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.

Contents

1. Introduction 2 2. Proof of Theorem 1.2 6 3. The p-closure of a field 7 4. The group C(G) 9 5. Proof of Theorem 1.3(a) 11 6. p-isogenies 14 7. Proof of Theorem 1.3(b) 16 8. An additivity theorem 17 9. Modules and lattices 18 10. Proof of Theorem 1.3(c) 20 11. Tori of essential dimension ≤ 1 22 12. Tori split by cyclic extensions of degree dividing p^{2} 25 Acknowledgments 28 References 28

2000 Mathematics Subject Classification. 20G15. Key words and phrases. Essential dimension, algebraic torus, twisted finite group,

lattice.

(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. Introduction

Throughout this paper p will denote a prime integer, k a base field of characteristic ≠ 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 Fields_{k} be the category of field extensions K/k, Sets be the category of sets, and F: Fields_{k} → Sets be a covariant functor. As usual, given a field extension k ⊂ K_{0} ⊂ K, we will denote the image of α ∈ F (K) under the natural map F (K) → F (L) by α_{L}.

An object α ∈ F (K) is said to descend to an intermediate field k ⊆ K_{0} ⊆ K if α is in the image of the induced map F (K_{0}) → F (K). The essential dimension ed_{k}(α) is defined as the minimum of the transcendence degrees trdeg_{k}(K) taken over all fields k ⊆ K_{0} ⊆ K such that α descends to K_{0}. The essential dimension ed_{k}(F) of the functor F is defined as the maximal value of ed_{k}(α), where the maximum is taken over all fields K/k and all α ∈ F (K).

Of particular interest to us will be the Galois cohomology functor F_{G} := H^{1}(∗, G), which associates to every K/k the set of isomorphism classes of G-torsors over Spec(K). The essential dimension of this functor is usually called the essential dimension of G and is denoted by the symbol 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 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 ℂ of complex numbers. The following related notion is often more accessible. Let F: Fields_{k} → Sets be a covariant functor and p be a prime integer, as above. The essential p-dimension of α ∈ F (K), denoted ed_{k}(α; p), is defined as the minimal value of ed_{k}(α_{K′}), where K′ ranges over all finite field extensions of K whose degree is prime to p. The essential p-dimension of F, ed_{k}(F; p) of F is once again, defined as the maximal value of ed_{k}(α; p), where the maximum is taken over all fields K/k and all α ∈ F (K), and once again we will write ed_{k}(G; p) in place of ed_{k}(F_{G}; p), where F_{G} := H^{1}(∗, G) is the Galois cohomology functor.

Note that ed_{k}(α), ed_{k}(F), ed_{k}(G), ed_{k}(α; p), etc., depend on k. We will write ed instead of 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, Me_{1}].

We also remark that in the case of the Galois cohomology functor F_{G}, the maximal value of ed_{k}(α) and ed_{k}(α; p) in the above definitions is attained in the case where α is a versal G-torsor in the sense of [GMS, Section I.5].

Since every generically free linear representation ρ: G → GL(V) gives rise to a versal G-torsor (see [GMS, Example I.5.4]), we obtain the inequality

(1)  ed_{k}(G; p) ≤ ed_{k}(G) ≤ dim(V) − dim(G) ;

see [Re, Therem 3.4] or [BF, Lemma 4.11]. (Recall that ρ is called generically free if there exists a G-invariant dense open subset U ⊂ 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 pth root of unity. Then

ed_{k}(G; p) = ed_{k}(G) = min dim(V) ,

where the minimum is taken over all faithful k-representations G ↪ 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 → C → G → Q → 1

be an exact sequence of algebraic groups over k 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), defined over k, such that ϕ(c) = χ(c) Id_{V} for every c ∈ C.

Theorem 1.2. Assume that k is a field of characteristic ≠ p containing a primitive pth root of unity. Suppose a sequence of k-groups of the form (2) satisfies the following condition:

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

for every character χ: C → μ_{p}. (Here, as usual, gcd stands for the greatest common divisor.) Then

ed_{k}(G; p) ≥ min dim(ρ) − dim G ,

where the minimum is taken over all finite-dimensional k-representations ρ of G such that ρ_{| 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 → T → G → F → 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 nontrivially on the separable points of G. For the sake of computing 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 ⊂ 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 ρ: G → 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 → T → G → F → 1 .

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

(a) ed_{k}(G; p) ≥ min dim(ρ)−dim G, where the minimum is taken over all p-faithful linear representations ρ 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 pdimension coincide:

ed_{k^{(p)}}(G_{k^{(p)}}) = ed_{k^{(p)}}(G_{k^{(p)}}; p) = ed_{k}(G; p).

If G is a p-group, a representation ρ 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 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 ≠ p. Suppose Γ = Gal(k_{sep}/k) acts on the character lattice X(T) via a finite quotient \overline{Γ}. Then

ed_{k}(T) = ed_{k}(T; p) = min rank(L) ,

where the minimum is taken over all exact sequences of ℤ_{(p)}\overline{Γ}-lattices of the form

(0) → L → P → X(T)_{(p)} → (0) ,

where P is permutation and X(T)_{(p)} stands for X(T) ⊗_{ℤ} ℤ_{(p)}.

In many cases Theorem 1.4 renders the value of 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 [Me_{2}], to show that

ed_{k}(PGL_{p^{r}}; p) ≥ (r − 1)p^{r} + 1

for any r ≥ 1. (For r = 2 the above inequality is the main result of [Me_{2}].) This represents dramatic improvement over the best previously known lower bounds on ed_{k}(PGL_{p^{r}}). The question of computing ed_{k}(PGL_{p^{r}}) is a longstanding open problem; for an overview, see [MR_{1}, MR_{2}].

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 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 μ − dim(G) ≤ ed(G; p) ≤ min dim ρ − dim G ,

where the two minima are taken over all p-faithful representations μ, and p-generically free representations ρ, respectively. Here we say that a representation ρ of G is p-generically free if the ker(ρ) is finite of order prime to p, and ρ descends to a generically free representation of G/ ker(ρ). 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 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 ≠ p. Then

ed(G; p) = min dim ρ − dim G,

where the minimum is taken over all p-generically free representations ρ 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 → Q is a p-isogeny then ed_{k}(G; p) = 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

ed_{k}(G_{1} × G_{2}; p) = ed_{k}(G_{1}; p) + ed_{k}(G_{2}; p) .

