# Essential p-dimension of algebraic tori

Roland Lötscher(1) ,  Mark MacDonald  ,  Aurel Meyer(2) and  Zinovy
Reichstein(3)

###### Abstract.

The essential dimension is a numerical invariant of an algebraic group $G$
which may be thought of as a measure of complexity of $G$-torsors over fields.
A recent theorem of N. Karpenko and A. Merkurjev gives a simple formula for
the essential dimension of a finite $p$-group. We obtain similar formulas for
the essential $p$-dimension of a broader class of groups, which includes all
algebraic tori.

###### Key words and phrases:

Essential dimension, algebraic torus, twisted finite group, lattice

###### 2000 Mathematics Subject Classification:

20G15

(1) Roland Lötscher was partially supported by the Swiss National Science
Foundation (Schweizerischer Nationalfonds).

(2) Aurel Meyer was partially supported by a University Graduate Fellowship at
the University of British Columbia

(3) Zinovy Reichstein was partially supported by NSERC Discovery and
Accelerator Supplement grants

###### Contents

  1. 1 Introduction
  2. 2 Proof of Theorem 1.2
  3. 3 The $p$-closure of a field
  4. 4 The group $C(G)$
  5. 5 Proof of Theorem 1.3(a)
  6. 6 $p$-isogenies
  7. 7 Proof of Theorem 1.3(b)
  8. 8 An additivity theorem
  9. 9 Modules and lattices
  10. 10 Proof of Theorem 1.3(c)
  11. 11 Tori of essential dimension $\leq 1$
  12. 12 Tori split by cyclic extensions of degree dividing $p^{2}$

##  1\. Introduction

Throughout this paper $p$ will denote a prime integer, $k$ a base field of
characteristic $\neq p$ and $G$ a (not necessarily smooth) algebraic group
defined over $k$. Unless otherwise specified, all fields are assumed to
contain $k$ and all morphisms between them are assumed to be
$k$-homomorphisms.

We begin by recalling the notion of essential dimension of a functor from
[BF]. Let $\operatorname{Fields}_{k}$ be the category of field extensions
$K/k$, $\operatorname{Sets}$ be the category of sets, and
$F\colon\operatorname{Fields}_{k}\to\operatorname{Sets}$ be a covariant
functor. As usual, given a field extension $k\subset K_{0}\subset K$, we will
denote the image of $\alpha\in F(K)$ under the natural map $F(K)\to F(L)$ by
$\alpha_{L}$.

An object $\alpha\in F(K)$ is said to _descend_ to an intermediate field
$k\subseteq K_{0}\subseteq K$ if $\alpha$ is in the image of the induced map
$F(K_{0})\to F(K)$. The _essential dimension_ $\operatorname{ed}_{k}(\alpha)$
is defined as the minimum of the transcendence degrees
$\operatorname{trdeg}_{k}(K)$ taken over all fields $k\subseteq K_{0}\subseteq
K$ such that $\alpha$ descends to $K_{0}$. The essential dimension
$\operatorname{ed}_{k}(F)$ of the functor $F$ is defined as the maximal value
of $\operatorname{ed}_{k}(\alpha)$, where the maximum is taken over all fields
$K/k$ and all $\alpha\in F(K)$.

Of particular interest to us will be the Galois cohomology functor
$F_{G}\colonequals H^{1}(*,G)$, which associates to every $K/k$ the set of
isomorphism classes of $G$-torsors over $\operatorname{Spec}(K)$. The
essential dimension of this functor is usually called the essential dimension
of $G$ and is denoted by the symbol $\operatorname{ed}_{k}(G)$. Informally
speaking, this number may be thought of a measure of complexity of $G$-torsors
over fields. For example, if $k$ is an algebraically closed field of
characteristic $0$ then groups $G$ of essential dimension $0$ are precisely
the so-called special groups, i.e., algebraic groups $G/k$ with the property
that every $G$-torsor over $\operatorname{Spec}(K)$ is split, for every field
$K/k$. These groups were classified by A. Grothendieck [Gro].

For many groups the essential dimension is hard to compute, even over the
field $\mathbb{C}$ of complex numbers. The following related notion is often
more accessible. Let $F\colon\operatorname{Fields}_{k}\to\operatorname{Sets}$
be a covariant functor and $p$ be a prime integer, as above. The essential
$p$-dimension of $\alpha\in F(K)$, denoted $\operatorname{ed}_{k}(\alpha;p)$,
is defined as the minimal value of
$\operatorname{ed}_{k}(\alpha_{K^{\prime}})$, where $K^{\prime}$ ranges over
all finite field extensions of $K$ whose degree is prime to $p$. The essential
$p$-dimension of $F$, $\operatorname{ed}_{k}(F;p)$ of $F$ is once again,
defined as the maximal value of $\operatorname{ed}_{k}(\alpha;p)$, where the
maximum is taken over all fields $K/k$ and all $\alpha\in F(K)$, and once
again we will write $\operatorname{ed}_{k}(G;p)$ in place of
$\operatorname{ed}_{k}(F_{G};p)$, where $F_{G}\colonequals H^{1}(\ast,G)$ is
the Galois cohomology functor.

Note that $\operatorname{ed}_{k}(\alpha)$, $\operatorname{ed}_{k}(F)$,
$\operatorname{ed}_{k}(G)$, $\operatorname{ed}_{k}(\alpha;p)$, etc., depend on
$k$. We will write $\operatorname{ed}$ instead of $\operatorname{ed}_{k}$ if
the reference to $k$ is clear from the context. For background material on
essential dimension we refer the reader to [BR, Re, RY, BF, Me1].

We also remark that in the case of the Galois cohomology functor $F_{G}$, the
maximal value of $\operatorname{ed}_{k}(\alpha)$ and
$\operatorname{ed}_{k}(\alpha;p)$ in the above definitions is attained in the
case where $\alpha$ is a versal $G$-torsor in the sense of [GMS, Section I.5].
Since every generically free linear representation $\rho\colon
G\to\operatorname{GL}(V)$ gives rise to a versal $G$-torsor (see [GMS, Example
I.5.4]), we obtain the inequality

(1)
$\operatorname{ed}_{k}(G;p)\leq\operatorname{ed}_{k}(G)\leq\dim(V)-\dim(G)\,;$

see [Re, Therem 3.4] or [BF, Lemma 4.11]. (Recall that $\rho$ is called
generically free if there exists a $G$-invariant dense open subset $U\subset
V$ such that the scheme-theoretic stabilizer of every point of $U$ is
trivial.)

N. Karpenko and A. Merkurjev [KM] recently showed that the inequality (1) is
in fact sharp for finite constant $p$-groups.

######  Theorem 1.1.

Let $G$ be a constant $p$-group and $k$ be a field containing a primitive
$p$th root of unity. Then

$\operatorname{ed}_{k}(G;p)=\operatorname{ed}_{k}(G)=\min\,\dim(V)\,,$

where the minimum is taken over all faithful $k$-representations
$G\hookrightarrow\operatorname{GL}(V)$.

The goal of this paper is to prove similar formulas for a broader class of
groups $G$. To state our first result, let

(2) $1\to C\to G\to Q\to 1$

be an exact sequence of algebraic groups over $k$ such that $C$ is central in
$G$ and isomorphic to $\mu_{p}^{r}$ for some $r\geq 0$. Given a character
$\chi\colon C\to\mu_{p}$, we will, following [KM], denote by
$\operatorname{Rep}^{\chi}$ the set of irreducible representations $\phi\colon
G\to\operatorname{GL}(V)$, defined over $k$, such that
$\phi(c)=\chi(c)\operatorname{Id}_{V}$ for every $c\in C$.

######  Theorem 1.2.

Assume that $k$ is a field of characteristic $\neq p$ containing a primitive
$p$th root of unity. Suppose a sequence of $k$-groups of the form (2)
satisfies the following condition:

$\gcd\\{\dim(\phi)\,|\,\phi\in\operatorname{Rep}^{\chi}\\}=\min\\{\dim(\phi)\,|\,\phi\in\operatorname{Rep}^{\chi}\\}$

for every character $\chi\colon C\to\mu_{p}$. (Here, as usual, $\gcd$ stands
for the greatest common divisor.) Then

$\operatorname{ed}_{k}(G;p)\geq\min\dim(\rho)-\dim G\,,$

where the minimum is taken over all finite-dimensional $k$-representations
$\rho$ of $G$ such that $\rho_{|\,C}$ is faithful.

Of particular interest to us will be extensions of finite $p$-groups by
algebraic tori, i.e., $k$-groups $G$ which fit into an exact sequence of the
form

(3) $1\to T\to G\to F\to 1\,,$

where $F$ is a finite $p$-group and $T$ is a torus over $k$. Note that in this
paper we will view finite groups $F$ as algebraic groups over $k$, and will
not assume they are constant, which is to say, the absolute Galois group of
$k$ may act non-trivially on the separable points of $G$. For the sake of
computing $\operatorname{ed}_{k}(G;p)$ we may assume that $k$ is a $p$-closed
field (as in Definition 3.1); see Lemma 3.3. In this situation we will show
that

(i) there is a natural choice of a split central subgroup $C\subset G$ in the
sequence (2) such that

(ii) the conditions of Theorem 1.2 are always satisfied.

(iii) Moreover, if $G$ is isomorphic to the direct product of a torus and a
finite twisted $p$-group, then a variant of (1) yields an upper bound,
matching the lower bound of Theorem 1.2.

This brings us to the main result of this paper. We will say that a
representation $\rho\colon G\to\operatorname{GL}(V)$ of an algebraic group $G$
is $p$-faithful if its kernel is finite and of order prime to $p$.

######  Theorem 1.3.

Let $G$ be an extension of a (twisted) finite $p$-group $F$ by an algebraic
torus $T$ defined over a field $k$ (of characteristic not $p$). In other
words, we have an exact sequence

$1\to T\to G\to F\to 1\,.$

Denote a $p$-closure of $k$ by $k^{(p)}$ (see Definition 3.1). Then

(a) $\operatorname{ed}_{k}(G;p)\geq\min\dim(\rho)-\dim G$, where the minimum
is taken over all $p$-faithful linear representations $\rho$ of $G_{k^{(p)}}$
over $k^{(p)}$.

Now assume that $G$ is the direct product of $T$ and $F$. Then

(b) equality holds in (a), and

(c) over $k^{(p)}$ the absolute essential dimension of $G$ and the essential
$p$-dimension coincide:

$\operatorname{ed}_{k^{(p)}}(G_{k^{(p)}})=\operatorname{ed}_{k^{(p)}}(G_{k^{(p)}};p)=\operatorname{ed}_{k}(G;p).$

If $G$ is a $p$-group, a representation $\rho$ is $p$-faithful if and only if
it is faithful. However, for an algebraic torus, “$p$-faithful” cannot be
replaced by “faithful”; see Remark 10.3.

Theorem 1.3 appears to be new even in the case where $G$ is a twisted cyclic
$p$-group, where it extends earlier work of Rost [Ro], Bayarmagnai [Ba] and
Florence [Fl]; see Corollary 9.3 and Remark 9.4.

If $G$ a direct product of a torus and an abelian $p$-group, the value of
$\operatorname{ed}_{k}(G;p)$ given by Theorem 1.3 can be rewritten in terms of
the character module $X(G)$; see Corollary 9.2. In particular, we obtain the
following formula for the essential dimension of a torus.

######  Theorem 1.4.

Let $T$ be an algebraic torus defined over a $p$-closed field $k=k^{(p)}$ of
characteristic $\neq p$. Suppose
$\Gamma=\operatorname{Gal}(k_{\operatorname{sep}}/k)$ acts on the character
lattice $X(T)$ via a finite quotient $\overline{\Gamma}$. Then

$\operatorname{ed}_{k}(T)=\operatorname{ed}_{k}(T;p)=\min\operatorname{rank}(L)\,,$

where the minimum is taken over all exact sequences of
$\mathbb{Z}_{(p)}\overline{\Gamma}$-lattices of the form

$(0)\to L\to P\to X(T)_{(p)}\to(0)\,,$

where $P$ is permutation and $X(T)_{(p)}$ stands for
$X(T)\otimes_{\mathbb{Z}}\mathbb{Z}_{(p)}$.

In many cases Theorem 1.4 renders the value of $\operatorname{ed}_{k}(T)$
computable by known representation-theoretic methods, e.g., from [CR]. We will
give several examples of such computations in Sections 11 and 12. Another
application was recently given by Merkurjev (unpublished), who used Theorem
1.4, in combination with techniques from [Me2], to show that

$\operatorname{ed}_{k}(\operatorname{PGL}_{p^{r}};p)\geq(r-1)p^{r}+1$

for any $r\geq 1$. (For $r=2$ the above inequality is the main result of
[Me2].) This represents dramatic improvement over the best previously known
lower bounds on $\operatorname{ed}_{k}(\operatorname{PGL}_{p^{r}})$. The
question of computing $\operatorname{ed}_{k}(\operatorname{PGL}_{p^{r}})$ is a
long-standing open problem; for an overview, see [MR1, MR2].

It is natural to try to extend the formula of Theorem 1.3(b) to all $k$-groups
$G$, whose connected component $G^{0}$ is a torus. For example, the normalizer
of a maximal torus in any reductive $k$-group is of this form. For the purpose
of computing $\operatorname{ed}_{k}(G;p)$ we may assume that $k$ is $p$-closed
and $G/G^{0}$ is a $p$-group; in other words, $G$ is as in Theorem 1.3(a).
Then

(4) $\min\dim\mu-\dim(G)\leq\operatorname{ed}(G;p)\leq\min\dim\rho-\dim G\,,$

where the two minima are taken over all $p$-faithful representations $\mu$,
and $p$-generically free representations $\rho$, respectively. Here we say
that a representation $\rho$ of $G$ is $p$-generically free if the
$\ker(\rho)$ is finite of order prime to $p$, and $\rho$ descends to a
generically free representation of $G/\ker(\rho)$. The upper bound in (4)
follows from (1), in combination with Theorem 6.1; the lower bound is Theorem
1.3(a). If $G$ is a direct product of a torus and a $p$-group, then every
$p$-generically free representation is $p$-faithful (see Lemma 7.1). In this
case the lower and upper bounds of (4) coincide, yielding the exact value of
$\operatorname{ed}_{k}(G;p)$ of Theorem 1.3(b). However, if we only assume $G$
is a $p$-group extended by a torus, then faithful $G$-representations no
longer need to be generically free. We do not know how to bridge the gap
between the upper and the lower bound in (4) in this generality; however, in
all of the specific examples we have considered, the upper bound turned out to
be sharp. We thus put forward the following conjecture.

