, let us first consider the ``toy" case,
where $G$ is finite.  Here~\eqref{e.ry} follows from~\eqref{e.subgroup},
but I will opt for a different argument below, with 
the view of using a variant of it in greater generality.

After birationally modifying $Y$,
we may assume that it is smooth and projective. (Note that this step
relies on $G$-equivariant resolution of singularities and thus
uses the characteristic $0$ assumption.)
Since $V$ has a smooth $A$-fixed point (namely, the origin), 
the Going Down Theorem~\ref{going-down} tells us that so does $Y$.
By Lemma~\ref{lem.fixed-pt},
$\dim(Y) \ge \rank(A)$, which proves \eqref{e.Y} in the case 
where $G$ is finite.

If $G$ is infinite, we can no longer hope to prove~\eqref{e.Y} 
by applying Lemma~\ref{lem.fixed-pt} to the $A$-action on $Y$. 
Instead, we will apply Lemma~\ref{lem.fixed-pt} to
a suitable $A$-invariant subvariety $Z \subset Y$.  This
subvariety $Z$ will be a cross-section for the $G$-action 
on $Y$, in the sense that a $G$-orbit in general position 
will intersect $Z$ in a finite number of points. Hence,
$\dim(Z) = \dim(Y) - \dim(G)$, and~\eqref{e.Y} reduces to
$\dim(Z) \ge \rank(A)$.  We will then proceed as in the previous 
paragraph: we will use Theorem~\ref{going-down} 
to find an $A$-fixed point on a smooth complete model of $Z$,
then use Lemma~\ref{lem.fixed-pt} to show that $\dim(Z) \ge \rank(A)$.

Let me now fill in the details. 
By~\cite{cgr} $Y$ is birationally isomorphic to $G \times^{S} Z$, 
where $S$ is a finite subgroup of $G$ and $Z$ is an algebraic 
variety equipped with a faithful $S$-action. 
(A priori $Z$ does not carry an $A$-action;
however, we will show below that some conjugate $A'$ of $A$ 
lies in $S$ and consequently, acts on $Z$. We will then replace $A$ by $A'$
and argue as above.)  We also note that
we are free to replace $Z$ by an ($S$-equivariantly) 
birationally isomorphic variety, so we may (and will) take it to be 
smooth and projective. 

Here, as usual, if $S$ acts on normal quasi-projective 
varieties $X$ and $Z$ then $X \times^{S} Z$ denotes the geometric quotient
of $X \times Z$ by the natural (diagonal) action of $S$.
Since $S$ is finite, there is no difficulty in forming the quotient map 
$\pi \colon X \times Z \to X \times^S Z$; cf.~\cite[Lemma 3.1]{gr}.
Moreover, if the $S$-action on $X$ extends to a $G \times S$-action,
then by the universal property of geometric quotients
$X \times^S Z$ inherits a $G$-action from $X \times Z$,
where $G$ acts on the first factor.
I will write $[x, z] \in X \times^S Z$ for the image 
of $(x, z)$ under $\pi$.

We now compactify $Y = G\times^S Z$ by viewing it as a $G$-invariant
open subset of the projective variety
$\overline{Y} := \overline{G} \times^S Z$,
where $\overline{G}$ is a so-called ``wonderful" (or ``regular") 
compactification of $G$.
Recall that $G \times G$ acts on $\overline{G}$, extending 
the right and left multiplication action of $G$ on itself,
The complement $\overline{G} \setminus G$
is a normal crossing divisor $D_1 \cup \dots \cup D_r$, where
each $D_i$ is irreducible, and
the intersection of any number of $D_i$ is the closure
of a single $G \times G$-orbit in $\overline{G}$.  
The compactification $\overline{G}$ has many wonderful 
properties; the only one we will need is Lemma~\ref{lem.parabolic} below. 
For a proof, see~\cite[Proposition A1]{brion}.
                       
\begin{lemma} \label{lem.parabolic} 
For every $x \in \overline{G}$,
$P = \pr_1 (\Stab_{G \times G} (x))$ is a parabolic subgroup
of $G$. Here $\pr_1$ is projection to the first factor.
Moreover, $P = G$ if and only if $x \in G$.
\end{lemma}             
                       
We are now ready to complete the proof of the inequality~\eqref{e.Y} 
(and thus of~\eqref{e.ry}) by showing that $S$ contains a conjugate 
$A'$ of $A$, and $A'$ has a fixed point in $Z$. In other words,
our goal is to show that some conjugate $A'$ of $A$ lies in $\Stab_S(z)$.
 
By the Going Down Theorem~\ref{going-down},
$\overline{Y}$ has an $A$-fixed point. 
Denote this point by $[x, z]$ for some 
$x \in \overline{G}$ and $z \in Z$. 
That is, for every $a \in A$, 
$[a x, z] = [x, z]$ 
in $\overline{Y}$. Equivalently,
\begin{equation}       
\label{e.wonderful1} \left\{ \begin{array}{l} ax = x s^{-1} \\
                           sz = z \end{array} \right.
\end{equation}         
for some $s \in S$. In other words, for every $a \in A$,
there exists an $s \in \Stab_S(z)$ such that
$(a^{-1}, s) \in \Stab_{G \times G}(x)$.
Equivalently, the image of the natural projection
$\pr_1 \colon \Stab_{G \times G}(x) \to G$ 
contains $A$. Since we are assuming that $C^0_G(A)$ is finite, $A$
cannot be contained in any proper parabolic subgroup of $G$.
Thus $x \in G$; see Lemma~\ref{lem.parabolic}.
Now the first equation in~\eqref{e.wonderful1} tells us that
$A':= x^{-1} A x \subset \Stab_S(z)$, as desired.
\qed                   

\begin{remark} \label{rem.char} The above argument 
proves Theorem~\ref{thm.gille-r} under two simplifying assumptions:
(i) $C_G(A)$ is finite and (ii) $\Char(k) = 0$. If assumption (i)
is removed, a variant of the same argument can still 
be used to prove Theorem~\ref{thm.gille-r}
in characteristic 0; see~\cite[Section 4]{gr}.
Assumption (ii) is more serious, because our argument 
heavily relies on resolution of singularities. Consequently,
the proof of Theorem~\ref{thm.gille-r} in prime characteristic
is considerably more complicated; see~\cite{gr}.
\end{remark}

\begin{cor} \label{cor.finite-abelian}
(a) $\ed(\SO_n) \ge n-1$ for any $n \ge 3$, (b) $\ed(\PGL_{p^s}) \ge 2s$,
                       
\smallskip             
(c) $\ed(\Spin_n) \ge \begin{cases}
\text{$[n/2]$ for any $n \ge 11$,} \\
\text{$[n/2] + 1$ if $n \equiv -1$, $0$ or $1$ modulo $8$,}
\end{cases}$           

\smallskip
(d) $\ed(\G_2) \ge 3$, (e) $\ed(\F_4) \geq 5$, (f) $\ed(\E_6^{sc}) \geq 4$. 

\smallskip             
(g) $\ed(\E_7^{sc}) \geq 7$, (h) $\ed(\E_7^{ad}) \geq 8$, 
(i) $\ed(\E_8) \geq 9$.

