ESSENTIAL 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 broad class of groups, which includes all algebraic tori.

1. Introduction

Throughout this paper p will denote a prime integer, k an arbitrary base field 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. Morphisms of algebraic groups over k are assumed to be defined over k.

Let K be a field and H^{1}(K, G) be the nonabelian cohomology set with respect to the finitely presented faithfully flat (fppf) topology. Equivalently H^{1}(K, G) is the set of isomorphism classes of G-torsors over Spec(K). If G is smooth then one may identify H^{1}(∗, G) with the first Galois cohomology functor. We say that α ∈ H^{1}(K, G) descends to an intermediate field k ⊂ K_{0} ⊂ K if it lies in the image of the natural map H^{1}(K_{0}, G) → H^{1}(K, G). The minimal transcendence degree trdeg_{k}(K_{0}), where α descends to K_{0}, is called the essential dimension of α and is denoted by the symbol ed(α). The essential dimension of the group G is the supremum of ed(α), as K ranges over all field extensions of k and α ranges over H^{1}(K, G). This numerical invariant of G has been extensively studied in recent years; see [BF, BR, Re, RY, Me_{1}].

For many groups G the essential dimension ed(G) is hard to compute, even over the field k = ℂ of complex numbers. Given a prime p, it is often easier to compute the essential p-dimension, ed(G; p), which is defined as follows. The essential p-dimension ed(α; p) of α ∈ H^{1}(K, G) is the minimal value of ed(α_{L}), as L ranges over all finite field extensions of K of degree prime to p. The essential p-dimension ed(G; p) of G is then the supremum of ed(α; p) taken over all fields K containing k and all α ∈ H^{1}(K, G). For details on this notion, see [RY] or [Me_{1}]. Clearly 0 ≤ ed(G; p) ≤ ed(G). It is also easy to check that if L/K is a finite extension of degree prime to p then

2000 Mathematics Subject Classification. 20G15, 11E72, 20C10.

Key words and phrases. Essential dimension, group of multiplicative type, algebraic torus, twisted finite group, lattice, Galois module.

(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.

(1)  ed(G; p) = ed(G_{L}; p) ;

see [Me_{1}, Proposition 1.5].

A representation ψ: G → GL(V) is called generically free if there exists a non-empty G-invariant open subset U ⊂ V such that the scheme-theoretic stabilizer of every point of U(k_{alg}) is trivial. Such a representation gives rise to an upper bound on the essential dimension,

(2)  ed(G; p) ≤ ed(G) ≤ dim(V) − dim(G) ;

see [Me_{1}, Theorem 4.1], [Re, Theorem 3.4], [BF, Lemma 4.11].

N. Karpenko and A. Merkurjev [KM] recently showed that the inequalities (2) are in fact sharp for finite constant p-groups, assuming that the base field k contains a primitive pth root of unity (note that this implies char k ≠ p). The purpose of this paper is to establish a similar result for a large class of groups which includes all algebraic tori.

For a field extension l/k, set G_{l} := G ×_{Spec k} Spec(l). Let k_{sep} be a fixed separable closure of k. Recall that an algebraic group G over a field k is called diagonalizable if it isomorphic to a closed subgroup of 𝔾_{m}^{n} for some n ≥ 0; G is said to be of multiplicative type if G_{k_{sep}} is diagonalizable, see, e.g., [Vos_{2}, Section 3.4]. Smooth connected groups of multiplicative type are precisely the algebraic tori.

Recall that the order of an algebraic group F is defined as |F | = dim_{k} k[F]; algebraic groups of finite order are called finite. We will say that a representation ψ: G → GL(V) of an algebraic group G is p-faithful if its kernel is finite and of order prime to p.

Theorem 1.1. Let G be a group of multiplicative type over an arbitrary field k. Assume that G has a Galois splitting field of p-power degree. Then

ed(G; p) = min dim(ψ) − dim G ,

where the minimum is taken over all p-faithful representations ψ of G. Moreover, if G is an extension of a p-group by a torus then

ed(G) = ed(G; p) .

