A LOWER BOUND ON THE ESSENTIAL DIMENSION OF A CONNECTED LINEAR GROUP

P. GILLE AND Z. REICHSTEIN

Abstract. Let G be a connected linear algebraic group defined over an algebraically closed field k and H be a finite abelian subgroup of G whose order does not divide char(k). We show that the essential dimension of G is bounded from below by rank(H)−rank C_{G}(H)^{0}, where rank C_{G}(H)^{0} denotes the rank of the maximal torus in the centralizer C_{G}(H). This inequality, conjectured by J.-P. Serre, generalizes previous results of Reichstein – Youssin (where char(k) is assumed to be 0 and C_{G}(H) to be finite) and Chernousov – Serre (where H is assumed to be a 2-group).

Contents

1. Introduction 1 2. Regular compactifications 3 3. Compactifications of homogeneous spaces 4 4. Proof of Theorem 1.2 in characteristic zero 6 5. The field of iterated Laurent series 8 6. Reduction of structure 10 7. Fixed points in homogeneous spaces 12 8. Proof of Theorem 1.2 14 9. Examples 16 Acknowledgements 20 References 20

1. Introduction

Let k be a base field, K/k be a field extension, G/k be a linear algebraic group and α ∈ H^{1}(K, G) be a G-torsor over Spec(K). We will say that α descends to a subfield K_{0} ⊂ K if α lies in the image of the natural map H^{1}(K_{0}, G) → H^{1}(K, G). The essential dimension ed_{k}(α) of α, is defined as the minimal value of trdeg_{k}(K_{0}), where α descends to K_{0} and k ⊂ K_{0}. (Throughout this paper we will work over a fixed algebraically closed field k;

1991 Mathematics Subject Classification. 11E72, 20G10, 14L30. Key words and phrases. Central simple algebra, algebraic group, group action, geomet-

ric action, tame action, semisimple algebra, Galois cohomology.

Z. Reichstein was partially supported by an NSERC research grant.

for this reason we will write ed(α) in place of ed_{k}.) We also define ed(α; l) as the minimal value of ed(α_{L}), as L/K ranges over all finite field extensions of degree prime to l. Here l is a prime integer. The essential dimension ed(G) of the group G (respectively, the essential dimension ed(G; l) of G at l) is defined as the maximal value of ed(α) (respectively, of ed(α; l)), as K/k ranges over all field extensions and α ranges over H^{1}(K, G).

Note that the numbers ed(α), ed(α; l), ed(G) and ed(G; l) all depend on the base field k, which will be assimed to be fixed (and algebraically closed) throughout this paper. For details on the notion of essential dimension, its various interpretations and numerous examples, see [Re], [RY_{1}] and [BF]. Many of the best known lower bounds on ed(G) and ed(G; l), for specific groups G, are deduced from the following inequality.

1.1. Theorem. ([RY_{1}, Theorem 7.8]) Let G be a connected semisimple linear algebraic group defined over an algebraically closed field k of characteristic zero and H be a finite abelian subgroup of G. Assume that the centralizer C_{G}(H) is finite. Then

(a) ed(G) ≥ rank(H) and

(b) if H is a l-group then ed(G; l) ≥ rank(H).

Here by the rank of a finite abelian group H we mean the smallest positive integer r such that H can be written as a direct product of r cyclic groups. Equivalently, rank(H) is the minimal dimension of a faithful complex linear representation of H.

The purpose of this paper is to prove the following more general inequality conjectured by J.-P. Serre (e-mail, July 25, 2005).

1.2. Theorem. Let G be a connected reductive linear algebraic group defined over an algebraically closed base field k. Suppose that H is a finite abelian subgroup of G and char(k) does not divide |H|. Then

(a) ed(G) ≥ rank H − rank C_{G}(H)^{0}.

(b) Moreover, if H is a l-group then ed(G; l) ≥ rank H − rank C_{G}(H)^{0}.

Here C_{G}(H)^{0} denotes the connected component of the centralizer of H in G, and by the rank of this connected group we mean the dimension of its maximal torus. In particular, if char(k) = 0 and the centralizer C_{G}(H) is finite (i.e., rank C_{G}(H)^{0} = 0) then Theorem 1.2 reduces to Theorem 1.1. Note, however, that even in this special case the proof we present here is simpler than the one in [RY_{1}]; in particular, it does not rely on resolution of singularities.

We also remark that our argument shows a bit more, namely that the essential dimension of a particular torsor, which we call a loop torsor, is ≥ rank(H) − rank C_{G}(H)^{0}. Here by a loop torsor we mean the image of a versal H-torsor under the natural map H^{1}( ∗ , H) → H^{1}( ∗ , G). (Such torsors come up in connection with loop algebras; see [GP].)

Chernousov and Serre [CS] used techniques from the theory of quadratic forms to show that, in the case where H is a 2-group, many of the bounds given by Theorem 1.1(b) remain valid over any algebraically closed field base field k of characteristic ≠ 2. The “incompressible” quadratic forms they construct are closely related to loop torsors; our arguments may thus be viewed as extending their approach to abelian subgroups H which are not necessarily 2-groups.

In order to clarify the exposition we will give two proofs of Theorem 1.2. The first one, presented in Section 4, is quite short but it relies on resolution of singularities and, in particular, only works in characteristic zero. The second proof, presented in Section 8 requires a bit more work. The advantage of this more elaborate argument is that it is entirely independent of resolution of singularities; in particular, it works in prime characteristic not dividing |H|. Both proofs rely, in a key way, on the existence results for wonderful (and regular) group compactifications from [Br] and [BK] (see Section 2) and on the “reduction of structure” Theorem 1.1 from [CGR]. The case where char(k) divides the order of the Weyl group of G is particularly delicate; here we use a refined version of [CGR, Theorem 1.1], which is proved in [CGR2].

The following symbols will be used for the remainder of the paper.

k  algebraically closed base field of characteristic p ≥ 0
G  connected reductive linear group defined over k
\overline{G}  a regular compactification of G
H  finite abelian group
ed  essential dimension over k
F_{n} = k((t_{1})) … ((t_{n})) iterated Laurent series field in n variables

2. Regular compactifications

Let G is a connected reductive algebraic group defined over k. Let B and B^{−} be opposite Borel subgroups of G, containing a maximal torus T. By the Bruhat decomposition, G has finitely many B × B^{−} orbits. Hence, by [BK, Proposition 6.5], G (viewed as a G × G-variety) has a “regular” compactification, in the sense of [BDP]; we will denote this compactification by \overline{G}. (Note that the terms “regular” and “smooth” are not interchangeable in this context; a regular compactification is smooth but not the other way around.) In particular, if G is adjoint then \overline{G} is the wonderful compactification of G constructed in [DP] (and in prime characteristic in [St]). Regular compactifications have many interesting special properties; most of them will not be used in the sequel. The only property of \overline{G} we will need is the following description of the stabilizers of points in \overline{G} from [Br, Proposition A.1].

Every G × G orbit O in a regular compactification \overline{G} of G has a unique point \overline{g} such that (BxB^{−})\overline{g} is open in O and \overline{g} is the limit of some 1parameter subgroup of T. We shall refer to \overline{g} as a special point. For example, the special point in the dense orbit O = G of \overline{G} is the identity element \overline{g} = e.