\smallskip             
\noindent
Here the superscript ${\, }^{sc}$ stands for ``simply connected"
and ${\, }^{ad}$ for ``adjoint".
\end{cor}      
                     
Each of these inequalities is proved by exhibiting a non-toral
abelian subgroup $A \subset G$ whose centralizer is finite.
For example,           
in part (a) we can take $A \simeq (\ZZ/2 \ZZ)^{n-1} $ to
be the subgroup of diagonal matrices of the form
\begin{equation} \label{e.A}
\text{$\operatorname{diag}(\epsilon_1, \dots, \epsilon_n)$,
 where each $\epsilon_i = \pm 1$ and
$\epsilon_1 \cdot \ldots \cdot \epsilon_n = 1.$}
\end{equation}         
The details are worked out in~\cite{ry}, with the exception 
of the first line in part (c), which was 
first proved by V.~Chernousov and J.-P.~Serre~\cite{cs},
by a different method. I later noticed that it can
be deduced from Theorem~\ref{thm.gille-r} as well; the finite
abelian subgroups one uses here can be found in~\cite{wood}.

\begin{remark} \label{rem.finite-abelian2}
The inequalities in parts (a), (b), (d), (e) and (f) can be recovered
by applying Lemma~\ref{lem.coh-inv} to suitable cohomological invariants.
For parts (b) and (d), this is done in Examples~\ref{ex.PGLn} 
and~\ref{ex.Pf}, respectively;
for parts (a), (e) and (f), see \cite[Example 12.7]{reichstein}, 
\cite[Example 12.10]{reichstein} and 
\cite[Remark 2.12]{garibaldi}.
        
It is not known whether or not 
parts (g), (h) and (i) can be proved in a similar manner, i.e.,
whether or not there exist cohomological invariants of
$E_7^{sc}$, $E_7^{ad}$ and $E_8$ of dimensions
$7$, $8$, and $9$, respectively. 
\end{remark}

\section{Central extensions}

In this section we will discuss another more recent method of proving
lower bounds on $\ed(G)$. This method does not apply as broadly 
as those described in the previous two sections, but in 
some cases it leads to much stronger bounds.  Let
\begin{equation} \label{e.exact-sequence}
1 \to C \to G \to \overline{G} \to 1
\end{equation}
be an exact sequence of algebraic groups over $k$ such that
$C$ is central in $G$ and isomorphic to $\mmu_p^r$ for
some $r \ge 1$.
Given a character $\chi \colon C \to \mmu_p$, we will, 
following \cite{km2}, denote by $\Rep^{\chi}$ the set of irreducible
representations $\phi \colon G \to \GL(V)$, defined
over $k$, such that $\phi(c) = \chi(c) \operatorname{Id}_V$ 
for every $c \in C$.

\begin{theorem} \label{thm.central}
Assume that $k$ is a field of characteristic $\ne p$ containing
a primitive $p$th root of unity.  Then
\begin{equation} \label{e.central}
\ed_k(G) \geq \min_{\langle \chi_1, \dots, \chi_r \rangle = C^*} 
\bigl( \sum_{i= 1}^r \gcd_{\rho_i \in \Rep^{\chi_i}} \dim(\rho_i) \bigr) - 
\dim G \, .
\end{equation}
Here $\gcd$ stands for the greatest common divisor and
the minimum is taken over all minimal generating sets 
$\chi_1, \dots, \chi_r$ of  $C^* \simeq (\ZZ/p \ZZ)^r$.
\end{theorem}

Theorem~\ref{thm.central} has two remarkable corollaries.

\begin{cor} \label{cor.km} {\rm (N.~Karpenko -- A.~Merkurjev~\cite{km2})}
Let $G$ be a finite $p$-group and
$k$ be a field containing a primitive $p$th root of unity.  Then
\begin{equation} \label{e.km} 
\ed_k(G) = \min \, \dim(\phi) \, , 
\end{equation}
where the minimum is taken over all faithful
$k$-representations $\phi$ of $G$.
\end{cor}

\begin{proof} 
We apply Theorem~\ref{thm.central} to the exact sequence 
$1 \to C \to G \to G/C \to 1$, where 
$C$ be the socle of $G$, i.e., 
$C := \{ g \in Z(G) \, | \, g^p = 1 \}$.
Since $\dim(\rho)$ is a power of $p$ for every irreducible representation
$\rho$ of $G$, we may replace $\gcd$ by $\min$
in~\eqref{e.central}. Choosing a minimal set of generators 
$\chi_1, \dots, \chi_r$ of $C^*$ so that the sum on the right hand side of
\eqref{e.central} has minimal value, and $\rho_i \in \Rep^{\chi_i}$
of minimal dimension, we see that~\eqref{e.central} reduces to 
$\ed_k(G) \geq \dim(\rho_1) + \dots + \dim(\rho_r)$.
Equivalently, $\ed_k(G) \ge \dim(\rho)$,
where $\rho := \rho_1 \oplus \dots \oplus \rho_r$ is faithful by
elementary $p$-group theory.
This shows that $\ed_k(G) \ge \min \, \dim(\phi)$ in~\eqref{e.km}.
The opposite inequality follows
from~\eqref{e.linear-rep}.
\end{proof}

\begin{cor} \label{cor.spin} Let $\Spin_n$ be the split spinor
group over a a field $k$ of characteristic $0$. Assume $n \ge 15$.
Then 
                                   
\smallskip                         
(a) $\ed(\Spin_{n}) = 2^{(n-1)/{2}} - \frac{n(n-1)}{2}$,
if $n$ is odd,                                
                                              
\smallskip   
(b) $\ed(\Spin_{n}) = 2^{(n-2)/2} - \frac{n(n-1)}{2}$,
if $n \equiv 2 \pmod{4}$, and                 
                                              
\smallskip                  
(c) $2^{(n-2)/2} - \frac{n(n-1)}{2} + 2^m \le \ed(\Spin_{n}) \le 
2^{(n-2)/2} - \frac{n(n-1)}{2} + n$, if $n \equiv 0 \pmod{4}$.                 
Here $2^m$ is the largest power of $2$ dividing $n$.
\end{cor}

We remark that M.~Rost and S.~Garibaldi have computed 
the essential dimension of $\Spin_n$ for every $n \le 14$;
see~\cite{rost2} and~\cite{garibaldi2}.

\begin{proof}[Proof outline]
The lower bounds (e.g.,  
$\ed(\Spin_{n}) \ge 2^{(n-1)/{2}} - \frac{n(n-1)}{2}$, in part (a)) are
valid whenever $\Char(k) \neq 2$; they can be deduced
either directly from Theorem~\ref{thm.central} or 
by applying the inequality~\eqref{e.subgroup}
to the finite $2$-subgroup $H$ of $G = \Spin_n$, where $H$ is the
inverse image of the diagonal subgroup $\mmu_2^{n-1} \subset \SO_n$,
as in~\eqref{e.A},
under the natural projection $\pi \colon \Spin_n \to \SO_n$. Here $\ed(H)$
is given by Corollary~\ref{cor.km}.

The upper bounds  (e.g., $\ed(\Spin_{n}) \le 2^{(n-1)/{2}} - \frac{n(n-1)}{2}$,
in part (a)) follow from the inequality~\eqref{e.linear-rep}, where
$V$ is spin representation
$V_{\rm spin}$ in part (a), the half-spin representation
$V_{\rm half-spin}$ in part (b), and to $V_{\rm half-spin} \oplus 
V_{\rm natural}$ in part (c), where 
$V_{\rm natural}$ is the natural $n$-dimensional representation of
$\SO_n$, viewed as a representation of $\Spin_n$ via $\pi$.
The delicate point here is to check that these representations are
generically free. In characteristic $0$ this is due to
E.~Andreev and V.~Popov~\cite{ap} for $n \ge 29$ and 
to A.~Popov~\cite{ampopov} in the remaining cases.