In Section 9 we restate and amplify Theorem 1.3(b) (with G abelian) in terms of Gal(k_{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^{∗} := Hom(C, μ_{p}) the character group of C. Let E → Spec K be a versal Q-torsor [GMS, Example 5.4], where K/k is some field extension, 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

(5)  ed_{k}(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(ρ) ,

where the greatest common divisor is taken over all (finite-dimensional) representations ρ of G such that ρ_{| 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 ρ_{i} of G with

ind β(χ_{i}) = dim(ρ_{i})

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

Set ρ := ρ_{1} ⊕ ⋯ ⊕ ρ_{r}. The inequality (5) can be written as

(6)  ed_{k}(G; p) ≥ dim(ρ) − dim G.

Since χ_{1}, …, χ_{r} forms a basis of C^{∗} the restriction of ρ 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_{alg} an algebraic closure. We will construct a field K^{(p)}/K in K_{alg} with all finite subextensions of K^{(p)}/K of degree prime to p and all finite subextensions of K_{alg}/K^{(p)} of degree a power of p.

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

Definition 3.1. We call the field

K^{(p)} = {a ∈ K_{alg}|a is purely inseparable over K_{sep}^{Φ}}

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_{alg} only up to the choice of a Sylow-p subgroup Φ in Γ. The notion of being p-closed does not depend on this choice.

Proposition 3.2.

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

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

(c) The cohomological dimension of Ψ = Gal(K_{alg}/K^{(p)}) is cd_{q}(Ψ) = 0 for any prime q ≠ p.

Proof. (a) First note that K_{sep} is the limit of the directed set {K_{sep}^{N}} over all normal subgroups N ⊂ Γ of finite index. Let

L = {K_{sep}^{NΦ}|N normal with finite index in Γ}.

This is a directed set, and since Φ is Sylow, the index of NΦ in Γ is prime to p. Therefore L consists of finite separable extensions of K of degree prime to p. Moreover, K_{sep}^{Φ} is the direct limit of fields L in L.

If char k = 0, K^{(p)} = K_{sep}^{Φ} and we are done. Otherwise suppose char k = q ≠ p. Let

E = {E ⊂ K_{alg}|E/L finite and purely inseparable for some L ∈ L}.

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 E forms a directed set.

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

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

(c) See [Se_{2}, Cor. 2, I. 3]. □

We call a covariant functor F: Fields /k → Sets limit-preserving if for any directed system of fields {K_{i}}, F(lim_{→} K_{i}) = lim_{→} F(K_{i}). For example if

G is an algebraic group, the Galois cohomology functor H^{1}(∗, G) is limitpreserving; see [Ma, 2.1].

Lemma 3.3. Let F be limit-preserving and α ∈ F(K) an object. Denote the image of α in F(K^{(p)}) by α_{K^{(p)}}.

(a) ed_{k}(α; p) = ed_{k}(α_{K^{(p)}}; p) = ed_{k}(α_{K^{(p)}}).

(b) ed_{k}(F; p) = ed_{k^{(p)}}(F; p).

Proof. (a) The inequalities ed(α; p) ≥ ed(α_{K^{(p)}}; p) = ed(α_{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 ed(α; p) ≤ ed(α_{K^{(p)}}). If L/K is finite of degree prime to p,

(7)  ed(α; p) = ed(α_{L}; p),

cf. [Me_{1}, 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} ⊂ K^{(p)} and α_{K^{(p)}} comes from an element β ∈ F(K_{0}), so that β_{K^{(p)}} = α_{K^{(p)}}. Write K^{(p)} = lim L, where L is a direct system of finite prime to p extensions of K. Then K_{0} = lim L_{0} with L_{0} = {L ∩ K_{0}|L ∈ L} and by assumption on F, F(K_{0}) = lim_{L′∈L_{0}} F(L′). Thus there

is a field L′ = L ∩ K_{0} (L ∈ L) and γ ∈ F(L′) such that γ_{K_{0}} = β. Since α_{L} and γ_{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

ed(α; p) = ed(α_{L}; p) = ed(γ_{L}; p) ≤ ed(γ_{L}) ≤ ed(α_{K^{(p)}}).

(b) This follows immediately from (a), taking α of maximal essential pdimension. □

Proposition 3.4. Let F, G: Fields /k → Sets be limit-preserving functors and F → G a natural transformation. If the map

F(K) → G(K)

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

ed(F; p) = ed(G; p) (resp. ed(F; p) ≥ ed(G; p)).

Proof. Assume the maps are surjective. By Proposition 3.2(a), the natural transformation is p-surjective, in the terminology of [Me_{1}], so we can apply [Me_{1}, Prop. 1.5] to conclude ed(F; p) ≥ ed(G; p).

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

Assume that β comes from β_{0} ∈ G(K_{0}) for some field K_{0} ⊂ K. Any finite prime to p extension of K_{0} is isomorphic to a subfield of K (cf. [Me_{1}, 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 F(K_{0}) → G(K_{0}) and F(K) → G(K) are bijective. The unique element α_{0} ∈ F(K_{0}) which maps to β_{0} must therefore map to α under the natural restriction map. This shows that ed(α) ≤ ed(β). The other inequality always holds and the claim follows.

Taking α maximal with respect to its essential dimension, we obtain ed(F; p) = ed(α; p) = ed(β; p) ≤ ed(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_{sep}} is diagonalizable over the separable closure k_{sep} of k; cf., e.g., [Vo, Section 3.4]. Here, as usual, G_{k′} := G ×_{Spec k} Spec(k′) for any field extension k′/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 Gal(l_{sep}/l) acts trivially on X(A).

• We will write A[p] for the p-torsion subgroup {a ∈ 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 Δ(A) be the Γ-invariant subgroup of X(A) generated by elements of the form x − γ(x), as x ranges over X(A) and γ ranges over Γ. Define

Split_{k}(A) = Diag(X(A)/Δ(A)) .

Here Diag denotes the anti-equivalence between continuous ℤΓ-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) := 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. (a) Split_{k}(A) is split over k,

(b) Split_{k}(A) = A if and only if A is split over k,

(c) If B is a k-subgroup of A then Split_{k}(B) ⊂ Split_{k}(A).

(d) For A = A_{1} × A_{2}, Split_{k}(A_{1} × A_{2}) = Split_{k}(A_{1}) × Split_{k}(A_{2}),

(e) If A[p] ≠ {1} and A is split over a Galois extension l/k, such that \overline{Γ} = Gal(l/k) is a p-group, then Split_{k}(A) ≠ \{1\}.

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

Proof of (e): By part (c), 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 finitedimensional 𝔽_{p}-vector space on which the p-group \overline{Γ} 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 ∈ \overline{Γ}. Our goal is to show that Δ(A) ≠ X(A). Indeed, every element of the form x − γ(x) is contained in the Γinvariant submodule Span(e_{2}, …, 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 ∩ C(G) = {1},

(iii) N ∩ Z(G)[p] = {1},

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

Proof. (i) ⟹ (ii) is obvious, since C(G) is a p-group.