The quantity min dim(ψ) which appears in the statement of the Theorem 1.1 can be conveniently described in terms of character modules; see Corollary 5.1. We give several applications of these results in Sections 5 and 6. Further applications of the Theorem 1.1, to the classical problem of computing essential dimensions of central simple algebras, can be found in [Me_{2}] and [BM].

Note that Theorem 1.1 allows us to compute ed(G; p) for any group G of multiplicative type over k. Indeed, we can always choose a finite field extension k′/k of degree prime to p such that G_{k′} has a Galois splitting field of p-power degree. In view of (1), ed(G; p) = ed(G_{k′}; p), and the latter number is given by Theorem 1.1.

In the last section we will prove analogous results for a finite (not necessarily abelian) algebraic group over k, assuming char k ≠ p; see Theorem 7.1 and Remark 7.2.

2. Preliminaries on groups of multiplicative type

Throughout this section, A will denote an algebraic group of multiplicative type over a field k, X(A) the character group of A, and Γ := Gal(k_{sep}/k) the absolute Galois group of k. Then X(A) is a continuous ℤΓ-module. Moreover, X(∗) defines an anti-equivalence between algebraic k-groups of multiplicative type and continuous ℤΓ-modules; see, e.g., [Wa, 7.3]. Let Diag denote the inverse of X, so that Diag(X(A)) ≃ A.

Given a field extension l/k, recall that A is called split over l if and only if the absolute Galois group Gal(l_{sep}/l) acts trivially on X(A). If a torsionfree ℤΓ-module P has a basis which is permuted by Γ, then it is called a permutation module, and Diag(P) is a quasi-split torus.

We will write A[p] for the p-torsion subgroup {a ∈ A | a^{p} = 1} of A. Clearly A[p] is defined over k. If A is a finite algebraic group of multiplicative type, then |A| = |X(A)| (by Cartier duality).

It is well known how to construct a maximal split subtorus of an algebraic torus, see for example [Wa, 7.4]. The following is a variant of this construction for algebraic groups of multiplicative type. Set

Split_{k}(A) := Diag(X(A)_{Γ}) ,

where X(A)_{Γ} is the module of co-invariants, defined as the largest quotient of X(A) with trivial Γ-action. Clearly Split_{k}(A) is split over k.

Lemma 2.1. If A[p] ≠ {1} and A is split over a Galois extension l/k of p-power degree, then Split_{k}(A) ≠ {1}.

Proof. If B is a k-subgroup of A then Split_{k}(B) ⊂ Split_{k}(A), so it suffices to show that Split_{k}(A[p]) ≠ {1}. Hence, we may assume that A = A[p] or equivalently, that X(A) is a finite-dimensional 𝔽_{p}-vector space on which the p-group Gal(l/k) acts. Any such action is upper-triangular, relative to some 𝔽_{p}-basis e_{1}, …, e_{n} of X(A); see, e.g., [Se_{1}, Proposition 26, p. 64]. That is,

γ(e_{i}) = e_{i}+ (𝔽_{p}-linear combination of e_{i+1}, …, e_{n})

for every i = 1, …, n and every γ ∈ Gal(l/k). The quotient of X(A) by the linear span of e_{2}, …, e_{n} has trivial Γ-action. Hence the module of coinvariants X(A)_{Γ} is non-trivial. Then Split_{k}(A) = Diag(X(A)_{Γ}) is nontrivial as well. □

Let G be an algebraic group whose centre Z(G) is of multiplicative type. Then we define C(G) := Split_{k}(Z(G)[p]). Note that this definition depends on the prime p, which we assume to be fixed throughout.

Lemma 2.2. Let N be a subgroup of A defined over k. Assume that A has a Galois splitting field l/k of p-power degree. Then N ∩ C(A) = {1} if and only if N is finite and its order is prime to p.