For details, see~\cite{brv2} and
(for the lower bound in part (c))~\cite[Theorem 4.9]{me2}. 
\end{proof}

To convey the flavor of the proof of Theorem~\ref{thm.central},
I will consider a special case, where $G$ is finite and $r = 1$. 
That is, I will start with 
a sequence
\begin{equation} \label{e.central-extension}
1 \to \mmu_p \to G \to \overline{G} \to 1 
\end{equation}
of finite groups and will aim to show that
\begin{equation} \label{e.central-simplified}
\ed_k(G) \geq \gcd_{\rho \in \Rep'} \dim(\rho) \, ,
\end{equation}
where $k$ contains a primitive $p$th root 
and $\Rep'$ denotes the set of irreducible representations
of $G$ whose restriction to $\mmu_p$ is non-trivial. The proof
relies on the following two results, which are of independent interest.

\begin{theorem} \label{thm.incompressible}
{\rm (Karpenko's Incompressibility Theorem; \cite[Theorem 2.1]{karpenko1})}
Let $X$ be a Brauer-Severi variety of prime power index $p^m$, over 
a field $K$ and let $f \colon X \dasharrow X$ be a rational map 
defined over $K$. Then $\dim_K \, \operatorname{Im}(f) \ge p^m - 1$.
\end{theorem}

\begin{theorem} \label{thm.index}
{\rm (Merkurjev's Index Theorem~\cite[Theorem 4.4]{km2}; cf. also~\cite{me1})}
Let $K/k$ be a field extension, and $\partial_K \colon H^1(K, \overline{G})
\to H^2(K, \mmu_p)$ be the connecting map induced by the short
exact sequence~\eqref{e.central-extension}. Then the maximal value of
the index of $\partial_K(a)$, as $K$ ranges over all field extension 
of $k$ and $a$ ranges over $H^1(K, \overline{G})$, equals 
$\gcd_{\rho \in \Rep'} \dim(\rho)$.
\end{theorem}

Recall that $H^2(K, \mmu_p)$ is naturally isomorphic to the $p$-torsion
subgroup of the Brauer group $\Br(K)$, so that it makes sense to talk
about the index.

I will now outline an argument, due to M.~Florence~\cite{florence},
which deduces the inequality~\eqref{e.central-simplified}
from these two theorems.
To begin with, let us choose a faithful representation $V$ of $G$,
where $C$ acts by scalar multiplication. In particular, we can
induce $V$ from a faithful 1-dimensional representation 
$\chi \colon C = \mmu_p \to k^*$. We remark that $\chi$
exists because we assume that $k$ 
contains a primitive $p$th root of unity 
and that we do not require $V$ to be irreducible. 
  
By~\eqref{e.linear-rep0}, there exists a non-zero $G$-equivariant 
rational map $f \colon V \dasharrow V$ defined over $k$ (or 
a {\em rational covariant}, for short) whose image has dimension $\ed(G)$. 
We will now replace $f$ by
a non-zero {\em homogeneous} rational covariant 
$f_{\rm hom} \colon V \dasharrow V$.  Here ``homogeneous" 
means that $f_{\rm hom}(tv) = t^d f_{\rm hom}(v)$ 
for some $d \ge 1$. Roughly speaking, $f_{\rm hom}$ 
is the ``leading term" of $f$, relative to some basis of $V$,
and it can be chosen so that 
\[ \dim \operatorname{Im}(f_{\rm hom}) \le 
\dim \operatorname{Im}(f) = \ed(G) \, ; \] 
see \cite[Lemma 2.1]{kls}. Since we no longer need the original 
covariant $f$, we will replace $f$ by $f_{hom}$ and thus assume that
$f$ is homogeneous. Note that $G$ may not act faithfully on 
the image of $f$ but this will not matter to us in the sequel.
Since $f$ is homogeneous and non-zero, it descends to 
an $\overline{G}$-equivariant rational map 
$\overline{f} \colon \PP(V) \dasharrow \PP(V)$ defined over $k$,
whose image has dimension $\le \ed_k(G) - 1$. 

Now, given a field extension $K/k$ and a $\overline{G}$-torsor 
$T \to \Spec(K)$ in $H^1(K, \overline{G})$, we can twist $\PP(V)$
by $T$. The resulting $K$-variety $^T \PP(V)$ is defined as the 
quotient of $\PP(V) \times_K T$ by the natural 
(diagonal) $\overline{G}$-action.
One can show, using the theory of descent, that this action is in fact 
free, i.e., the natural projection map 
$\PP(V) \times_K T \to ^T \PP(V)$ is a $\overline{G}$-torsor; 
see~\cite[Proposition 2.12 and Remark 2.13]{florence}.
Note that we have encountered a variant of this construction
in the previous section, where we wrote 
$\PP(V) \times^{\overline{G}} T$ in place of $^T \PP(V)$.

We also remark that $^T \PP(V)$ is a $K$-form of $\PP(V)$, i.e., 
is a Brauer-Severi variety defined over $K$. Indeed, if 
a field extension $L/K$ splits $T$ then it is easy to see that
$^T \PP(V)$ is isomorphic to $\PP(V)$ over $L$.  One can now show 
that the index of this Brauer-Severi variety equals the index of
$\partial_K(T) \in H^2(K, \mmu_p)$; in particular, it is a power of $p$.
By Theorem~\ref{thm.index} we can choose $K$ and $T$ so that 
\[ \ind(^T \PP(V)) = \gcd_{\rho \in \Rep'} \dim(\rho) \, . \]
\indent
The $\overline{G}$-equivariant rational map $\overline{f} \colon 
\PP(V) \dasharrow \PP(V)$ induces a $\overline{G}$-equivariant
rational map $\overline{f} \times \id \colon \PP(V) \times T
\dasharrow \PP(V) \times T$, which, in turn, descends to 
a $K$-rational map $^T \overline{f} \colon ^T \PP(V) \dasharrow ^T \PP(V)$. 
Since the dimension of the image of $\overline{f}$
is $\le \ed_k(G) - 1$, the dimension of the image of $\overline{f} \times \id$
is $\le \ed_k(G) - 1 + \dim(G)$, and thus the dimension of the image of
$^T \overline{f}$ is $\le \ed_k(G) - 1$. By Theorem~\ref{thm.incompressible},
\[ \ed_k(G) - 1 \ge \dim_K (\operatorname{Im}(^T \overline{f})) \ge 
\ind(^T \PP(V)) - 1 = \gcd_{\rho \in \Rep'} \dim(\rho) -1 \, , \]
and~\eqref{e.central-simplified} follows.
\qed
 
\begin{remark} \label{rem.roland}
Now suppose $G$ is finite but $r \ge 1$ is arbitrary.
The above argument has been modified by R.~L\"otscher~\cite{lotscher}
to prove Theorem~\ref{thm.central} 
in this more general setting. The proof relies on
Theorem~\ref{thm.index} and a generalization of 
Theorem~\ref{thm.incompressible} to the case where $X$ is 
the direct product of Brauer-Severi varieties 
$X_1 \times \dots \times X_r$, such that $\ind(X_i)$ 
is a power of $p$ for each $i$; see \cite[Theorem 2.1]{km2}.

Here we choose our faithful $k$-representation $V$
so that $V = V_1 \times \dots \times V_r$, 
where $C$ acts on $V_i$ by scalar multiplication via a multiplicative
character $\chi_i \in C^*$, and $\chi_1, \dots, \chi_r$ generate
$C^*$. Once again, there exists a $G$-equivariant rational map
$f \colon V \dasharrow V$ whose image 
has dimension $\ed(G)$. To make the rest of the 
argument go through in this setting one needs 
to show that $f$ can be chosen to be multi-homogeneous,
so that it will descend to a $\overline{G}$-equivarint rational
map 
\[ \overline{f} \colon \PP(V_1) \times \dots \PP(V_r) \dasharrow
\colon \PP(V_1) \times \dots \PP(V_r) \, . \]
If $G$ is finite, this is done in~\cite{lotscher}. 
The rest of the argument goes through unchanged.
 
In his (still unpublished) Ph.~D. thesis L\"otscher has 
extended this proof of Theorem~\ref{thm.central} 
to the case where $G$ is no longer 
assumed to be finite. His only requirement on $G$ is that it
should have a completely reducible faithful $k$-representation.  
The only known proof of Theorem~\ref{thm.central} in full generality 
uses the notion of essential dimension for an algebraic stack, 
introduced in \cite{brv1}; cf. also~\cite{brv3}. For details, see
\cite[Theorem 4.8 and Example 3.7]{me2}, in combination with
\cite[Theorem 4.4 and Remark 4.5]{km2}.
\end{remark}

\section{Essential dimension at $p$ and two types of problems}
\label{sect.type}

Let $p$ be a prime integer.  I will say that a field extension $L/K$ 
is {\em prime-to-}$p$ if $[L:K]$ is finite and not divisible by $p$.  