######  Conjecture 1.5.

Let $G$ be an extension of a $p$-group by a torus, defined over a field $k$ of
characteristic $\neq p$. Then

$\operatorname{ed}(G;p)=\min\dim\rho-\dim G,$

where the minimum is taken over all $p$-generically free representations
$\rho$ of $G_{k^{(p)}}$ over $k^{(p)}$.

The rest of the paper is structured as follows. Theorem 1.2 is proved in
Section 2. Section 3 is devoted to preliminary material on the $p$-closure of
a field. Theorem 1.3(a) is proved in Sections 4 and 5. In Section 6 we will
show that if $G\to Q$ is a $p$-isogeny then
$\operatorname{ed}_{k}(G;p)=\operatorname{ed}_{k}(Q;p)$. This result playes a
key role in the proof of Theorem 1.3(b) in Section 7. At the end of Section 7
we prove a formula for the essential $p$-dimension of any finite group $G$ by
passing to a Sylow $p$-subgroup defined over $k$; see Corollory 7.2. In
Section 8 we prove the following Additivity Theorem 8.1: If $G_{1}$ and
$G_{2}$ are direct products of tori and $p$-groups, then

$\operatorname{ed}_{k}(G_{1}\times
G_{2};p)=\operatorname{ed}_{k}(G_{1};p)+\operatorname{ed}_{k}(G_{2};p)\,.$

In Section 9 we restate and amplify Theorem 1.3(b) (with $G$ abelian) in terms
of $\operatorname{Gal}(k_{\operatorname{sep}}/k)$-modules; in particular,
Theorem 1.4 stated above is a special case of Corollary 9.2 which is proved
there. In Section 10 we prove Theorem 1.3(c) by using Theorem 1.3(b),
additivity, and the lattice perspective from Section 9. The last two sections
are intended to illustrate our results by computing essential dimensions of
specific algebraic tori. In Section 11 we classify algebraic tori $T$ of
essential $p$-dimension $0$ and $1$; see Theorems 11.1 and 11.5. In Section 12
we compute the essential $p$-dimension of all tori $T$ over a $p$-closed field
$k$, which are split by a cyclic extension $l/k$ of degree dividing $p^{2}$.

##  2\. Proof of Theorem 1.2

Denote by $C^{\ast}\colonequals\operatorname{Hom}(C,\mu_{p})$ the character
group of $C$. Let $E\to\operatorname{Spec}K$ be a versal $Q$-torsor [GMS,
Example 5.4], where $K/k$ is some field extension, and let $\beta\colon
C^{\ast}\to\operatorname{Br}_{p}(K)$ denote the homomorphism that sends
$\chi\in C^{\ast}$ to the image of $E\in H^{1}(K,Q)$ in
$\operatorname{Br}_{p}(K)$ under the map

$H^{1}(K,Q)\to
H^{2}(K,C)\stackrel{{\scriptstyle\chi_{\ast}}}{{\to}}H^{2}(K,\mu_{p})=\operatorname{Br}_{p}(K)$

given by composing the connecting map with $\chi_{\ast}$. Then there exists a
basis $\chi_{1},\dotsc,\chi_{r}$ of $C^{\ast}$ such that

(5)
$\operatorname{ed}_{k}(G;p)\geq\sum_{i=1}^{r}\operatorname{ind}\beta(\chi_{i})-\dim
G,$

see [Me1, Theorem 4.8, Example 3.7]. Moreover, by [KM, Theorem 4.4, Remark
4.5]

$\operatorname{ind}\beta(\chi_{i})=\gcd\dim(\rho)\,,$

where the greatest common divisor is taken over all (finite-dimensional)
representations $\rho$ of $G$ such that $\rho_{|\,C}$ is scalar multiplication
by $\chi_{i}$. By our assumption, $\gcd$ can be replaced by $\min$. Hence, for
each $i\in\\{1,\dotsc,r\\}$ we can choose a representation $\rho_{i}$ of $G$
with

$\operatorname{ind}\beta(\chi_{i})=\dim(\rho_{i})$

such that $(\rho_{i})_{|\,C}$ is scalar multiplication by $\chi_{i}$.

Set $\rho\colonequals\rho_{1}\oplus\dotsb\oplus\rho_{r}$. The inequality (5)
can be written as

(6) $\operatorname{ed}_{k}(G;p)\geq\dim(\rho)-\dim G.$

Since $\chi_{1},\dotsc,\chi_{r}$ forms a basis of $C^{\ast}$ the restriction
of $\rho$ to $C$ is faithful. This proves the theorem. ∎

##  3\. The $p$-closure of a field

Let $K$ be a field extension of $k$ and $K_{\operatorname{alg}}$ an algebraic
closure. We will construct a field $K^{(p)}/K$ in $K_{\operatorname{alg}}$
with all finite subextensions of $K^{(p)}/K$ of degree prime to $p$ and all
finite subextensions of $K_{\operatorname{alg}}/K^{(p)}$ of degree a power of
$p$.

Fix a separable closure $K_{\operatorname{sep}}\subset K_{\operatorname{alg}}$
of $K$ and denote $\Gamma=\operatorname{Gal}(K_{\operatorname{sep}}/K)$.
Recall that $\Gamma$ is profinite and has Sylow-$p$ subgroups which enjoy
similar properties as in the finite case, see for example [RZ] or [Wi]. Let
$\Phi$ be a Sylow-$p$ subgroup of $\Gamma$ and $K_{\operatorname{sep}}^{\Phi}$
its fixed field.

######  Definition 3.1.

We call the field

$K^{(p)}=\\{a\in K_{\operatorname{alg}}|a\mbox{ is purely inseparable over
}K_{\operatorname{sep}}^{\Phi}\\}$

a $p$-closure of $K$. A field $K$ will be called $p$-closed if $K$=$K^{(p)}$.

Note that $K^{(p)}$ is unique in $K_{\operatorname{alg}}$ only up to the
choice of a Sylow-$p$ subgroup $\Phi$ in $\Gamma$. The notion of being
$p$-closed does not depend on this choice.

######  Proposition 3.2.

  1. (a)

$K^{(p)}$ is a direct limit of finite extensions $K_{i}/K$ of degree prime to
$p$.

  2. (b)

Every finite extension of $K^{(p)}$ is separable of degree a power of $p$; in
particular, $K^{(p)}$ is perfect.

  3. (c)

The cohomological dimension of
$\Psi=\operatorname{Gal}(K_{\operatorname{alg}}/K^{(p)})$ is ${\rm
cd}_{q}(\Psi)=0$ for any prime $q\neq p$.

###### Proof.

(a) First note that $K_{\operatorname{sep}}$ is the limit of the directed set
$\\{K_{\operatorname{sep}}^{N}\\}$ over all normal subgroups $N\subset\Gamma$
of finite index. Let

$\mathcal{L}=\\{K_{\operatorname{sep}}^{N\Phi}|N\mbox{ normal with finite
index in $\Gamma$}\\}.$

This is a directed set, and since $\Phi$ is Sylow, the index of $N\Phi$ in
$\Gamma$ is prime to $p$. Therefore $\mathcal{L}$ consists of finite separable
extensions of $K$ of degree prime to $p$. Moreover,
$K_{\operatorname{sep}}^{\Phi}$ is the direct limit of fields $L$ in
$\mathcal{L}$.

If $\operatorname{char}k=0$, $K^{(p)}=K_{\operatorname{sep}}^{\Phi}$ and we
are done. Otherwise suppose $\operatorname{char}k=q\neq p$. Let

$\mathcal{E}=\\{E\subset K_{\operatorname{alg}}|E/L\mbox{ finite and purely
inseparable for some }L\in\mathcal{L}\\}.$

$\mathcal{E}$ consists of finite extensions of $K$ of degree prime to $p$,
because a purely inseparable extension has degree a power of $q$. One can
check that $\mathcal{E}$ forms a directed set.

Finally note that if $a$ is purely inseparable over
$K_{\operatorname{sep}}^{\Phi}$ with minimal polynomial $x^{q^{n}}-l$ (so that
$l\in K_{\operatorname{sep}}^{\Phi}$), then $l$ is already in some
$L\in\mathcal{L}$ since $K_{\operatorname{sep}}^{\Phi}$ is the limit of
$\mathcal{L}$. Thus $a\in E=L(a)$ which is in $\mathcal{E}$ and we conclude
that $K^{(p)}$ is the direct limit of $\mathcal{E}$.

(b) $K^{(p)}$ is the purely inseparable closure of
$K_{\operatorname{sep}}^{\Phi}$ in $K_{\operatorname{alg}}$ and
$K_{\operatorname{alg}}/K^{(p)}$ is separable, see [Win, 2.2.20]. Moreover,
$\operatorname{Gal}(K_{\operatorname{alg}}/K^{(p)})\simeq\operatorname{Gal}(K_{\operatorname{sep}}/K_{\operatorname{sep}}^{\Phi})=\Phi$
is a pro-$p$ group and so every finite extension of $K^{(p)}$ is separable of
degree a power of $p$.

(c) See [Se2, Cor. 2, I. 3]. ∎

We call a covariant functor
$\mathcal{F}\colon\operatorname{Fields}/k\to\operatorname{Sets}$ limit-
preserving if for any directed system of fields $\\{K_{i}\\}$,
$\displaystyle{\mathcal{F}(\lim_{\rightarrow}K_{i})=\lim_{\rightarrow}\mathcal{F}(K_{i})}$.
For example if $G$ is an algebraic group, the Galois cohomology functor
$H^{1}(*,G)$ is limit-preserving; see [Ma, 2.1].

######  Lemma 3.3.

Let $\mathcal{F}$ be limit-preserving and $\alpha\in\mathcal{F}(K)$ an object.
Denote the image of $\alpha$ in $\mathcal{F}(K^{(p)})$ by $\alpha_{K^{(p)}}$.

  1. (a)

$\operatorname{ed}_{k}(\alpha;p)=\operatorname{ed}_{k}(\alpha_{K^{(p)}};p)=\operatorname{ed}_{k}(\alpha_{K^{(p)}})$.

  2. (b)

$\operatorname{ed}_{k}(\mathcal{F};p)=\operatorname{ed}_{k^{(p)}}(\mathcal{F};p)$.

###### Proof.

(a) The inequalities
$\operatorname{ed}(\alpha;p)\geq\operatorname{ed}(\alpha_{K^{(p)}};p)=\operatorname{ed}(\alpha_{K^{(p)}})$
are clear from the definition and Proposition 3.2(b) since $K^{(p)}$ has no
finite extensions of degree prime to $p$. It remains to prove
$\operatorname{ed}(\alpha;p)\leq\operatorname{ed}(\alpha_{K^{(p)}})$. If $L/K$
is finite of degree prime to $p$,

(7) $\operatorname{ed}(\alpha;p)=\operatorname{ed}(\alpha_{L};p),$

cf. [Me1, Proposition 1.5] and its proof. For the $p$-closure $K^{(p)}$ this
is similar and uses (7) repeatedly:

Suppose there is a subfield $K_{0}\subset K^{(p)}$ and $\alpha_{K^{(p)}}$
comes from an element $\beta\in\mathcal{F}(K_{0})$, so that
$\beta_{K^{(p)}}=\alpha_{K^{(p)}}$. Write $K^{(p)}=\lim\mathcal{L}$, where
$\mathcal{L}$ is a direct system of finite prime to $p$ extensions of $K$.
Then $K_{0}=\lim\mathcal{L}_{0}$ with $\mathcal{L}_{0}=\\{L\cap
K_{0}|L\in\mathcal{L}\\}$ and by assumption on $\mathcal{F}$,
$\displaystyle{\mathcal{F}(K_{0})=\lim_{L^{\prime}\in\mathcal{L}_{0}}\mathcal{F}(L^{\prime})}$.
Thus there is a field $L^{\prime}=L\cap K_{0}$ ($L\in\mathcal{L}$) and
$\gamma\in\mathcal{F}(L^{\prime})$ such that $\gamma_{K_{0}}=\beta$. Since
$\alpha_{L}$ and $\gamma_{L}$ become equal over $K^{(p)}$, after possibly
passing to a finite extension, we may assume they are equal over $L$ which is
finite of degree prime to $p$ over $K$. Combining these constructions with (7)
we see that

$\operatorname{ed}(\alpha;p)=\operatorname{ed}(\alpha_{L};p)=\operatorname{ed}(\gamma_{L};p)\leq\operatorname{ed}(\gamma_{L})\leq\operatorname{ed}(\alpha_{K^{(p)}}).$

(b) This follows immediately from (a), taking $\alpha$ of maximal essential
$p$-dimension. ∎

######  Proposition 3.4.

Let
$\mathcal{F},\mathcal{G}\colon\operatorname{Fields}/k\to\operatorname{Sets}$
be limit-preserving functors and $\mathcal{F}\to\mathcal{G}$ a natural
transformation. If the map

$\mathcal{F}(K)\to\mathcal{G}(K)$

is bijective (resp. surjective) for any $p$-closed field extension $K/k$ then

$\operatorname{ed}(\mathcal{F};p)=\operatorname{ed}(\mathcal{G};p)\quad(\mbox{resp.
}\operatorname{ed}(\mathcal{F};p)\geq\operatorname{ed}(\mathcal{G};p)).$

###### Proof.

Assume the maps are surjective. By Proposition 3.2(a), the natural
transformation is $p$-surjective, in the terminology of [Me1], so we can apply
[Me1, Prop. 1.5] to conclude
$\operatorname{ed}(\mathcal{F};p)\geq\operatorname{ed}(\mathcal{G};p)$.

Now assume the maps are bijective. Let $\alpha$ be in $\mathcal{F}(K)$ for
some $K/k$ and $\beta$ its image in $\mathcal{G}(K)$. We claim that
$\operatorname{ed}(\alpha;p)=\operatorname{ed}(\beta;p)$. First, by Lemma 3.3
we can assume that $K$ is $p$-closed and it is enough to prove that
$\operatorname{ed}(\alpha)=\operatorname{ed}(\beta)$.

Assume that $\beta$ comes from $\beta_{0}\in\mathcal{G}(K_{0})$ for some field
$K_{0}\subset K$. Any finite prime to $p$ extension of $K_{0}$ is isomorphic
to a subfield of $K$ (cf. [Me1, Lemma 6.1]) and so also any $p$-closure of
$K_{0}$ (which has the same transcendence degree over $k$). We may therefore
assume that $K_{0}$ is $p$-closed. By assumption
$\mathcal{F}(K_{0})\rightarrow\mathcal{G}(K_{0})$ and
$\mathcal{F}(K)\rightarrow\mathcal{G}(K)$ are bijective. The unique element
$\alpha_{0}\in\mathcal{F}(K_{0})$ which maps to $\beta_{0}$ must therefore map
to $\alpha$ under the natural restriction map. This shows that
$\operatorname{ed}(\alpha)\leq\operatorname{ed}(\beta)$. The other inequality
always holds and the claim follows.