(ii) ⟹ (iii). Assume the contrary: A := N∩Z(G)[p] ≠ {1}. By Lemma 4.3

{1} ≠ C(A) ⊂ N ∩ C(Z(G)[p]) = N ∩ C(G) ,

contradicting (ii).

Our proof of the implication (iii) ⟹ (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 ∩ Z(G)[p] ≠ {1}.

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

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

(8)  1 → N ∩ T → N → \overline{N} → 1 ,

where \overline{N} is the image of N in G/T, shows that N is finite. Now observe that for every r ≥ 1, the commutator (N, T [p^{r}]) is a p-subgroup of N ∩ T. Thus (N, T [p^{r}]) = {1} for every r ≥ 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: N × T → G sending (n, t) to the commutator ntn^{−1}t^{−1} descends to \overline{c}: \overline{N} ×T → G (indeed, N ∩T clearly commutes with T). Since |\overline{N}| is a power of p and char(k) ≠ p, \overline{N} is smooth over k, and we can pass to the separable closure k_{sep} and apply the usual Zariski density argument to show that the image of \overline{c} is trivial.

We thus conclude that N ∩ T is central in N. Since gcd(|N ∩ T |, \overline{N}) = 1, by [Sch_{2}, Corollary 5.4] the extension (8) splits, i.e., N ≃ (N ∩ T) × \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 ∩ 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 ρ of G defined over k is p-faithful if and only ρ_{| 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 char(k) ≠ p, this implies that k contains a primitive pth root of unity. (Indeed, if ζ is a p-th root of unity in k_{sep} then d = [k(ζ) : 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 → C(G) → G → Q → 1 .

This yields

(10)  ed(G; p) ≥ min dim(ρ) − dim(G) ,

where the minimum is taken over all representations ρ: G → GL(V) such that ρ_{|C(G)} is faithful. Corollary 4.4 now tells us that ρ_{|C(G)} is faithful if and only if ρ 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 := k[G]^{∗} 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 Amodule and let W ⊆ V ⊗_{k} l be an irreducible A ⊗_{k} l-submodule. Then by [Ka, Theorem 5.22] there exists a divisor e of [l : k] such that

V ⊗ l ≃ e (⨁_{i=1}^{r}^{σ_{i}}W ) ,

where σ_{i} ∈ Gal(l/k) and {^{σ_{i}}W | 1 ≤ i ≤ 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}(^{σ_{1}}W) = … = dim_{l}(^{σ_{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 ⊗ l = e(dim_{l} ^{σ_{1}}W + ⋯ + dim_{l} ^{σ_{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} ⊆ F_{2} ⊆ ⋯ ⊂ G

be an ascending sequence of finite k-subgroups whose union ∪_{n≥1}F_{n} is Zariski dense in G. If ρ: G → GL(V) is an irreducible representation of G defined over k then ρ_{| F_{i}} is irreducible for sufficiently large integers i.

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

X^{(d)} = ∩_{i≥0}X_{i}^{(d)} .

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

Since V does not have any G-invariant d-dimensional k-subspaces, we know that X^{(d)}(k) = ∅. Thus, X_{m_{d}}^{(d)}(k) = ∅, i.e., V does not have any F_{m_{d}}- invariant d-dimensional k-subspaces. Setting m := max{m_{1}, …, m_{dim(V)−1}}, we see that ρ_{| 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} ⊆ F_{2} ⊆ ⋯ ⊂ G

defined over k whose union ∪_{n≥1}F_{n} is Zariski dense in G.

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

The following lemma, which establishes the existence of F′, is thus the final step in our proof of Proposition 5.1 (and hence, of Theorem 1.3(a)).

Lemma 5.4. Let 1 → T → G ⟶^{π} F → 1 be an extension of a p-group F by a torus T over k. Then G has a finite p-subgroup F′ with π(F′) = 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 Ẽx^{1}(F, T) the group of equivalence classes of extensions of F by T. We claim that Ẽx^{1}(F, T) is torsion. Let Ex^{1}(F, T) ⊂ Ẽx^{1}(F, T) be the classes of extensions which have a scheme-theoretic section (i.e. G(K) → F (K) is surjective for all K/k). There is a natural isomorphism Ex^{1}(F, T) ≃ H^{2}(F, T), where the latter one denotes Hochschild cohomology, see [DG, III. 6.2, Proposition]. By [Sch_{3}] the usual restriction-corestriction arguments can be applied in Hochschild cohomology and in particular, m · H^{2}(F, T) = 0 where m is the order of F. Now recall that M ↦ Ẽx^{i}(F, M) and M ↦ Ex^{i}(F, M) are both derived functors of the crossed homomorphisms M ↦ 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 [Sch_{1}, Satz 1.2 & Satz 3.3] there is an exact sequence of F -module schemes 1 → T → M_{1} → M_{2} → 1 and an exact sequence Ex^{0}(F, M_{1}) → Ex^{0}(F, M_{2}) → Ẽx^{1}(F, T) → H^{2}(F, M_{1}) ≃ Ex^{1}(F, M_{1}). The F -module sequence also induces a long exact sequence on Ex(F, ∗) and we have a diagram

\xymatrix{ &  & Ẽx^{1}(F, T) \ar[dr] \\ Ex^{0}(F, M_{1}) \ar[r] & Ex^{0}(F, M_{2}) \ar[ur] \ar[dr] &  & Ex^{1}(F, M_{1}) \\  &  & Ex^{1}(F, T) \ar[ur] \ar@{^{(}->}[uu]}

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

Now let us consider the exact sequence 1 → N → T ⟶^{×m^{2}} T → 1, where N is the kernel of multiplication by m^{2}. Clearly N is finite and we have an induced exact sequence

Ẽx^{1}(F, N) → Ẽx^{1}(F, T) ⟶^{×m^{2}} Ẽx^{1}(F, T)

which shows that the given extension G comes from an extension F′ of F by N. Then G is the pushout of F′ by N → T and we can identify F′ with a subgroup of G. □

6. p-isogenies

An isogeny of algebraic groups is a surjective morphism G → 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 pisogenous 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 → Q is a p-isogeny of algebraic groups over k. Then

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

(b) ed_{k}(G; p) = 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 ≠ 2 and 3 respectively, then

ed_{k}(E_{6}^{sc}; 2) = 3 and ed_{k}(E_{7}^{sc}; 3) = 3.

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

ed_{k}(E_{6}^{ad}; 2) = 3 and ed_{k}(E_{7}^{ad}; 3) = 3.

We will need two lemmas.