This section is mostly ``metamathematical"; the main point 
I would like to convey is that some problems in Galois cohomology
and related areas are sensitive to prime-to-$p$ extensions and
some aren't. Loosely speaking, I will refer to such problems 
as being of ``Type 2" and ``Type 1", respectively. 

More precisely, suppose we are given a functor 
$\mathcal{F} \colon \Fields_k \to \Sets$ 
and we would like to show that some (or every) 
$\alpha \in \mathcal{F}(K)$ has a certain property.
For example, this property may be that $\ed(\alpha) \le d$ 
for a given $d$. If our functor is 
$\mathcal{F}(K) = H^1(K, \Orth_n)$, we may want to show
that the quadratic form representing $\alpha$ 
is isotropic over $K$. If our functor is 
$\mathcal{F}(K) = H^1(K, \PGL_n)$, we may ask
if the central simple algebra representing $\alpha$
is a crossed product. Note that in many interesting examples,
including the three examples above, the property 
in question is functorial, i.e., if $\alpha \in \mathcal{F}(K)$
has it then so does $\alpha_L$ for every field extension $L/K$.

The problem of whether or not $\alpha \in \mathcal{F}(K)$ has 
a property we are interested in can be broken into two steps. 
For the first step we 
choose a prime $p$ and ask whether or not $\alpha_L$ has the desired 
property for some prime-to-$p$ extension $L/K$. This is what I call 
a {\em Type 1 problem}. If the answer is ``no" for some $p$ then 
we are done: we have solved the original problem in the negative. 
If the answer is ``yes" for every prime $p$, then the remaining
problem is to determine whether or not $\alpha$ itself has 
the desired property. I refer to problems of this type 
as {\em Type 2 problems}.
Let me now explain what this means in the context of essential 
dimension.

Let $\mathcal{F} \colon \Fields_k \to \Sets$ be 
a functor and $a \in \mathcal{F}(K)$
for some field $K/k$. The essential dimension
$\ed(a; p)$ of $a$ at a prime integer $p$ is defined
as the minimal value of $\ed(a_L)$, as $L$ ranges
over all finite field extensions $L/K$ such that
$p$ does not divide the degree $[L:K]$.
The essential dimension $\ed(\mathcal{F}; p)$ is then defined as
the maximal value of $\ed(a; p)$, as $K$ ranges over
all field extensions of $k$ and $a$ ranges over $\mathcal{F}(K)$.
                
As usual, in the case where $\mathcal{F}(K) = H^1(K, G)$ for some algebraic
group $G$ defined over $k$, we will write $\ed(G; p)$
in place of $\ed(\mathcal{F}; p)$. Clearly,
$\ed(a; p) \le \ed(a)$, $\ed(\mathcal{F}; p) \le \ed(\mathcal{F})$,
and $\ed(G; p) \le \ed(G)$ for every prime $p$.
                
In the previous three sections we proved a number of lower bounds 
of the form $\ed(G) \ge d$, 
where $G$ is an algebraic group and $d$ is a positive integer.
A closer look reveals that in every single case the argument 
can be modified to show that $\ed(G; p) \ge d$,
for a suitable prime $p$. (Usually $p$ is a
so-called ``exceptional prime" for $G$;  
see, e.g.,~\cite{steinberg} or~\cite{me2}. 
Sometimes there is more than one such prime.)
In particular, the arguments we used 
in Examples~\ref{ex.Pf},~\ref{ex.O_n},~\ref{ex.abelian} 
and~\ref{ex.PGLn} 
show that $\ed(\G_2; 2) = 3$, $\ed(\Orth_n; 2) = n$, $\ed(\mmu_p^r; p) = r$
and $\ed(\PGL_{p^s}; p) \ge 2s$, respectively. In 
Theorem~\ref{thm.gille-r} we may replace $\ed(G)$ by $\ed(G; p)$,
as long as $A$ is a $p$-group;~see~\cite[Theorem 1.2(b)]{gr}.
Consequently, in Corollary~\ref{cor.finite-abelian} 
$\ed(G)$ can be replaced by $\ed(G; p)$,
where $p = 2$ in parts (a), (c), (d), (e), (g), (h), (i) and
$p = 3$ in part (f). Theorem~\ref{thm.central} remains 
valid with $\ed(G)$ replaced by $\ed(G; p)$.\footnote{At 
the moment the only known proof of this
relies on the stack-theoretic approach;
see the references at the end of Remark~\ref{rem.roland}. 
The more elementary ``homogenization" argument 
I discussed in the previous section has not (yet?) 
yielded a lower bound on $\ed(G; p)$.}
Consequently,
Corollary~\ref{cor.km} remains valid if
$\ed(G)$ is replaced by $\ed(G; p)$ (see~\cite{km2}),
and Corollary~\ref{cor.spin}
remains valid if $\ed(\Spin_n)$ is replaced by $\ed(\Spin_n; 2)$
(see~\cite{brv2}).
                                                  
The same is true of virtually all existing methods
for proving lower bounds on $\ed(\mathcal{F})$ and, 
in particular, on $\ed(G)$: they are well suited to address 
Type 1 problems 
and poorly suited for Type 2 problems. In this context 
a Type 1 problem is the problem of computing
$\ed(\mathcal{F}; p)$ for various primes $p$ and a Type 2 problem
is the problem of computing $\ed(\mathcal{F})$, 
assuming $\ed(\mathcal{F}; p)$ is known for all $p$. 
 
I will now make an (admittedly vague) claim that this phenomenon 
can be observed in a broader context and illustrate it with three
examples not directy related to essential dimension.  

\begin{observation} \label{ob.type}
Most existing methods in Galois cohomology and related areas 
apply to Type 1 problems only. On the other hand, many long-standing 
open problems are of Type 2.
\end{observation}

\begin{example} \label{ex.crossed-products} {\bf The crossed product problem.}
Recall that a central simple algebra $A/K$ of degree $n$
is a crossed product if it contains a commutative
Galois subalgebra $L/K$ of degree $n$.  We will
restrict our attention to the case where $n = p^r$ is a prime power;
the general case reduces to this one by the primary decomposition
theorem. In 1972 Amitsur~\cite{amitsur}
showed that for $r \ge 3$ a generic division algebra
$U(p^r)$ of degree $p^r$ is not a crossed product, solving a long-standing
open problem.  L.~H.~Rowen and D.~J.~Saltman~\cite[Theorem 2.2]{rs} 
modified Amitsur's argument to show that, in fact, $UD(p^r)_L$ 
is a non-crossed product for any prime-to-$p$ extension $L$ 
of the center of $\UD(p^r)$.

For $r = 1, 2$ it is not known whether or not every central 
smple algebra $A$ of degree $p^r$ is a crossed product.
It is, however, known that every such algebra becomes 
a crossed product after a prime-to-$p$ extension of the center;
see~\cite[Section 1]{rs}. In other words, the Type 1 part of
the crossed product problem has been completely solved, and
the remaining open questions,
for algebras of degree $p$ and $p^2$, are of Type 2. 
\end{example}

\begin{example} {\bf The torsion index.} 
Let $G$ be an algebraic group defined over $k$ and
$K/k$ be a field extension.  The torsion index $n_{\alpha}$
of $\alpha \in H^1(K, G)$ was defined by Grothendieck as the 
greatest common divisor of the degrees $[L:K]$,
where $L$ ranges over all finite splitting fields $L/K$.
The torsion index $n_G$ of $\mathcal{G}$ is then the least common 
multiple of $n_{\alpha}$ taken over all $K/k$ and all 
$\alpha \in H^1(K, G)$.
One can show that $n_G = n_{\alpha_{\rm ver}}$, where
$\alpha_{\rm ver} \in H^1(K_{versal}, G)$ is a versal $G$-torsor.
One can also show, using a theorem of J.~Tits~\cite{serre-bourbaki}, that
the prime divisors of $n_G$ are precisely the exceptional primes of $G$.