Taking $\alpha$ maximal with respect to its essential dimension, we obtain
$\operatorname{ed}(\mathcal{F};p)=\operatorname{ed}(\alpha;p)=\operatorname{ed}(\beta;p)\leq\operatorname{ed}(\mathcal{G};p)$.
∎

##  4\. The group $C(G)$

As we indicated in the Introduction, our proof of Theorem 1.3(a) will rely on
Theorem 1.2. To apply Theorem 1.2, we need to construct a split central
subgroup $C$ of $G$. In this section, we will explain how to construct this
subgroup (we will call it $C(G)$) and discuss some of its properties.

Recall that an algebraic group $G$ over a field $k$ is said to be of
multiplicative type if $G_{k_{\operatorname{sep}}}$ is diagonalizable over the
separable closure $k_{\operatorname{sep}}$ of $k$; cf., e.g., [Vo, Section
3.4]. Here, as usual, $G_{k^{\prime}}\colonequals
G\times_{\operatorname{Spec}{k}}\operatorname{Spec}(k^{\prime})$ for any field
extension $k^{\prime}/k$. Connected groups of multiplicative type are
precisely the algebraic tori.

We will use the following common conventions in working with an algebraic
group $A$ of multiplicative type over $k$.

  * •

We will denote the character group of $A$ by $X(A)$.

  * •

Given a field extension $l/k$, $A$ is split over $l$ if and only if the
absolute Galois group $\operatorname{Gal}(l_{\operatorname{sep}}/l)$ acts
trivially on $X(A)$.

  * •

We will write $A[p]$ for the $p$-torsion subgroup $\\{a\in A\,|\,a^{p}=1\\}$
of $A$. Clearly $A[p]$ is defined over $k$.

Let $T$ be an algebraic torus. It is well known how to construct a maximal
split subtorus of $T$, see for example [Bo, 8.15] or [Wa, 7.4]. The following
definition is a variant of this.

######  Definition 4.1.

Let $A$ be an algebraic group of multiplicative type over $k$. Let $\Delta(A)$
be the $\Gamma$-invariant subgroup of $X(A)$ generated by elements of the form
$x-\gamma(x)$, as $x$ ranges over $X(A)$ and $\gamma$ ranges over $\Gamma$.
Define

$\operatorname{Split}_{k}(A)=\operatorname{Diag}(X(A)/\Delta(A))\,.$

Here $\operatorname{Diag}$ denotes the anti-equivalence between continuous
$\mathbb{Z}\Gamma$-modules and algebraic groups of multiplicative type, cf.
[Wa, 7.3].

######  Definition 4.2.

Let $G$ be an extension of a finite $p$-group by a torus, defined over a field
$k$, as in (3). Then

$C(G)\colonequals\operatorname{Split}_{k}(Z(G)[p])\,,$

where $Z(G)$ denotes the centre of $G$.

######  Lemma 4.3.

Let $A$ be an algebraic group of multiplicative type over $k$.

  1. (a)

$\operatorname{Split}_{k}(A)$ is split over $k$,

  2. (b)

$\operatorname{Split}_{k}(A)=A$ if and only if $A$ is split over $k$,

  3. (c)

If $B$ is a $k$-subgroup of $A$ then
$\operatorname{Split}_{k}(B)\subset\operatorname{Split}_{k}(A)$.

  4. (d)

For $A=A_{1}\times A_{2}$, $\operatorname{Split}_{k}(A_{1}\times
A_{2})=\operatorname{Split}_{k}(A_{1})\times\operatorname{Split}_{k}(A_{2})$,

  5. (e)

If $A[p]\neq\\{1\\}$ and $A$ is split over a Galois extension $l/k$, such that
$\overline{\Gamma}=\operatorname{Gal}(l/k)$ is a $p$-group, then
$\operatorname{Split}_{k}(A)\neq\\{1\\}$.

###### Proof.

Parts (a), (b), (c) and (d) easily follow from the definition.

Proof of (e): By part (c), it suffices to show that
$\operatorname{Split}_{k}(A[p])\neq\\{1\\}$. Hence, we may assume that
$A=A[p]$ or equivalently, that $X(A)$ is a finite-dimensional
$\mathbb{F}_{p}$-vector space on which the $p$-group $\overline{\Gamma}$ acts.
Any such action is upper-triangular, relative to some $\mathbb{F}_{p}$-basis
$e_{1},\dots,e_{n}$ of $X(A)$; see, e.g., [Se1, Proposition 26, p.64]. That
is,

$\gamma(e_{i})=e_{i}+$ ($\mathbb{F}_{p}$-linear combination of
$e_{i+1},\ldots,e_{n}$)

for every $i=1,\dots,n$ and every $\gamma\in\overline{\Gamma}$. Our goal is to
show that $\Delta(A)\neq X(A)$. Indeed, every element of the form
$x-\gamma(x)$ is contained in the $\Gamma$-invariant submodule
$\operatorname{Span}(e_{2},\dots,e_{n})$. Hence, these elements cannot
generate all of $X(A)$. ∎

######  Proposition 4.4.

Suppose $G$ is an extension of a $p$-group by a torus, defined over a
$p$-closed field $k$. Suppose $N$ is a normal subgroup of $G$ defined over
$k$. Then the following conditions are equivalent:

(i) $N$ is finite of order prime to $p$,

(ii) $N\cap C(G)=\\{1\\}$,

(iii) $N\cap Z(G)[p]=\\{1\\}$,

In particular, taking $N=G$, we see that $C(G)\neq\\{1\\}$ if $G\neq\\{1\\}$.

###### Proof.

(i) $\Longrightarrow$ (ii) is obvious, since $C(G)$ is a $p$-group.

(ii) $\Longrightarrow$ (iii). Assume the contrary: $A\colonequals N\cap
Z(G)[p]\neq\\{1\\}$. By Lemma 4.3

$\\{1\\}\neq C(A)\subset N\cap C(Z(G)[p])=N\cap C(G)\,,$

contradicting (ii).

Our proof of the implication (iii) $\Longrightarrow$ (i), will rely on the
following

Claim: Let $M$ be a non-trivial normal finite $p$-subgroup of $G$ such that
the commutator $(G^{0},M)=\\{1\\}$. Then $M\cap Z(G)[p]\neq\\{1\\}$.

To prove the claim, note that $M(k_{\operatorname{sep}})$ is non-trivial and
the conjugation action of $G(k_{\operatorname{sep}})$ on
$M(k_{\operatorname{sep}})$ factors through an action of the $p$-group
$(G/G^{0})(k_{\operatorname{sep}})$. Thus each orbit has $p^{n}$ elements for
some $n\geq 0$; consequently, the number of fixed points is divisible by $p$.
The intersection $(M\cap Z(G))(k_{\operatorname{sep}})$ is precisely the fixed
point set for this action; hence, $M\cap Z(G)[p]\neq\\{1\\}$. This proves the
claim.

We now continue with the proof of the implication (iii) $\Longrightarrow$ (i).
For notational convenience, set $T\colonequals G^{0}$. Assume that
$N\triangleleft G$ and $N\cap Z(G)[p]=\\{1\\}$. Applying the claim to the
normal subgroup $M\colonequals(N\cap T)[p]$ of $G$, we see that $(N\cap
T)[p]=\\{1\\}$, i.e., $N\cap T$ is a finite group of order prime to $p$. The
exact sequence

(8) $1\to N\cap T\to N\to\overline{N}\to 1\,,$

where $\overline{N}$ is the image of $N$ in $G/T$, shows that $N$ is finite.
Now observe that for every $r\geq 1$, the commutator $(N,T[p^{r}])$ is a
$p$-subgroup of $N\cap T$. Thus $(N,T[p^{r}])=\\{1\\}$ for every $r\geq 1$. We
claim that this implies $(N,T)=\\{1\\}$ by Zariski density. If $N$ is smooth,
this is straightforward; see [Bo, Proposition 2.4, p. 59]. If $N$ is not
smooth, note that the map $c\colon N\times T\to G$ sending $(n,t)$ to the
commutator $ntn^{-1}t^{-1}$ descends to $\overline{c}\colon\overline{N}\times
T\to G$ (indeed, $N\cap T$ clearly commutes with $T$). Since $|\overline{N}|$
is a power of $p$ and $\operatorname{char}(k)\neq p$, $\overline{N}$ is smooth
over $k$, and we can pass to the separable closure $k_{\operatorname{sep}}$
and apply the usual Zariski density argument to show that the image of
$\overline{c}$ is trivial.

We thus conclude that $N\cap T$ is central in $N$. Since $\gcd(|N\cap
T|,\overline{N})=1$, by [Sch2, Corollary 5.4] the extension (8) splits, i.e.,
$N\simeq(N\cap T)\times\overline{N}$. This turns $\overline{N}$ into a
subgroup of $G$ satisfying the conditions of the claim. Therefore
$\overline{N}$ is trivial and $N=N\cap T$ is a finite group of order prime to
$p$, as claimed. ∎

For future reference, we record the following obvious consequence of the
equivalence of conditions (i) and (ii) in Proposition 4.4.

######  Corollary 4.5.

Let $k=k^{(p)}$ be a $p$-closed field and $G$ be an extension of a $p$-group
by a torus, defined over $k$, as in (3). A finite-dimensional representation
$\rho$ of $G$ defined over $k$ is $p$-faithful if and only $\rho_{|\,C(G)}$ is
faithful. ∎

##  5\. Proof of Theorem 1.3(a)

The key step in our proof will be the following proposition.

######  Proposition 5.1.

Let $k$ be a $p$-closed field, and $G$ be an extension of a $p$-group by a
torus, as in (3). Then the dimension of every irreducible representation of
$G$ over $k$ is a power of $p$.

Assuming Proposition 5.1 we can easily complete the proof of Theorem 1.3(a).
Indeed, by Proposition 3.4 we may assume that $k=k^{(p)}$ is $p$-closed. In
particular, since we are assuming that $\operatorname{char}(k)\neq p$, this
implies that $k$ contains a primitive $p$th root of unity. (Indeed, if $\zeta$
is a $p$-th root of unity in $k_{\operatorname{sep}}$ then $d=[k(\zeta):k]$ is
prime to $p$; hence, $d=1$.) Proposition 5.1 tells us that Theorem 1.2 can be
applied to the exact sequence

(9) $1\to C(G)\to G\to Q\to 1\,.$

This yields

(10) $\operatorname{ed}(G;p)\geq\min\;\dim(\rho)-\dim(G)\,,$

where the minimum is taken over all representations $\rho\colon
G\to\operatorname{GL}(V)$ such that $\rho_{|C(G)}$ is faithful. Corollary 4.4
now tells us that $\rho_{|C(G)}$ is faithful if and only if $\rho$ is
$p$-faithful, and Theorem 1.3(a) follows. ∎

The rest of this section will be devoted to the proof of Proposition 5.1. We
begin by settling it in the case where $G$ is a finite $p$-group.

######  Lemma 5.2.

Proposition 5.1 holds if $G$ is a finite $p$-group.

###### Proof.

Choose a finite Galois field extension $l/k$ such that (i) $G$ is constant
over $l$ and (ii) every irreducible linear representation of $G$ over $l$ is
absolutely irreducible. Since $k$ is assumed to be $p$-closed, $[l:k]$ is a
power of $p$.

Let $A\colonequals k[G]^{\ast}$ be the dual Hopf algebra of the coordinate
algebra of $G$. By [Ja, Section 8.6] a $G$-module structure on a $k$-vector
space $V$ is equivalent to an $A$-module structure on $V$. Now assume that $V$
is an irreducible $A$-module and let $W\subseteq V\otimes_{k}l$ be an
irreducible $A\otimes_{k}l$-submodule. Then by [Ka, Theorem 5.22] there exists
a divisor $e$ of $[l:k]$ such that

$V\otimes l\simeq e\left(\bigoplus_{i=1}^{r}{}^{\sigma_{i}}W\right)\,,$

where $\sigma_{i}\in\operatorname{Gal}(l/k)$ and $\\{{}^{\sigma_{i}}W\mid
1\leq i\leq r\\}$ are the pairwise non-isomorphic Galois conjugates of $W$. By
our assumption on $k$, $e$ and $r$ are powers of $p$ and by our choice of $l$,
$\dim_{l}W=\dim_{l}({}^{\sigma_{1}}W)=\ldots=\dim_{l}({}^{\sigma_{r}}W)$ is
also a power of $p$, since it divides the order of $G_{l}$. Hence, so is
$\dim_{k}(V)=\dim_{l}V\otimes
l=e(\dim_{l}{}^{\sigma_{1}}W+\dotsb+\dim_{l}{}^{\sigma_{r}}W)$. ∎

Our proof of Proposition 5.1 in full generality will based on leveraging Lemma
5.2 as follows.

######  Lemma 5.3.

Let $G$ be an algebraic group defined over a field $k$ and

$F_{1}\subseteq F_{2}\subseteq\dots\subset G$

be an ascending sequence of finite $k$-subgroups whose union $\cup_{n\geq
1}F_{n}$ is Zariski dense in $G$. If $\rho\colon G\to\operatorname{GL}(V)$ is
an irreducible representation of $G$ defined over $k$ then $\rho_{|\,F_{i}}$
is irreducible for sufficiently large integers $i$.

###### Proof.

For each $d=1,...,\dim(V)-1$ consider the $G$-action on the Grassmannian
$\operatorname{Gr}(d,V)$ of $d$-dimensional subspaces of $V$. Let
$X^{(d)}=\operatorname{Gr}(d,V)^{G}$ and
$X_{i}^{(d)}=\operatorname{Gr}(d,V)^{F_{i}}$ be the subvariety of
$d$-dimensional $G$\- (resp. $F_{i}$-)invariant subspaces of $V$. Then
$X_{1}^{(d)}\supseteq X_{2}^{(d)}\supseteq\ldots$ and since the union of the
groups $F_{i}$ is dense in $G$,

$X^{(d)}=\cap_{i\geq 0}X_{i}^{(d)}\,.$

By the Noetherian property of $\operatorname{Gr}(d,V)$, we have
$X^{(d)}=X_{m_{d}}^{(d)}$ for some $m_{d}\geq 0$.