Lemma 6.3. Let N be a finite algebraic group over k (char k ≠ p). The following are equivalent:

(a) p does not divide the order of N.

(b) p does not divide the order of N(k_{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.

(a′) N[p](k_{alg}) = {1}.

(b′) N[p](k^{(p)}) = {1}.

Proof. (a) ⟺ (b): Let N^{∘} be the connected component of N and N^{et} = N/N^{∘} 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^{∘}||N^{et}|, see for example [Ta]. If char k = 0, N^{∘} is trivial, if char k = q ≠ p is positive, |N^{∘}| 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^{∘} is connected and finite, N^{∘}(k_{alg}) = {1} and so N(k_{alg}) is of order prime to p if and only if the group N^{et}(k_{alg}) is. Then |N^{et}| = dim_{k} k[N^{et}] = |N^{et}(k_{alg})|, cf. [Bou, V.29 Corollary].

(b) ⟺ (a′) ⇒ (b′) are clear.

(a′) ⇐ (b′): Suppose N[p](k_{alg}) is nontrivial. The Galois group Γ = Gal(k_{alg}/k^{(p)}) is a pro-p group and acts on the p-group N[p](k_{alg}). The image of Γ in Aut(N[p](k_{alg})) is again a (finite) p-group and the size of every Γ-orbit in N[p](k_{alg}) is a power of p. Since Γ fixes the identity in N[p](k_{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)} ∩ k_{sep}) = {1}, but we won’t need this in the sequel.

Lemma 6.5. Let Γ be a profinite group, G an (abstract) finite Γ-group and |Γ|, |G| coprime. Then H^{1}(Γ, G) = {1}.

The case where Γ is finite and G abelian is classical. In the generality we stated, this lemma is also known [Se_{2}, I.5, ex. 2].

Proof of Theorem 6.1. (a) Let N be the kernel of G → Q and K = K^{(p)} be a p-closed field over k. Since K_{sep} = K_{alg} (see Proposition 3.2(b)), the sequence of K_{sep}-points 1 → N(K_{sep}) → G(K_{sep}) → Q(K_{sep}) → 1 is exact. By Lemma 6.3, the order of N(K_{sep}) is not divisible by p and therefore coprime to the order of Ψ = Gal(K_{sep}/K). Thus H^{1}(K, N) = {1} (Lemma 6.5). Similarly, if_{c}N is the group N twisted by a cocycle c : Ψ → G, cN(K_{sep}) = N(K_{sep}) is of order prime to p and H^{1}(K, _{c}N) = {1}. It follows that H^{1}(K, G) → H^{1}(K, Q) is injective, cf. [Se_{2}, I.5.5].

Surjectivity is a consequence of [Se_{2}, I. Proposition 46] and the fact that the q-cohomological dimension of Ψ is 0 for any divisor q of |N(K_{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 × F, where T is a torus and F is a finite p-group, defined over k. Our goal is to show that

(11)  ed_{k}(G; p) ≤ dim(ρ) − dim G ,

where ρ is a p-faithful representation of G defined over k.

Lemma 7.1. If a representation ρ: G → GL(V) is p-faithful, then G/ ker(ρ) → GL(V) is generically free. In other words, ρ is p-generically free.

Proof. Since ker(ρ) has order prime to p, its image under the projection map G = T × F → F is trivial. Hence ker(ρ) ⊂ T and T/N is again a torus. So without loss of generality, we may assume ρ is faithful.

Let V_{1} ⊊ V be a closed subset of V such that T acts freely on V \ 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 ≠ g ∈ T [n] × F. Here as usual, T [n] denotes the n-torsion subgroup of T. Since ρ is faithful none of these fixed point sets are all of V, hence U := V \ (V_{1} ∪ V_{2}) is a dense open subset of V.

We claim that Stab_{G}(v) = {1} for every v ∈ U. Indeed, assume 1 ≠ g = (t, f) ∈ Stab_{G}(v). Since v ∉ V_{2}, t^{n} ≠ 1. Then 1 ≠ g^{n} = (t^{n}, 1) lies in both T and Stab_{G}(v). Since v ∉ V_{1}, this is a contradiction. □

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

ed_{k}(G/N; p) ≤ dim(ρ) − dim(G/ ker(ρ)) = dim(ρ) − dim(G) .

By Theorem 6.1

ed_{k}(G; p) = ed(G/N; p) ≤ dim(ρ) − 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

ed_{k}(G; p) = ed_{k}(G_{p}; p) = ed_{k}(G_{p}) = min dim(ρ)

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

Proof. By assumption, Γ = Gal(k_{sep}/k) is a pro-p group. It acts on the set of Sylow-p subgroups of G(k_{sep}). Since the number of such subgroups is prime to p, Γ 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 ed_{k}(G; p) = ed_{k}(G_{p}; p) is shown in [MR_{1}, 4.1] (the reference is for smooth groups but can be generalized to the non-smooth case as well). The minimal G_{p}-representation ρ from Theorem 1.3(b) is faithful and thus ed_{k}(G_{p}) ≤ dim(ρ), 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 ed_{k}(G_{1} × G_{2}; p) = ed_{k}(G_{1}; p) + 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 ρ of G (defined over k) such that ρ_{| 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 μ_{p}^{r_{i}} over k for some r_{1}, r_{2} ≥ 0. Then

f(G_{1} × G_{1}; C_{1} × 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 π_{i}: G_{1} × G_{2} → G_{i} be the natural projection and ϵ_{i}: G_{i} → G_{1} × G_{2} be the natural inclusion.

If ρ_{i} is a d_{i}-dimensional k-representation of G_{i} whose restriction to C_{i} is faithful, then clearly ρ_{1} ∘ π_{1} ⊕ ρ_{2} ∘ π_{2} is a d_{1} + d_{2}-dimensional representation of G_{1} × G_{2} whose restriction to C_{1} × C_{2} is faithful. This shows that

f(G_{1} × G_{1}; C_{1} × C_{2}) ≤ f(G_{1}; C_{1}) + f(G_{2}; C_{2}) .