Proof. If the order of N ⊆ A is finite and prime to p then clearly N ∩C(A) = {1}, because C(A) is a p-group. Conversely, suppose the order of N is either infinite or is finite but divisible by p. Then N[p] ≠ {1}, and N[p] is split by l. By Lemma 2.1, {1} ≠ Split_{k}(N[p]) ⊆ Split_{k}(A[p]) = C(A), as desired. □

Now suppose l/k be a Galois splitting field of A and ψ: A → GL(V) is a krepresentation. Then we can decompose V_{l} = ⨁_{χ∈Λ} V (χ), where Λ ⊆ X(A) is the set of weights and V (χ) ⊂ V is the weight space associated to χ ∈ Λ, i.e., the subspace of V, where A acts via χ. The Galois group Γ = Gal(l/k) permutes Λ and weight spaces V (χ).

Lemma 2.3. Let d_{χ} = dim_{l} V (χ). Then there exists an l-basis

Δ = {e_{j}^{χ} | χ ∈ Λ, j = 1, …, d_{χ}}

of V_{l} such that γe_{j}^{χ} = e_{j}^{γχ} for every γ ∈ Γ.

Proof. We may assume that Γ acts transitively on Λ. Then d = dim_{l} V (χ) is independent of χ ∈ Λ.

Choose a weight χ_{0} ∈ Λ. The stabilizer Γ_{0} of χ_{0} in Γ acts semi-linearly on the l-vector space V (χ_{0}). By the no-name lemma [Sh, Appendix 3] there exists a basis e_{1}, ..., e_{d} of V (χ_{0}) such that each e_{i} is preserved by Γ_{0}. Now for χ ∈ Λ and j = 1, …, d, set e_{j}^{χ} := γ(e_{j}), where γ ∈ Γ takes χ_{0} to χ. It is now easy to see that the e_{j}^{χ} are well defined and form an l-basis of V_{l} with the desired property. □

Corollary 2.4. Suppose A is split by a Galois extension l/k and ψ is an irreducible representation of A. Then dim ψ divides [l : k].

Proof. By our construction Γ = Gal(l/k) permutes the l-basis Δ of V_{l}. Since V is k-irreducible, this permutation action is transitive. Hence, |Δ| = dim ψ divides |Γ| = [l : k]. □

Now consider the k-torus T := Diag(ℤ[Δ]), which is split over l and quasi-split over k. By our construction T is equipped with a representation

ι: T ↪ GL(V) .

In the basis Δ of V_{l}, this representation is given by ι(t) · e_{j}^{χ} = χ(t)e_{j}^{χ}. Note that by Galois descent, ι is defined over k. One easily checks that ι is generically free (this can be done over l).

We also remark that the original representation ψ: A → GL(V) can be written as a composition ψ = ι ∘ ψ̂, where ψ̂: A → T is induced by the map ℤ[Δ] → X(A) of Γ-modules, sending e_{j}^{χ} to χ.

Lemma 2.5. Every faithful representation ψ: A → GL(V) of A is generically free.

Proof. As we saw above, ψ = ι ∘ ψ̂, where ι: T → GL(V) is generically free. If ψ is faithful then ψ̂: A → T is faithful, and hence, ψ is generically free. □

Lemma 2.6. Let N be a closed subgroup of A, l/k be a Galois splitting field of A and Γ = Gal(l/k). Then

min dim ψ = min rank(P)

where the minimum on the left hand side is taken over all k-representations ψ of A with kernel N, and the minimum on the right is taken over all homomorphisms f: P → X(A) of ℤΓ-modules, with P permutation and cokernel(f) = X(N).

Proof. Given ψ: A → GL(V) with kernel N, write ψ: A →^{ψ̂} T ↪^{ι} GL(V) as above, where T is a quasi-split k-torus of dimension dim T = rank X(T) = dim ψ which splits over l. Then ker ψ̂ = N and the cokernel of the induced map X(ψ̂): X(T) → X(A) of ℤΓ-modules is X(N).