The stabilizer of a special point \overline{g} in G × G has the following form. Let P be the projection of Stab_{G×G}(\overline{g}) to the first factor of G × G and Q be the projection to the second factor. Then P and Q are opposite parabolic subgroups. Denote the unipotent radicals of P and Q by P_{u} and Q_{u} and their common Levi subgroup P ∩ Q by L. The stabilizer Stab_{G×G}(\overline{g}) is then given by

(2.1) Stab_{G×G}(\overline{g}) = \{(p_{u}lz, q_{u}l) | p_{u} ∈ P_{u}, q_{u} ∈ Q_{u}, l ∈ L, z ∈ Z(L)^{0}\} .

Of course, the stabilizer of any other point in O is conjugate to this subgroup.

2.2. Proposition. Let \overline{g} be a special point in the regular compactification \overline{G} of G and Γ be a finite subgroup of Stab_{G×G}(\overline{g}) whose order is prime to char(k). Then | rank(π_{2}(Γ)) − rank(π_{1}(Γ)) | ≤ rank(Z(L)^{0}).

Here π_{1}, π_{2} : G × G → G denote the projection to the first factor and second factors respectively, P = π_{1} Stab_{G×G}(\overline{g}), Q = π_{2} Stab_{G×G}(\overline{g}) and L = P ∩ Q as above, and rank(Γ) denotes the maximal value of rank(A), as A ranges over the abelian subgroups of a finite group Γ.

Proof. The proof is based on tracing Γ through the diagram of natural projections

\xymatrix{ & Stab_{G×G}(\overline{g}) \ar[dl]_{p_{1}} \ar[dr]^{p_{2}} \\ P \ar[dr]_{α_{P}} &  & Q \ar[dl]^{α_{Q}} \\  & L \ar[d] \\  & L/Z^{0}(L),}

where α_{P} (p_{u}l) = l and α_{Q}(q_{u}l) = l for any p_{u} ∈ P_{u}, q_{u} ∈ Q_{u} and l ∈ L. Since the kernels P_{u} and Q_{u} of α_{P} and α_{Q} are unipotent, we see that α_{P} and α_{Q} project the finite groups π_{1}(Γ) and π_{2}(Γ) isomorphically onto subgroups of L, which we will denote by Γ_{1} and Γ_{2}, respectively. By (2.1), Γ_{1} and Γ_{2} have the same image in L/Z(L)^{0}, which we will denote by Γ_{0}. Since the natural projection ϕ_{|Γ_{2}} : Γ_{2} → Γ_{0} is surjective with kernel Z_{2} ⊂ Z^{0}(L), we have

rank(Γ_{2}) ≤ rank(Γ_{1}) + rank(Z_{2}) ≤ rank(Γ_{1}) + rank Z^{0}(L) .

By symmetry, we also have the reverse inequality rank(Γ_{1}) − rank(Γ_{2}) ≤ rank Z^{0}(L), and the proposition follows. □

3. Compactifications of homogeneous spaces

The following lemma is well known in characteristic zero (see [RY_{2}, Lemma 2.1]).

3.1. Lemma. Let Γ be a finite group and let X be a normal quasiprojective Γ-variety. Then

(a) X is covered by affine open Γ-invariants subsets. Assume moreover that the order of Γ is invertible in k. Then

(b) there is a geometric quotient map π : X → X/Γ.

(c) Moreover, if X is projective, then so is X/Γ.

Recall that we are assuming throughout that the base field k is algebraically closed.

Proof. (a) The proof of loc. cit is characteristic free.

(b) Recall first that the group Γ is linearly reductive, see [MFK, §1]. If X is affine, part (b) is proved in [MFK, Theorem 1.1 and Amplification 1.3]. The general case follows from part (a) and the characteristic free glueing assertion in [PV, Theorems 4.14].

(c) See [N, Theorem 3.14]. □

Let G/k be a connected reductive linear algebraic group, \overline{G} be a regular compactification, and Γ be a finite subgroup of G of order prime to char(k). We shall denote by \overline{G}/Γ the geometric quotient of \overline{G} for the action of the finite group Γ (on the right). It may be viewed as a (possibly singular) compactification of the homogeneous space G/Γ. By the properties of geometric quotients

(i) the fibers of the natural projection \overline{G} → \overline{G}/Γ are the Γ-orbits of the right action of Γ on \overline{G}, and

(ii) the left action of G on \overline{G} descends to \overline{G}/Γ.

3.2. Proposition. Let G be a connected reductive group, \overline{G} be a regular compactification and H_{1}, H_{2} be finite abelian subgroups of G whose orders are prime to char(k). If \overline{G}/H_{2} has an H_{1}-fixed point then

rank(H_{1}) − rank(H_{2}) ≤ rank C_{G}(H_{1})^{0} .

Proof. Let y be an H_{1}-fixed point of \overline{G}/H_{2} and x be a point of \overline{G} lying above y. After translating x by a suitable element (g_{1}, g_{2}) ∈ G × G and replacing H_{1}, H_{2} by g_{1}H_{1}g_{1}^{−1}, g_{2}H_{2}g_{2}^{−1} respectively, we may assume that x is a special point. Then P = π_{1}(Stab_{G×G}(x)), Q = π_{2}(Stab_{G×G}(x)) is a pair of opposite parabolics, as in the previous section. Letting L = P ∩Q be their common Levi subgroup, we see that Stab_{G×G}(x) is as in (2.1). (Here, as before, π_{i}: G × G → G denotes the projection to the ith factor, wheere i = 1 or 2.)

Let H = Stab_{H_{1}×H_{2}}(x). The fact that y is H_{1}-fixed means that for every h_{1} ∈ H_{1}, there is an h_{2} ∈ H_{2} such that (h_{1}, h_{2}) ∈ Stab_{G×G}(x). In other words, π_{1}(H) = H_{1}. By Proposition 2.2 applied to the group H,

rank(H_{1}) − rank(π_{2}(H)) ≤ rank Z(L)^{0} .

Since rank(H_{2}) ≥ rank(π_{2}(H)), we have

rank(H_{1}) − rank(H_{2}) ≤ rank Z(L)^{0} .

It remains to show that

(3.3)  rank Z(L)^{0} ≤ rank C_{G}(H_{1})^{0} .

By the Levi decomposition, P is isomorphic to the semidirect product P_{u}◁L; see, e.g., [H, 30.2]. Denote the projection of H_{1} to L ≃ P/P_{u} by H_{L}. Then clearly Z(L) ⊂ C_{L}(H_{L}) ⊂ C_{G}(H_{1}), and the inequality (3.3) follows. □

4. Proof of Theorem 1.2 in characteristic zero

The following lemma is well known; we supply a short proof for lack of a direct reference.

4.1. Lemma. Consider a faithful action of a finite abelian group A on an irreducible algebraic variety X, defined over a field k. Assume char(k) does not divide |A|. If A fixes a smooth k-point in X then dim(X) ≥ rank(A).