Since $V$ does not have any $G$-invariant $d$-dimensional $k$-subspaces, we
know that $X^{(d)}(k)=\emptyset$. Thus, $X_{m_{d}}^{(d)}(k)=\emptyset$, i.e.,
$V$ does not have any $F_{m_{d}}$-invariant $d$-dimensional $k$-subspaces.
Setting $m\colonequals\max\\{m_{1},\dots,m_{\dim(V)-1}\\}$, we see that
$\rho_{|\,F_{m}}$ is irreducible. ∎

We now proceed with the proof of Proposition 5.1. By Lemmas 5.2 and 5.3, it
suffices to construct a sequence of finite $p$-subgroups

$F_{1}\subseteq F_{2}\subseteq\dots\subset G$

defined over $k$ whose union $\cup_{n\geq 1}F_{n}$ is Zariski dense in $G$.

In fact, it suffices to construct one $p$-subgroup $F^{\prime}\subset G$,
defined over $k$ such that $F^{\prime}$ surjects onto $F$. Indeed, once
$F^{\prime}$ is constructed, we can define $F_{i}\subset G$ as the subgroup
generated by $F^{\prime}$ and $T[p^{i}]$, for every $i\geq 0$. Since
$\cup_{n\geq 1}F_{n}$ contains both $F^{\prime}$ and $T[p^{i}]$, for every
$i\geq 0$ it is Zariski dense in $G$, as desired.

The following lemma, which establishes the existence of $F^{\prime}$, is thus
the final step in our proof of Proposition 5.1 (and hence, of Theorem 1.3(a)).

######  Lemma 5.4.

Let $1\to T\to G\xrightarrow{\pi}F\to 1$ be an extension of a $p$-group $F$ by
a torus $T$ over $k$. Then $G$ has a finite $p$-subgroup $F^{\prime}$ with
$\pi(F^{\prime})=F$.

In the case where $F$ is split and $k$ is algebraically closed this is proved
in [CGR, p. 564]; cf. also the proof of [BS, Lemme 5.11].

###### Proof.

Denote by $\widetilde{\operatorname{Ex}}^{1}(F,T)$ the group of equivalence
classes of extensions of $F$ by $T$. We claim that
$\widetilde{\operatorname{Ex}}^{1}(F,T)$ is torsion. Let
$\operatorname{Ex}^{1}(F,T)\subset\widetilde{\operatorname{Ex}}^{1}(F,T)$ be
the classes of extensions which have a scheme-theoretic section (i.e. $G(K)\to
F(K)$ is surjective for all $K/k$). There is a natural isomorphism
$\operatorname{Ex}^{1}(F,T)\simeq H^{2}(F,T)$, where the latter one denotes
Hochschild cohomology, see [DG, III. 6.2, Proposition]. By [Sch3] the usual
restriction-corestriction arguments can be applied in Hochschild cohomology
and in particular, $m\cdot H^{2}(F,T)=0$ where $m$ is the order of $F$. Now
recall that $M\mapsto\widetilde{\operatorname{Ex}}^{i}(F,M)$ and
$M\mapsto\operatorname{Ex}^{i}(F,M)$ are both derived functors of the crossed
homomorphisms $M\mapsto\operatorname{Ex}^{0}(F,M)$, where in the first case
$M$ is in the category of $F$-module sheaves and in the second, $F$-module
functors, cf. [DG, III. 6.2]. Since $F$ is finite and $T$ an affine scheme, by
[Sch1, Satz 1.2 & Satz 3.3] there is an exact sequence of $F$-module schemes
$1\to T\to M_{1}\to M_{2}\to 1$ and an exact sequence
$\operatorname{Ex}^{0}(F,M_{1})\to\operatorname{Ex}^{0}(F,M_{2})\to\widetilde{\operatorname{Ex}}^{1}(F,T)\to
H^{2}(F,M_{1})\simeq\operatorname{Ex}^{1}(F,M_{1})$. The $F$-module sequence
also induces a long exact sequence on $\operatorname{Ex}(F,*)$ and we have a
diagram

$\textstyle{\widetilde{\operatorname{Ex}}^{1}(F,T)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Ex}^{0}(F,M_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Ex}^{0}(F,M_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Ex}^{1}(F,M_{1})}$$\textstyle{\operatorname{Ex}^{1}(F,T)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

An element in $\widetilde{\operatorname{Ex}}^{1}(F,T)$ can thus be killed
first in $\operatorname{Ex}^{1}(F,M_{1})$ so it comes from
$\operatorname{Ex}^{0}(F,M_{2})$. Then kill its image in
$\operatorname{Ex}^{1}(F,T)\simeq H^{2}(F,T)$, so it comes from
$\operatorname{Ex}^{0}(F,M_{1})$, hence is $0$ in
$\widetilde{\operatorname{Ex}}^{1}(F,T)$. In particular we see that
multiplying twice by the order $m$ of $F$,
$m^{2}\cdot\widetilde{\operatorname{Ex}}^{1}(F,T)=0$. This proves the claim.

Now let us consider the exact sequence $1\to N\to T\xrightarrow{\times
m^{2}}T\to 1$, where $N$ is the kernel of multiplication by $m^{2}$. Clearly
$N$ is finite and we have an induced exact sequence

$\widetilde{\operatorname{Ex}}^{1}(F,N)\to\widetilde{\operatorname{Ex}}^{1}(F,T)\xrightarrow{\times
m^{2}}\widetilde{\operatorname{Ex}}^{1}(F,T)$

which shows that the given extension $G$ comes from an extension $F^{\prime}$
of $F$ by $N$. Then $G$ is the pushout of $F^{\prime}$ by $N\to T$ and we can
identify $F^{\prime}$ with a subgroup of $G$. ∎

##  6\. $p$-isogenies

An isogeny of algebraic groups is a surjective morphism $G\to Q$ with finite
kernel. If the kernel is of order prime to $p$ we say that the isogeny is a
$p$-isogeny. In this section we will prove Theorem 6.1 which says that
$p$-isogenous groups have the same essential $p$-dimension. This result will
play a key role in the proof of Theorem 1.3(b) in Section 7.

######  Theorem 6.1.

Suppose $G\to Q$ is a $p$-isogeny of algebraic groups over $k$. Then

  1. (a)

For any $p$-closed field $K$ containing $k$ the natural map $H^{1}(K,G)\to
H^{1}(K,Q)$ is bijective.

  2. (b)

$\operatorname{ed}_{k}(G;p)=\operatorname{ed}_{k}(Q;p)$.

######  Example 6.2.

Let $E_{6}^{sc},E_{7}^{sc}$ be simply connected simple groups of type
$E_{6},E_{7}$ respectively. In [GR, 9.4, 9.6] it is shown that if $k$ is an
algebraically closed field of characteristic $\neq 2$ and $3$ respectively,
then

$\operatorname{ed}_{k}(E_{6}^{sc};2)=3$ and
$\operatorname{ed}_{k}(E_{7}^{sc};3)=3$.

For the adjoint groups $E_{6}^{ad}=E_{6}^{sc}/\mu_{3}$,
$E_{7}^{ad}=E_{7}^{sc}/\mu_{2}$ we therefore have

$\operatorname{ed}_{k}(E_{6}^{ad};2)=3$ and
$\operatorname{ed}_{k}(E_{7}^{ad};3)=3$.

We will need two lemmas.

######  Lemma 6.3.

Let $N$ be a finite algebraic group over $k$ ($\operatorname{char}k\neq p$).
The following are equivalent:

  1. (a)

$p$ does not divide the order of $N$.

  2. (b)

$p$ does not divide the order of $N(k_{\operatorname{alg}})$.

If $N$ is also assumed to be abelian, denote by $N[p]$ the $p$-torsion
subgroup of $N$. The following are equivalent to the above conditions.

  1. (a′)

$N[p](k_{\operatorname{alg}})=\\{1\\}$.

  2. (b′)

$N[p](k^{(p)})=\\{1\\}$.

###### Proof.

(a)$\iff$(b): Let $N^{\circ}$ be the connected component of $N$ and
$N^{et}=N/N^{\circ}$ the étale quotient. Recall that the order of a finite
algebraic group $N$ over $k$ is defined as $|N|=\dim_{k}k[N]$ and
$|N|=|N^{\circ}||N^{et}|$, see for example [Ta]. If $\operatorname{char}k=0$,
$N^{\circ}$ is trivial, if $\operatorname{char}k=q\neq p$ is positive,
$|N^{\circ}|$ is a power of $q$. Hence $N$ is of order prime to $p$ if and
only if the étale algebraic group $N^{et}$ is. Since $N^{\circ}$ is connected
and finite, $N^{\circ}(k_{\operatorname{alg}})=\\{1\\}$ and so
$N(k_{\operatorname{alg}})$ is of order prime to $p$ if and only if the group
$N^{et}(k_{\operatorname{alg}})$ is. Then
$|N^{et}|=\dim_{k}k[N^{et}]=|N^{et}(k_{\operatorname{alg}})|$, cf. [Bou, V.29
Corollary].

(b)$\iff$(a′) $\Rightarrow$ (b′) are clear.

(a′) $\Leftarrow$ (b′): Suppose $N[p](k_{\operatorname{alg}})$ is nontrivial.
The Galois group $\Gamma=\operatorname{Gal}(k_{\operatorname{alg}}/k^{(p)})$
is a pro-$p$ group and acts on the $p$-group $N[p](k_{\operatorname{alg}})$.
The image of $\Gamma$ in $\operatorname{Aut}(N[p](k_{\operatorname{alg}}))$ is
again a (finite) $p$-group and the size of every $\Gamma$-orbit in
$N[p](k_{\operatorname{alg}})$ is a power of $p$. Since $\Gamma$ fixes the
identity in $N[p](k_{\operatorname{alg}})$, this is only possible if it also
fixes at least $p-1$ more elements. It follows that $N[p](k^{(p)})$ contains
at least $p$ elements, a contradiction. ∎

######  Remark 6.4.

Part (b′) could be replaced by the slightly stronger statement that
$N[p](k^{(p)}\cap k_{\operatorname{sep}})=\\{1\\}$, but we won’t need this in
the sequel.

######  Lemma 6.5.

Let $\Gamma$ be a profinite group, $G$ an (abstract) finite $\Gamma$-group and
$|\Gamma|,|G|$ coprime. Then $H^{1}(\Gamma,G)=\\{1\\}$.

The case where $\Gamma$ is finite and $G$ abelian is classical. In the
generality we stated, this lemma is also known [Se2, I.5, ex. 2].

###### Proof of Theorem 6.1.

(a) Let $N$ be the kernel of $G\to Q$ and $K=K^{(p)}$ be a $p$-closed field
over $k$. Since $K_{\operatorname{sep}}=K_{\operatorname{alg}}$ (see
Proposition 3.2(b)), the sequence of $K_{\operatorname{sep}}$-points $1\to
N(K_{\operatorname{sep}})\to G(K_{\operatorname{sep}})\to
Q(K_{\operatorname{sep}})\to 1$ is exact. By Lemma 6.3, the order of
$N(K_{\operatorname{sep}})$ is not divisible by $p$ and therefore coprime to
the order of $\Psi=\operatorname{Gal}(K_{\operatorname{sep}}/K)$. Thus
$H^{1}(K,N)=\\{1\\}$ (Lemma 6.5). Similarly, if ${}_{c}N$ is the group $N$
twisted by a cocycle $c:\Psi\to G$,
${}_{c}N(K_{\operatorname{sep}})=N(K_{\operatorname{sep}})$ is of order prime
to $p$ and $H^{1}(K,\,_{c}N)=\\{1\\}$. It follows that $H^{1}(K,G)\to
H^{1}(K,Q)$ is injective, cf. [Se2, I.5.5].

Surjectivity is a consequence of [Se2, I. Proposition 46] and the fact that
the $q$-cohomological dimension of $\Psi$ is $0$ for any divisor $q$ of
$|N(K_{\operatorname{sep}})|$ (Proposition 3.2).

This concludes the proof of part (a). Part (b) immediately follows from (a)
and Proposition 3.4. ∎

##  7\. Proof of Theorem 1.3(b)

Let $k$ be a closed field and $G=T\times F$, where $T$ is a torus and $F$ is a
finite $p$-group, defined over $k$. Our goal is to show that

(11) $\operatorname{ed}_{k}(G;p)\leq\dim(\rho)-\dim G\,,$

where $\rho$ is a $p$-faithful representation of $G$ defined over $k$.

######  Lemma 7.1.

If a representation $\rho\colon G\to\operatorname{GL}(V)$ is $p$-faithful,
then $G/\ker(\rho)\to\operatorname{GL}(V)$ is generically free. In other
words, $\rho$ is $p$-generically free.

###### Proof.

Since $\ker(\rho)$ has order prime to $p$, its image under the projection map
$G=T\times F\to F$ is trivial. Hence $\ker(\rho)\subset T$ and $T/N$ is again
a torus. So without loss of generality, we may assume $\rho$ is faithful.

Let $V_{1}\subsetneq V$ be a closed subset of $V$ such that $T$ acts freely on
$V\setminus V_{1}$. Let $n=p^{r}$ be the order of $F$ and $V_{2}$ be the
(finite) union of the fixed point sets of $1\neq g\in T[n]\times F$. Here as
usual, $T[n]$ denotes the $n$-torsion subgroup of $T$. Since $\rho$ is
faithful none of these fixed point sets are all of $V$, hence $U\colonequals
V\setminus(V_{1}\cup V_{2})$ is a dense open subset of $V$.

We claim that $\operatorname{Stab}_{G}(v)=\\{1\\}$ for every $v\in U$. Indeed,
assume $1\neq g=(t,f)\in\operatorname{Stab}_{G}(v)$. Since $v\not\in V_{2}$,
$t^{n}\neq 1$. Then $1\neq g^{n}=(t^{n},1)$ lies in both $T$ and
$\operatorname{Stab}_{G}(v)$. Since $v\not\in V_{1}$, this is a contradiction.
∎

Now suppose $\rho$ is any $p$-faithful representation of $G$. Then (1) yields

$\operatorname{ed}_{k}(G/N;p)\leq\dim(\rho)-\dim(G/\ker(\rho))=\dim(\rho)-\dim(G)\,.$

By Theorem 6.1

$\operatorname{ed}_{k}(G;p)=\operatorname{ed}(G/N;p)\leq\dim(\rho)-\dim(G)\,,$

as desired. This completes the proof of (11) and thus of Theorem 1.3(b). ∎

######  Corollary 7.2.

Let $G$ be a finite algebraic group over a $p$-closed field $k=k^{(p)}$. Then
$G$ has a Sylow-$p$ subgroup $G_{p}$ defined over $k$ and

$\operatorname{ed}_{k}(G;p)=\operatorname{ed}_{k}(G_{p};p)=\operatorname{ed}_{k}(G_{p})=\min\dim(\rho)$

where the minimum is taken over all faithful representations of $G_{p}$ over
$k$.