Conversely, if P is a permutation ℤΓ-module then we can embed the torus Diag(P) in GL_{n}, where n = rk P [Vos_{2}, Section 6.1]. A map f: P → X(A) of ℤΓ-modules with cokernel X(N) then yields a representation A → Diag(P) ↪ GL_{n} with kernel N. □

3. A lower bound on essential p-dimension

Consider an exact sequence of algebraic groups over k

(3)  1 → C → G → Q → 1

such that C is central in G and isomorphic to μ_{p}^{r} for some r ≥ 0. Given a character χ: C → μ_{p}, we will, following [KM], denote by Rep^{χ} the set of irreducible representations ϕ: G → GL(V), such that ϕ(c) = χ(c) Id for every c ∈ C.

Theorem 3.1. Suppose a sequence of k-groups of the form (3) satisfies the following condition:

gcd{dim(ϕ) | ϕ ∈ Rep^{χ}} = min{dim(ϕ) | ϕ ∈ Rep^{χ}}

for every character χ: C → μ_{p}. Then

ed(G; p) ≥ min dim(ψ) − dim G ,

where the minimum is taken over all finite-dimensional representations ψ of G such that ψ_{| C} is faithful.

Proof. Denote by C^{∗} := Hom(C, μ_{p}) the character group of C. Let V be a generically free Q-module, and U ⊆ V an open dense Q-invariant subvariety such that U → U/Q is a Q-torsor. Then let E → Spec K be the generic fibre of this torsor, and let β: C^{∗} → Br_{p}(K) denote the homomorphism that sends χ ∈ C^{∗} to the image of E ∈ H^{1}(K, Q) in Br_{p}(K) under the map

H^{1}(K, Q) → H^{2}(K, C) →^{χ_{∗}} H^{2}(K, μ_{p}) = Br_{p}(K)

given by composing the connecting map with χ_{∗}. Then there exists a basis χ_{1}, …, χ_{r} of C^{∗} such that

(4)  ed(G; p) ≥ ∑_{i=1}^{r} ind β(χ_{i}) − dim G,

see [Me_{1}, Theorem 4.8, Example 3.7]. Moreover, by [KM, Theorem 4.4, Remark 4.5]

ind β(χ_{i}) = gcd dim(ψ) ,

where the greatest common divisor is taken over all (finite-dimensional) representations ψ of G such that ψ_{| C} is scalar multiplication by χ_{i}. By our assumption, gcd can be replaced by min. Hence, for each i ∈ {1, …, r} we can choose a representation ψ_{i} of G with

ind β(χ_{i}) = dim(ψ_{i})

such that (ψ_{i})_{| C} is scalar multiplication by χ_{i}.

Set ψ := ψ_{1} ⊕ ⋯ ⊕ ψ_{r}. The inequality (4) can be written as

(5)  ed(G; p) ≥ dim(ψ) − dim G.

Since χ_{1}, …, χ_{r} form a basis of C^{∗} the restriction of ψ to C is faithful. This proves the theorem. □

4. Proof of the main result

The following lemma generalizes [MR, Lemma 4.1].

Lemma 4.1. Let A be an algebraic group of multiplicative type over a field k, and let B ⊂ A a closed subgroup of (finite) index prime to p. Then ed(A; p) = ed(B; p).

Proof. The inequality ed(B; p) ≤ ed(A; p) is clear, since dim A = dim B; see [Me_{1}, Corollary 4.3].

To prove the opposite inequality, set Q := A/B. In view of the exact sequence H^{1}(K, B) → H^{1}(K, A) → H^{1}(K, Q) it suffices to show that every Q-torsor X → Spec(K) splits over a finite prime to p extension of K. (Here K is assumed to be an arbitrary field extension of k.)

First suppose char k = p. In this case X is étale over Spec(K) (since Q is étale over Spec(K), see [Wa, 14.4]). The proof now proceeds as in [MR, Lemma 4.1]. That is, X is K-isomorphic to a direct product Spec(K_{1} ×⋯× K_{n}), where each K_{i}/K is a finite separable field extension. One of the fields K_{i} has degree prime to p over K, and we get a K_{i}-point of X from the map Spec(K_{i}) → X, induced by the projection K[X] → K_{i}. This implies that X splits over K_{i}.