Proof. Let x ∈ X(k) be a smooth A-fixed k-point. Then A acts on the regular local ring R = O_{x}(X) and on its maximal ideal M = M_{x}(X).

Assume the contrary; dim(X) = d and rank(A) > d. Then the Arepresentation on the d-dimensional contangent space T_{x}(X)^{∗} = M/M^{2} cannot be faithful; denote its kernel by A_{0} ≠ {1}. Since |A| is prime to char(k), the map M → M/M^{2} of A-representations splits. Thus R has a system of local parameters t_{1}, …, t_{d} ∈ M such that each t_{i} is fixed by A_{0}. Then A_{0} acts trivially on the completion R̂ = k[[t_{1}, …, t_{d}]], hence, on R ⊂ R̂. Since X is irreducible, A_{0} acts trivially on X. This contradicts our assumption that the A-action on X is faithful. □

For the remainder of this section we will assume that k is an algebraically closed field of characteristic zero. Before proceeding with the proof of Theorem 1.2, we recall that every Γ-variety is birationally isomorphic to a smooth projective Γ-variety; cf. [RY_{2}, Proposition 2.2]. Here Γ is an arbitrary linear algebraic group (not necessarily connected). This fact, whose proof relies on equivariant resolution of singularities and thus requires the assumption that char(k) = 0, will be used repeatedly for the remainder of this section.

4.2. Lemma. Every irreducible generically free G-variety is birationally isomorphic to a projective G-variety X of the form (\overline{G} × Y)/S, where

(a) S is a finite subgroup of G,

(b) Y is a smooth irreducible smooth projective S-variety,

(c) S acts on \overline{G} × Y by s · (h, y) = (hs^{−1}, s · y), and (\overline{G} × Y)/S is the geometric quotient for this action.

Note that in (c) S is a finite group acting on the smooth projective variety \overline{G} × Y. In this situation a geometric quotient exists and is projective; see, e.g., [PV, Theorem 4.14] or Lemma 3.1.

Proof. Let Γ is a linear algebraic group defined over k. and K/k be a finitely generated field extension. Recall that elements of H^{1}(K, Γ) are in a natural 1-1 correspondence with birational isomorphism classes of a generically free primitive Γ-varieties Z, with K = k(Z)^{Γ} see, e.g., [Po, Section 1.3]. Here by saying that Z is a primitive Γ-variety, we mean that Γ transitively permutes the irreducible components of Y. Denote the class of the generically free Γ-variety Z in H^{1}(K, Γ) by [Z].

Now let K = k(X)^{G}. By [CGR, Theorem 1.1] there exists a finite subgroup S ⊂ G and a generically free primitive S-variety Y such that that [X] is the image of some [Y] ∈ H^{1}(K, S) under the natural map H^{1}(K, S) → H^{1}(K, G). In other words, X is birationally isomorphic to (G × Y)/S. Since the S-variety Y is only defined up to birational isomorphism, we may assume without loss of generality that it is smooth and projective. Moreover, (G × Y)/S is birationally isomorphic to (\overline{G} × Y)/S, as in (c).

In general, the above construction produces only a primitive S-variety Y. However, it can be slightly modified to ensure that Y is irreducible, as follows. Let Y_{0} be an irreducible component of Y and S_{0} be the subgroup of S consisting of elements that leave Y_{0} invariant. Then it is easy to see that [Y] is the image of [Y_{0}] under the natural map H^{1}(K, S_{0}) → H^{1}(K, S); cf. e.g., [Re, Example 2.10]. Thus after replacing S by S_{0} and Y by Y_{0}, we may assume that Y is irreducible. □

4.3. Lemma. Let G be a reductive group, H ⊂ G be a finite abelian subgroup, and X be a generically free G-variety. If H fixes a smooth k-point of X then

dim k(X)^{G} ≥ rank(H) − rank C_{G}(H)^{0} .

Proof. By Lemma 4.2 there is a birational G-equivariant isomorphism X −−→^{≃} (\overline{G} × Y)/S, where Y is a smooth complete S-variety and S ⊂ G is a finite subgroup, as in Lemma 4.2. Note that dim(Y) = trdeg_{k} k(X)^{G}; we will denote this number by d. By the Going Down Theorem [RY_{1}, Proposition A.2] (\overline{G} × Y)/S also has an H-fixed k-point; denote it by x = [\overline{g}, y]. The fiber of the natural projection X = (\overline{G} × Y)/S → Y/S containing x is easily seen to be G-equivariantly isomorphic to \overline{G}/S_{0}, where S_{0} := Stab_{S}(y). Now observe that since y is a smooth k-point of Y, S_{0} can contain no abelian subgroup of rank ≥ d. In other words, rank(S_{0}) ≤ d. By Proposition 3.2,

rank(H) − rank(S_{0}) ≤ rank C_{G}(H)^{0}

and thus

d ≥ rank(S_{0}) ≥ rank(H) − rank C_{G}(H)^{0} .

This completes the proof of Lemma 4.3. □

We are now ready to proceed with the proof of Theorem 1.2. Let V be a generically free linear k-representation of G.

(a) The essential dimension ed(G) is the minimal value of trdeg_{k} k(X)^{G}, where the minimum is taken over all dominant rational G-equivariant maps

V −−→ X ,

such that X is a generically free G-variety; see [Re, Section 3]. Thus our goal is to show that

(4.4)  trdeg_{k} k(X)^{G} ≥ rank(H) − rank C_{G}(H)^{0} .

After replacing X by a birationally equivalent G-variety, we may assume that X is smooth and projective. Since V has a smooth H-fixed k-point (namely the origin), the Going Down Theorem [RY_{1}, Proposition A.2] tells us that X also has an H-fixed k-point (which is smooth, because every k-point of X is smooth). The inequality (4.4) now follows from Lemma 4.3.

(b) It suffices to show that the inequality (4.4) holds if there is a diagram of dominant rational G-equivariant maps of the form

\xymatrix{V′ \ar@{-->}[d] \ar@{->>}[dr] \\ V & X ,}

where X is a generically free G-variety, dim(V′) = dim(V) and [k(V′) : k(V)] is prime to l. (Note that the G-variety V′ is not required to be linear.) Once again, we may assume without loss of generality that V′ and X are smooth and complete. Since H fixes the origin in V, the Going Up Theorem [RY_{1}, Proposition A.4] tells us that V′ has an H-fixed k-point. Now by the Going Down Theorem [RY_{1}, Proposition A.2], X has an H-fixed k-point as well, and Lemma 4.3 completes the proof. □

5. The field of iterated Laurent series

In this section we will describe the structure of the iterated Laurent polynomial field F_{n} = k((t_{1})) … ((t_{n})). Our proof of Theorem 1.2 in full generality (i.e., without assuming that char(k) = 0) will make use of these results.

We begin by describing the absolute Galois group Gal(F_{n}). Since F_{n} = F_{n−1}((t_{n})) is a complete valuated field with residue field F_{n−1}, we have an exact sequence

1 → I_{n} → Gal(F_{n}) →^{π_{n}} Gal(F_{n−1}) → 1

where I_{n} stands for the inertia group; see [GMS, §II.7]. Let ℤ̂′ be the prime to p part of ℤ̂, i.e ℤ̂′ = ∏_{q≠p} ℤ̂_{q}. In particular, if p = 0 then ℤ̂′ = ℤ̂.

