arXiv:1801.00089v3 [] 27 Mar 2018

ESSENTIAL DIMENSION OF FINITE GROUPS
IN PRIME CHARACTERISTIC
ZINOVY REICHSTEIN† AND ANGELO VISTOLI‡
Abstract. Let F be a field of characteristic p > 0 and G be a smooth finite algebraic
group over F . We compute the essential dimension edF (G; p) of G at p. That is, we
show that
(
1, if p divides |G|, and
edF (G; p) =
0, otherwise.
Résumé. Soit F un corps de caractéristique p > 0, et soit G un groupe algébrique fini
étale sur F . On calcule la dimension essentielle de G en p, que l’on note edF (G; p). Plus
précisément, on démontre que
(
1, si p divise |G|,
edF (G; p) =
0, sinon.

1. Introduction
Let F be a field and G be an algebraic group over F . We begin by recalling the
definition of the essential dimension of G.
Let K be a field containing F and τ : T → Spec(K) be a G-torsor. We will say that τ
descends to an intermediate subfield F ⊂ K0 ⊂ K if τ is the pull-back of some G-torsor
τ0 : T0 → Spec(K0 ), i.e., if there exists a Cartesian diagram of the form
T
/

T0
τ0

τ



Spec(K)
/



Spec(K0 )
/

Spec(F ).

The essential dimension of τ , denoted by edF (τ ), is the smallest value of the transcendence
degree trdeg(K0 /F ) such that τ descends to K0 . The essential dimension of G, denoted
by edF (G), is the maximal value of edF (τ ), as K ranges over all fields containing F and
τ ranges over all G-torsors T → Spec(K).
Now let p be a prime integer. A field K is called p-closed if the degree of every finite
extension L/K is a power of p. Equivalently, Gal(K s /K) is a pro-p-group, where K s is a
2010 Mathematics Subject Classification. 20G15, 14G17.
Key words and phrases. Essential dimension, versal torsor, Nottingham group, reduction of structure,
Serre’s Conjecture I.
†
Partially supported by National Sciences and Engineering Research Council of Canada Discovery
grant 253424-2017.
‡
Partially supported by research funds from the Scuola Normale Superiore.
The authors are grateful to the Collaborative Research Group in Geometric and Cohomological Methods in Algebra at the Pacific Institute for the Mathematical Sciences for their support of this project.
1

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REICHSTEIN AND VISTOLI

separable closure of K. For example, the field of real numbers is 2-closed. The essential
dimension edF (G; p) of G at p is the maximal value of edF (τ ), where K ranges over pclosed fields K containing F , and τ ranges over the G-torsors T → Spec(K). For an
overview of the theory of essential dimension, we refer the reader to the surveys [Rei10]
and [Me13].
The case where G is a finite group (viewed as a constant group over F ) is of particular
interest. A theorem of N. A. Karpenko and A. S. Merkurjev [KM08] asserts that in this
case
(1)

edF (G; p) = edF (Gp ; p) = edF (Gp ) = rdimF (Gp ) ,

provided that F contains a primitive p-th root of unity ζp . Here Gp is any Sylow psubgroup of G, and rdimF (Gp ) denotes the minimal dimension of a faithful representation
of Gp defined over F . For example, assuming that ζp ∈ F , edF (G) = ed(G; p) = r if
G = (Z/pZ)r , and ed(G) = ed(G; p) = p if G is a non-abelian group of order p3 . Further
examples can be found in [MR10].
Little is known about essential dimension of finite groups over a field F of characteristic
p > 0. A. Ledet [Le04] conjectured that
(2)