To prove the opposite inequality, let ρ: G_{1}×G_{2} → GL(V) be a k-representation such that ρ_{| C_{1}×C_{2}} is faithful, and of minimal dimension

d = f(G_{1} × G_{1}; C_{1} × C_{2})

with this property. Let ρ_{1}, ρ_{2}, …, ρ_{n} denote the irreducible decomposition factors in a decomposition series of ρ. Since C_{1} × C_{2} is central in G_{1} × G_{2}, each ρ_{i} restricts to a multiplicative character of C_{1}×C_{2} which we will denote by χ_{i}. Moreover since C_{1} × C_{2} ≃ μ_{p}^{r_{1}+r_{2}} is linearly reductive ρ_{| C_{1}×C_{2}} is a direct sum χ_{1}^{⊕d_{1}} ⊕ ⋯ ⊕ χ_{n}^{⊕d_{n}} where d_{i} = dim V_{i}. It is easy to see that the following conditions are equivalent:

(i) ρ_{| C_{1}×C_{2}} is faithful,

(ii) χ_{1}, …, χ_{n} generate (C_{1} × C_{2})^{∗} as an abelian group. In particular we may assume that ρ = ρ_{1} ⊕ ⋯ ⊕ ρ_{n}. Since C_{i} is isomorphic to μ_{p}^{r_{i}}, we will think of (C_{1} × C_{2})^{∗} as a 𝔽_{p}-vector space of dimension r_{1} + r_{2}. Since (i) ⇔ (ii) above, we know that χ_{1}, …, χ_{n} span (C_{1} × C_{2})^{∗}. In fact, they form a basis of (C_{1} × 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 ρ_{1} ⊕ ⋯ ⊕ ρ_{n}, so that the restriction of the resulting representation to C_{1} × C_{2} would still be faithful, contradicting the minimality of dim(ρ).

We claim that it is always possible to replace each ρ_{j} by ρ′_{j}, where ρ′_{j} is either ρ_{j} ∘ ϵ_{1} ∘ π_{1} or ρ_{j} ∘ ϵ_{2} ∘ π_{2} such that the restriction of the resulting representation ρ′ = ρ′_{1} ⊕⋯⊕ρ′_{n} to C_{1} ×C_{2} remains faithful. Since dim(ρ_{i}) = dim(ρ′_{i}), we see that dim(ρ′) = dim(ρ). Moreover, ρ′ will then be of the form α_{1} ∘ π_{1} ⊕ α_{2} ∘ π_{2}, where α_{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

f(G_{1} × G_{1}; C_{1} × C_{2}) = dim(ρ) = dim(ρ′) = dim(α_{1}) + dim(α_{2})
≥ f(G_{1}, C_{1}) + f(G_{2}, C_{2}) ,

as desired.

To prove the claim, we will define ρ′_{j} recursively for j = 1, …, n. Suppose ρ′_{1}, …, ρ′_{j−1} have already be defined, so that the restriction of

ρ′_{1} ⊕ ⋯ ⊕ ρ′_{j−1} ⊕ ρ_{j} ⋯ ⊕ ρ_{n}

to C_{1} × C_{2} is faithful. For notational simplicity, we will assume that ρ_{1} = ρ′_{1}, …, ρ_{j−1} = ρ′_{j−1}. Note that

χ_{j} = (χ_{j} ∘ ϵ_{1} ∘ π_{1}) + (χ_{j} ∘ ϵ_{2} ∘ π_{2}) .

Since χ_{1}, …, χ_{n} form a basis (C_{1}×C_{2})^{∗} as an 𝔽_{p}-vector space, we see that (a) χ_{j} ∘ϵ_{1}∘π_{1} or (b) χ_{j} ∘ϵ_{2}∘π_{2} does not lie in Span_{𝔽_{p}}(χ_{1}, …, χ_{j−1}, χ_{j+1}, …, χ_{n}). Set

ρ′_{j} := \begin{cases}ρ_{j} ∘ ϵ_{1} ∘ π_{1} in case (a), and \\ ρ_{j} ∘ ϵ_{2} ∘ π_{2}, otherwise. \end{cases}

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

ρ_{1} ⊕ ⋯ ⊕ ρ_{j−1} ⊕ ρ′_{j} ⊕ ρ_{j+1}, ⋯ ⊕ ρ_{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)

ed(G; p) = f(G, C(G)) − dim G;

cf. Corollary 4.5. Furthermore, we have C(G_{1} × G_{2}) = C(G_{1}) × 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 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 = ℤ and R = 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 = ℤΓ (Γ a group) this is as usual a free abelian group of finite rank with an action of Γ. Particular cases of RΓ-lattices are permutation lattices L = R[Λ] where Λ is a Γ-set.

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

Recall that ℤ_{(p)} denotes the localization of the ring ℤ at the prime ideal (p). For a ℤ-module M we also write M_{(p)} := ℤ_{(p)} ⊗ M.

When Γ = Gal(k_{sep}/k) we will often pass from ℤΓ-lattices to ℤ_{(p)}Γlattices. This corresponds to identifying p-isogeneous tori:

Lemma 9.1. Let Γ = Gal(k_{sep}/k) and let M, L be ℤΓ-lattices. Then the following statements are equivalent:

(a) L_{(p)} ≃ M_{(p)}.

(b) There exists an injective map ϕ: L → M of ℤΓ-modules with cokernel Q finite of order prime to p.

(c) There exists a p-isogeny Diag(M) → Diag(L).

Proof. The equivalence (b) ⇔ (c) is clear from the anti-equivalence of Diag.

The implication (b) ⇒ (a) follows from Q_{(p)} = 0 and that tensoring with ℤ_{(p)} is exact.

For the implication (a) ⇒ (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 α: L_{(p)} → M_{(p)} of ℤ_{(p)}Γ-modules lands in \frac{1}{m}M for some m ∈ ℕ (prime to p) and the index of α(L) in \frac{1}{m}M is finite and prime to p if α is surjective. Since \frac{1}{m}M ≃ M as ℤΓ-modules the claim follows. □

Corollary 9.2. Let G be an abelian group which is an extension of a pgroup by a torus over k and Γ := Gal(k_{sep}/k) be the absolute Galois group of k = k^{(p)}. Let Γ act through a finite quotient \overline{Γ} on X(G). Then

ed_{k}(G; p) = min rk L − dim G ,

where the minimum is taken over all permutation ℤ\overline{Γ}-lattices L which admit a map of ℤ\overline{Γ}-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 ℤ_{(p)}\overline{Γ}-lattices L which admit a surjective map of ℤ_{(p)}\overline{Γ}-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 ℤ\overline{Γ}-module L which admits a map to X(G) with cokernel finite of order prime to p.

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

Conversely let ρ: G → GL(V) be a p-faithful representation of G. Since G_{sep} is diagonalizable, there exist characters χ_{1}, …, χ_{n} ∈ X(G) such that G acts on V_{sep} via diagonal matrices with entries χ_{1}(g), …, χ_{n}(g) (for g ∈ G) with respect to a suitable basis of V_{sep}. Moreover \overline{Γ} permutes the set Λ := \{χ_{1}, …, χ_{n}\}. Define a map ϕ: ℤ[Λ] → X(G) of ℤ\overline{Γ}-modules by sending the basis element χ_{i} ∈ Λ of L := ℤ[Λ] to itself. Then the p-faithfulness of ρ implies that the cokernel of ϕ is finite and of order prime to p. Moreover rk L = |Λ| ≤ n = dim V.

