The kernel of the Rost invariant, Serre’s Conjecture II and the Hasse principle for quasi-split groups ^{3,6}D_{4}, E_{6}, E_{7}

V. Chernousov ^{∗}

Forschungsinstitut für Mathematik ETH–Zentrum CH-8092 Zürich Switzerland

Abstract

We prove that for a simple simply connected quasi-split group of type^{3,6}D_{4}, E_{6}, E_{7} defined over a perfect field F of characteristic ≠ 2, 3 the Rost invariant has trivial kernel. In certain cases we give a formula for the Rost invariant. It follows immediately from the result above that if cd F ≤ 2 (resp. vcd F ≤ 2) then Serre’s Conjecture II (resp. the Hasse principle) holds for such a group. For a (C_{2})-field, in particular ℂ(x, y), we prove the stronger result that Serre’s Conjecture II holds for all (not necessary quasi-split) exceptional groups of type ^{3,6}D_{4}, E_{6}, E_{7}.

1 Introduction

This paper grew out of the letters [Ch98, Ch00] where we sketched how Harder’s proof [H65, H66] of the Hasse principle for exceptional groups ^{3,6}D_{4}, E_{6}, E_{7} over number fields can be carried over to the case of quasi-split groups defined over a perfect field of cohomological dimension ≤ 2 and how the same ideas can be applied to describe in particular the kernel of the Rost invariant for ^{2}E_{6}.

In this paper we give full proofs of all these results. The main ones are the following. Let G_{0} be a quasi-split simple simply connected exceptional group of type ^{3,6}D_{4}, E_{6}, E_{7} defined over a perfect field F of characteristic ≠ 2, 3. Then

• the kernel of the Rost invariant of G_{0} is trivial;

• if cd F ≤ 2, then Serre’s Conjecture II holds for G_{0};

• if vcd F ≤ 2, then the Hasse principle Conjecture II holds for G_{0};

• if F is a (C_{2})-field, then Serre’s Conjecture II holds for an arbitrary simple simply connected group (not necessary quasi-split) of the same type as G_{0}.

∗The author gratefully acknowledge the support of TMR ERB FMRX CT-97-0107 and Forschungsinstitut für Mathematik, ETH in Zürich.

In [G97] Gille proved that a group G of inner type E_{6} or E_{7} defined over F is F -split iff there exist finite field extensions E_{1}, …, E_{s} of F splitting G and such that the

g.c.d.{ [E_{1} : F], …, [E_{s} : F] } = 1.

As an easy corollary of our results we obtain that the same quasi-splitting criterion is true for outer forms of type ^{3,6}D_{4} or ^{2}E_{6}.

Another corollary says that for any F -group G (not necessary quasi-split) of type 3,6D_{4}, ^{1,2}E_{6} or E_{7} there exists a chain of cyclic extensions

L_{0} = F ⊂ L_{1} ⊂ ⋯ ⊂ L_{n}

of degrees 2 or 3 such that G splits over L_{n}.

Recall that for all classical groups and groups of type G_{2}, F_{4} Serre’s Conjecture II and the Hasse principle Conjecture II were proved by E. Bayer-Fluckiger and R. Parimala [BP95, BP98]. J. Ferrar [Fer69] essentially proved the Hasse principle Conjecture II for inner groups E_{6} with the Tits algebras of index ≤ 3. Note also that our results related to Serre’s Conjecture II (items 2, 4) were obtained independently by P. Gille [G01] using different methods.

S. Garibaldi [Gar01] also proved independently the triviality of the kernel of the Rost invariant for quasi-split groups E_{6}, E_{7}. His argument is based on consideration of explicit geometric realizations of quasi-split E_{6}, E_{7} and studying properties of algebra structures which occur in his constructions. Since items 2, 3 immediately follow from the first one, Garibaldi’s result gives another proof of Serre’s Conjecture II and the Hasse principle Conjecture II for quasi-split groups E_{6}, E_{7}.