5.1. Lemma. There is a split exact sequence

(5.2)  1 → J_{n} → Gal(F_{n}) → (ℤ̂′)^{n} → 1.

such that (1) J_{n} = 1 if p = 0,

(2) J_{n} is a free pro-p-group if p > 0.

Proof. We proceed by induction on n. The group I_{n} fits in an exact sequence

0 → I_{n}^{wild} → I_{n} → ℤ̂′ → 1 ,

where I_{n}^{wild} is the wild inertia group (it is a pro-p-group). Define F_{n,m} :=

k(( \sqrt[m]{t}_{1})) … (( \sqrt[m]{t}_{n})) and F_{n,∞} := \underrightarrow{lim}_{(m,p)=1} F_{n,m}. Since

Gal(F_{n,∞}/F_{n−1,∞}((t_{n}))) = ℤ̂′ ,

we have the following commutative diagram of profinite groups

1  1  1
↓  ↓  ↓
1 ⟶  I_{n}^{wild}  ⟶  J_{n}  ⟶  J_{n−1}  ⟶ 1
↓  ↓  ↓
1 ⟶  I_{n}  ⟶ Gal(F_{n}) ⟶  Gal(F_{n−1})  ⟶ 1
↓  ↓  ↓
1 ⟶ Gal(F_{n,∞}/F_{n−1,∞}((t_{n}))) ⟶ Gal(F_{n,∞}/F_{n}) ⟶ Gal(F_{n−1,∞}/F_{n−1}) ⟶ 1.
↓  ↓  ↓
1  1  1

If p = 0, it follows that J_{n} = 1. If p > 0, we see by induction that J_{n} is pro-p-group. The group J_{n} is the absolute Galois group of the field F_{n,∞}, so cd_{p}(J_{n}) ≤ 1 ([Se], II.3.1, proposition 7) and J_{n} is then a free pro-p-group (ibid, I.4.2, Corollary 2).

Since (ℤ̂′)^{n} is a prime-to-p group, we conclude that the sequence (5.2) splits; see [Se, I.5.9, Corollary 1]. □

We will now show that every finite field extension of F_{n} is k-isomorphic to F_{n}. Recall that the lexicographic order ≺ on Z^{n} is defined as follows:

(m_{1}, ⋯ m_{n}) ≺ (m′_{1}, ⋯ m′_{n})

if m_{i} < m′_{i} for the smallest subscript i with m_{i} ≠ m′_{i}. A valuation v = (v^{(n)}, ⋯, v^{(1)}) : E^{×} → Γ on a field E is called n-discrete if the group Γ is isomorphic to a (lexicographically ordered) subgroup of Z^{n}; see [F, 1.1.3]. Then E is a 1-discrete valuation with respect to the first component v(^{n}) of v and the residue field E_{n−1} is a (n−1)-discrete valuation via (v^{(n−1)}, ⋯, v^{(1)}). In this way we obtain a sequence of fields E = E_{n}, E_{n−1}, ... , E_{1} such that E_{i} is the residue field of E_{i+1} with respect to a 1-discrete valuation. The residue field E_{0} of E_{1} then coincides with the residue field \overline{E}_{v}.

The definition of completeness for E is inductive as follows [F, 1.2.1]. The field E is a complete n-discrete field if E_{n} is complete to respect to v^{(n)} and E_{n−1} is complete. Assume from now on that E is a complete n-discrete field. Then according to [W, 3.1], E is henselian, i.e its valuation ring is a henselian ring. In particular, given a finite extension E′/E the valuation v extends uniquely to v′ : E′^{×} → \frac{1}{[E′:E]}Z^{n}, the formula being v′ = \frac{1}{[E′:E]} v ∘ N_{E′/E}. Then E′ is a n-discrete field which is complete (by induction).

The field F_{n} = k((t_{1}))((t_{2})) ⋯ ((t_{n})) over iterated Laurent series over a base field k is n-complete; see [W, §3]. Here the valuation v on F_{n} is defined by

v(∑_{i_{1}} ⋯ ∑_{i_{n}} c_{i_{1},...,i_{n}} t_{1}^{i_{1}} … t_{n}^{i_{n}}) = Min{ (i_{1}, ..., i_{n}) | c_{i_{1},...,i_{n}} ≠ 0}.

5.3. Proposition. Let E be a complete n-discrete k-field. Then (1) E is isomorphic to \overline{E}_{v}((t_{1}))((t_{2})) ⋯ ((t_{n}));

(2) Suppose the valuation v is trivial on a perfect subfield K of E. Then E is K–isomorphic to \overline{E}_{v}((t_{1}))((t_{2})) ⋯ ((t_{n})).

Proof. (1) immediately follows from (2) if we take K to be the prime subfield of E.

To prove (2), first assume that n = 1. In this case Cohen’s structure theorem [Co, Theorem 10] shows that the valuation ring of E is K-isomorphic to \overline{E}_{v}[[t_{1}]]. Thus E is K-isomorphic to \overline{E}_{v}((t_{1})).

Now suppose n ≥ 2. Let E = E_{n}, E_{n−1},..., E_{1} the sequence of fields constructed above. By induction, we may assume that E_{n−1} is K–isomorphic to \overline{E}_{v}((t_{1}))((t_{2})) ⋯ ((t_{n−1})). Since we know part (2) holds for n = 1, we conclude that E = E_{n} is K-isomorphic to \overline{E}_{v}((t_{1}))((t_{2})) ⋯ ((t_{n})). □

5.4. Corollary. Assume that k is algebraically closed. Then any finite extension of F_{n} = k((t_{1}))...((t_{n})) is k–isomorphic to F_{n}.

Proof. A finite extension E of F_{n} is a complete n-discrete field. Its residue field E_{v} is a finite extension of k, since k is algebraically closed, we conclude that E_{v} = k. Proposition 5.3(2) now shows that E is k–isomorphic to F_{n}. □

6. Reduction of structure

In this section G/k will denote a linear algebraic group defined over an algebraically closed field k of characteristic ≥ 0, whose identity component G^{0} is reductive. F_{n} will denote the iterated power series field k((t_{1}))((t_{2}))....((t_{n})) in variables t_{1}, …, t_{n}, as in the previous section. As usual, we will say that γ ∈ H^{1}(K, G)

descends to a subfield K_{0} ⊂ K if γ lies in the image of the restriction map H^{1}(K_{0}, G) → H^{1}(K, G)

admits reduction of structure to a subgroup A ⊂ G if γ lies in the image of the natural map H^{1}(K, A) → H^{1}(K, G).

6.1. Proposition. Suppose γ ∈ H^{1}(F_{n}, G) descends to a subfield K ⊂ F_{n} such that trdeg_{k}(K) = d < ∞.

(a) Assume that char(k) = 0. Then γ admits reduction of structure to a finite abelian subgroup A ⊂ G of rank ≤ d.