edF (Z/pr Z) = r

for every r > 1. This conjecture remains open for every r > 3. In this paper we will prove
the following surprising result.
Theorem 1. Let F be a field of characteristic p > 0 and G be a smooth finite algebraic
group over F . Then
(
1, if p divides |G|, and
edF (G; p) =
0, otherwise.
In particular, Ledet’s conjecture (2) fails dramatically if essential dimension is replaced
by essential dimension at p. On the other hand, Theorem 1 fails if ed(G; p) is replaced by
ed(G); see [Le07].
Before proceeding with the proof of Theorem 1, we remark that the condition that G
is smooth cannot be dropped. Indeed, it is well known that edF (µrp ; p) = r for any r > 0.
More generally, if G is a group scheme of finite type over a field F of characteristic p
(not necessarily finite or smooth), then edF (G; p) > dim(G) − dim(G), where G is the Lie
algebra of G; see [TV13, Theorem 1.2].
Acknowledgments: We are grateful to the referee for a thorough reading of the
paper and numerous constructive suggestions, including an alternate proof of Lemma 3.
We would also like to thank D. Tossici and J.-P. Serre for helpful comments.
2. Versality
Let G be an algebraic group and X be an irreducible G-variety (i.e., a variety with
a G-action) over F . We will say that the G-variety X is generically free if there exists
a dense open subvariety U of X such that the scheme-theoretic stabilizer Gu of every
geometric point u of X is trivial. Equivalently, there exists a G-invariant dense open
subvariety U ′ of X, which is the total space of a G-torsor; see [Se03, Section 5].

ESSENTIAL DIMENSION IN PRIME CHARACTERISTIC

3

Following [Se03, Section 5] and [DR15, Section 1], we will say that X is weakly versal
(respectively, weakly p-versal), if for every infinite field (respectively, every p-closed field)
E, and every G-torsor T → Spec(E) there is a G-equivariant F -morphism T → X. We
will say that X versal (respectively, p-versal), if every G-invariant dense open subvariety
of X is weakly versal (respectively, weakly p-versal).
It readily follows from these definitions that ed(G) (respectively, ed(G; p)) is the minimal dimension dim(X)−dim(G), where the minimum is taken over all versal (respectively
p-versal) generically free G-varieties X; see [Se03, Section 5.7], [DR15, Remark 2.6 and
Section 8]. Our proof of Theorem 1 will be based on the the following facts.
(i) ([DR15, Proposition 2.2]) Every G-variety X with a G-fixed F -point is weakly versal.
(ii) ([DR15, Theorem 8.3]) Let X be a smooth geometrically irreducible G-variety. Then
X is weakly p-versal if and only if X is p-versal.
Combining (i) and (ii), we obtain:
Proposition 2. ([DR15, Corollary 8.6(b)]) Let G be a finite smooth algebraic group over
F . If there exists a faithful geometrically irreducible G-variety X with a smooth G-fixed
F -point, then ed(G; p) 6 dim(X).
If we replace “p-versal" by “versal", then (ii) fails: a weakly versal G-variety does not
need to be versal. This is the underlying reason why both Proposition 2 and Theorem 1
fail if ed(G; p) is replaced by ed(G).
3. Proof of Theorem 1
In this section we will prove Theorem 1, assuming Lemmas 3 and 4 below. We will
defer the proofs of these lemmas to sections 4 and 5, respectively.
By [MR10, Lemma 4.1], if G′ ⊂ G is a subgroup of index prime to p, then
(3)

edF (G; p) = edF (G′ ; p).

In particular, if p does not divide |G|, then taking G′ = {1}, we conclude that edF (G; p) =
0. On the other hand, if p divides |G|, then edF (G; p) > 1; see [Me09, Proposition 4.4]
or [LMMR13, Lemma 10.1]. Our goal is thus to show that edF (G; p) 6 1.
First let us consider the case where G is a finite group, viewed as a constant algebraic
group over F . After replacing G by a Sylow p-subgroup, we may assume that G is a pgroup. Moreover, since Fp ⊂ F , edF (G; p) 6 edFp (G; p). Thus, for the purpose of proving
the inequality edF (G; p) 6 1, we may assume that F = Fp . In view of Proposition 2 it
suffices to prove the following.
Lemma 3. For every finite constant p-group G there exists a faithful G-curve defined
over Fp with a smooth G-fixed Fp -point.
Now consider the general case, where G is a smooth finite algebraic group over F . In
other words, G = τ Γ, where Γ is a constant finite group, A = Autgrp (Γ) is the group of
automorphisms of Γ and τ is a cocycle representing a class in H 1 (F, A).
Lemma 4. (a) edF (G) 6 edF (Γ ⋊ A),

(b) edF (G; p) 6 ed(Γ ⋊ A; p).