The problem of computing $n_G$ and more generally, of $n_{\alpha}$ for
$\alpha \in H^1(K, G)$ can thus be rephrased as follows. Given
an exceptional prime $p$ for $G$, find the highest exponent $d_p$
such that $p^{d_p}$ divides $[L:K]$ for every splitting extension $L/K$.
It is easy to see that this is a Type $1$ problem; $d$ does not change
if we replace $\alpha$ by $\alpha_{K'}$, where $K'/K$ is a prime-to-p
extension. This torsion index $n_G$ has been computed 
Tits and B.~Totaro, for all simple groups $G$ that are either 
simply connected or adjoint; for details and further references, 
see~\cite{tits, totaro2}.
        
The remaining Type $2$ problem consists of finding 
the possible values of $e_1, \dots, e_r$ such that $\alpha_{\rm ver}$ 
is split by a field extension $L/K$ of degree $p_1^{e_1} \dots p_r^{e_r}$, 
where $p_1, \dots, p_r$ are the exceptional primes for $G$. 
This problem is open for many groups $G$. It is particularly natural 
for those $G$ with only one exceptional prime, e.g., $G = \Spin_n$.
\end{example}

\begin{example} \label{cd}
{\bf Canonical dimension.}
Let $G$ be a connected linear algebraic group defined over $k$,
$K/k$ be a field extension, and $X$ be a $G$-torsor over $K$.
Recall that the {\em canonical dimension} $\cd(X)$ of $X$
is the minimal value of $\dim_K(\operatorname{Im}(f))$,
where the minimum is taken over all rational maps
$f \colon X \dasharrow X$ defined over $K$.
In particular, $X$ is split if and only if $\cd(X) = 0$.
The maximal possible value of $\cd(X)$, as $X$ ranges
over all $G$-torsors over $K$ and $K$ ranges over all field
extensions of $k$, is called the canonical dimension of $G$
and is denoted by $\cd(G)$. 
Clearly $0 \le \cd(G) \le \dim(G)$ and
$\cd(G) = 0$ if and only if $G$ is special.
For a detailed discussion of the notion of canonical dimension,
we refer the reader to~\cite{ber-r},~\cite{km1} and~\cite{me2}.

Computing the canonical dimension $\cd(G)$ of an algebraic group $G$
is a largely open Type 2 problem.  The associated Type 1 problem 
of computing the canonical $p$-dimension $\cd(G; p)$ has been solved by
Karpenko-Merkurjev~\cite{km1} and K.~Zainoulline~\cite{zainoulline}.
\end{example}

\section{Finite groups of low essential dimension}
\label{sect.type2}

Suppose we would like to determine the essential dimension of
a finite group $G$. To keep things simple, we will assume throughout
this section that, unless otherwise specified, the base
field $k$ is algebraically closed and of characteristic $0$.
Let us break up the problem of computing $\ed(G)$ into a Type 1 part
and a Type 2 part, as we did in the previous section.

The Type 1 problem is to determine $\ed(G; p)$ for a prime $p$.
It is not difficult to show
that $\ed(G; p) = \ed(G_p; p)$, where $p$ is a prime and
$G_p$ is a $p$-Sylow subgroup of $G$; see~\cite[Lemma 4.1]{mr1}
or~\cite[Proposition 5.1]{me1}.  The value of $\ed(G_p; p)$
is given by Corollary~\ref{cor.km}. So, to the extent that 
we are able to compute the dimension
of the smallest faithful representation of $G_p$,
our Type 1 problem has been completely solved, i.e.,  
we know $\ed(G; p)$ for every prime $p$.

Now our best hope of computing $\ed(G)$ is to obtain a strong 
upper bound $\ed(G) \le n$, e.g., by constructing an 
explicit $G$-equivariant dominant rational map $V \dasharrow Y$, 
as in~\eqref{e.compression1}, with $\dim(Y) = n$. 
If $n = \ed(G; p)$ then we conclude that
$\ed(G) = \ed(G; p)$, i.e., the remaining Type 2 problem 
is trivial, and we are done.
In particular, this is what happens if $G$ is a $p$-group.

If the best upper bound we can prove is $\ed(G) \le n$, where $n$
  is strictly greater than
$\ed(G; p)$ for every $p$ then we are entering rather murky waters.
Example~\ref{ex.pq} below shows that it is indeed possible for
$\ed(G)$ to be strictly greater than $\ed(G; p)$ for every
prime $p$. On the other hand, there is no general method 
for computing $\ed(G)$ in such cases. The only ray of light 
in this situation is that it may be possible to prove 
a lower bound of the form $\ed(G) > d$, where $d = 1$ or 
(with more effort) $2$ and sometimes even $3$.

Let us start with the simplest case where $d = 1$.

\begin{lemma} \label{lem.ed=1}
\rm{(cf.~\cite[Theorem 6.2]{bu-r})}
Let $G$ be a finite group.  Then 

\smallskip
(a) $\ed(G) = 0$ if and only if $G = \{ 1 \}$,

\smallskip
(b) $\ed(G) = 1$ if and only if $G \neq \{ 1 \}$ is either cyclic 
or odd dihedral.
\end{lemma}

\begin{proof} Let $V$ be a faithful linear representation of $G$.
By~\eqref{e.linear-rep0}
there exists a dominant $G$-equivariant rational map
$V \dasharrow X$, where $G$ acts faithfully on $X$ and $\dim(X) = \ed(G)$.

\smallskip
(a) If $\ed(G) = 0$ then $X$ is a point. This forces $G$ to be trivial.

\smallskip
(b) If $\ed(G) = 1$ then $X$ is a rational curve, by a theorem of L\"uroth.
We may assume that $X$ is
smooth and complete, i.e., we may assume that
$X = \PP^1$. Consequently, $G$ is isomorphic to
a subgroup of $\PGL_2$. By a theorem of Klein~\cite{klein}, $G$ is cyclic,
dihedral or is isomorphic to $\Sym_4$, $\Alt_4$ or $\Sym_5$. If
$G$ is an even dihedral group, $\Sym_4$, $\Alt_4$ or $\Sym_5$ 
then $G$ contains a copy of $\ZZ/2 \times \ZZ/ 2\ZZ \simeq 
\mmu_2 \times \mmu_2$. Hence, 
\[ \ed(G) \ge \ed(\mmu_2^2)  = 2 \, ; \]
see Example~\ref{ex.abelian}.  
This means that if $\ed(G) = 1$ then $G$ 
is cyclic or odd dihedral. 