Our approach is based on different ideas which, as we have already mentioned, come back to G. Harder. In contrast to Garibaldi’s paper we focus on studying intrinsic properties of groups splitting over small extensions of the ground field F. The main body of the paper are Sections 4, 5 where we study properties of groups splitting over an extension K/F of degree p = 2, 3. In this part F is an arbitrary field of characteristic ≠ 2, 3. We show that the F -structure of such groups can be described completely by certain numerical invariants. In the case of quadratic extensions we follow the author proof [Cher89] of the Platonov-Margulis conjecture on the projective simplicity of groups of rational points splitting over a quadratic extension. The results obtained in these two sections allow us to give a formula for the Rost invariant of strongly inner or outer forms of type E_{6}, E_{7} splitting over an extension of degree p = 2, 3 in the case where the Galois group Gal(F^{s}/F) is a pro-p-group (see 5.12 and 6.2.3).

Gille’s splitting criterion [G97] for groups of inner type E_{6}, E_{7} combined with the results described in Sections 4,5 gives proofs of the main theorems more or less quickly. This is done in Sections 6,7,8.

Finally we note that the same methods together with the results of [Ch94, Ch89] prove Serre’s Conjecture II for E_{8} in each of the following cases:

• the nilpotent closure of the basic field F has cohomological dimension 1;

• Gille’s splitting criterion [G97] holds for E_{8}.

Acknowledgements. I am grateful to J.-P. Serre and S. Garibaldi for useful comments and discussions of the first version of the paper.

Notation

The letter F denotes a perfect field and F^{s} denotes a separable closure of F. If L/F is a separable extension, then R_{L/F} denotes the Weil functor of restriction of scalars.

We say that a reductive F -subgroup G′ ⊂ G is standard if there is a maximal F -torus T of G normalizing G′.

If G is a reductive algebraic group and T ⊂ G is a maximal torus, we let Σ(G, T) denote the root system of G with respect to T.

If S ⊂ G is a maximal F -split torus then the semisimple part of the centralizer C_{G}(S) is called the semisimple F -anisotropic kernel of G.

We number the simple roots of exceptional groups as in [Bourb68].

2 The Rost invariant and its properties

In the 90s, for a simple simply connected linear algebraic group G defined over a field F, M. Rost [R] constructed a cohomological H^{3}(ℚ/ℤ(2))–invariant of G which nowadays is called the Rost invariant of the group G. Thus, for any field extension K/F the Rost invariant associates a canonical map of pointed sets

R_{G}^{K} : H^{1}(K, G) → H^{3}(K, ℚ/ℤ(2)).

By abuse of language, instead of R_{G}^{K} we will use the abbreviated notation R_{G} for the Rost invariant whenever there is no danger of confusion.

In this section we will state (without proofs) properties of the Rost invariant used in Section 6. For the proofs we refer to [R], [KMRT98] (§31), [EKLV98], [G00], [M01] (Appendix A and B). We note that an explicit formula for R_{G} is known only in a few cases. But fortunately that will do for us to prove triviality of the kernel of the Rost invariant for quasi-split groups E_{6}, E_{7}.

2.1 Inner type A_{n}

Let A be a central simple algebra over F of exponent e and degree n+1. Let G = SL_{1}(A). Assume that the characteristic of F doesn’t divide n+1. Then H^{1}(F, G) = F^{×}/ Nrd (A^{×}) and the formula for the Rost invariant R_{G} is given (up to a sign) by

R_{G} : F^{×}/Nrd (A^{×}) ⟶ H^{3}(F, μ_{e}^{⊗2}), R_{G} (x Nrd (A^{×})) = (x) ∪ [A].

Theorem 2.1 ([MS82], Theorem 12.2) If n + 1 is square-free, then R_{G} is injective.

2.2 Spinor groups

Let f be a quadratic form over a field F of characteristic ≠ 2 of dimension at least 5 and let G = Spin(f). The set H^{1}(F, G) fits into the exact sequence

H^{1}(F, Spin(f)) ⟶^{π} H^{1}(F, SO(f)) ⟶ _{2}BrF.

It follows that Im π consists of classes of quadratic forms having the same dimension, discriminant and Hasse-Witt invariant as f. Therefore, for any cocycle ξ ∈ Z^{1}(F, Spin(f)) we obtain that π([ξ]) − [f] ∈ I^{3}F and the formula for the Rost invariant is given by

R_{Spin(f)} : H^{1}(F, Spin(f)) ⟶ H^{3}(F, μ_{2}^{⊗2}), R_{Spin(f)}([ξ]) = e_{3}(π([ξ]) − [f]). (1)

Here e_{3} is the Arason invariant [Ar75].

Proposition 2.2 Let f be a quadratic form of dimension ≤ 12 over a field F such that G = Spin(f) is quasi-split over F, i.e. f is of maximal Witt index. Then R_{G} has trivial kernel.