Now consider the case where G is a torus. Assume we have a surjective map α: L → X(G)_{(p)} of ℤ_{(p)}\overline{Γ}-modules where L = ℤ_{(p)}[Λ] is permutation, Λ a \overline{Γ}-set. Then α(Λ) ⊆ \frac{1}{m}X(G) for some m ∈ 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 ℤ[Λ] → \frac{1}{m}X(G) ≃ 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 Γ := Gal(l/k). Then

ed(A; p) = |Γ|.

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 ed(A; p) is equal to the least cardinality of a Γ-set Λ such that there exists a map ϕ: ℤ[Λ] → X(A) of ℤΓ-modules with cokernel finite of order prime to p. The group X(A) is a (cyclic) p-group, hence ϕ must be surjective. Moreover Γ acts faithfully on X(A). Surjectivity of ϕ implies that some element λ ∈ Λ maps to a generator a of X(A). Hence |Λ| ≥ |Γλ| ≥ |Γa| = |Γ|. Conversely we have a surjective homomorphism ℤ[Γa] → 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{Γ} be a finite group. Two ℤ\overline{Γ}-lattices M, N are said to be in the same genus if M_{(p)} ≃ N_{(p)} for all primes p, cf. [CR, 31A]. It is sufficient to check this condition for divisors p of the order of \overline{Γ}. 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 ℤ\overline{Γ}-lattice L in the genus of the free ℤ\overline{Γ}-lattice of rank one such that M ⊕ ℤ\overline{Γ} ≃ N ⊕ L. This has the following consequence for essential dimension:

Proposition 10.1. Let T, T′ be k-tori. If the lattices X(T), X(T′) belong to the same genus then

ed_{k}(T) = ed_{k}(T) and ed_{k}(T; ℓ) = ed_{k}(T; ℓ) for all primes ℓ.

Proof. Let Gal(k_{sep}/k) act through a finite quotient \overline{Γ} on X(T) and X(T′). By assumption there exists a ℤ\overline{Γ}-lattice L in the genus of ℤ\overline{Γ} such that X(T) ⊕ ℤ\overline{Γ} ≃ X(T′) ⊕ L. The torus S = Diag(ℤ\overline{Γ}) has a generically free representation of dimension dim S, hence ed_{k}(S) = 0. Since L is a direct summand of ℤ\overline{Γ} ⊕ ℤ\overline{Γ} the torus S′ := Diag(L) has ed_{k}(S′) ≤ ed_{k}(S × S) ≤ 0 as well, where the first inequality follows from [BF, Remarks 1.16 (b)]. Therefore

ed_{k}(T) ≤ ed_{k}(T × S) = ed_{k}(T′ × S′) ≤ ed_{k}(T′) + ed_{k}(S′) = ed_{k}(T′)

and similarly ed_{k}(T′) ≤ ed_{k}(T). Hence ed_{k}(T) = ed_{k}(T′).

A similar argument shows that ed_{k}(T; ℓ) = ed_{k}(T′; ℓ) for any prime ℓ. This concludes the proof. □

Corollary 10.2. Let k = k^{(p)} be a p-closed field and T a k-torus. Then

ed_{k}(T) = ed_{k}(T; p) = min dim(ρ) − 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 ed_{k}(T; p) ≤ ed_{k}(T) is clear. Hence it suffices to show ed_{k}(T) ≤ ed_{k}(T; p). Let ρ: T → GL(V) be a p-faithful representation of minimal dimension so that ed_{k}(T; p) = dim ρ − dim T. The representation ρ can be considered as a faithful representation of the torus T′ = T/N where N := ker ρ is finite of order prime to p. By construction the character lattices X(T) and X(T′) are isomorphic after localization at p. Since Gal(k_{sep}/k) is a (profinite) pgroup it follows that X(T) and X(T′) belong to the same genus. Hence by Proposition 10.1 we have ed_{k}(T′) = ed_{k}(T). Moreover ed_{k}(T′) ≤ dim ρ − dim T′, since ρ is a generically free representation of T′. This finishes the proof. □

Proof of Theorem 1.3(b). The equality ed_{k^{(p)}}(G_{k^{(p)}}; p) = ed_{k}(G; p) follows from Lemma 3.3. Now we are assuming G = T ×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 ed_{k}(F; p) = ed_{k}(F). Combining this with Corollary 10.2 and the additivity Theorem 8.1, we see

ed(T ×F) ≤ ed(T)+ed(F) = ed(T; p)+ed(F; p) = ed(T ×F; p) ≤ ed(T ×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 ℚ(ζ_{p}) is nontrivial (this applies to all but finitely many primes, e.g. to p = 23). This means that the subring R = ℤ[ζ_{p}] ⊆ ℚ(ζ_{p}) of algebraic integers has nonprincipal ideals. Let k be a field which admits a Galois extension l of degree p and let Γ := Gal(k_{sep}/k), \overline{Γ} := Gal(l/k) ≃ Γ/Γ_{l} ≃ C_{p} where Γ_{l} = Gal(k_{sep}/l) and C_{p} denotes the cyclic group of order p.

We endow the ring R with a ℤΓ-module structure through the quotient map Γ → \overline{Γ} by letting a generator of \overline{Γ} act on R via multiplication by ζ_{p}. The k-torus Q := Diag(R) is isomorphic to the Weil restriction R_{l/k}(𝔾_{m}) and has a p-dimensional faithful representation. We will construct a k-torus G with a p-isogeny G → 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 ℤΓ-module and set G := Diag(I). We first show that I and R become isomorphic as ℤΓ-modules after localization at p. For this purpose let I^{∗} = {x ∈ ℚ(ζ_{p}) | xI ⊆ R} denote the inverse fractional ideal. We have I ⊕ I^{∗} ≃ R ⊕ R by [CR, Theorem 34.31]. The Krull-Schmidt Theorem [CR, Theorem 36.1] for ℤ_{(p)}C_{p}-lattices implies I_{(p)} ≃ R_{(p)}, hence the claim. Therefore by Lemma 9.1 there exists a p-isogeny G → 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 ℤΓ-lattices ℤ\overline{Γ} → I. However such a map cannot exist since I is non-principal, hence non-cyclic as a ℤΓ-module.

11. Tori of essential dimension ≤ 1

Theorem 11.1. Let T be a torus over k, k^{(p)} a p-closure and Γ = Gal(k_{alg}/k^{(p)}). The following are equivalent:

(a) ed_{k}(T; p) = 0.