The semidirect product Γ ⋊ A is a constant finite group. Hence, as we showed above,
edF (Γ ⋊ A; p) 6 1. Theorem 1 now follows from Lemma 4(b).

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REICHSTEIN AND VISTOLI

4. Proof of Lemma 3
We will give two proofs: our original proof, extracted from the literature, and a selfcontained proof suggested to us by the referee.
Proof. Recall that the Nottingham group Aut0 (Fp [[t]]) is the group of automorphisms σ
of the algebra Fp [[t]] of formal power series such that σ(t) = t + a2 t2 + a2 t3 + . . ., for
some a2 , a3 , . . . ∈ Fp . By a theorem of of Leedham-Green and Weiss [C97, Theorem 3],
every finite p-group G embeds into Aut0 (Fp [[t]]). Fix an embedding φ : G ֒→ Aut0 (Fp [[t]]).
By [Ka86, Theorem 1.4.1], there exists a smooth G-curve X over Fp , with an Fp -point
x ∈ X fixed by G, such that the G-action in the formal neighborhood of x is given by φ;
see also [Ha80, Section 2] and [BCPS17, Theorem 4.8]. Since φ is injective, the G-action
on X is faithful.
♠
Alternative proof. First consider the case, where G = (Z/pZ)n is an elementary abelian
p-group. Here we can construct X as the cover of P1 (with function field Fp (s)) given by
the compositum of n linearly disjoint Artin-Schreier extensions Fp (s, ti )/Fp (s) given by
tpi − ti = fi (s) (e.g., taking fi (s) = spi+1).
Now consider a general finite p-group G. Denote the Frattini subgroup of G by Φ
and the quotient G/Φ by (Z/pZ)n . Let Y be the smooth curve and Y → P1 be a
G/Φ = (Z/pZ)n -cover constructed in the previous paragraph, totally ramified at a point
y ∈ Y (Fp ) above ∞ ∈ P1 . Let E/Fp (s) be the (Z/pZ)n -Galois extension associated to
this cover. By [Se97, Proposition II.2.2.3], the cohomological dimension of Fp (s) at p is
6 1. Consequently by [Se97, Propositions I.3.4.16], E/Fp (s) lifts to a G-Galois extension
K/Fp (s) such that K Φ = E. Let X be the smooth curve associated to K and x ∈ X(Fp )
is a point above y:
x
∈
X


y



∈

Y

∞



∈

P1Fp .



We claim that x is fixed by G; in particular, this will imply that x ∈ X(Fp ). Let H be
the stabilizer of x in G. Since Φ acts transitively on the fiber above y in X, we have
Φ · H = G. By Frattini’s theorem (see, e.g., [Ro96, Theorem 5.2.12]), Φ is the set of
non-generators of G. We conclude that H = G, as claimed.
♠
5. Proof of Lemma 4
We will make use of the following description of edF (G) and edF (G; p) in the case where
G is a finite algebraic group over F . Let G → GL(V ) be a faithful representation. A
compression (respectively, a p-compression) of V is a dominant G-equivariant rational
map V 99K X (respectively, a dominant G-equivariant correspondence V
X of degree
prime to p), where G acts faithfully on X. Here by a correspondence we mean a Gequivariant subvariety V ′ of V × X such that the G transitively permutes the irreducible