(b) Assume that char(k) = p > 0. Then there exists a finite field extension F′/F_{n} such that [F′ : F_{n}] is a power of p and γ_{F′} admits reduction of structure to a finite abelian subgroup A ⊂ G of rank ≤ d, whose order |A| is prime to p.

Our proof of Proposition 6.1 will make use of the following two simple lemmas.

6.2. Lemma. Suppose K ⊂ E is a field extension such that K is algebraically closed in E. Then

(a) for every finite Galois field extension K′/K, K′_{E} = K′ ⊗_{K} E is a field.

(b) The absolute Galois group Gal(K) is a quotient of Gal(E).

Proof. (a) By the primitive element theorem we can write K′ as K[x]/(p(x)) for some irreducible monic polynomial p(x) ∈ K[x]. Then K′_{E} = K′ ⊗_{K} E = E[x]/(p(x)), and we need to show that p(x) remains irreducible over E.

We argue by contradiction. Suppose p(x) = p_{1}(x)p_{2}(x) for some nonconstant monic polynomials p_{1}(x), p_{2}(x) ∈ E[x]. The coefficients of p_{i}(x) are then polynomials in the roots of p(x). (Here i = 1 or 2.) In particular, they are algebraic over K. Since K is algebraically closed in E, we conclude that p_{i}(x) ∈ K[x]. Thus p(x) is reducible over K, a contradiction.

(b) Let \overline{K} be the algebraic closure of K. Then Gal(K) = Gal(\overline{K}/K) = Gal(\overline{K}_{E}/E), and \overline{K}_{E} = \overline{K} ⊗_{K} E is an E-subfield of \overline{E} by part (a). □

6.3. Lemma. Let d ≥ 0 be an integer. Let Γ be a finitely generated abelian profinite group such that cd(Γ) ≤ d. Then Γ is a direct summand of ℤ̂^{d}.

Proof. Without loss of generality, we may assume that Γ is a p-profinite group for a prime p. Since Γ is finitely generated and abelian, Γ ≅ ℤ_{p}^{m} for some integer m; see [RZ, Theorem 4.3.4.(a)]. Thus cd(Γ) = m. Since we are assuming r ≤ d, this shows that Γ is a direct summand of ℤ̂^{d}. □

Proof of Proposition 6.1. Let W be the Weyl group of G. We recall that there exists a finite subgroup S ↪ G such that every prime factor of |S| divides |W | and S has the following property.

(i) If Char(k) does not divide |W |, then the map H^{1}(K, S) → H^{1}(K, G) is surjective for every field K/k; see [CGR].

(ii) If char(k) divides |W |, the above map is surjective for every perfect field K/k; see [CGR2].

We fix the finite subgroup S with these properties for the rest of the proof. Let ρ : (ℤ̂′)^{n} → Gal(F_{n}) be a splitting of the exact sequence (5.2) in

Lemma 5.1. Denote the extension of F_{n} associated by the Galois correspondence to the image of ρ by E_{n}/F_{n} and its perfect closure by E_{n}^{perf}/F_{n}. Note that if char(k) = 0 then ρ is an isomorphism and E_{n}^{perf} = E_{n} = F_{n}. If char(k) = p then the degree of any finite subextension of E_{n}/F_{n} is a power of p and Gal(E_{n}) = (ℤ̂′)^{n}. The same is true for the perfect closure E_{n}^{perf}/F_{n}.

Since Galois cohomology commutes with direct limits of fields, in order to establish parts (a) and (b) of the proposition, it suffices to show that γ_{E_{n}^{perf}} admits reduction of structure to some abelian subgroup A ⊂ S of rank ≤ trdeg_{k}(K) (where |A| is prime to p, if p = char(k) > 0). After replacing K by its algebraic closure in E_{n}^{perf}, we may assume that K is algebraically closed in E_{n}^{perf}. In particular, K is perfect.

By our assumption γ descends to some γ_{K} ∈ H^{1}(K, G). On the other hand, by (ii) γ_{K} is the image of some δ_{K} ∈ H^{1}(K, S). The class δ_{K} is represented by a continuous homomorphism ψ: Gal(K) → S. Clearly δ_{K} (and hence, γ_{K} and β_{E_{n}^{perf}}) admit reduction of structure to the subgroup A = Im(ψ) of S. It remains to show that A is an abelian group of rank ≤ d whose order is prime to p.

By Lemma 6.2 we can identify Gal(K) with a quotient of Gal(E_{n}^{perf}) = (ℤ̂′)^{n}. In particular, Gal(K) is finitely generated, abelian, and the order of every finite quotient of Gal(K) is prime to p. Moreover, by Tsen’s theorem, cd(Gal(K)) ≤ d; cf. [Se, II.4.2]. Thus Lemma 6.3 enables us to conclude that Gal(K) is a direct summand of (ℤ̂′)^{d}. Hence, the finite quotient A of Gal(K) is an abelian group of rank ≤ d whose order is prime to p. □

6.4. Remark. A minor modification of the above argument (in particular, using (i) instead of (ii)) shows that the assertion of Proposition 6.1(a) holds whenever Char(k) does not divide the order of the Weyl group W of G. In other words, in this case we can take F′ to be F_{n} in part (b). Since we will not use this result in the sequel, we leave the details of its proof as an exercise for an interested reader.

7. Fixed points in homogeneous spaces

Let k is an algebraically closed field of characteristic p ≥ 0, t_{1}, … t_{n} are independent variables over k, and H = (ℤ/mℤ)^{n}. If p > 0, we will assume that m is prime to p. We we will continue to denote the iterated power series field k((t_{1}))((t_{2})) … ((t_{n})) by F_{n}.

The purpose of this section is to establish the following fixed point result which will be used in the proof of Theorem 1.2. For notational convenience, we will consider an arbitrary (not necessarily injective) morphism

(7.1)  ϕ: H = (ℤ/mℤ)^{n} → G

of algebraic groups. This is slightly more general than considering a finite abelian subgroup of G. We will assume that G, H and ϕ are fixed throughout this section.

7.2. Proposition. Assume that (1) F′/F_{n} is a finite field extension of degree prime to |H|,

(2) β ∈ H^{1}(F_{n}, H) is represented by an H-Galois field extension E/F, and

(3) ϕ_{∗}(β)_{F′} ∈ H^{1}(F′, G) admits reduction of structure to a finite subgroup S of G.

Then ϕ(H) has a fixed k-point in any G-equivariant compactification Y of G/S.

Here by a G-equivariant compactification of G/S we mean a complete (but not necessarily smooth) G-variety, which contains G/S as a dense open G-subvariety.

Proof. By Corollary 5.4, F′ is k-isomorphic to F_{n}. Thus, after replacing F_{n} by F′ and β by β_{F′}, we may assume that F′ = F_{n}. (Note that β_{F′} ∈ H^{1}(F′, H) is represented by the H-Galois algebra E_{F′}/F′, where E_{F′} = E ⊗_{F} F′. Since E is a field and [F′_{n} : F_{n}] is prime to |H| = [E : F], E_{F′} is again a field.)