Now suppose char k ≠ p. By [EKM, Prop 101.16] there exists an algebraic field extension K^{(p)}/K such that every finite extension of K^{(p)} has degree a power of p and every finite sub-extension L/K of K^{(p)}/K has degree prime to p. It is easy to see that K^{(p)} is a perfect field and Γ = Gal(K_{alg}/K^{(p)}) is a profinite p-group. Since Q(K_{alg}) has order prime to p the group H^{1}(K^{(p)}, Q) = H^{1}(Γ, Q(K_{alg})) is trivial by [Se_{2}, I.5, ex. 2]. Thus X splits over K^{(p)} and hence over a finite sub-extension L/K of K^{(p)}/K. □

Proposition 4.2. Let G be an algebraic group of multiplicative type over k, T its maximal k-torus, and l/k a minimal Galois splitting field of T. Let N ⊂ G be a finite k-subgroup whose order is coprime to both [l : k] and |G/T |. Let π: G → G/N be the natural projection. Then

π_{∗}: H^{1}(K, G) → H^{1}(K, G/N)

is bijective, for any field extension K/k. In particular, ed(G) = ed(G/N).

The following argument, simplifying our earlier proof, was suggested to us by Merkurjev.

Proof. We claim that H^{1}(K, G) is m-torsion, where m = [l : k] · |G/T |. Indeed, since T_{K} is split by a Galois extension of degree dividing [l : k], restricting and corestricting in Galois cohomology yields [l : k] · H^{1}(K, T) = (0). On the other hand, since |G/T | · H^{1}(K, G/T) = (0), the exact sequence

H^{1}(K, T) → H^{1}(K, G) → H^{1}(K, G/T)

shows that H^{1}(K, G) is m-torsion, as claimed. Note that N is contained in T and the quotient of G/N by its maximal torus T/N is isomorphic to G/T. So the group H^{1}(K, G/N) is m-torsion as well.

Now let n = |N| and p_{n}: G → G be given by g → g^{n}. The induced

map H^{1}(K, G) ⟶^{(p_{n})_{∗}} H^{1}(K, G) is multiplication by n. Since H^{1}(K, G) is m-torsion and by assumption n and m coprime, (p_{n})_{∗} is an isomorphism. Moreover, N lies in the kernel of p_{n} and so (p_{n})_{∗} factors through π_{∗}:

(p_{n})_{∗}: H^{1}(K, G) ⟶^{π_{∗}} H^{1}(K, G/N) → H^{1}(K, G) .

In particular, π_{∗} is injective. A similar argument shows that composing these maps in the opposite order,

H^{1}(K, G/N) → H^{1}(K, G) ⟶^{π_{∗}} H^{1}(K, G/N) ,

we get an isomorphism as well. This shows that π_{∗} is surjective and hence, bijective, as desired. □

Proof of the Theorem 1.1. We will first prove ed(G; p) ≥ min dim(ψ)−dim G, where the minimum is over p-faithful representations. Since G is split by a Galois extension of p-power degree, Corollary 2.4 tells us that for any character χ of C(G) and any ϕ ∈ Rep^{χ}, dim(ϕ) is a power of p. By Theorem 3.1, ed(G; p) ≥ min dim(ψ) − dim G, where ψ ranges over representations of G whose restriction to C(G) is faithful. By Lemma 2.2 representations with this property are precisely the p-faithful representations.

We will now show that ed(G; p) ≤ dim ψ − dim G for any p-faithful representation ψ of G. We will proceed in two steps.

Step 1. Suppose G is an extension of a p-group F by a torus T. Since N := ker ψ is finite of order prime to p, Proposition 4.2 yields ed(G) = ed(G/N). Now ψ can be considered as a faithful representation of G/N. By Lemma 2.5, this representation of G/N is generically free. By (2),

ed(G; p) ≤ ed(G) = ed(G/N) ≤ dim ψ − dim(G/N) = dim ψ − dim(G) ,

as desired.

Taking ψ to be of minimal dimension, we also see that in this case we have ed(G; p) = ed(G), as asserted in the statement of the theorem.