Proof. The statement follows from (1) and the Arason-Pfister Hauptsatz [L73], X.3.1.

2.3 Outer forms A_{n}

Let K/F be a quadratic extension, V a vector space of dimension n + 1 over K and h a hermitian form on V. The group SU(V, h) of isometries with determinant 1 is an almost simple simply connected group of type A_{n} defined over F and splitting over K. The hermitian form h corresponds to the quadratic form q on the vector space V viewed as an F -vector space: q is given by the formula q(v) = h(v, v). It easily follows that we have a natural embedding SU(V, h) ↪ SO(V, q) which can be lifted to an embedding SU(V, h) ↪ Spin(V, q), since SU(V, h) is simply connected. It turns out that the Rost invariant for SU(V, h) is just the restriction of R_{Spin}.

Proposition 2.3 Let h be a quasi-split hermitian form of dimension ≤ 6. Then the Rost invariant R_{SU(V,h)} has trivial kernel.

Proof. Apply the same argument as in Proposition 2.2.

2.4 The Rost numbers

Let G, H be almost simple simply connected linear algebraic groups over F and let ρ : H ↪ G be an F -embedding. The restriction R_{G} at ρ(H) gives a cohomological invariant of H, hence is equal to n_{ρ}R_{H} for a positive integer n_{ρ} (see [KMRT98], §31). The smallest such integer is called the Rost number of embedding ρ.

Proposition 2.4 Let ρ : H ↪ G be an F -embedding such that ρ(H) is a standard subgroup. Assume that all roots of the root system Σ = Σ(G, T) of G with respect to a maximal torus T have the same length. Then n_{ρ} = 1; in particular, for any [ξ] ∈ H^{1}(F, H) one has R_{G}(ρ([ξ])) = R_{H}([ξ]).

Proof. Let T be a maximal F -torus of G normalizing ρ(H). Then ρ(H) · T is a reductive subgroup of G. Since T is a maximal torus of ρ(H) · T and all maximal tori are conjugate, we easily obtain that the connected component S of the intersection T ∩ρ(H) is a maximal torus of ρ(H). Since T normalizes ρ(H), there is a natural embedding

Σ(ρ(H), S) ↪ Σ(G, T).

Hence the coroots of ρ(H) are also the coroots of G (we used the fact that all roots of the root system Σ = Σ(G, T) have the same length). The rest of the proof follows from [M01], Appendix B.

3 Steinberg’s theorem

We state now two theorems which are due to Steinberg [St65]. Although they are not formulated explicitly in [St65], their proofs can be easily obtained from arguments contained in [St65], §10 (see also [PR94], Propositions 6.18, 6.19, p. 338–339).

Let G_{0} be a simple (not necessary simply connected) linear algebraic group split or quasi-split over F. Let ξ ∈ Z^{1}(F, G_{0}) be a cocycle and let G = ^{ξ}G_{0} be the corresponding twisted group.

Theorem 3.1 For any maximal torus S ⊂ G over F there is an F -embedding S ↪ G_{0} such that the class [ξ] lies in the image of H^{1}(F, S) → H^{1}(F, G_{0}).

Theorem 3.2 In the notation above assume that G_{0} is a simple simply connected group split over F and that G is isotropic over F. Then ξ is equivalent to a cocycle with coefficients in a proper semisimple simply connected F -split subgroup of G_{0} which is standard and isomorphic over F^{s} to a semisimple anisotropic kernel of G.

Proof. Let S_{1} ⊂ G be a maximal F -split torus. Then the semisimple part of its centralizer C_{G}(S_{1}) is a semisimple simply connected F -anisotropic subgroup of G (see [T66]). Let S ⊂ G be a maximal torus over F containing S_{1}. By Theorem 3.1, there is an F -embedding ϕ : S ↪ G_{0} such that

[ξ] ∈ Im [H^{1}(F, S) → H^{1}(F, G_{0})].

The centralizer C_{G_{0}}(ϕ(S_{1})) ⊂ G_{0} is a reductive subgroup of G_{0} and it follows from the construction of ϕ (see the proof of Proposition 6.18 in [PR94], p. 339) that the groups C_{G_{0}}(ϕ(S_{1})) and C_{G}(S_{1}) are isomorphic over F^{s}. In particular,

H = [C_{G_{0}}(ϕ(S_{1})), C_{G_{0}}(ϕ(S_{1}))]

is a simply connected semisimple algebraic group over F.