###### Proof.

By assumption, $\Gamma=\operatorname{Gal}(k_{\operatorname{sep}}/k)$ is a
pro-$p$ group. It acts on the set of Sylow-$p$ subgroups of
$G(k_{\operatorname{sep}})$. Since the number of such subgroups is prime to
$p$, $\Gamma$ fixes at least one of them and by Galois descent one obtains a
subgroup $G_{p}$ of $G$. By Lemma 6.3, $G_{p}$ is a Sylow-$p$ subgroup of $G$.
The first equality $\operatorname{ed}_{k}(G;p)=\operatorname{ed}_{k}(G_{p};p)$
is shown in [MR1, 4.1] (the reference is for smooth groups but can be
generalized to the non-smooth case as well). The minimal
$G_{p}$-representation $\rho$ from Theorem 1.3(b) is faithful and thus
$\operatorname{ed}_{k}(G_{p})\leq\dim(\rho)$, see for example [BF, Prop.
4.11]. The Corollary follows. ∎

######  Remark 7.3.

Two Sylow-$p$ subgroups of $G$ defined over $k=k^{(p)}$ do not need to be
isomorphic over $k$.

##  8\. An additivity theorem

The purpose of this section is to prove the following:

######  Theorem 8.1.

Let $G_{1}$ and $G_{2}$ be direct products of tori and $p$-groups over a field
$k$. Then $\operatorname{ed}_{k}(G_{1}\times
G_{2};p)=\operatorname{ed}_{k}(G_{1};p)+\operatorname{ed}_{k}(G_{2};p)$.

Let $G$ be an algebraic group defined over $k$ and $C$ be a $k$-subgroup of
$G$. Denote the minimal dimension of a representation $\rho$ of $G$ (defined
over $k$) such that $\rho_{|\,C}$ is faithful by $f(G,C)$.

######  Lemma 8.2.

For $i=1,2$ let $G_{i}$ be an algebraic group defined over $k$ and $C_{i}$ be
a central $k$-subgroup of $G_{i}$. Assume that $C_{i}$ is isomorphic to
$\mu_{p}^{r_{i}}$ over $k$ for some $r_{1},r_{2}\geq 0$. Then

$f(G_{1}\times G_{1};C_{1}\times C_{2})=f(G_{1};C_{1})+f(G_{2};C_{2})\,.$

Our argument is a variant of the proof of [KM, Theorem 5.1], where $G$ is
assumed to be a (constant) finite $p$-group and $C$ is the socle of $G$.

###### Proof.

For $i=1,2$ let $\pi_{i}\colon G_{1}\times G_{2}\to G_{i}$ be the natural
projection and $\epsilon_{i}\colon G_{i}\to G_{1}\times G_{2}$ be the natural
inclusion.

If $\rho_{i}$ is a $d_{i}$-dimensional $k$-representation of $G_{i}$ whose
restriction to $C_{i}$ is faithful, then clearly
$\rho_{1}\circ\pi_{1}\oplus\rho_{2}\circ\pi_{2}$ is a
$d_{1}+d_{2}$-dimensional representation of $G_{1}\times G_{2}$ whose
restriction to $C_{1}\times C_{2}$ is faithful. This shows that

$f(G_{1}\times G_{1};C_{1}\times C_{2})\leq f(G_{1};C_{1})+f(G_{2};C_{2})\,.$

To prove the opposite inequality, let $\rho\colon G_{1}\times
G_{2}\to\operatorname{GL}(V)$ be a $k$-representation such that
$\rho_{|\,C_{1}\times C_{2}}$ is faithful, and of minimal dimension

$d=f(G_{1}\times G_{1};C_{1}\times C_{2})$

with this property. Let $\rho_{1},\rho_{2},\dotsc,\rho_{n}$ denote the
irreducible decomposition factors in a decomposition series of $\rho$. Since
$C_{1}\times C_{2}$ is central in $G_{1}\times G_{2}$, each $\rho_{i}$
restricts to a multiplicative character of $C_{1}\times C_{2}$ which we will
denote by $\chi_{i}$. Moreover since $C_{1}\times
C_{2}\simeq\mu_{p}^{r_{1}+r_{2}}$ is linearly reductive $\rho_{|\,C_{1}\times
C_{2}}$ is a direct sum $\chi_{1}^{\oplus
d_{1}}\oplus\dotsb\oplus\chi_{n}^{\oplus d_{n}}$ where $d_{i}=\dim V_{i}$. It
is easy to see that the following conditions are equivalent:

(i) $\rho_{|\,C_{1}\times C_{2}}$ is faithful,

(ii) $\chi_{1},\dots,\chi_{n}$ generate $(C_{1}\times C_{2})^{*}$ as an
abelian group.

In particular we may assume that $\rho=\rho_{1}\oplus\dotsb\oplus\rho_{n}$.
Since $C_{i}$ is isomorphic to $\mu_{p}^{r_{i}}$, we will think of
$(C_{1}\times C_{2})^{*}$ as a $\mathbb{F}_{p}$-vector space of dimension
$r_{1}$ \+ $r_{2}$. Since (i) $\Leftrightarrow$ (ii) above, we know that
$\chi_{1},\dots,\chi_{n}$ span $(C_{1}\times C_{2})^{*}$. In fact, they form a
basis of $(C_{1}\times C_{2})^{*}$, i.e., $n=r_{1}+r_{2}$. Indeed, if they
were not linearly independent we would be able to drop some of the terms in
the irreducible decomposition $\rho_{1}\oplus\dots\oplus\rho_{n}$, so that the
restriction of the resulting representation to $C_{1}\times C_{2}$ would still
be faithful, contradicting the minimality of $\dim(\rho)$.

We claim that it is always possible to replace each $\rho_{j}$ by
$\rho_{j}^{\prime}$, where $\rho_{j}^{\prime}$ is either
$\rho_{j}\circ\epsilon_{1}\circ\pi_{1}$ or
$\rho_{j}\circ\epsilon_{2}\circ\pi_{2}$ such that the restriction of the
resulting representation
$\rho^{\prime}=\rho_{1}^{\prime}\oplus\dots\oplus\rho_{n}^{\prime}$ to
$C_{1}\times C_{2}$ remains faithful. Since
$\dim(\rho_{i})=\dim(\rho_{i}^{\prime})$, we see that
$\dim(\rho^{\prime})=\dim(\rho)$. Moreover, $\rho^{\prime}$ will then be of
the form $\alpha_{1}\circ\pi_{1}\oplus\alpha_{2}\circ\pi_{2}$, where
$\alpha_{i}$ is a representation of $G_{i}$ whose restriction to $C_{i}$ is
faithful. Thus, if we can prove the above claim, we will have

$\displaystyle f(G_{1}\times G_{1};C_{1}\times C_{2})$
$\displaystyle=\dim(\rho)=\dim(\rho^{\prime})=\dim(\alpha_{1})+\dim(\alpha_{2})$
$\displaystyle\geq f(G_{1},C_{1})+f(G_{2},C_{2})\,,$

as desired.

To prove the claim, we will define $\rho_{j}^{\prime}$ recursively for
$j=1,\dots,n$. Suppose $\rho_{1}^{\prime},\dots,\rho_{j-1}^{\prime}$ have
already be defined, so that the restriction of

$\rho_{1}^{\prime}\oplus\dots\oplus\rho_{j-1}^{\prime}\oplus\rho_{j}\dots\oplus\rho_{n}$

to $C_{1}\times C_{2}$ is faithful. For notational simplicity, we will assume
that $\rho_{1}=\rho_{1}^{\prime},\dots,\rho_{j-1}=\rho_{j-1}^{\prime}$. Note
that

$\chi_{j}=(\chi_{j}\circ\epsilon_{1}\circ\pi_{1})+(\chi_{j}\circ\epsilon_{2}\circ\pi_{2})\,.$

Since $\chi_{1},\dots,\chi_{n}$ form a basis $(C_{1}\times C_{2})^{*}$ as an
$\mathbb{F}_{p}$-vector space, we see that (a)
$\chi_{j}\circ\epsilon_{1}\circ\pi_{1}$ or (b)
$\chi_{j}\circ\epsilon_{2}\circ\pi_{2}$ does not lie in
$\operatorname{Span}_{\mathbb{F}_{p}}(\chi_{1},\dots,\chi_{j-1},\chi_{j+1},\dots,\chi_{n})$.
Set

$\rho_{j}^{\prime}\colonequals\begin{cases}\text{$\rho_{j}\circ\epsilon_{1}\circ\pi_{1}$
in case (a), and}\\\ \text{$\rho_{j}\circ\epsilon_{2}\circ\pi_{2}$,
otherwise.}\end{cases}$

Using the equivalence of (i) and (ii) above, we see that the restriction of

$\rho_{1}\oplus\dots\oplus\rho_{j-1}\oplus\rho_{j}^{\prime}\oplus\rho_{j+1},\dots\oplus\rho_{n}$

to $C$ is faithful. This completes the proof of the claim and thus of Lemma
8.2. ∎

###### Proof of Theorem 8.1.

We can pass to a $p$-closure $k^{(p)}$ by Lemma 3.3. Let $C(G)$ be as in
Definition 4.2. By Theorem 1.3(b)

$\operatorname{ed}(G;p)=f(G,C(G))-\dim G\,;$

cf. Corollary 4.5. Furthermore, we have $C(G_{1}\times G_{2})=C(G_{1})\times
C(G_{2})$; cf. Lemma 4.3(d). Applying Lemma 8.2 finishes the proof. ∎

##  9\. Modules and lattices

In this section we rewrite the value of $\operatorname{ed}_{k}(G;p)$ in terms
of the character module $X(G)$ for an abelian group $G$ which is an extension
of a $p$-group and a torus. Moreover we show that tori with locally isomorphic
character lattices have the same essential dimension. We need the following
preliminaries.

Let $R$ be a commutative ring (we use $R=\mathbb{Z}$ and $R=\mathbb{Z}_{(p)}$
mostly) and $A$ an $R$-algebra. An $A$-module is called an $A$-lattice if it
is finitely generated and projective as an $R$-module. For
$A=\mathbb{Z}\Gamma$ ($\Gamma$ a group) this is as usual a free abelian group
of finite rank with an action of $\Gamma$. Particular cases of
$R\Gamma$-lattices are permutation lattices $L=R[\Lambda]$ where $\Lambda$ is
a $\Gamma$-set.

For $\Gamma=\operatorname{Gal}(k_{\operatorname{sep}}/k)$ the absolute Galois
group of $k$ we tacitly assume that our $R\Gamma$-lattices are continuous,
i.e. $\Gamma$ acts through a finite quotient $\overline{\Gamma}$. Under the
anti-equivalence $\operatorname{Diag}$ a $\mathbb{Z}\Gamma$-lattice
corresponds to an algebraic $k$-torus. A torus $S$ is called quasi split if it
corresponds to a permutation lattice. Equivalently $S\simeq
R_{E/k}(\operatorname{\mathbb{G}}_{m})$ where $E/k$ is étale and $R_{E/k}$
denotes Weil restriction.

Recall that $\mathbb{Z}_{(p)}$ denotes the localization of the ring
$\mathbb{Z}$ at the prime ideal $(p)$. For a $\mathbb{Z}$-module $M$ we also
write $M_{(p)}\colonequals\mathbb{Z}_{(p)}\otimes M$.

When $\Gamma=\operatorname{Gal}(k_{\operatorname{sep}}/k)$ we will often pass
from $\mathbb{Z}\Gamma$-lattices to $\mathbb{Z}_{(p)}\Gamma$-lattices. This
corresponds to identifying $p$-isogeneous tori:

######  Lemma 9.1.

Let $\Gamma=\operatorname{Gal}(k_{\operatorname{sep}}/k)$ and let $M,L$ be
$\mathbb{Z}\Gamma$-lattices. Then the following statements are equivalent:

  1. (a)

$L_{(p)}\simeq M_{(p)}$.

  2. (b)

There exists an injective map $\phi\colon L\to M$ of
$\mathbb{Z}\Gamma$-modules with cokernel $Q$ finite of order prime to $p$.

  3. (c)

There exists a $p$-isogeny $\operatorname{Diag}(M)\to\operatorname{Diag}(L)$.

###### Proof.

The equivalence (b) $\Leftrightarrow$ (c) is clear from the anti-equivalence
of $\operatorname{Diag}$.

The implication (b) $\Rightarrow$ (a) follows from $Q_{(p)}=0$ and that
tensoring with $\mathbb{Z}_{(p)}$ is exact.

For the implication (a) $\Rightarrow$ (b) we use that $L$ and $M$ can be
considered as subsets of $L_{(p)}$ (resp. $M_{(p)}$). The image of $L$ under a
map $\alpha\colon L_{(p)}\to M_{(p)}$ of $\mathbb{Z}_{(p)}\Gamma$-modules
lands in $\frac{1}{m}M$ for some $m\in\mathbb{N}$ (prime to $p$) and the index
of $\alpha(L)$ in $\frac{1}{m}M$ is finite and prime to $p$ if $\alpha$ is
surjective. Since $\frac{1}{m}M\simeq M$ as $\mathbb{Z}\Gamma$-modules the
claim follows. ∎

######  Corollary 9.2.

Let $G$ be an abelian group which is an extension of a $p$-group by a torus
over $k$ and $\Gamma\colonequals\operatorname{Gal}(k_{\operatorname{sep}}/k)$
be the absolute Galois group of $k=k^{(p)}$. Let $\Gamma$ act through a finite
quotient $\overline{\Gamma}$ on $X(G)$. Then

$\operatorname{ed}_{k}(G;p)=\min\operatorname{rk}L-\dim G\,,$

where the minimum is taken over all permutation
$\mathbb{Z}\overline{\Gamma}$-lattices $L$ which admit a map of
$\mathbb{Z}\overline{\Gamma}$-modules to $X(G)$ with cokernel finite of order
prime to $p$.

If $G$ is a torus, then the minimum can also be taken over all
$\mathbb{Z}_{(p)}\overline{\Gamma}$-lattices $L$ which admit a surjective map
of $\mathbb{Z}_{(p)}\overline{\Gamma}$-modules to $X(G)_{(p)}$.

###### Proof.

Let us prove the first claim. In view of Theorem 1.3(a) it suffices to show
that the least dimension of a $p$-faithful representation of $G_{k^{(p)}}$
over $k^{(p)}$ is equal to the least rank of a permutation
$\mathbb{Z}\overline{\Gamma}$-module $L$ which admits a map to $X(G)$ with
cokernel finite of order prime to $p$.

Assume we have such a map $L\to X(G)$. Using the anti-equivalence
$\operatorname{Diag}$ we obtain a $p$-isogeny $G\to\operatorname{Diag}(L)$. We
can embed the quasi-split torus $\operatorname{Diag}(L)$ in
$\operatorname{GL}_{n}$ where $n=\operatorname{rk}L$ [Vo, Section 6.1]. This
yields a $p$-faithful representation of $G$ of dimension $\operatorname{rk}L$.