Step 2. Let G be an arbitrary group of multiplicative type. Let T be the maximal torus of G, and F′ be the Sylow p-subgroup of the multiplicative finite group F := G/T. Recall that F′ is defined as Diag(X(F)/Y), where Y is the submodule of elements of order prime to p.

Now denote the preimage of F′ under the projection G → F = G/T by G′. Since G′ is an extension of a p-group by a torus, we know from Step 1 that

ed(G′; p) ≤ dim ψ|_{G′} − dim G′ = dim ψ − dim G .

The index of G′ in G is finite and prime to p, hence ed(G; p) = ed(G′; p) by Lemma 4.1 and the desired inequality, ed(G; p) ≤ dim ψ−dim G follows. □

5. Main theorem in the language of character modules

Let G be of multiplicative type over k and let l/k be a Galois splitting field of G. We will call a map of ℤ Gal(l/k)-modules P → X(G) a p-presentation if P is permutation, and the cokernel is finite of order prime to p.

We now restate our Theorem 1.1 in a way that is often more convenient to use.

Corollary 5.1. Let G be a group of multiplicative over k, l/k be a finite Galois splitting field of G, and Γ_{p} be a Sylow p-subgroup of Gal(l/k). Then

ed(G; p) = min rk ker ϕ,

where the minimum is taken over all p-presentations ϕ: P → X(G) of X(G), viewed as a ℤΓ_{p}-module.

Proof. Let k′ = l^{Γ_{p}}. Then Gal(l/k′) = Γ_{p}. Since [k′ : k] is finite and prime to p, (1) tells us that ed(G; p) = ed(G_{k′}; p). By Theorem 1.1 ed(G_{k′}; p) = min dim(ψ) − dim G, where the minimum is taken over all p-faithful representations ψ of G_{k′}. By Lemma 2.6

min dim(ψ) − dim G = min rank(P) − dim G = min rk ker ϕ,

where the minimum on the right is taken over all p-presentations ϕ: P → X(G), as in the statement of the theorem. □

Example 5.2. Let T be a torus of dimension < p − 1. Then ed(T; p) = 0, because there is no non-trivial integral representation of dimension < p − 1 of any p-group [AP, Satz].

Example 5.3. Assume char k = 0, and let Γ = S_{p^{r}} denote the symmetric group for some r ≥ 1. The generic torus T of PGL_{n}, defined in [Vos_{2}, §4.1– 4.2], is of dimension p^{r} − 1 and has character lattice

X(T) = {a ∈ ℤ^{p^{r}}|a_{1} + ⋯ + a_{p^{r}} = 0}

with the natural action of Γ on it; see [Vos_{1}]. Let Γ_{p} be a Sylow p-subgroup of Γ. In [MR, Prop. 7.2] it is shown that the minimal rank of a permutation module with a p-presentation to X(T) is p^{2r−1}. Thus by Corollary 5.1, ed(T; p) = p^{2r−1} − p^{r} + 1.

6. Forms of μ_{n}

Proposition 6.1. Let A be a twisted form of μ_{p^{n}} over k and l/k a minimal Galois splitting field. Then ed(A; p) = p^{r}, where p^{r} is the highest power of p dividing [l : k].

Proof. Let Γ_{p} be a Sylow p-subgroup of Gal(l/k) and ϕ: P → X(A) be a ppresentation. Since ϕ has prime to p cokernel and X(A) is a cyclic p-group, ϕ must be surjective. Thus, if Λ is a basis of P, permuted by Γ_{p}, some element λ ∈ Λ maps to a generator a of X(A). Moreover, Γ_{p} acts faithfully on X(A) and |Λ| ≥ |Γ_{p}λ| ≥ |Γ_{p}a| = |Γ_{p}|. Conversely we have a surjective homomorphism ℤ[Γ_{p}a] → X(A) that sends a to itself. So the minimal value of rk P is |Γ_{p}|. Now apply Corollary 5.1. □

Remark 6.2. For char k ≠ p, Proposition 6.1 was previously known in the following special cases:

For twisted cyclic groups of order 4 it is due to M. Rost [Ro] and in the case of cyclic groups of order 8 to G. Bayarmagnai [Ba]. The case of constant cyclic groups of arbitrary prime power order is due to M. Florence [Fl].