The group H is clearly standard. It is split over F, since any F -split subtorus in G_{0}, and in particular ϕ(S_{1}), lies in a maximal F -split torus. Furthermore, from the exact sequence

H^{1}(F, H) ⟶ H^{1}(F, C_{G_{0}}(ϕ(S_{1}))) ⟶ H^{1}(F, C_{G_{0}}(ϕ(S_{1}))/H) = 1

we see that ξ is equivalent to a cocycle with coefficients in H.

4 Groups splitting over a quadratic extension

Throughout this section F denotes a field of characteristic ≠ 2, K = F (\sqrt{d}) its quadratic extension and τ the non-trivial automorphism of K/F.

Let G be a simple simply connected algebraic group of rank n defined over F and splitting over K. By Lemma 6.17 in [PR94], p. 329, there is a Borel subgroup over K such that B∩τ(B) = T is a maximal torus. We remark that this lemma in [PR94] is proved under the condition that char(F) = 0. However the same proof with trivial modifications works in the case of positive characteristic.

The torus T is F -defined and splitting over K, since any K-torus in B splits over K. For our purposes it suffices to treat only the case where T is an F -anisotropic torus.

4.1 The structure of G(K)

Let 𝔤 be the Lie algebra of G and let Σ = Σ(T, G) be the root system of G relative to T. The Borel subgroup B determines an ordering of Σ, hence the system of simple roots Π = {α_{1}, …, α_{n}}. Let Σ^{+} (resp. Σ^{−}) be the set of positive (resp. negative) roots and let B^{−} be the Borel subgroup opposite to B with respect to T. We pick a Chevalley basis [St67]

{H_{α_{1}}, … H_{α_{n}}, X_{α}, α ∈ Σ}

in 𝔤 corresponding to the pair (T, B). This basis is unique up to signs and automorphisms of 𝔤 which preserve B and T (see [St67], §1, Remark 1).

Since G is a Chevalley (in other words, quasi-split) group over K, its K-structure is well known. For details and proofs of all standard facts about G(K) used in this section we refer to [St67]. In particular, we refer to [St67] for the definition of the operators X_{α}^{n} /n! in the case char(F) = p > 0 (see also [SGA3]). Recall only that G(K) is generated by the so-called root subgroups U_{α} = ⟨x_{α}(u) | u ∈ K⟩, where α ∈ Σ and

x_{α}(u) = ∑_{n=0}^{∞} u^{n}X_{α}^{n} /n! ,

and T is generated by the one-parameter subgroups

T_{α} = T ∩ G_{α} = ⟨ h_{α}(t) | t ∈ K^{∗} ⟩.

Here G_{α} is the subgroup generated by U_{±α} and

h_{α}(t) = w_{α}(t) w_{α}(1)^{−1}, w_{α}(t) = x_{α}(t) x_{−α}(−t^{−1}) x_{α}(t).

Furthermore, since G is a simply connected group, the following relations hold in G (cf. [St67], Lemma 28 b), Lemma 20 c) ):

(A) T = T_{α_{1}} × ⋯ × T_{α_{n}};

(B) for any two roots α, β ∈ Σ we have

h_{α}(t) x_{β}(u) h_{α}(t)^{−1} = x_{β}(t^{⟨β,α⟩}u)

where ⟨β, α⟩ = 2 (β, α)/(α, α).

If Δ ⊂ Σ^{+} is a subset, we let G_{Δ} denote the subgroup generated by U_{±α}, α ∈ Δ.

4.2 Galois descent data

We shall now describe explicitly the F -structure of G, i.e. the action of τ on the generators {x_{α}(u), α ∈ Σ}.

Lemma 4.1 τ(α) = −α for any α ∈ Σ.

Proof. The character α + τ(α) of T is F -defined, hence it is zero, since by our assumption T is F -anisotropic.

Lemma 4.2 τ(B) = B^{−}.

Proof. The statement follows immediately from Lemma 4.1.

Lemma 4.3 T_{α} ≃ R_{K/F}^{(1)} (G_{m,K}).

Proof. T_{α} is a one-dimensional F -torus splitting over K. According to Lemma 4.1, τ acts on its character lattice by multiplication by −1. So the result follows.