Conversely let $\rho\colon G\to\operatorname{GL}(V)$ be a $p$-faithful
representation of $G$. Since $G_{\operatorname{sep}}$ is diagonalizable, there
exist characters $\chi_{1},\dotsc,\chi_{n}\in X(G)$ such that $G$ acts on
$V_{\operatorname{sep}}$ via diagonal matrices with entries
$\chi_{1}(g),\dotsc,\chi_{n}(g)$ (for $g\in G$) with respect to a suitable
basis of $V_{\operatorname{sep}}$. Moreover $\overline{\Gamma}$ permutes the
set $\Lambda\colonequals\\{\chi_{1},\dotsc,\chi_{n}\\}$. Define a map
$\phi\colon\mathbb{Z}[\Lambda]\to X(G)$ of
$\mathbb{Z}\overline{\Gamma}$-modules by sending the basis element
$\chi_{i}\in\Lambda$ of $L\colonequals\mathbb{Z}[\Lambda]$ to itself. Then the
$p$-faithfulness of $\rho$ implies that the cokernel of $\phi$ is finite and
of order prime to $p$. Moreover $\operatorname{rk}L=|\Lambda|\leq n=\dim V$.

Now consider the case where $G$ is a torus. Assume we have a surjective map
$\alpha\colon L\to X(G)_{(p)}$ of $\mathbb{Z}_{(p)}\overline{\Gamma}$-modules
where $L=\mathbb{Z}_{(p)}[\Lambda]$ is permutation, $\Lambda$ a
$\overline{\Gamma}$-set. Then $\alpha(\Lambda)\subseteq\frac{1}{m}X(G)$ for
some $m\in\mathbb{N}$ prime to $p$ (note that $\frac{1}{m}X(G)$ can be
considered as a subset of $X(G)_{(p)}$ since $X(G)$ is torsion free). By
construction the induced map $\mathbb{Z}[\Lambda]\to\frac{1}{m}X(G)\simeq
X(G)$ becomes surjective after localization at $p$, hence its cokernel is
finite of order prime to $p$. ∎

######  Corollary 9.3.

Let $A$ be a finite (twisted) cyclic $p$-group over $k$. Let $l/k$ be a
minimal Galois splitting field of $A$, and
$\Gamma\colonequals\operatorname{Gal}(l/k)$. Then

$\operatorname{ed}(A;p)=|\Gamma|.$

###### Proof.

Since $[l:k]$ is a power of $p$, $l^{(p)}/k^{(p)}$ is a Galois extension of
the same degree and the same Galois group as $l/k$. So we can assume
$k=k^{(p)}$.

By Corollary 9.2 $\operatorname{ed}(A;p)$ is equal to the least cardinality of
a $\Gamma$-set $\Lambda$ such that there exists a map
$\phi\colon\mathbb{Z}[\Lambda]\to X(A)$ of $\mathbb{Z}\Gamma$-modules with
cokernel finite of order prime to $p$. The group $X(A)$ is a (cyclic)
$p$-group, hence $\phi$ must be surjective. Moreover $\Gamma$ acts faithfully
on $X(A)$. Surjectivity of $\phi$ implies that some element
$\lambda\in\Lambda$ maps to a generator $a$ of $X(A)$. Hence
$|\Lambda|\geq|\Gamma\lambda|\geq|\Gamma a|=|\Gamma|$. Conversely we have a
surjective homomorphism $\mathbb{Z}[\Gamma a]\to X(A)$ that sends $a$ to
itself. Hence the claim follows.

∎

######  Remark 9.4.

In the case of twisted cyclic groups of order $4$ Corollary 9.3 is due to Rost
[Ro] (see also [BF, Theorem 7.6]), and in the case of cyclic groups of order
$8$ to Bayarmagnai [Ba]. The case of constant groups of arbitrary prime power
order is due to Florence [Fl]; it is now a special case of the Karpenko-
Merkurjev Theorem 1.1.

##  10\. Proof of Theorem 1.3(c)

We will prove Theorem 1.3(c) by using the lattice point of view from Section 9
and the additivity theorem from Section 8.

Let $\overline{\Gamma}$ be a finite group. Two
$\mathbb{Z}\overline{\Gamma}$-lattices $M,N$ are said to be in the same genus
if $M_{(p)}\simeq N_{(p)}$ for all primes $p$, cf. [CR, 31A]. It is sufficient
to check this condition for divisors $p$ of the order of $\overline{\Gamma}$.
By a theorem of A.V. Roǐter [CR, Theorem 31.28] $M$ and $N$ are in the same
genus if and only if there exists a $\mathbb{Z}\overline{\Gamma}$-lattice $L$
in the genus of the free $\mathbb{Z}\overline{\Gamma}$-lattice of rank one
such that $M\oplus\mathbb{Z}\overline{\Gamma}\simeq N\oplus L$. This has the
following consequence for essential dimension:

######  Proposition 10.1.

Let $T,T^{\prime}$ be $k$-tori. If the lattices $X(T),X(T^{\prime})$ belong to
the same genus then

$\operatorname{ed}_{k}(T)=\operatorname{ed}_{k}(T)\text{ and
}\operatorname{ed}_{k}(T;\ell)=\operatorname{ed}_{k}(T;\ell)\text{ for all
primes }\ell.$

###### Proof.

Let $\operatorname{Gal}(k_{\operatorname{sep}}/k)$ act through a finite
quotient $\overline{\Gamma}$ on $X(T)$ and $X(T^{\prime})$. By assumption
there exists a $\mathbb{Z}\overline{\Gamma}$-lattice $L$ in the genus of
$\mathbb{Z}\overline{\Gamma}$ such that
$X(T)\oplus\mathbb{Z}\overline{\Gamma}\simeq X(T^{\prime})\oplus L$. The torus
$S=\operatorname{Diag}(\mathbb{Z}\overline{\Gamma})$ has a generically free
representation of dimension $\dim S$, hence $\operatorname{ed}_{k}(S)=0$.
Since $L$ is a direct summand of
$\mathbb{Z}\overline{\Gamma}\oplus\mathbb{Z}\overline{\Gamma}$ the torus
$S^{\prime}\colonequals\operatorname{Diag}(L)$ has
$\operatorname{ed}_{k}(S^{\prime})\leq\operatorname{ed}_{k}(S\times S)\leq 0$
as well, where the first inequality follows from [BF, Remarks 1.16 (b)].
Therefore

$\operatorname{ed}_{k}(T)\leq\operatorname{ed}_{k}(T\times
S)=\operatorname{ed}_{k}(T^{\prime}\times
S^{\prime})\leq\operatorname{ed}_{k}(T^{\prime})+\operatorname{ed}_{k}(S^{\prime})=\operatorname{ed}_{k}(T^{\prime})$

and similarly $\operatorname{ed}_{k}(T^{\prime})\leq\operatorname{ed}_{k}(T)$.
Hence $\operatorname{ed}_{k}(T)=\operatorname{ed}_{k}(T^{\prime})$.

A similar argument shows that
$\operatorname{ed}_{k}(T;\ell)=\operatorname{ed}_{k}(T^{\prime};\ell)$ for any
prime $\ell$. This concludes the proof. ∎

######  Corollary 10.2.

Let $k=k^{(p)}$ be a $p$-closed field and $T$ a $k$-torus. Then

$\operatorname{ed}_{k}(T)=\operatorname{ed}_{k}(T;p)=\min\dim(\rho)-\dim T,$

where the minimum is taken over all $p$-faithful representations of $T$.

###### Proof.

The second equality follows from Theorem 1.3(a) and the inequality
$\operatorname{ed}_{k}(T;p)\leq\operatorname{ed}_{k}(T)$ is clear. Hence it
suffices to show $\operatorname{ed}_{k}(T)\leq\operatorname{ed}_{k}(T;p)$. Let
$\rho\colon T\to\operatorname{GL}(V)$ be a $p$-faithful representation of
minimal dimension so that $\operatorname{ed}_{k}(T;p)=\dim\rho-\dim T$. The
representation $\rho$ can be considered as a faithful representation of the
torus $T^{\prime}=T/N$ where $N\colonequals\ker\rho$ is finite of order prime
to $p$. By construction the character lattices $X(T)$ and $X(T^{\prime})$ are
isomorphic after localization at $p$. Since
$\operatorname{Gal}(k_{\operatorname{sep}}/k)$ is a (profinite) $p$-group it
follows that $X(T)$ and $X(T^{\prime})$ belong to the same genus. Hence by
Proposition 10.1 we have
$\operatorname{ed}_{k}(T^{\prime})=\operatorname{ed}_{k}(T)$. Moreover
$\operatorname{ed}_{k}(T^{\prime})\leq\dim\rho-\dim T^{\prime}$, since $\rho$
is a generically free representation of $T^{\prime}$. This finishes the proof.
∎

###### Proof of Theorem 1.3(b).

The equality
$\operatorname{ed}_{k^{(p)}}(G_{k^{(p)}};p)=\operatorname{ed}_{k}(G;p)$
follows from Lemma 3.3. Now we are assuming $G=T\times F$ for a torus $T$ and
a $p$-group $F$ over $k$, which is $p$-closed. Notice that a minimal
$p$-faithful representation of $F$ from Theorem 1.3(a) is also faithful, and
therefore $\operatorname{ed}_{k}(F;p)=\operatorname{ed}_{k}(F)$. Combining
this with Corollary 10.2 and the additivity Theorem 8.1, we see

$\operatorname{ed}(T\times
F)\leq\operatorname{ed}(T)+\operatorname{ed}(F)=\operatorname{ed}(T;p)+\operatorname{ed}(F;p)=\operatorname{ed}(T\times
F;p)\leq\operatorname{ed}(T\times F).$

This completes the proof. ∎

######  Remark 10.3.

The following example shows that “$p$-faithful” cannot be replaced by
“faithful” in the statement of Theorem 1.3(a) (and Corollary 10.2), even in
the case where $G$ is a torus.

Let $p$ be a prime number such that the ideal class group of
$\mathbb{Q}(\zeta_{p})$ is non-trivial (this applies to all but finitely many
primes, e.g. to $p=23$). This means that the subring
$R=\mathbb{Z}[\zeta_{p}]\subseteq\mathbb{Q}(\zeta_{p})$ of algebraic integers
has non-principal ideals. Let $k$ be a field which admits a Galois extension
$l$ of degree $p$ and let
$\Gamma\colonequals\operatorname{Gal}(k_{\operatorname{sep}}/k)$,
$\overline{\Gamma}\colonequals\operatorname{Gal}(l/k)\simeq\Gamma/\Gamma_{l}\simeq
C_{p}$ where $\Gamma_{l}=\operatorname{Gal}(k_{\operatorname{sep}}/l)$ and
$C_{p}$ denotes the cyclic group of order $p$.

We endow the ring $R$ with a $\mathbb{Z}\Gamma$-module structure through the
quotient map $\Gamma\to\overline{\Gamma}$ by letting a generator of
$\overline{\Gamma}$ act on $R$ via multiplication by $\zeta_{p}$. The
$k$-torus $Q\colonequals\operatorname{Diag}(R)$ is isomorphic to the Weil
restriction $R_{l/k}(\operatorname{\mathbb{G}}_{m})$ and has a $p$-dimensional
faithful representation. We will construct a $k$-torus $G$ with a $p$-isogeny
$G\to Q$, such that $G$ does not have a $p$-dimensional faithful
representation.

Let $I$ be a non-principal ideal of $R$. We may consider $I$ as a
$\mathbb{Z}\Gamma$-module and set $G\colonequals\operatorname{Diag}(I)$. We
first show that $I$ and $R$ become isomorphic as $\mathbb{Z}\Gamma$-modules
after localization at $p$. For this purpose let
$I^{\ast}=\\{x\in\mathbb{Q}(\zeta_{p})\mid xI\subseteq R\\}$ denote the
inverse fractional ideal. We have $I\oplus I^{\ast}\simeq R\oplus R$ by [CR,
Theorem 34.31]. The Krull-Schmidt Theorem [CR, Theorem 36.1] for
$\mathbb{Z}_{(p)}C_{p}$-lattices implies $I_{(p)}\simeq R_{(p)}$, hence the
claim. Therefore by Lemma 9.1 there exists a $p$-isogeny $G\to Q$, which shows
in particular that $G$ has a $p$-faithful representation of dimension $p$.

Assume that $G$ has a $p$-dimensional faithful representation. Similarly as in
the proof of Corollary 9.2 this would imply the existence of a surjective map
of $\mathbb{Z}\Gamma$-lattices $\mathbb{Z}\overline{\Gamma}\to I$. However
such a map cannot exist since $I$ is non-principal, hence non-cyclic as a
$\mathbb{Z}\Gamma$-module.

##  11\. Tori of essential dimension $\leq 1$

######  Theorem 11.1.

Let $T$ be a torus over $k$, $k^{(p)}$ a $p$-closure and
$\Gamma=\operatorname{Gal}(k_{\operatorname{alg}}/k^{(p)})$. The following are
equivalent:

  1. (a)

$\operatorname{ed}_{k}(T;p)=0$.

  2. (b)

$\operatorname{ed}_{k^{(p)}}(T;p)=0$.

  3. (c)

$\operatorname{ed}_{k^{(p)}}(T)=0$

  4. (d)

$H^{1}(K,T)=\\{1\\}$ for any $p$-closed field $K$ containing $k$.

  5. (e)

$X(T)_{(p)}$ is a $\mathbb{Z}_{(p)}\Gamma$-permutation module.

  6. (f)

$X(T)$ is an invertible $\mathbb{Z}\Gamma$-lattice (i.e a direct summand of a
permutation lattice).

  7. (g)

There is a torus $S$ over $k^{(p)}$ and an isomorphism

$T_{k^{(p)}}\times
S\simeq\operatorname{R}_{E/k^{(p)}}(\operatorname{\mathbb{G}}_{m}),$

for some étale algebra $E$ over $k^{(p)}$.

######  Remark 11.2.

A prime $p$ for which any of these statements fails is called a torsion prime
of $T$.

###### Proof.

(a) $\Leftrightarrow$ (b) is Lemma 3.3.

(a) $\Leftrightarrow$ (d) follows from [Me1, Proposition 4.4].

(c) $\Rightarrow$ (b) is clear.

(b) $\Rightarrow$ (e): This follows from Corollary 9.2. Indeed,
$\operatorname{ed}_{k}(T;p)=0$ implies the existence of a
$\mathbb{Z}_{(p)}\Gamma$-permutation lattice $L$ together with a surjective
homomorphism $\alpha:L\to X(T)_{(p)}$ and
$\operatorname{rk}L=\operatorname{rk}X(T)_{(p)}$. It follows that $\alpha$ is
injective and $X(T)_{(p)}\simeq L$.