Example 6.3. Let char k = p. D. Tossici and A. Vistoli [TV, Question 4.1 (2)] asked if the essential dimension of every algebraic k-group of order p^{n} is ≤ n. The following example, with n = 2 and p > 2, answers this question in the negative.

Let l/k be a cyclic extension of order p; set Γ := Gal(l/k). (For example, we can take k and l to be finite fields of orders p and p^{p}, respectively.) Now let M ≃ ℤ/p^{2}ℤ be the Γ-module obtained by identifying Γ with the unique subgroup of Aut(ℤ/p^{2}ℤ) ≃ ℤ/p(p − 1)ℤ of order p. By construction G = Diag(M) is a form of μ_{p^{2}} defined over k, whose minimal Galois splitting field is l. Proposition 6.1 now tells us that ed(G) = ed(G; p) = [l : k] = p > 2. □

7. Twisted p-groups

In this section we will use Theorem 3.1 to generalize the Karpenko– Merkurjev theorem to arbitrary (possibly twisted) finite p-groups over a field k, assuming that char k ≠ p and k contains a primitive pth root of unity.

Theorem 7.1. Let G be an algebraic group over k such that G_{L} is a constant group of order p^{n} for some n ≥ 1 and some Galois extension L/k of p-power degree. Then

ed(G) = ed(G; p) = min dim ψ ,

where ψ runs through all faithful representations of G.

Proof. The inequalities ed(G; p) ≤ ed(G) ≤ min dim ψ follow from (2). Hence it suffices to show that ed(G; p) ≥ min dim ψ.

Since char k ≠ p the centre of G is of multiplicative type, the subgroup C(G) = Split_{k}(Z(G)[p]) is well-defined (as in Section 2) and is isomorphic to μ_{p}^{r} for some r ≥ 1.

We claim that every irreducible representation ψ of G has dimension equal to a power of p. Denote by ζ a primitive root of unity of order equal to the exponent of G(L). Since k contains a primitive pth root of unity, L′ := L(ζ) is Galois over k and of p-power degree, and ψ decomposes over L′ as a direct sum of absolutely irreducible representations of the abstract p-group G(L′) = G(L). All direct summands in this decomposition have the same dimension, equal to a power of p. By [Ka, Theorem 5.22] the number of direct summands in this decomposition is also a power of p, and the claim follows.

Therefore, Theorem 3.1 can be applied, i.e., ed(G; p) ≥ min dim ψ taken over all representations ψ of G whose restriction to C(G) is faithful. Let N be the kernel of such a representation. We claim that N ∩ C(G) = {1} implies that N is trivial. If G is constant we have C(G) = Z(G)[p] since k contains a primitive pth root of unity and the claim is a standard elementary fact about p-groups. The general case follows from Lemma 2.1 applied to A = Z(G)[p] ∩ N. □

Remark 7.2. Theorem 7.1 allows one to compute ed(G; p), at least in principle, for any étale algebraic group G over k, provided char(k) ≠ p.

To carry out this computation, we first pass to a suitable Galois extension L/k of degree prime to p such that L contains a primitive pth root of unity and G_{L} becomes constant over a Galois extension E/L of p-power degree.

We claim that G_{L} has a Sylow p-subgroup S defined over L. Indeed, the p-group Gal(E/L) permutes the Sylow subgroups of G(E). By the Sylow theorems, the number of such subgroups is prime to p. Thus one of them is fixed by the p-group Gal(E/L). This proves the claim.

Now we have ed(G; p) = ed(G_{L}; p) = ed(S; p), and ed(S; p) is given by Theorem 7.1.

Acknowledgments

The authors are grateful to A. S. Merkurjev for numerous constructive comments on earlier versions of this paper, and to A. Auel and A. Vistoli for helpful discussions.

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