By Lemma 5.1, we may assume that E = k((s_{1})) … ((s_{n})), where s_{i}^{m} = t_{i} and there exists a minimal set of generators τ_{1}, ..., τ_{n} of H such that H acts on k((s_{1})) … ((s_{n})) by

(7.3)  τ_{i}(s_{j}) = \begin{cases}ζ s_{j}, if i = j and \\ s_{j}, if i ≠ j, \end{cases}

where ζ is a primitive mth root of unity (independent of i and j). In the sequel we will denote E by F_{n,m}; note that we previously encountered this field in the proof of Lemma 5.1.

Set γ = ϕ_{∗}(β) ∈ H^{1}(F_{n}, G) and consider the twisted F_{n}-variety _{γ}Y which is a compactification of the twisted variety _{γ}(G/S). By our asumption γ(G/S) has a F_{n}-point, so a fortiori

γY (F_{n,m}) = {y ∈ Y (F_{n,m}) | γ(σ).^{σ}y = y ∀σ ∈ H} ≠ ∅.

Since Y is complete, this implies _{γ}Y (F_{n−1,m}[[ \sqrt[m]{t}_{n}]]) ≠ ∅. Specializing t_{n} to 0, we see that

{y ∈ Y (F_{n−1,m}) | γ(σ).^{σ}y = y ∀σ ∈ H} ≠ ∅,

where the Galois action of H on Y (F_{n−1,m}) is induced by the canonical projection H → Gal(F_{n,m}/F_{n}) → Gal(F_{n−1,m}/F_{n−1}). Repeating this process, we finally obtain

{y ∈ Y (k) | γ(σ).^{σ}y = y ∀σ ∈ H} ≠ ∅.

Since k is algebraically closed, we conclude that ϕ(H) fixes some k-point of Y. □

7.4. Corollary. Let k be an algebraically closed field and G/k be a connected reductive group. Suppose there exists a class β ∈ H^{1}(F_{n}, H) such that β is represented by an H-Galois field extension of F_{n}. If ϕ_{∗}(β) ∈ H^{1}(F_{n}, G) descends to a k-subfield K ⊂ F_{n} then

trdeg_{k}(K) ≥ rank ϕ(H) − rank C_{G}(ϕ(H))^{0} .

Here H and ϕ are as in (7.1).

Proof. Let trdeg_{k}(K) = d. By Proposition 6.1 there exists a finite extension F′/F_{n} and a finite abelian subgroup A ⊂ G of rank ≤ d such that

|A| is prime to char(k),

F′ = F_{n} if char(k) = 0, and [F′ : F_{n}] is a power of p if char(k) = p, and ϕ_{∗}(β)_{F′} admits reduction of structure to A. Let \overline{G} be a regular compactification of G. By Proposition 7.2 Y = \overline{G}/A

has a ϕ(H)-fixed point. Now Proposition 3.2, with H_{1} = ϕ(H) and H_{2} = A, tells us that

rank ϕ(H) − rank A ≤ rank C_{G}(ϕ(H))^{0} .

Consequently,

d ≥ rank A ≥ rank ϕ(H) − rank C_{G}(ϕ(H))^{0} ,

as claimed. □

8. Proof of Theorem 1.2

In the statement of Theorem 1.2, we assume that H is a subgroup of G, where as in the previous section we worked with a homomorphism ϕ: H → G instead. For notational consistency, we will restate Theorem 1.2 in the following (clearly equivalent) form.

8.1. Theorem. Let G be a connected reductive linear algebraic group defined over an algebraically closed base field k, H ≃ (ℤ/mℤ)^{n} and ϕ: H → G be a (not necessarily injective) homomorphism of algebraic groups. Assume char(k) does not divide m. Then

(a) ed(G) ≥ rank ϕ(H) − rank C_{G}(ϕ(H))^{0}.

(b) Moreover, if H is a l-group (i.e., m is a power of a prime integer l) then ed(G; l) ≥ rank ϕ(H) − rank C_{G}(ϕ(H))^{0}.

Let t_{1}, …, t_{n} be independent variables over k, K_{n} = k(t_{1}, …, t_{n}) and K_{n,m} = k(s_{1}, …, s_{n}), where s_{i}^{m} = t_{i}. The H-Galois field extension K_{n,m}/K_{n} gives rise to a class α ∈ H^{1}(K_{n}, H). We will be interested in the class ϕ_{∗}(α) ∈ H^{1}(K_{n}, G), which we will sometimes refer to as a loop torsor. (Such torsors naturally come up in connection with loop algebras; see [GP].) We are now ready to proceed with the proof of Theorem 8.1(a).

(a) By the definition of ed(G), it suffices to show that

ed ϕ_{∗}(α) ≥ rank ϕ(H) − rank C_{G}(ϕ(H))^{0} .

Let d = ed(ϕ_{∗}(α)). Let β = α_{F_{n}} ∈ H^{1}(F_{n}, H). Then β is represented by the field extension F_{n,m}/F_{n}. Moreover, ϕ_{∗}(β) descends to ϕ^{∗}(α) ∈ H^{1}(K_{n}, G), which, by our assumption, further descends to a k-subfield of K_{n} of transcendence degree d. Corollary 7.4 now tells us that

d ≥ rank ϕ(H) − rank C_{G}(ϕ(H))^{0} .

This completes the proof of Theorem 8.1(a).

Our proof of Theorem 8.1(b) will rely on the following lemma.

8.2. Lemma. Let l be a prime integer, E/K be a finite field extension of degree prime to l and F/K be an arbitrary (not necessarily finitely generated) field extension. Assume char(K) ≠ l. Then there exists a field extension F′/F and a K-embedding E ↪ F′ such that the diagram

[figure: E; prime-to-l; F′; prime-to-l; F; K]

commutes.

Proof. We may assume without loss of generality that E = K(α). Let P be the minimal polynomial of α and let P = Q_{1}^{e_{1}}...Q_{r}^{e_{r}} be the prime decomposition of P as F -polynomial. We have

F ⊗_{K} E = F ⊗_{K} \frac{K[X]}{P} ≅ \frac{F [X]}{Q_{1}^{e_{1}}} ⊕ ⋯ ⊕ \frac{F [X]}{Q_{r}^{e_{r}}}.

Denote by F_{i} = F [X]/Q_{i}(X) the residue field of the prime polynomial Q_{i}. Since

[K′ : K] = [F_{1} : F]^{e_{1}} + ⋯ + [F_{r} : F]^{e_{r}}

is prime to l, one of the degrees [F_{i} : F] is prime to l for some i = 1, …, r. We can now take F′ = F_{i}. □

We are now ready to finish the proof of Theorem 8.1(b). Once again, we will denote the class of the H-Galois field extension K_{n,m}/K_{n} in H^{1}(K_{n}, H) by α and consider the loop torsor ϕ_{∗}(α) ∈ H^{1}(K_{n}, G). By the definition of ed(G; l) it suffices to show that

ed(ϕ_{∗}(α); l) ≥ rank H − rank C_{G}(H)^{0} .

Equivalently, we want to show that

ed(ϕ_{∗}(α)_{E}) ≥ rank H − rank C_{G}(H)^{0}

for every finite extension E/K_{n} of degree prime to l. Suppose E/K_{n} is such an extension and ed(ϕ_{∗}(α)_{E}) = d.