Conversely, if $G$ is cyclic or odd dihedral then one can easily check
that, under our assumption on $k$, $\ed_k(G) = 1$.
\end{proof}

\begin{example} \label{ex.pq} Suppose $q$ and $r$ are odd primes and $q$ divides
$r-1$. Let $G = \ZZ/r \ZZ \rtimes \ZZ/ q \ZZ$ be a non-abelian 
group of order $rq$. Clearly all Sylow subgroups of $G$ are cyclic;
hence, $\ed(G; p) \le 1$ for every prime $p$. On the other hand,
since $G$ is neither cyclic nor odd dihedral, 
Lemma~\ref{lem.ed=1} tells us that $\ed(G) \ge 2$.
\qed
\end{example} 

Similar reasoning can sometimes be used to show that $\ed(G) > 2$.
Indeed, assume that $\ed(G) = 2$. Then there is a faithful
representation $V$ of $G$ and a dominant rational $G$-equivariant map
\begin{equation} \label{e.versal}
V \dasharrow X \, ,
\end{equation}
where $G$ acts faithfully on $X$ and $\dim(X) = 2$. 
By a theorem of G.~Castelnuovo, $X$ is a rational surface.
Furthermore, we may assume that $X$ is smooth, complete, and
is minimal with these properties (i.e., does not allow 
any $G$-equivariant blow-downs $X \to X_0$, with $X_0$ smooth). 
Such surfaces (called minimal rational $G$-surfaces)
have been classified by Yu.~Manin and V.~Iskovskikh,
following up on classical work of F.~Enriques; for details 
and further references, see~\cite{duncan1}.  This classification 
is significantly more complicated than Klein's classification 
of rational curves but one can use it to determine, at least 
in principle, which finite groups $G$ can act on 
a rational surface and describe all such actions; cf.~\cite{di}. 
Once all minimal rational $G$-surfaces $X$ are accounted for, one then 
needs to decide, for each $X$, whether or not the $G$-action 
is versal, i.e., whether or not a dominant rational 
$G$-equivariant map~\eqref{e.versal} can exist for some
faithful linear representation $G \to \GL(V)$.
Note that by the Going Down Theorem~\ref{going-down}
if some abelian subgroup $A$ of $G$ acts on $X$ without fixed points 
then the $G$-action on $X$ cannot be versal.  
If every minimal rational $G$-surface $X$ can be ruled out this way
(i.e., is shown to be non-versal) then one can conclude that $\ed(G) > 2$.

This approach was used by Serre to show that
$\ed(\Alt_6) > 2$; see~\cite[Proposition 3.6]{serre-bourbaki2}.
Since the upper bound 
$\ed(\Alt_6) \le \ed(\Sym_6) \le 3$ was previously known 
(cf.~\eqref{e.upperSym} and the references there) 
this implies $\ed(\Alt_6) = 3$. Note that   
\[ \ed(\Alt_6; p) = \begin{cases}
\text{$2$, if $p = 2$ or $3$,} \\
\text{$1$, if $p = 5$, and} \\
\text{$0$, otherwise;} \end{cases} \]
see~\eqref{e.Sym}.
A. Duncan~\cite{duncan1} has recently refined this approach to give 
the following complete classification of groups of essential 
dimension $\le 2$. 

\begin{theorem} \label{thm.duncan1}
Let $k$ be an algebraically closed field of characteristic $0$ and
$T = \GG_m^2$ be the $2$-dimensional torus over $k$. A finite group
$G$ has essential dimension $\le 2$ if and only if
it is isomorphic to a subgroup of one of the following groups:

\smallskip
(1) The general linear group $\GL_2 (k)$, 

\smallskip
(2) $\operatorname{PSL}_2 (\FF_7)$, the simple group of order 168,

\smallskip
(3) $\Sym_5$, the symmetric group on $5$ letters,

\smallskip
(4) $T \rtimes G_1$, where $|G \cap T|$ is coprime to $2$ and $3$
and 
\[ G_1 = \left\langle \begin{array}{ll} 
\begin{pmatrix} 1 & -1 \\ 1 & 0 \end{pmatrix}, &
   \begin{pmatrix} 0 & 1 \\  1 & 0 \end{pmatrix} 
\end{array} \right\rangle 
\simeq D_{12} , \]

\smallskip
(5) $T \rtimes G_2$, where $|G \cap T|$ is coprime to $2$
and 
\[ G_2 = \left\langle \begin{array}{ll} 
\begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}, \; \;
   \begin{pmatrix} 0 & 1 \\  1 & 0 \end{pmatrix} 
\end{array} \right\rangle 
\simeq D_8 , \]

\smallskip
(6) $T \rtimes G_3$, where $|G \cap T|$ is coprime to $3$
and 
\[ G_3 = \left\langle \begin{array}{ll} 
\begin{pmatrix} 0 & -1 \\ 1 & -1 \end{pmatrix}, \; \;
   \begin{pmatrix} 0 & -1 \\  -1 & 0 \end{pmatrix} 
\end{array} \right\rangle 
\simeq \Sym_3 , \]

\smallskip
(7) $T \rtimes G_4$, where $|G \cap T|$ is coprime to $3$
and 
\[ G_4 = \left\langle \begin{array}{ll} 
\begin{pmatrix} 0 & -1 \\ 1 & -1 \end{pmatrix}, \; \;
   \begin{pmatrix} 0 & 1 \\  1 & 0 \end{pmatrix} 
\end{array} \right\rangle 
\simeq \Sym_3 . \]
\end{theorem}

If one would like to go one step further and show that $\ed(G) > 3$
by this method, for a particular finite group $G$,
the analysis becomes considerably more complicated. 
First of all, while $X$ in~\eqref{e.versal} is still unirational, 
if $\dim(X) \ge 3$, we can no longer conclude that it is rational. 
Secondly, there is no analogue of the Enriques-Manin-Iskovskikh 
classification of rational surfaces in higher dimensions. 
Nevertheless, in dimension $3$ one can sometimes use Mori theory to
get a handle on $X$.  In particular, Yu.~Prokhorov~\cite{prokhorov} 
recently classified the finite simple groups with faithful
actions on rationally connected threefolds.  This classification was
used by Duncan~\cite{duncan2} to prove the following theorem,
which is out of the reach of all previously existing methods.

\begin{theorem} \label{thm.S7}
Let $k$ be a field of characteristic $0$. Then
$\ed_k(A_7) = \ed_k(S_7) = 4$.
\end{theorem}

\noindent
Note that $\ed(\Alt_7; p) \le \ed(\Sym_7; p) \le 3$ for 
every prime $p$; cf.~\cite[Corollary 4.2]{mr1}.

\section{Open problems}
\label{sect.problems}

\subsection{Strongly incompressible elements}
\label{subsect.incompressible}
Let $\mathcal{F} \colon \Fields_k \to \Sets$ be a covariant functor. 
We say that an object $\alpha \in \mathcal{F}(K)$ 
is {\em strongly incompressible} if $\alpha$ does not descend to any
proper subfield of $K$.

Examples of strongly incompressible elements in the case 
where $G$ is a finite group, $K$ is the function field 
of an algebraic curve $Y$ over $k$, and
$\mathcal{F} = H^1(\, \ast \, , G)$, are given in~\cite{compressions}.
In these examples $\alpha$ is represented by a (possibly ramified) 
$G$-Galois cover $X \to Y$.
I do not know any such examples in higher dimensions.

\begin{problem} \label{pr1}
Does there exist a finitely generated field extension 
$K/k$ of transcendence degree $\ge 2$ and a finite 
group $G$ (or an algebraic group $G$ defined over $k$)
such that $H^1(K, G)$ has a strongly incompressible element?
\end{problem}

For $G = \operatorname{O}_n$ Problem~\ref{pr1} 
is closely related to the questions of existence 
of a minimal field of definition of a quadratic form
posed at the beginning of Section~\ref{sect.def}.

It is easy to see that if an element of $H^1(K, \PGL_n)$ 
represented by a non-split central simple algebra $A$ is 
strongly incompressible and $\trdeg_k(K) \ge 2$
then $A$ cannot be cyclic.  In particular,
if $n = p$ is a prime then the existence of 
a strongly incompressible element in $H^1(K, \PGL_n)$ 
would imply the existence of a non-cyclic algebra
of degree $p$ over $K$, thus solving (in the negative)
the long-standing cyclicity conjecture of Albert.