(e) $\Rightarrow$ (f): Let $L$ be a $\mathbb{Z}\Gamma$-permutation lattice
such that $L_{(p)}\simeq X(T)_{(p)}$. Then by [CR, Corollary 31.7] there is a
$\mathbb{Z}\Gamma$-lattice $L^{\prime}$ such that $L\oplus L\simeq X(T)\oplus
L^{\prime}$.

(g) $\Rightarrow$ (c): The torus
$R=\operatorname{R}_{E/k^{(p)}}(\operatorname{\mathbb{G}}_{m})$ has a faithful
representation of dimension $\dim R$ (over $k^{(p)}$) and hence
$\operatorname{ed}_{k^{(p)}}(R)=0$. Since $T_{k^{(p)}}$ is a direct factor of
$R$ we must have $\operatorname{ed}_{k^{(p)}}(T)\leq 0$ by [BF, Remarks 1.16
b)].

(f) $\Leftrightarrow$ (g): A permutation lattice $P$ can be written as

$P=\bigoplus_{i+1}^{m}\mathbb{Z}[\Gamma/\Gamma_{L_{i}}],$

for some (separable) extensions $L_{i}/k^{(p)}$ and
$\Gamma_{L_{i}}=\operatorname{Gal}(k_{\operatorname{alg}}/L_{i})$. Set
$E=L_{1}\times\cdots\times L_{m}$. The torus corresponding to $P$ is exactly
$\operatorname{R}_{E/k^{(p)}}(\operatorname{\mathbb{G}}_{m})$, cf. [Vo, 3.
Example 19]. ∎

######  Example 11.3.

Let $T$ be a torus over $k$ of rank $<p-1$. Then
$\operatorname{ed}_{k}(T;p)=0$. This follows from the fact that there is no
non-trivial integral representation of dimension $<p-1$ of any $p$-group, see
for example [AP, Satz]. Thus any finite quotient of
$\Gamma=\operatorname{Gal}(k_{\operatorname{alg}}/k^{(p)})$ acts trivially on
$X(T)$ and so does $\Gamma$.

######  Remark 11.4.

The equivalence of parts (d) and (f) in Theorem 11.1 can also be deduced from
[CTS, Proposition 7.4].

######  Theorem 11.5.

Let $p$ be an odd prime, $T$ an algebraic torus over $k$, and
$\Gamma=\operatorname{Gal}(k_{\operatorname{alg}}/k^{(p)})$.

  1. (a)

$\operatorname{ed}(T;p)\leq 1$ iff there exists a $\Gamma$-set $\Lambda$ and
an $m\in\mathbb{Z}[\Lambda]$ fixed by $\Gamma$ such that
$X(T)_{(p)}\cong\mathbb{Z}_{(p)}[\Lambda]/\langle m\rangle$ as
$\mathbb{Z}_{(p)}\Gamma$-lattices.

  2. (b)

$\operatorname{ed}(T;p)=1$ iff $m=\sum a_{\lambda}\lambda$ from part (a) is
not $0$ and for any $\lambda\in\Lambda$ fixed by $\Gamma$, $a_{\lambda}=0\mod
p$.

  3. (c)

If $\operatorname{ed}(T;p)=1$ then $T_{k^{(p)}}\cong T^{\prime}\times S$ where
$\operatorname{ed}_{k^{(p)}}(S;p)=0$ and $X(T^{\prime})_{(p)}$ is an
indecomposable $\mathbb{Z}_{(p)}\Gamma$-lattice, and
$\operatorname{ed}_{k^{(p)}}(T^{\prime};p)=1$.

###### Proof.

(a) If $\operatorname{ed}(T;p)=1$, then by Corollary 9.2 there is a map of
$\mathbb{Z}\Gamma$-lattices from $\mathbb{Z}[\Lambda]$ to $X(T)$ which becomes
surjective after localization at $p$ and whose kernel is generated by one
element. Since the kernel is stable under $\Gamma$, any element of $\Gamma$
sends a generator $m$ to either itself or its negative. Since $p$ is odd, $m$
must be fixed by $\Gamma$.

The $\operatorname{ed}(T;p)=0$ case and the converse follows from Theorem 1.4
or Corollary 9.2.

(b) Assume we are in the situation of (a), and say $\lambda_{0}\in\Lambda$ is
fixed by $\Gamma$ and $a_{\lambda_{0}}$ is not $0\mod p$. Then
$X(T)_{(p)}\cong\mathbb{Z}_{(p)}[\Lambda-\\{\lambda_{0}\\}]$, so by Theorem
11.1 we have $\operatorname{ed}(T;p)=0$.

Conversely, assume $\operatorname{ed}(T;p)=0$. Then by Theorem 11.1, we have
an exact sequence $0\to\langle
m\rangle\to\mathbb{Z}_{(p)}[\Lambda]\to\mathbb{Z}_{(p)}[\Lambda^{\prime}]\to
0$ for some $\Gamma$-set $\Lambda^{\prime}$ with one fewer element than
$\Lambda$. We have

$\operatorname{Ext}^{1}_{\Gamma}(\mathbb{Z}_{(p)}[\Lambda^{\prime}],\mathbb{Z}_{(p)})=(0)$

by [CTS, Key Lemma 2.1(i)] together with the Change of Rings Theorem [CR,
8.16]; therefore this sequence splits. In other words, there exists a
$\mathbb{Z}_{(p)}\Gamma$-module homomorphism
$f\colon\mathbb{Z}_{(p)}[\Lambda]\to\mathbb{Z}_{(p)}[\Lambda]$ such that the
image of $f$ is $\langle m\rangle$ and $f(m)=m$. Then we can define
$c_{\lambda}\in\mathbb{Z}_{(p)}$ by $f(\lambda)=c_{\lambda}m$. Note that
$f(\gamma(\lambda))=f(\lambda)$ and thus

(12) $c_{\gamma(\lambda)}=c_{\lambda}$

for every $\lambda\in\Lambda$ and $\gamma\in\Gamma$. If
$m=\sum_{\lambda\in\Lambda}a_{\lambda}\lambda$, as in the statement of the
theorem, then $f(m)=m$ translates into

$\sum_{\lambda\in\Lambda}c_{\lambda}a_{\lambda}=1\,.$

Since every $\Gamma$-orbit in $\Lambda$ has a power of $p$ elements, reducing
modulo $p$, we obtain

$\sum_{\lambda\in\Lambda^{\Gamma}}c_{\lambda}a_{\lambda}=1\pmod{p}\,.$

This shows that $a_{\lambda}\neq 0$ modulo $p$, for some
$\lambda\in\Lambda^{\Gamma}$, as claimed.

(c) Decompose $X(T)_{(p)}$ uniquely into a direct sum of indecomposable
$\mathbb{Z}_{(p)}\Gamma$-lattices by the Krull-Schmidt theorem [CR, Theorem
36.1]. Since $\operatorname{ed}(T;p)=1$, and the essential $p$-dimension of
tori is additive (Thm. 8.1), all but one of these summands are permutation
$\mathbb{Z}_{(p)}\Gamma$-lattices. Now by [CR, 31.12], we can lift this
decomposition to $X(T)\cong X(T^{\prime})\oplus X(S)$, where
$\operatorname{ed}(T^{\prime};p)=1$ and $\operatorname{ed}(S;p)=0$. ∎

######  Example 11.6.

Let $E$ be an étale algebra over $k$. It can be written as
$E=L_{1}\times\cdots\times L_{m}$ with some separable field extensions
$L_{i}/k$. The kernel of the norm
$\operatorname{R}_{E/k}(\operatorname{\mathbb{G}}_{m})\to\operatorname{\mathbb{G}}_{m}$
is denoted by $\operatorname{R}_{E/k}^{(1)}(\operatorname{\mathbb{G}}_{m})$.
It is a torus with lattice

$\bigoplus_{i=1}^{m}\mathbb{Z}[\Gamma/\Gamma_{L_{i}}]\;/\;\langle
1,\cdots,1\rangle,$

where $\Gamma=\operatorname{Gal}(k_{\operatorname{sep}}/k)$ and
$\Gamma_{L_{i}}=\operatorname{Gal}(k_{\operatorname{sep}}/L_{i})$. Let
$\Lambda$ be the disjoint union of the cosets $\Gamma/\Gamma_{L_{i}}$ Passing
to a $p$-closure $k^{(p)}$ of $k$, $\Gamma_{k^{(p)}}$ fixes a $\lambda$ in
$\Lambda$ iff $[L_{i}:k]$ is prime to $p$ for some $i$. We thus have

$\operatorname{ed}_{k}(\operatorname{R}_{E/k}^{(1)}(\operatorname{\mathbb{G}}_{m});p)=\left\\{\begin{array}[]{ll}1,&\mbox{$[L_{i}:k]$
is divisible by $p$ for all $i=1,...,m$}\\\ 0,&\mbox{$[L_{i}:k]$ is prime to
$p$ for some $i$.}\end{array}\right.$

##  12\. Tori split by cyclic extensions of degree dividing $p^{2}$

In this section we assume $k=k^{(p)}$ is $p$-closed. Over $k=k^{(p)}$ every
torus is split by a Galois extension of $p$-power order. We wish to compute
the essential dimension of all tori split by a Galois extension with a (small)
fixed Galois group $G$. The following theorem tells us for which $G$ this is
feasible:

######  Theorem 12.1 (A. Jones [Jo]).

For a $p$-group $G$ there are only finitely many genera of indecomposable
$\mathbb{Z}G$-lattices if and only if $G$ is cyclic of order dividing $p^{2}$.

######  Remark 12.2.

For $G=C_{2}\times C_{2}$ a classification of the (infinitely many) different
genera of $\mathbb{Z}G$-lattices has been worked out by [NA]. In contrast for
$G=C_{p^{3}}$ or $G=C_{p}\times C_{p}$ and $p$ odd (in the latter case) no
classification is known.

Hence in this section we consider tori $T$ whose minimal splitting field is
cyclic of degree dividing $p^{2}$. Its character lattice $X(T)$ is then a
$\mathbb{Z}G$-lattice where $G=\langle g|g^{p^{2}}=1\rangle$ denotes the cylic
group of order $p^{2}$. Heller and Reiner [HR], (see also [CR, 34.32])
classified all indecomposable $\mathbb{Z}G$-lattices. Our goal consists in
computing the essential dimension of $T$. By Corollary 10.2 we have
$\operatorname{ed}_{k}(T)=\operatorname{ed}_{k}(T;p)$, hence by the additivity
Theorem 8.1 it will be enough to find the essential $p$-dimension of the tori
corresponding to indecomposable $\mathbb{Z}G$-lattices. Recall that two
lattices are in the same genus if their $p$-localization (or equivalently
$p$-adic completion) are isomorphic. By Proposition 10.1 tori with character
lattices in the same genus have the same essential $p$-dimension, which
reduces the task to calculating the essential $p$-dimension of tori
corresponding to the $4p+1$ cases in the list [CR, 34.32].

Denote by $H=\langle\nolinebreak h|h^{p}=1\nolinebreak\rangle$ the group of
order $p$. We can consider $\mathbb{Z}H$ as a $G$-lattice with the action
$g\cdot h^{i}=h^{i+1}$. Let

$\delta_{G}=1+g+\ldots+g^{p^{2}-1}\quad\delta_{H}=1+h+\ldots+h^{p-1}$

be the “diagonals” in $\mathbb{Z}G$ and $\mathbb{Z}H$ and

$\epsilon=1+g^{p}+\ldots+g^{p^{2}-p}.$

The following $\mathbb{Z}G$-lattices represent all genera of indecomposable
$\mathbb{Z}G$-lattices (by $\langle*\rangle$ we mean the
$\mathbb{Z}G$-sublattice generated by $*$):

$\begin{array}[]{lcll}M_{1}&=&\mathbb{Z}&\\\ M_{2}&=&\mathbb{Z}H&\\\
M_{3}&=&\mathbb{Z}H/\langle\delta_{H}\rangle\\\ M_{4}&=&\mathbb{Z}G\\\
M_{5}&=&\mathbb{Z}G/\langle\delta_{G}\rangle\\\
M_{6}&=&\mathbb{Z}G\oplus\mathbb{Z}/\langle\delta_{G}-p\rangle&\\\
M_{7}&=&\mathbb{Z}G/\langle\epsilon\rangle\\\
M_{8}&=&\mathbb{Z}G/\langle\epsilon-g\epsilon\rangle&\\\
M_{9,r}&=&\mathbb{Z}G\oplus\mathbb{Z}H/\langle\epsilon-(1-h)^{r}\rangle&1\leq
r\leq p-1\\\
M_{10,r}&=&\mathbb{Z}G\oplus\mathbb{Z}H/\langle\epsilon(1-g)-(1-h)^{r+1}\rangle&1\leq
r\leq p-2\\\
M_{11,r}&=&\mathbb{Z}G\oplus\mathbb{Z}H/\langle\epsilon-(1-h)^{r},\delta_{H}\rangle&1\leq
r\leq p-2\\\
M_{12,r}&=&\mathbb{Z}G\oplus\mathbb{Z}H/\langle\epsilon(1-g)-(1-h)^{r+1},\delta_{H}\rangle&1\leq
r\leq p-2\\\ \end{array}$

In the sequel we will refer to the above list as $(\mathbb{L})$.

In $(\mathbb{L})$ we describe $\mathbb{Z}G$-lattices as quotients of
permutation lattices of minimal possible rank, whereas [CR, 34.32] describes
these lattices as certain extensions $1\to L\to M\to N\to 1$ of
$\mathbb{Z}[\zeta_{p^{2}}]$-lattices by $\mathbb{Z}H$-lattices. Therefore
these two lists look differently. Nevertheless they represent the same
$\mathbb{Z}G$-lattices. We show in the example of the lattice $M_{10,r}$ how
one can translate from one list to the other.

Let $\mathbb{Z}x$ be a $\mathbb{Z}G$-module of rank $1$ with trivial
$G$-action. We have an isomorphism

$M_{10,r}=\mathbb{Z}G\oplus\mathbb{Z}H/\langle\epsilon(1-g)-(1-h)^{r+1}\rangle\simeq\mathbb{Z}G\oplus\mathbb{Z}H\oplus\mathbb{Z}x/\langle\epsilon-(1-h)^{r}-x\rangle$

induced by the inclusion
$\mathbb{Z}G\oplus\mathbb{Z}H\hookrightarrow\mathbb{Z}G\oplus\mathbb{Z}H\oplus\mathbb{Z}x$.