ESSENTIAL DIMENSION IN PRIME CHARACTERISTIC

5

components of V ′ , and the dimension of each component equals the dimension of V . The
degree of this correspondence is defined as the degree of the projection V ′ → V to the
first factor.
Recall that edF (G) (respectively, edF (G; p)) equals the minimal value of dim(X) taken
over all compressions V 99K X (respectively all p-compressions V
X). In particular,
these numbers depend only on G and F and not on the choice of the generically free
representation V . For details, see [Rei10].
We are now ready to proceed with the proof of Lemma 4. To prove part (a), let V be a
generically free representation of Γ ⋊ A and let f : V 99K X be a Γ ⋊ A-compression, with
X of minimal possible dimension. That is, dimF (X) = edF (Γ ⋊ A). Twisting by τ , we
obtain a G = τ Γ-equivariant map τ f : τ V 99K τ X; see e.g., [FR17, Proposition 2.6(a)].
Now observe that by Hilbert’s Theorem 90, τ V is a vector space with a linear action of
G = τ Γ and τ f : τ V 99K τ X is a compression. (To see that the G-action on τ V and
τ
X are faithful, we may pass to the algebraic closure F of F . Over F , τ is split, so that
G = Γ, τ V = V , τ X = X and τ f = f , and it becomes obvious that the G-actions on τ V
and τ X are faithful.) We conclude that edF (G) 6 dimF ( τ X) = dimF (X) = edF (Γ ⋊ A),
as desired.
The proof of part (b) proceeds along the same lines. The starting point is a pcompression f : V
X with X of minimal possible dimension, dimF (X) = edF (Γ ⋊ A; p).
τ
We twist f by τ to obtain a p-compression τ f : τ V
X of the linear action of G = τ Γ
τ
on V . The rest of the argument is the same as in part (a). This completes the proof of
Lemma 4 and thus of Theorem 1.
♠
6. An application
In this section G will denote a connected reductive linear algebraic group over a field
F . It is shown in [CGR06, Theorem 1.1(c)] that there exists a finite F -subgroup S ⊂ G
such that every G-torsor over every field K/F admits reduction of structure to S; see
also [CGR08, Corollary 1.4]. In other words, the map H 1 (K, S) → H 1 (K, G) is surjective
for every field K containing F . If this happens, we will say that “G admits reduction of
structure to S ”.
We will now use Theorem 1 to show that if char(F ) = p > 0 and p is a torsion
prime for G, then S cannot be smooth. For the definition of torsion primes, a discussion of their properties and further references, see [Se00]. Note that by a theorem of
A. Grothendieck [Gr58], if G is not special (i.e., if H 1 (K, G) 6= {1} for some field K
containing F ), then G has at least one torsion prime; see also [Se00, 1.5.1].
Corollary 5. Let G be a connected reductive linear algebraic group over an algebraically
closed field F of characteristic p > 0.
(a) If S is a smooth finite subgroup of G defined over F , then the natural map
fK : H 1 (K, S) → H 1 (K, G)
is trivial for any p-closed field K containing F . In other words, fK sends every α ∈
H 1 (K, S) to 1 ∈ H 1 (K, G).
(b) If p is a torsion prime for G, then G does not admit reduction of structure to any
smooth finite subgroup.

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REICHSTEIN AND VISTOLI

Proof. (a) Let α ∈ H 1 (K, S) and β = fK (α) ∈ H 1 (K, G). By Theorem 1, α descends
to α0 ∈ H 1 (K0 , S) for some intermediate field F ⊂ K0 ⊂ K, where trdeg(K0 /F ) 6 1.
Since F is algebraically closed, dim(K0 ) 6 1; see [Se97, Sections II.3.1-3]. By Serre’s
Conjecture I (proved by R. Steinberg [St65] for a perfect field K0 and by A. Borel and
T. A. Springer [BS68, §8.6] for an arbitrary K0 of dimension 6 1), H 1 (K0 , G) = {1}.
Tracing through the diagram
fK0

H 1 (K0 , S)
α❴0 ✤

/



H 1 (K0 , G) = {1}

1❴



α✤

H 1 (K, S)

/

/

β

fK


/

H 1 (K, G),

we see that β = 1, as desired.
(b) If p is a torsion prime for G, then H 1 (K, G) 6= {1} for some p-closed field K
containing F ; see [Me09, Proposition 4.4]. In view of part (a), this implies that fK is not
surjective.
♠
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(Reichstein) Department of Mathematics, University of British Columbia, Vancouver,
B.C., Canada V6T 1Z2
E-mail address: reichst@math.ubc.ca
(Vistoli) Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy
E-mail address: angelo.vistoli@sns.it