\subsection{Symmetric groups}

\begin{problem} What is the essential dimension 
of the symmetric group $\Sym_n$? of the alternating group $\Alt_n$?
\end{problem}

Let us assume that $\Char(k)$ does not divide $n!$.
Then in the language of Section~\ref{sect.type}, 
the above problem is of Type 2.
The associated Type 1 problem has been solved:
$\ed(\Sym_n; p) = [n/p]$ (see \cite[Corollary 4.2]{mr1}) and similarly
\begin{equation} \label{e.Sym} \text{ 
$\ed(\Alt_n; p)$} = \begin{cases} 
\text{$2[\frac{n}{4}]$, if $p = 2$, and} \\
\text{$[\frac{n}{p}]$, otherwise.} 
\end{cases} 
\end{equation}
It is shown in~\cite{bu-r} that $\ed(\Sym_{n + 2}) \ge \ed(\Sym_n) + 1$,
$\ed(\Alt_{n + 4}) \ge \ed(\Alt_n) + 2$, and
\begin{equation} \label{e.upperSym}
\ed(\Alt_n) \le \ed(\Sym_n) \le n - 3 \, . 
\end{equation}
I believe the true value of $\ed(\Sym_n)$ is closer to $n-3$ than
to $[n/2]$; the only piece of evidence for this is Theorem~\ref{thm.S7}.

\subsection{Cyclic groups}

\begin{problem} \label{pr.cyclic}
What is the essential dimension $\ed_k(\ZZ/n \ZZ)$?
\end{problem}

Let us first consider the case where $\Char(k)$ is prime to $n$.
Under further assumptions that $n = p^r$ is a prime power 
and $k$ contains a primitive $p$th root of unity $\zeta_p$, 
Problem~\ref{pr.cyclic} has been solved by Florence~\cite{florence}.
It is now a special case of Corollary~\ref{cor.km}:
\begin{equation} \label{e.cyclic}
\ed_k(\ZZ/p^r \ZZ) = \ed_k(\ZZ/p^r \ZZ; p) = [k(\zeta_{p^r}): k] \, ; 
\end{equation}
see~\cite[Corollary 5.2]{km2}. 
This also settles the (Type 1) problem of computing
$\ed_k(\ZZ/ n \ZZ; p)$ for every integer $n \ge 1$ and every 
prime $p$. Indeed, \[ \ed_k(\ZZ/ n \ZZ; p) = \ed_k(\ZZ/ p^r \ZZ; p) \, , \]
where $p^r$ is the largest power of $p$ dividing $n$.
Also, since $[k(\zeta_p):k]$ is prime to $p$, for the purpose
of computing $\ed_k(\ZZ/ n \ZZ; p)$ we are allowed to 
replace $k$ by $k(\zeta_p)$; then formula \eqref{e.cyclic} applies.

If we do not assume that $\zeta_p \in k$ then the best currently
known upper bound on $\ed_k(\ZZ/ p^r \ZZ)$, due to A. Ledet~\cite{ledet1}, 
is $\ed_k(\ZZ/p^r \ZZ) \le \varphi(d) p^e$.
Here $[k(\zeta_{p^r}): k] = d p^e$, where $d$ divides $p-1$,
and $\varphi$ is the Euler $\varphi$-function.

Now let us suppose $\Char(k) = p > 0$. Here it is easy to see that
$\ed_k(\ZZ/p^r \ZZ) \le r$; Ledet~\cite{ledet2} conjectured that 
equality holds. This seems to be out of reach at the moment,
at least for $r \ge 5$.  
More generally, essential dimension of finite (but not necessarily 
smooth) group schemes over a field $k$ of prime characteristic is 
poorly understood; some interesting results in this direction can 
be found in~\cite{tv}. 

\subsection{Quadratic forms}

Let us assume that $\Char(k) \ne 2$. The following question 
is due to J.-P.~Serre (private communication, April 2003).

\begin{problem} \label{pr.Serre} 
If $q$ is a quadratic form over $K/k$, is it true 
that $\ed(q;2) = ed(q)$?
\end{problem} 

 A similar question for central
simple algebras $A$ of prime power degree $p^r$ is also open:
is it true that $\ed(A; p) = \ed(A)$? 

Here is another natural essential dimension question 
in the context of quadratic form theory.

\begin{problem} \label{pr.Witt}
Assume $\Char(k) \ne 2$.  If $q$ and $q'$ are Witt equivalent
quadratic forms over a field $K/k$, is it true 
that $\ed_k(q) = \ed_k(q')$?
\end{problem}

The analogous question for central simple algebras, with Witt
equivalence replaced by Brauer equivalence, has a negative answer.
Indeed, assume $k$ contains a primitive $4$th root of unity
and $D = UD_k(4)$ is a universal division algebra of degree $4$.
Then $\ed(D) = 5$ (see \cite[Corollary 1.2]{me3},
cf.~also~\cite{rost}) while
$\ed \, \Mat_2(D) = 4$ (see~\cite[Corollary 1.4]{lrrs}).

\subsection{Canonical dimension of Brauer-Severi varieties}

Let $X$ be a smooth complete variety defined over a field $K/k$.
The canonical dimension $\cd(X)$ is the minimal dimension of the
image of a $K$-rational map $X \dasharrow X$.
For a detailed discussion of this notion, see~\cite{km1} and~\cite{me2}.

\begin{conjecture} {\rm (Colliot-Th\'el\`ene, 
Karpenko, Merkurjev~\cite{ckm})}
Suppose $X$ is a Brauer-Severi variety of index $n$.
If $n = p_1^{e_1} \dots p_r^{e_r}$ is the prime decomposition of $n$
then $\cd(X) = p_1^{e_1} +  \dots + p_r^{e_r} - r$. 
\end{conjecture}

This is a Type 2 problem. The associated Type $1$ question
is completely answered by Theorem~\ref{thm.incompressible}:
$\cd(X; p_i) = p_i^{e_1} -1$. Also, by
Theorem~\ref{thm.incompressible} the conjecture is true if $r = 1$.
The only other case 
where this conjecture has been proved is $n = 6$; see~\cite{ckm}.
The proof is similar in spirit to the results of Section~\ref{sect.type2}; 
it relies on the classification of rational 
surfaces over a non-algebraically closed field. 
For other values of $n$ the conjecture has not even been 
checked for one particular $X$. 

Note that the maximal value of 
$\cd(X)$, as $X$ ranges over the Brauer-Severi varieties 
of index $n$, equals $\cd(\PGL_n)$. As I mentioned in
Example~\ref{cd}, computing the canonical dimension $\cd(G)$ 
of a linear algebraic (and in particular, simple) group $G$ is 
a largely open Type 2 problem.  In particular, the exact value of
$\cd(\PGL_n)$ is only known if $n = 6$ or a prime power.
 
\subsection{Essential dimension of $\PGL_n$}
\label{subsect.pgln}

\begin{problem} \label{pr.pgln}
What is $\ed(\PGL_n; p)$? $\ed(\PGL_n)$?
\end{problem}

As I mentioned in Section~\ref{sect.def}, this problem originated
in the work of Procesi~\cite{procesi}; for 
a more detailed history, see~\cite{mr1, mr2}. 
The second question appears to be out of reach at the moment,
except for a few small values of $n$.  However, there has been a great 
deal of progress on the first (Type 1) question in the past year.
By primary decomposition $\ed(\PGL_n; p) = \ed(\PGL_{p^r}; p)$,
where $p^r$ is the highest power of $p$ dividing $n$.
Thus we may assume that $n = p^r$. As I mentioned 
in Example~\ref{ex.crossed-products},
every central simple algebra $A$ of degree $p$ becomes cyclic after 
a prime-to-$p$ extension. Hence, $\ed(\PGL_p; p) = 2$; 
cf.~\cite[Lemma 8.5.7]{ry}.
For $r \ge 2$ we have
\[ (r-1)p^r + 1 \le \ed(\PGL_{p^r}; p) \le p^{2r-2} + 1 \, . 
\]
The lower bound is due to Merkurjev~\cite{me4}; the upper bound
is proved in a recent preprint of A.~Ruozzi~\cite{ruozzi}.
(A weaker upper bound, $\ed(\PGL_n; p) \le 2p^{2r-2} - p^r + 1$, 
is proved in~\cite{mr2}.) In particular,   
$\ed(\PGL_{p^2}; p) = p^2 + 1$; see~\cite{me3}.