(b) ed_{k^{(p)}}(T; p) = 0.

(c) ed_{k^{(p)}}(T) = 0

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

(e) X(T)_{(p)} is a ℤ_{(p)}Γ-permutation module.

(f) X(T) is an invertible ℤΓ-lattice (i.e a direct summand of a permutation lattice).

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

T_{k^{(p)}} × S ≃ R_{E/k^{(p)}}(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) ⇔ (b) is Lemma 3.3.

(a) ⇔ (d) follows from [Me_{1}, Proposition 4.4].

(c) ⇒ (b) is clear.

(b) ⇒ (e): This follows from Corollary 9.2. Indeed, ed_{k}(T; p) = 0 implies the existence of a ℤ_{(p)}Γ-permutation lattice L together with a surjective homomorphism α : L → X(T)_{(p)} and rk L = rk X(T)_{(p)}. It follows that α is injective and X(T)_{(p)} ≃ L.

(e) ⇒ (f): Let L be a ℤΓ-permutation lattice such that L_{(p)} ≃ X(T)_{(p)}. Then by [CR, Corollary 31.7] there is a ℤΓ-lattice L′ such that L ⊕ L ≃ X(T) ⊕ L′.

(g) ⇒ (c): The torus R = R_{E/k^{(p)}}(𝔾_{m}) has a faithful representation of dimension dim R (over k^{(p)}) and hence ed_{k^{(p)}}(R) = 0. Since T_{k^{(p)}} is a direct factor of R we must have ed_{k^{(p)}}(T) ≤ 0 by [BF, Remarks 1.16 b)].

(f) ⇔ (g): A permutation lattice P can be written as

P = ⨁_{i+1}^{m} ℤ[Γ/Γ_{L_{i}}],

for some (separable) extensions L_{i}/k^{(p)} and Γ_{L_{i}} = Gal(k_{alg}/L_{i}). Set E = L_{1} × ⋯ × L_{m}. The torus corresponding to P is exactly R_{E/k^{(p)}}(𝔾_{m}), cf. [Vo,

3. Example 19]. □

Example 11.3. Let T be a torus over k of rank < p−1. Then 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 Γ = Gal(k_{alg}/k^{(p)}) acts trivially on X(T) and so does Γ.

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 Γ = Gal(k_{alg}/k^{(p)}).

(a) ed(T; p) ≤ 1 iff there exists a Γ-set Λ and an m ∈ ℤ[Λ] fixed by Γ such that X(T)_{(p)} ≅ ℤ_{(p)}[Λ]/⟨m⟩ as ℤ_{(p)}Γ-lattices.

(b) ed(T; p) = 1 iff m = ∑ a_{λ}λ from part (a) is not 0 and for any λ ∈ Λ fixed by Γ, a_{λ} = 0 mod p.

(c) If ed(T; p) = 1 then T_{k^{(p)}} ≅ T′ × S where ed_{k^{(p)}}(S; p) = 0 and X(T′)_{(p)} is an indecomposable ℤ_{(p)}Γ-lattice, and ed_{k^{(p)}}(T′; p) = 1.

Proof. (a) If ed(T; p) = 1, then by Corollary 9.2 there is a map of ℤΓlattices from ℤ[Λ] to X(T) which becomes surjective after localization at p and whose kernel is generated by one element. Since the kernel is stable under Γ, any element of Γ sends a generator m to either itself or its negative. Since p is odd, m must be fixed by Γ.

The 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 λ_{0} ∈ Λ is fixed by Γ and a_{λ_{0}} is not 0 mod p. Then X(T)_{(p)} ≅ ℤ_{(p)}[Λ − {λ_{0}}], so by Theorem 11.1 we have ed(T; p) = 0.

Conversely, assume ed(T; p) = 0. Then by Theorem 11.1, we have an exact sequence 0 → ⟨m⟩ → ℤ_{(p)}[Λ] → ℤ_{(p)}[Λ′] → 0 for some Γ-set Λ′ with one fewer element than Λ. We have

Ext_{Γ}^{1}(ℤ_{(p)}[Λ′], 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 ℤ_{(p)}Γmodule homomorphism f: ℤ_{(p)}[Λ] → ℤ_{(p)}[Λ] such that the image of f is ⟨m⟩ and f(m) = m. Then we can define c_{λ} ∈ ℤ_{(p)} by f(λ) = c_{λ}m. Note that f(γ(λ)) = f(λ) and thus

(12)  c_{γ(λ)} = c_{λ}

for every λ ∈ Λ and γ ∈ Γ. If m = ∑_{λ∈Λ} a_{λ}λ, as in the statement of the theorem, then f(m) = m translates into

∑_{λ∈Λ} c_{λ}a_{λ} = 1 .

Since every Γ-orbit in Λ has a power of p elements, reducing modulo p, we obtain

∑_{λ∈Λ^{Γ}} c_{λ}a_{λ} = 1 (mod p) .

This shows that a_{λ} ≠ 0 modulo p, for some λ ∈ Λ^{Γ}, as claimed.

(c) Decompose X(T)_{(p)} uniquely into a direct sum of indecomposable ℤ_{(p)}Γ-lattices by the Krull-Schmidt theorem [CR, Theorem 36.1]. Since ed(T; p) = 1, and the essential p-dimension of tori is additive (Thm. 8.1), all but one of these summands are permutation ℤ_{(p)}Γ-lattices. Now by [CR, 31.12], we can lift this decomposition to X(T) ≅ X(T′) ⊕ X(S), where ed(T′; p) = 1 and ed(S; p) = 0. □

Example 11.6. Let E be an étale algebra over k. It can be written as E = L_{1} × ⋯ × L_{m} with some separable field extensions L_{i}/k. The kernel of the norm R_{E/k}(𝔾_{m}) → 𝔾_{m} is denoted by R_{E/k}^{(1)}(𝔾_{m}). It is a torus with lattice

⨁_{i=1}^{m} ℤ[Γ/Γ_{L_{i}}] / ⟨1, ⋯, 1⟩,

where Γ = Gal(k_{sep}/k) and Γ_{L_{i}} = Gal(k_{sep}/L_{i}). Let Λ be the disjoint union of the cosets Γ/Γ_{L_{i}} Passing to a p-closure k^{(p)} of k, Γ_{k^{(p)}} fixes a λ in Λ iff [L_{i} : k] is prime to p for some i. We thus have

ed_{k}(R_{E/k}^{(1)}(𝔾_{m}); p) = \{ \begin{matrix}1, & [L_{i} : k] is divisible by p for all i = 1, ..., m \\ 0, & [L_{i} : k] is prime to p for some i. \end{matrix}

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 ℤG-lattices if and only if G is cyclic of order dividing p^{2}.