By Lemma 8.2 there exists a finite field extension F′/F_{n} of degree prime to l and a K_{n}-embedding : E ↪ F′ such that the diagram

[figure: E; prime-to-l; F′; prime-to-l; F_{n}; K_{n}]

commutes. We want to conclude that d ≥ rank H − rank C_{G}(H)^{0} by applying Corollary 7.4 to β = α_{F′} ∈ H^{1}(F′, H). Since F′ is k-isomorphic to F_{n} (see Corollary 5.4), Corollary 7.4 can be applied to this β, as long as we can show that

(i) ϕ_{∗}(β) descends to a k-subfield of F′, of transcendence degree d over k, and

(ii) β is represented by an H-Galois field extension of F′.

(i) is clear since ϕ^{∗}(β) ∈ H^{1}(F′, G) descends to ϕ^{∗}(α) ∈ H^{1}(E, G), which, by our assumption, descends to a k-subfield E_{0} ⊂ E with trdeg_{k}(E_{0}) = d. To prove (ii), note that α_{F_{n}} is represented by the field extension F_{n,m}/F_{n}. Thus β = α_{F′} is represented by the H-Galois algebra F_{n,m} ⊗_{F_{n}} F′ over F′. Since [F′ : F_{n}] is a finite and prime to l, F_{n,m}⊗_{F_{n}} F′ is a field. This concludes the proof of (ii) and thus of Theorem 8.1. □

9. Examples

This section contains five examples illustrating Theorem 1.2(b).

9.1. Example. If char(k) ≠ 2 then ed(GO_{n}; 2) ≥ n − 1.

Proof. Let H ≃ (ℤ/2ℤ)^{n} be the subgroup of diagonal matrices in O_{n}. Viewing H as a subgroup of GO_{n}, we easily see that C_{GO_{n}}^{0}(H) = the center of GO_{n}, has rank 1. Applying Theorem 1.2(b) to this subgroup we obtain the desired bound. □

9.2. Example. If p is a prime and char(k) ≠ p then

(a) ed(SL_{p^{r}}/μ_{p^{s}}; p) ≥ \begin{cases}2s + 1, if s < r, \\ 2s, if s = r \end{cases} and

(b) ed(GL_{p^{r}}/μ_{p^{s}}; p) ≥ \begin{cases}2s, if s < r, \\ 2s − 1, if s = r. \end{cases}

Proof. (a) The group SL_{p^{r}}/μ_{s} has a self-centralizing subgroup H ≃ (ℤ/pℤ)^{r}× ℤ/p^{r−s}ℤ; see [RY_{1}, Lemma 8.12]. Now apply Theorem 1.2(b) to this group.

(b) We now view H as a subgroup of GL_{p^{r}}/μ_{p^{s}}. The centralizer C_{GL_{p^{r}} /μ_{s}}^{0}(H) is the center of GL_{p^{r}}/μ_{s}; it is isomorphic to a 1-dimensional torus. Part (b) now follows from Theorem 1.2(b). □

The non-vanishing of the Rost invariant H^{1}(∗ , G) → H^{3}(∗ , μ_{p}) for a group G and a prime p implies that ed(G; p) ≥ 3; cf. [Re, Theorem 12.14]. In particular, one can show that ed(F_{4}; 3), ed(E_{6}; 2) and ed(E_{7}; 3) ≥ 3 in this way. In Examples 9.3–9.6 below we will deduce these inequalities directly from Theorem 1.2(b) and show that equality holds in each case.

9.3. Example. If char(k) ≠ 3 then ed(F_{4}; 3) = 3.

Proof. F_{4} has a self-centralizing subgroup isomorphic to (μ_{3})^{3}; see [Gr, Theorem 7.4]. Theorem 1.2 now tells us that ed(F_{4}; 3) ≥ 3. To prove the opposite inequality, recall that H^{1}(K, E_{6}) classifies the exceptional 27dimensional Jordan algebras J/K. After a quadratic extension K′/K, J ⊗_{K} K′ is given by the first Tits construction. Without loss of generality, we may assume that J = (A, ν) where A is a central simple K-algebra of degree 3 and ν is a scalar in K. Since A is a symbol algebra (a, b)_{3}, we conclude that that J ⊗_{K} K′ descends to the subfield k(a, b, ν) of K of transcendence degree ≤ 3. We conclude that ed(J; 3) ≤ 3 and thus ed(F_{4}; 3) ≤ 3, as claimed. □

9.4. Example. If char(k) ≠ 2 then ed(E_{6}; 2) = 3.

Here E_{6} denotes the simply connected simple groups of type E_{6}. By abuse of notation we will also write E_{6} for the Dynkin diagram of E_{6}.

Proof. By [Gr, Table II], E_{6} has a unique non-toral subgroup H isomorphic to (ℤ/2ℤ)^{5}. To compute the rank of its centralizer, we make use of its Witt–Tits index I(H) ⊂ E_{6} which describes the type of a minimal parabolic subgroup containing H; see [GP, Section 3]. The Dynkin diagram for E_{6} is

[figure: α_{1}; α_{3}; α_{4}; α_{5}; α_{6}; α_{2}]

Set I = {α_{2}, α_{3}, α_{4}, α_{5}}. Let P_{I} be a standard parabolic subgroup and L_{I} = Z_{G}(T_{I}) its standard Levi subgroup. Then DL_{I} = Spin_{8}. Since Spin_{8} has a maximal non-toral 2-elementary abelian subgroup of rank 5 (see [Gr, Table I]), we may assume that H ⊂ Spin_{8}. Moreover, C_{Spin_{8}}(H) is finite, so H is irreducible in L_{I}. It follows that P_{I} is a minimal parabolic subgroup of E_{6} containing H; the Witt–Tits index of H is then I. By [GP, Proposition 3.19], we have

rank C_{E_{6}}(H) = |E_{6} \ I| = 2 .

Theorem 1.2(b) now tells us that ed(E_{6}; 2) ≥ 5 − 2 = 3.

To prove the opposite inequality, suppose α ∈ H^{1}(K, E_{6}), where K is a field containing k. Let L ⊂ E_{6} be the Levi subgroup of the parabolic E_{6} \ {α_{6}}. We observe that the finite groups N_{L}(T)/T and N_{E_{6}}(T)/T have isomorphic 2-Sylow subgroups (of order 2^{7}). By [Gi, Lemme 3.a], it follows that there exists a finite odd degree extension K′/K such that α_{K′} belongs to the image of H^{1}(K′, L) → H^{1}(K′, E_{6}). Hence the class α_{K′} is isotropic with respect to the root α_{6}. By the list of Witt–Tits indices in [T], the class α_{K′} is isotropic with respect to I. So α_{K′} belongs to the image of H^{1}(K′, L_{I}) ≅ H^{1}(K′, P_{I}) → H^{1}(K′, E_{6}).

It thus remains to show that

ed(L_{I}, 2) ≤ 3 .