This allows us to write $M_{10,r}$ as the pushout

$\textstyle{\mathbb{Z}H\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h\mapsto\epsilon}$$\scriptstyle{h\mapsto(1-h)^{r}+x}$$\textstyle{\mathbb{Z}G\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{Z}H\oplus\mathbb{Z}x\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M_{10,r}}$

Completing both lines on the right we see that $M_{10,r}$ is an extension

$0\to\mathbb{Z}H\oplus\mathbb{Z}x\to M_{10,r}\to\mathbb{Z}G/\mathbb{Z}H\to 0$

with extension class determined by the vertical map $h\mapsto(1-h)^{r}+x$ cf.
[CR, 8.12] and we identify (the $p$-adic completion of) $M_{10,r}$ with one of
the indecomposable lattices in the list [CR, 34.32].

Similarly, $M_{1},\ldots,M_{12,r}$ are representatives of the genera of
indecomposable $\mathbb{Z}G$-lattices.

######  Theorem 12.3.

Every indecomposable torus $T$ over $k$ split by $G$ has character lattice
isomorphic to one of the $\mathbb{Z}G$-lattices $M$ in the list $(\mathbb{L})$
after $p$-localization and
$\operatorname{ed}(T)=\operatorname{ed}(T;p)=\operatorname{ed}(\operatorname{Diag}(M);p)$.
Their essential dimensions are given in the tables below.

$\begin{array}[]{c|c|c}M&\operatorname{rk}M&\operatorname{ed}(T)\\\ \hline\cr
M_{1}&1&0\\\ M_{2}&p&0\\\ M_{3}&p-1&1\\\ M_{4}&p^{2}&0\\\ M_{5}&p^{2}-1&1\\\
M_{6}&p^{2}&1\end{array}\quad\quad\begin{array}[]{c|c|c}M&\operatorname{rk}M&\operatorname{ed}(T)\\\
\hline\cr M_{7}&p^{2}-p&p\\\ M_{8}&p^{2}-p+1&p-1\\\ M_{9,r}&p^{2}&p\\\
M_{10,r}&p^{2}+1&p-1\\\ M_{11,r}&p^{2}-1&p+1\\\ M_{12,r}&p^{2}&p\end{array}$

###### Proof of Proposition 12.3.

We will assume $p>2$ in the sequel. For $p=2$ the Theoerem is still true but
some easy additional arguments are needed which we leave out here.

The essential $p$-dimension of tori corresponding to $M_{1}\ldots,M_{6}$
easily follows from the discussion in section 11. Let $M$ be one of the
lattices $M_{7},\ldots,M_{12,r}$ and $T=\operatorname{Diag}M$ the
corresponding torus. We will determine the minimal rank of a permutation
$\mathbb{Z}G$-lattice $P$ admitting a homomorphism $P\to M$ which becomes
surjective after localization at $p$. Then we conclude
$\operatorname{ed}(T;p)=\operatorname{rk}P-\operatorname{rk}M$ with Corollary
9.2.

We have the bounds

(13) $\operatorname{rk}M\leq\operatorname{rk}P\leq p^{2}\;(\mbox{or
}p^{2}+p),$

where the upper bound holds since every $M$ is given as a quotient of
$\mathbb{Z}G$ (or $\mathbb{Z}G\oplus\mathbb{Z}H$). Let
$C=\operatorname{Split}_{k}(T[p])$ the finite constant group used in the proof
of Theorem 1.3. The rank of $C$ determines exactly the number of direct
summands into which $P$ decomposes. Moreover each indecomposable summand has
rank a power of $p$.

As an example, we show how to find $C$ for $M=M_{11,r}$: The relations
$g^{j}\cdot(\epsilon-(1-h)^{r});\delta_{H}$ are written out as

$\sum_{i=0}^{p-1}g^{pi+j}-\sum_{\ell=0}^{r}\binom{r}{\ell}(-1)^{\ell}h^{\ell+j},\;0\leq
j\leq p-1;\quad\sum_{i=0}^{p-1}h^{i}$

and the $k_{\operatorname{sep}}$-point of the torus are

$\begin{array}[]{ll}T(k_{\operatorname{sep}})=&{\Big{\\{}}(t_{0},\dotsc,t_{p^{2}-1},s_{0},\dotsc,s_{p-1})\mid\\\
&\prod_{i=0}^{p-1}t_{pi+j}=\prod_{\ell=0}^{r}s_{\ell+j}^{(-1)^{\ell}\binom{r}{\ell}},\;0\leq
j\leq p-1;\quad\prod_{i=0}^{p-1}s_{i}=1{\Big{\\}}}\end{array}$

and $C$ is the constant group of fixed points of the $p$-torsion $T[p]$:

$C(k)=\left\\{\left(\zeta_{p}^{i},\dotsc,\zeta_{p}^{i},\zeta_{p}^{j},\dotsc,\zeta_{p}^{j}\right)\mid\
0\leq i,j\leq p-1\right\\}\simeq\mu_{p}^{2}.$

(Note that the primitive $p$th root of unity $\zeta_{p}$ is in $k$ by our
assumption that $k$ is $p$-closed). For other lattices this is similar: $C$ is
equal to
$\operatorname{Split}_{k}(\operatorname{Diag}(P)[p])\simeq\mu_{p}^{r}$ where
$M$ is presented as a quotient $P/N$ of a permutation lattice $P$ (of minimal
rank) as in $(\mathbb{L})$ and where $r$ denotes the number of summands in a
decomposition of $P$.

$\begin{array}[]{c|c|c|c}M&\mbox{rank $C$}&\mbox{rank $M$}&\mbox{possible
$\operatorname{rk}P$}\\\ \hline\cr M_{7}&1&p^{2}-p&p^{2}\\\
M_{8}&1&p^{2}-p+1&p^{2}\\\ M_{9,r}&2&p^{2}&p^{2}+1\mbox{ or }p^{2}+p\\\
M_{10,r}&2&p^{2}+1&p^{2}+1\mbox{ or }p^{2}+p\\\
M_{11,r}&2&p^{2}-1&p^{2}+1\mbox{ or }p^{2}+p\\\ M_{12,r}&2&p^{2}&p^{2}+1\mbox{
or }p^{2}+p\end{array}$

We need to exclude the possibility $\operatorname{rk}P=p^{2}+1$ for the
lattices $M=M_{9,r},\ldots,M_{12,r}$. We can only have the value $p^{2}+1$ if
there exists a character in $M$ which is fixed under the Galois group and
nontrivial on $C$. The following Lemma 12.4 tells us, that such characters do
not exist in either case. Hence the minimal dimension of a $p$-faithful
representation of all these tori is $p^{2}+p$. ∎

######  Lemma 12.4.

For $i=9,\ldots,12$ and $r\geq 1$ every character $\chi\in M_{i,r}$ fixed
under $G$ has trivial restriction to $C$.

###### Proof.

By [Hi] the cohomology group $H^{0}(G,M_{i,r})=M_{i,r}^{G}$ of $G$-fixed
points in $M_{i,r}$ is trivial for $i=11$, has rank $1$ for $i=9,12$ and rank
$2$ for $i=10$, respectively. They are represented by $\mathbb{Z}\delta_{H}$
in $M_{9,r}$, by $\mathbb{Z}(\epsilon-(1-h)^{r})$ in $M_{12,r}$ and by
$\mathbb{Z}(\epsilon-(1-h)^{r})\oplus\mathbb{Z}\delta_{H}$ in $M_{10,r}$,
respectively. Since all these characters are trivial on

$C=\operatorname{Split}_{k}(\operatorname{Diag}(\mathbb{Z}G\oplus\mathbb{Z}H)[p]),$

the claim follows. ∎

## Acknowledgments

The authors are grateful to A. Auel, A. Merkurjev and A. Vistoli for helpful
comments and conversations.

## References

  * [AP] H. Abold, W. Plesken, Ein Sylowsatz für endliche $p$-Untergruppen von ${\rm GL}(n,Z)$, Math. Ann. 232 (1978), no. 2, 183–186.
  * [Ba] G. Bayarmagnai, Essential dimension of some twists of $\mu_{p^{n}}$, Proceedings of the Symposium on Algebraic Number Theory and Related Topics, 145–151, RIMS K$\hat{\rm o}$ky$\hat{\rm u}$roku Bessatsu, B4, Res. Inst. Math. Sci. (RIMS), Kyoto (2007).
  * [BF] G. Berhuy, G. Favi, Essential Dimension: A Functorial Point of View (after A. Merkurjev), Doc. Math. 8:279–330 (electronic) (2003).
  * [Bo] A. Borel Linear Algebraic Groups, Benjamin (1969).
  * [BS] A. Borel, J.-P. Serre, Théorèmes de finitude en cohomologie galoisienne, Comment. Math. Helv. 39 (1964), 111–164.
  * [Bou] N. Bourbaki, Algebra. II. Chapters 4–7. Translated from the French by P. M. Cohn and J. Howie. Elements of Mathematics. Springer-Verlag, Berlin, (1990).
  * [BR] J. Buhler, Z. Reichstein, On the essential dimension of a finite group, Compositio Mathematica 106:159–179.(1997).
  * [CGR] V. Chernousov, Ph. Gille, Z. Reichstein, Resolving $G$-torsors by abelian base extensions, J. Algebra 296 (2006), no. 2, 561–581.
  * [CTS] J.-L. Colliot-Thélène, J. J. Sansuc, Principal Homogeneous Spaces under Flasque Tori: Applications, J. Algebra 106 (1087), 148–205.
  * [CR] C. W. Curtis, I. Reiner, Methods of representation theory ,vol. 1, Wiley (Interscience), 1981.
  * [DG] M. Demazure, P. Gabriel, Groupes algébriques. Tome I, Masson & Cie, Paris; North-Holland Publishing Co., Amsterdam, 1970.
  * [Fl] M. Florence, On the essential dimension of cyclic $p$-groups, Inventiones Mathematicae, 171 (2007), 175-189.
  * [GMS] S. R. Garibaldi, A. Merkurjev, J.-P. Serre: Cohomological Invariants in Galois Cohomology, University Lecture Series, Vol. 28, American Mathematical Society, Providence, RI, (2003).
  * [GR] Ph. Gille, Z. Reichstein, A lower bound on the essential dimension of a connected linear group, Comment. Math. Helv. 4, no. 1 (2009), 189–212.
  * [Gro] A. Grothendieck, La torsion homologique et les sections rationnelles, Exposé 5, Séminaire C. Chevalley, Anneaux de Chow et applications, IHP, (1958).
  * [HR] A. Heller, I. Reiner: Representations of cyclic groups in rings of integers I, Annals of Math, 76 (1962), 73-92.
  * [Hi] H. Hiller, Flat Manifolds with $\mathbb{Z}/p^{2}$ Holonomy, L’Enseignement Mathématique, 31 (1985), 283–297.
  * [Ja] J. C. Jantzen, Representations of Algebraic Groups. Pure and Applied Mathematics, 131. Academic Press, Orlando, Florida, (1987).
  * [Jo] A. Jones, Groups with a finite number of indecomposable integral representations, Mich. Math. J, 10 (1963), 257-261.
  * [Ka] G. Karpilovsky, Clifford Theory for Group Representations. Mathematics Studies, 156. North-Holland, Netherlands, (1989).
  * [KM] N. Karpenko, A. Merkurjev, Essential dimension of finite p-groups, Inventiones Mathematicae, 172 (2008), 491–508.
  * [Ma] B. Margaux, Passage to the limit in non-abelian Čech cohomology. J. Lie Theory 17, no. 3 (2007), 591–596.
  * [Me1] A. Merkurjev, Essential dimension, in Quadratic forms – algebra, arithmetic, and geometry (R. Baeza, W.K. Chan, D.W. Hoffmann, and R. Schulze-Pillot, eds.), Contemporary Mathematics 493 (2009), 299–326.
  * [Me2] A. Merkurjev Essential dimension of PGL($p^{2}$), preprint, available at http://www.math.ucla.edu/~merkurev/publicat.htm.
  * [MR1] A. Meyer, Z. Reichstein, The essential dimension of the normalizer of a maximal torus in the projective linear group, Algebra and Number Theory, 3, no. 4 (2009), 467–487.
  * [MR2] A. Meyer, Z. Reichstein, An upper bound on the essential dimension of a central simple algebra, to appear in Journal of Algebra, 10.1016/j.jalgebra.2009.09.019, preprint available at arXiv:0907.4496
  * [NA] L. A. Nazarova, Unimodular representations of the four group, Dokl. Akad. Nauk SSSR, 140 (1961), 1011–1014.
  * [Re] Z. Reichstein, On the Notion of Essential Dimension for Algebraic Groups, Transformation Groups, 5, 3 (2000), 265-304.
  * [RY] Z. Reichstein, B. Youssin, Essential Dimensions of Algebraic Groups and a Resolution Theorem for $G$-varieties, with an appendix by J. Kollar and E. Szabo, Canadian Journal of Mathematics, 52, 5 (2000), 1018–1056.
  * [RZ] L. Ribes, P. Zalesskii, Profinite Groups. Springer-Verlag, Berlin, 2000.
  * [Ro] M. Rost, Essential dimension of twisted $C_{4}$, available at http://www.math.uni-bielefeld.de/~rost/ed.html.
  * [Sch1] H.-J. Schneider, Zerlegbare Erweiterungen affiner Gruppen J. Algebra 66, no. 2 (1980), 569–593.
  * [Sch2] H.-J. Schneider, Decomposable Extensions of Affine Groups, in Lecture Notes in Mathematics 795, Springer Berlin/Heidelberg (1980), 98–115.
  * [Sch3] H.-J. Schneider, Restriktion und Corestriktion für algebraische Gruppen J. Algebra, 68, no. 1 (1981), 177–189.
  * [Se1] J.-P. Serre, Linear representations of finite groups, Graduate Texts in Mathematics, 42, Springer-Verlag, 1977.
  * [Se2] J.-P. Serre, Galois cohomology. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2002.
  * [Ta] J. Tate, Finite flat group schemes. Modular forms and Fermat’s last theorem (Boston, MA, 1995), 121–154, Springer, New York, 1997.
  * [Vo] V. E. Voskresenskiĭ, Algebraic Groups and Their Birational Invariants, American Mathematical Society, Providence, RI, 1998.
  * [Wa] W. C. Waterhouse, Introduction to affine group schemes. Springer-Verlag, New York-Berlin, 1979.
  * [Wi] J. S. Wilson, Profinite Groups. London Math. Soc. Monographs 19, Oxford University Press, New York, 1998.
  * [Win] D. Winter, The structure of fields. Graduate Texts in Mathematics, no. 16. Springer-Verlag, New York-Heidelberg, 1974.