Remark 12.2. For G = C_{2} × C_{2} a classification of the (infinitely many) different genera of ℤG-lattices has been worked out by [NA]. In contrast for G = C_{p^{3}} or G = C_{p} × 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 ℤG-lattice where G = ⟨g|g^{p^{2}} = 1⟩ denotes the cylic group of order p^{2}. Heller and Reiner [HR], (see also [CR, 34.32]) classified all indecomposable ℤG-lattices. Our goal consists in computing the essential dimension of T. By Corollary 10.2 we have ed_{k}(T) = 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 ℤG-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 pdimension of tori corresponding to the 4p + 1 cases in the list [CR, 34.32].

Denote by H = ⟨h|h^{p} = 1⟩ the group of order p. We can consider ℤH as a G-lattice with the action g · h^{i} = h^{i+1}. Let

δ_{G} = 1 + g + … + g^{p^{2}−1} δ_{H} = 1 + h + … + h^{p−1}

be the “diagonals” in ℤG and ℤH and

ϵ = 1 + g^{p} + … + g^{p^{2}−p}.

The following ℤG-lattices represent all genera of indecomposable ℤGlattices (by ⟨∗⟩ we mean the ℤG-sublattice generated by ∗):

M_{1} = ℤ
M_{2} = ℤH
M_{3} = ℤH/⟨δ_{H}⟩
M_{4} = ℤG
M_{5} = ℤG/⟨δ_{G}⟩
M_{6} = ℤG ⊕ ℤ/⟨δ_{G} − p⟩
M_{7} = ℤG/⟨ϵ⟩
M_{8} = ℤG/⟨ϵ − gϵ⟩
M_{9,r} = ℤG ⊕ ℤH/⟨ϵ − (1 − h)^{r}⟩  1 ≤ r ≤ p − 1
M_{10,r} = ℤG ⊕ ℤH/⟨ϵ(1 − g) − (1 − h)^{r+1}⟩  1 ≤ r ≤ p − 2
M_{11,r} = ℤG ⊕ ℤH/⟨ϵ − (1 − h)^{r}, δ_{H}⟩  1 ≤ r ≤ p − 2
M_{12,r} = ℤG ⊕ ℤH/⟨ϵ(1 − g) − (1 − h)^{r+1}, δ_{H}⟩ 1 ≤ r ≤ p − 2

In the sequel we will refer to the above list as (𝕃).

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

Let ℤx be a ℤG-module of rank 1 with trivial G-action. We have an isomorphism

M_{10,r} = ℤG⊕ℤH/⟨ϵ(1−g)−(1−h)^{r+1}⟩ ≃ ℤG⊕ℤH ⊕ℤx/⟨ϵ−(1−h)^{r} −x⟩

induced by the inclusion ℤG ⊕ ℤH ↪ ℤG ⊕ ℤH ⊕ ℤx.

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

\xymatrix{ℤH \ar[r]^{h↦ϵ} \ar[d]_{h↦(1−h)^{r}+x} & ℤG \ar[d] \\ ℤH ⊕ ℤx \ar[r] & M_{10,r}}

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

0 → ℤH ⊕ ℤx → M_{10,r} → ℤG/ℤH → 0

with extension class determined by the vertical map h ↦ (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}, …, M_{12,r} are representatives of the genera of indecomposable ℤG-lattices.

Theorem 12.3. Every indecomposable torus T over k split by G has character lattice isomorphic to one of the ℤG-lattices M in the list (𝕃) after p-localization and ed(T) = ed(T; p) = ed(Diag(M); p). Their essential dimensions are given in the tables below.

\begin{matrix}M & rk M & ed(T) \\ 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{matrix}  \begin{matrix}M & rk M & ed(T) \\ 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{matrix}

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} …, M_{6} easily follows from the discussion in section 11. Let M be one of the lattices M_{7}, …, M_{12,r} and T = Diag M the corresponding torus. We will determine the minimal rank of a permutation ℤG-lattice P admitting a homomorphism P → M which becomes surjective after localization at p. Then we conclude ed(T; p) = rk P − rk M with Corollary 9.2.

We have the bounds

(13)  rk M ≤ rk P ≤ p^{2} (or p^{2} + p),

where the upper bound holds since every M is given as a quotient of ℤG (or ℤG ⊕ ℤH). Let C = 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} · (ϵ − (1 − h)^{r}); δ_{H} are written out as

∑_{i=0}^{p−1} g^{pi+j} − ∑_{ℓ=0}^{r} \binom{r}{ℓ}(−1)^{ℓ}h^{ℓ+j}, 0 ≤ j ≤ p − 1; ∑_{i=0}^{p−1} h^{i}

and the k_{sep}-point of the torus are

\begin{aligned}T (k_{sep}) = & \{(t_{0}, …, t_{p^{2}−1}, s_{0}, …, s_{p−1}) | \\ & ∏_{i=0}^{p−1} t_{pi+j} = ∏_{ℓ=0}^{r} s_{ℓ+j}^{(−1)^{ℓ}\binom{r}{ℓ}}, 0 ≤ j ≤ p − 1; ∏_{i=0}^{p−1} s_{i} = 1\} \end{aligned}

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

C(k) = {(ζ_{p}^{i}, …, ζ_{p}^{i}, ζ_{p}^{j}, …, ζ_{p}^{j}) | 0 ≤ i, j ≤ p − 1} ≃ μ_{p}^{2}.

(Note that the primitive pth root of unity ζ_{p} is in k by our assumption that k is p-closed). For other lattices this is similar: C is equal to Split_{k}(Diag(P)[p]) ≃ μ_{p}^{r} where M is presented as a quotient P/N of a permutation lattice P (of minimal rank) as in (𝕃) and where r denotes the number of summands in a decomposition of P.

M rank C rank M  possible rk P
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 or p^{2} + p
M_{10,r}  2  p^{2} + 1 p^{2} + 1 or p^{2} + p
M_{11,r}  2  p^{2} − 1 p^{2} + 1 or p^{2} + p
M_{12,r}  2  p^{2}  p^{2} + 1 or p^{2} + p

We need to exclude the possibility rk P = p^{2} + 1 for the lattices M = M_{9,r}, …, 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, …, 12 and r ≥ 1 every character χ ∈ 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 ℤδ_{H} in M_{9,r}, by ℤ(ϵ − (1 − h)^{r}) in M_{12,r} and by ℤ(ϵ − (1 − h)^{r}) ⊕ ℤδ_{H} in M_{10,r}, respectively. Since all these characters are trivial on

C = Split_{k}(Diag(ℤG ⊕ ℤH)[p]),

the claim follows. □

Acknowledgments

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

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