To prove this inequality, we need an explicit description of the group L_{I}. Recall that there is a natural inclusion μ_{2}×μ_{2} = C(DL_{I}) ⊂ T_{I} = 𝔾_{m}×𝔾_{m}; see [CP, Proof of Proposition 14.a]. Hence we have the following commutative exact diagram

1  1
↓  ↓
0 ⟶ C(DL_{I}) = μ_{2} × μ_{2} ⟶ DL_{I} = Spin_{8} ⟶ PSO_{8} ⟶ 1
↓  ↓  ||
0 ⟶ T_{I} = 𝔾_{m} × 𝔾_{m} ⟶  L_{I}  ⟶ PSO_{8} ⟶ 1.
(2,2)↓  ↓
𝔾_{m} × 𝔾_{m}  =  𝔾_{m} × 𝔾_{m}
↓  ↓
1  1

Taking Galois cohomology of the right square over a field F/k, we obtain the following commutative exact diagram of pointed sets

H^{1}(F, Spin_{8}) ⟶ H^{1}(F, PSO_{8})
↓  ||
1 ⟶ H^{1}(F, L_{I}) ⟶ H^{1}(F, PSO_{8}).

↓

1

So H^{1}(F, L_{I}) = Im(H^{1}(F, Spin_{8}) → H^{1}(F, PSO_{8})). It is well known that

H^{1}(F, L_{I}) classifies the similarity classes of 8-dimensional quadratic F -forms whose class belong to I^{3}(F); cf., e.g., [KMRT, pp. 409 and 437]. By the Arason-Pfister Theorem, every 8-dimensional quadratic form q ∈ I^{3}(F) is similar to a 3-fold Pfister form ⟨⟨a, b, c⟩⟩. Thus the similarity class of q is defined over k(a, b, c). This shows that ed(L_{I}, 2) ≤ 3, as claimed. □

9.5. Remark. One can show that for every α ∈ H^{1}(K, E_{6}) there is an odd degree field extension L/K such that α_{L} lies in the image of the natural map H^{1}(L, G_{2}) → H^{1}(L, E_{6}); see [GMS, Exercise 22.9]. Since ed(G_{2}) = 3, this leads to an alternative proof of the inequality ed(E_{6}; 2) ≤ 3.

9.6. Example. If char(k) ≠ 3 then ed(E_{7}; 3) = 3.

Here E_{7} denotes the simply connected simple groups of type E_{7}. By abuse of notation we will sometimes also write E_{7} for the Dynkin diagram of E_{7}.

Proof. By [Gr, Table III], E_{7} has a unique non-toral subgroup H isomorphic to (ℤ/3ℤ)^{5}. To compute the rank of its centralizer, we make use of its Witt– Tits index I(H) ⊂ E_{7} [GP, section 3], where E_{7} is the following Dynkin diagram

[figure: α_{7}; α_{6}; α_{5}; α_{4}; α_{3}; α_{1}; α_{2}]

Set I = E_{7} \ {α_{7}} and let P_{I} be the standard parabolic subgroup. Denote by L_{I} = Z_{G}(T_{I}) its standard Levi subgroup. Then DL_{I} = E_{6}, where E_{6} denotes a simply connected group of type E_{6}, Since E_{6} has a maximal nontoral 3-elementary abelian subgroup of rank 4 (see [Gr, Table III]), we may assume that H ⊂ E_{6}. Moreover, C_{E_{6}}(H) is finite, so H is irreducible in L_{I}. It follows that P_{I} is a minimal parabolic subgroup of E_{7} containing H and thus the Witt-Tits index of H is I. By [GP, Proposition 3.19], the group C_{E_{7}}(H) is of rank 1. Theorem 1.2(b) now tells us that ed(E_{7}; 3) ≥ 4−1 = 3.

To prove the opposite inequality, consider α ∈ H^{1}(K, E_{7}). By [Ga, Example 3.5], the natural map

H^{1}(K, E_{6} ⋊ μ_{4}) → H^{1}(K, E_{7})

is surjective. Here, once again, E_{6} stands for the simply connected group of type E_{6}. It follows that there exists a quartic extension K′/K such that α_{K′} admits reduction of structure to E_{6} (i.e., lies in the image of the map H^{1}(K′, E_{6}) → H^{1}(K′, E_{7})). Thus we may assume without loss of generality that α comes from E_{6}. Now recall that the natural map H^{1}(K, F_{4} ⋊ μ_{3}) → H^{1}(K, E_{6}) is surjective (see [Ga, Example 3.5]); here μ_{3} = C(E_{6}). Thus there exists a β ∈ H^{1}(K, F_{4} ⋊ μ_{3}) mapping to α by the composite map

H^{1}(K, F_{4} ⋊ μ_{3}) → H^{1}(K, E_{6}) → H^{1}(K, E_{7}).

We claim that α ∈ H^{1}(K, E_{7}) admits further reduction of structure to F_{4} (i.e., α depends only of the F_{4}-component of β). If this claim is established the desired inequality ed(α; 3) ≤ 3 will immediately follow from Example 9.3.

To prove the claim we again view E_{6} inside E_{7} as E_{6} = DL_{I}. We have L_{I} = C_{E_{7}}(T_{I}) = DL_{I}.T_{I}, where T_{I} = 𝔾_{m} is the standard torus associated to I and C(L_{I}) = T_{I}.C(E_{7}). Since C(E_{7}) = μ_{2}, it follows that μ_{3} = C(E_{6}) ⊂ T_{I} ⊂ L_{I}. We consider the commutative diagram of pairings

H^{1}(K, μ_{3})×H^{1}(K, E_{6}) ⟶ H^{1}(K, E_{6})
↓  ↓  ↓
H^{1}(K, T_{I})×H^{1}(K, L_{I}) ⟶ H^{1}(K, L_{I}).

From the vanishing of H^{1}(K, T_{I}), it follows that the map H^{1}(K, E_{6}) → H^{1}(K, L_{I}) is H^{1}(K, μ_{3})-invariant. A fortiori, the image of the map

H^{1}(K, F_{4}) → H^{1}(K, E_{6}) → H^{1}(K, L_{I})

is H^{1}(K, μ_{3})-invariant. We conclude that the image of β ∈ H^{1}(F_{4} × μ_{3}) in H^{1}(K, L_{I}) depends only of its F_{4} component, as claimed. □

9.7. Remark. One can show that if G → G′ is a central isogeny of degree d then ed(G; p) = ed(G′; p) for any prime p not dividing d. In particular, the equalities ed(E_{6}; 2) = 3 and ed(E_{7}; 3) = 3 are valid for adjoint E_{6} and E_{7} as well. We leave the details of this argument to the reader.

Acknowledgements

We would like to thank J.-P. Serre for sharing his conjecture with us and for numerous insights related to the subject matter of this paper. We are also grateful to M. Brion, P. Brosnan, V. Chernousov and O. Gabber for helpful comments.

The second author would like to thank Laboratoire de mathématiques at the Université Paris-Sud for their hospitality during the week of June 19–25, 2006, when the work on this collaborative project began.

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UMR 8552 du CNRS, DMA, Ecole Normale Supérieure, F-75005 Paris, France

E-mail address: Philippe.Gille@ens.fr

Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2, Canada

E-mail address: reichst@math.ubc.ca