Note that the argument in~\cite{me4} shows that if $A$ is 
a generic $(\ZZ /p\ZZ)^r$-crossed product then 
$\ed(A; p) = (r-1)p^r + 1$.  As mentioned in 
Example~\ref{ex.crossed-products}, 
for $r \ge 3$ a general division algebra $A/K$ of degree $p^r$
is not a crossed product and neither is $A_L = A \otimes_K L$ 
for any prime-to-$p$ extension $L/K$. Thus 
for $r \ge 3$ it is reasonable to expect
the true value of $\ed(\PGL_{p^r}; p)$ to be strictly 
greater than $(r-1)p^r + 1$.

\subsection{Spinor groups}

\begin{problem} \label{pr.spin1}
Does Corollary~\ref{cor.spin} remain valid over 
an algebraically closed field of characteristic $p > 2$?
\end{problem}

As I mentioned at the beginning of the proof of Corollary~\ref{cor.spin},
the lower bound in each part remains valid over any field of
characteristic $> 2$. Consequently, Problem~\ref{pr.spin1}
concerns only the upper bounds. It would, in fact, suffice to show
that the spin representation $V_{\rm spin}$ and the half-spin
representation $V_{\rm half-spin}$ of $\Spin_n$ are generically 
free, if $n$ is odd or $n \equiv 2$ (mod $4$), respectively;
see~\cite[Lemma 3-7 and Remark 3-8]{brv2}.

\begin{problem} \label{pr.spin2}
What is $\ed_k(\Spin_{4m}; 2)$? $\ed_k(\Spin_{4m})$? Here $m \ge 5$ is
an integer.
\end{problem}

Corollary~\ref{cor.spin} answers this question in the case where $m$ 
is a power of $2$. In the other cases there is a gap between the upper 
and the lower bound in that corollary, even for $k = \CC$.

\subsection{Exceptional groups}

\begin{problem} \label{pr.exceptional} Let $G$ be an exceptional simple group
and $p$ be an exceptional prime for $G$. What is $\ed_k(G; p)$? $\ed_k(G)$?
Here we assume that $k$ is an algebraically closed field of characteristic
$0$ (or at least, $\Char(k)$ is not an exceptional prime for $G$).
\end{problem} 

For the exceptional group $G = \G_2$ we know that $\ed(\G_2) = 
\ed(\G_2; 2) = 3$; see Example~\ref{ex.Pf}. 

For $G = \F_4$, the Type 1 problem has been completely solved:
$\ed(\F_4; 2) = 5$ (see~\cite[Section 5]{macdonald}),
$\ed(\F_4; 3) = 3$ (see~\cite[Example 9.3]{gr}),
and $\ed(\F_4; p) = 0$ for all other primes.
It is claimed in~\cite{kordonskii}
that $\ed(\F_4) = 5$. However, the argument there appears
to be incomplete, so the (Type 2) problem of 
computing $\ed(\F_4)$ remains open.

The situation is similar for the simply connected group $\E_6^{sc}$.
The Type 1 problem has been solved,
\[ \ed(\E_6^{sc}; p) = \begin{cases} 
\text{$3$, if $p = 2$ (see~\cite[Example 9.4]{gr}),} \\  
\text{$4$, if $p = 3$ (see~\cite[Theorem 8.19.4 and Remark 8.20]{ry}),} 
\\
\text{$0$, if $p \ge 5$.} 
\end{cases} \]
(For the upper bound on the second line, cf.~also~\cite[11.1]{garibaldi2}.)
The Type 2 problem of computing $\ed(\E_6^{sc})$ remains open.

For the other exceptional groups, $\E_6^{ad}$, $\E_7^{ad}$ $\E_7^{sc}$
and $\E_8$, even the Type 1 problem 
of computing $\ed(G; p)$ is only partially solved.
It is known that $\ed(\E_6^{ad}; 2) = 3$ (see~\cite[Remark 9.7]{gr}), 
$\ed(\E_7^{ad}; 3) = \ed(\E_7^{sc}; 3) = 3$
(see~\cite[Example 9.6 and Remark 9.7]{gr}; 
cf.~also~\cite[Lemma 13.1]{garibaldi2}) and 
$\ed(\E_8; 5) = 3$ (see~\cite[Theorem 18.19.9]{ry} and
\cite[Proposition 14.7]{garibaldi2}).
% Here the superscript ${\, }^{sc}$ stands for ``simply connected"
% and ${\, }^{ad}$ for ``adjoint", as in Corollary~\ref{cor.finite-abelian}.
On the other hand,
the values of $\ed(\E_6^{ad}; 3)$, $\ed(\E_7^{ad}; 2)$, 
$\ed(\E_7^{sc}; 2)$, $\ed(\E_8; 3)$
and $\ed(\E_8; 2)$ are wide open, even for $k = \mathbb{C}$.  For example, 
the best known lower bound on $\ed_{\mathbb{C}}(\E_8; 2)$ is $9$ 
(see Corollary~\ref{cor.finite-abelian}(i)) 
but the best upper bound I know is $\ed_{\mathbb{C}}(\E_8; 2) \le 120$. 
The essential dimension $\ed(G)$ for these groups is largely 
uncharted territory, beyond the upper bounds in~\cite{lemire}.

\subsection{Groups whose connected component is a torus}

Let $G$ be an algebraic group over $k$ and $p$ be a prime.
We say that a linear representation $\phi \colon G \to \GL(V)$ 
is $p${\em-faithful} (respectively, $p${\em-generically free}) if 
$\Ker(\phi)$ is a finite group of order prime to $p$ and $\phi$
descends to a faithful (respectively, generically free) 
representation of $G/\Ker(\phi)$.

Suppose the connected component $G^0$ of $G$ is a $k$-torus.
One reason such groups are of interest is that the normalizer
$G$ of a maximal torus in a reductive $k$-group $\Gamma$ 
is of this form  
and $\ed(G)$ (respectively $\ed(G; p)$) is an upper bound on 
$\ed(\Gamma)$ (respectively, $\ed(\Gamma; p)$).
The last assertion follows from \cite[III.4.3, Lemma 6]{serre-gc}, in 
combination with Lemma~\ref{lem.surjective}.

For the sake of computing $\ed(G; p)$ we may 
assume that $G/G^0$ is a $p$-group and $k$ is $p$-closed, 
i.e., the degree of every finite field extension $k'/k$ is 
a power of $p$; see~\cite[Lemma 3.3]{lmmr}.
It is shown in~\cite{lmmr} that
\begin{equation} \label{e.conjecture}
\min \dim \nu - \dim(G) \le \ed(G;p) \le \min \dim \rho - \dim G \, ,
\end{equation}
where the two minima are taken over all $p$-faithful
representations $\nu$, and 
$p$-generically free representations $\rho$,
respectively. In the case where $G = T$ is a torus or $G = F$ is
a finite $p$-group or, more generally, $G$ is a direct product
$T \times F$, a faithful representation is automatically 
generically free. Thus in these cases the lower and upper 
bounds of~\eqref{e.conjecture} coincide, yielding the exact 
value of $\ed_k(G; p)$. If we only assume that $G^0$ is a torus,
I do not know how to close the gap 
between the lower and the upper bound in~\eqref{e.conjecture}.
However, in every example I have been able to work out the upper bound 
in~\eqref{e.conjecture} is, in fact, sharp. 

\begin{conjecture} \label{conjecture}
{\rm (\cite{lmmr})}
Let $G$ be an extension of a $p$-group by a torus, defined over
a $p$-closed field $k$ of characteristic $\ne p$.
Then $\ed(G;p)=\min \dim \rho - \dim G$,
where the minimum is taken over all $p$-generically free
$k$-representations $\rho$ of $G$.
\end{conjecture}                 
              

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\end{thebibliography}
\end{document}