Let α ∈ Σ. Since, by Lemma 4.1, τ(α) = −α, there exists a constant c_{α} ∈ K^{×} such that τ(X_{α}) = c_{α}X_{−α}. It follows that the action of τ on G(K) is determined completely by the family {c_{α}, α ∈ Σ}. We call these constants structure constants of G with respect to T. We summarize their properties in the following two lemmas.

Lemma 4.4 Let α ∈ Σ. Then we have

(i) c_{−α} = c_{α}^{−1}; (ii) c_{α} ∈ F^{×}; (iii) if β ∈ Σ is a root such that α + β ∈ Σ, then c_{α+β} = ±c_{α} c_{β}; in particular, the family {c_{α}, α ∈ Σ} is determined completely by its subfamily {c_{α_{1}}, …, c_{α_{n}}}.

Proof. (i) Apply τ to the equality [X_{α}, X_{−α}] = H_{α} and use the fact that τ(H_{α}) = −H_{α}. (ii) One has X_{α} = τ^{2}(X_{α}) = τ(c_{α})c_{−α}X_{α}, hence τ(c_{α})c_{−α} = 1. Substituting c_{−α} = c_{α}^{−1} we obtain τ(c_{α}) = c_{α}, as required. (iii) Apply τ to the equality [X_{α}, X_{β}] = ±(r + 1)X_{α+β}. Here r is an integer depending on α, β only.

Remark 4.5 The number r above is the biggest integer such that

β, β − α, …, β − rα

are roots, but β − (r + 1)α is not a root. Therefore, this number is equal to 0 or 1, since, by our assumption, α + β is a root. It follows, in particular, that r + 1 ≠ 0, since we assumed that char(F) ≠ 2.

Lemma 4.6 τ[ x_{α}(u) ] = x_{−α}(c_{α}τ(u)) for any u ∈ K and α ∈ Σ.

Proof. This follows from the equality x_{α}(u) = exp (uX_{α}).

4.3 Comparison of different Galois descent data

The family {c_{α}, α ∈ Σ} determining the action of τ on G(K) depends on the chosen Borel subgroup B and the corresponding Chevalley basis. Given another Borel subgroup and Chevalley basis we get another family of constants and we want now to describe the relation between the old ones and the new ones.

Let B′ ⊂ G be a Borel subgroup over K such that the intersection T′ = B′ ∩ τ(B′) is a maximal and F -anisotropic torus. Both tori T and T′ are isomorphic over F since, by Property (A) in 4.1 and Lemma 4.3, they are the direct products of n copies of the torus R_{K/F}^{(1)} (G_{m,K}). Furthermore, there exists an F -isomorphism T → T′ preserving positive roots, i.e. which takes (Σ′)^{+} = Σ(G, T′)^{+} into Σ^{+} = Σ(G, T)^{+}. Any such isomorphism can be extended to an inner automorphism

i_{g} : G ⟶ G, x → g x g^{−1}

for some g ∈ G(F^{s}), which takes B into B′ ( see [Hum75], Theorem 32.1 ).

Lemma 4.7 g can be chosen in G(K).

Proof. Since the restriction i_{g} |_{T} is an F -defined isomorphism we easily get that t_{σ} = g^{−1+σ} ∈ T (F^{s}) for any σ ∈ Gal (F^{s}/F). Consider the cocycle ξ = (t_{σ}) ∈ Z^{1}(F, T). Since T splits over K, res_{K}(ξ) viewed as a cocycle in T is trivial, by Hilbert’s Theorem 90. It follows that there is z ∈ T (F^{s}) such that t_{σ} = z^{1−σ}, σ ∈ Gal (F^{s}/K). Then the element g′ = gz is stable under Gal (F^{s}/K) and clearly g′B(g′)^{−1} = B′.

Let g be an element from Lemma 4.7 and let t = g^{−1+τ}. Since t belongs to T (K), it can be written (cf. 4.1) as a product t = h_{α_{1}}(t_{1}) ⋯ h_{α_{n}}(t_{n}), where t_{1}, …, t_{n} ∈ K^{×} are some parameters. Using the equality t τ(t) = 1 and the fact that τ acts on characters of T as multiplication by −1 one can easily see that t_{1}, …, t_{n} ∈ F^{×}. Consider the set

{H′_{α_{1}} = gH_{α_{1}}g^{−1}, …, H′_{α_{n}} = gH_{α_{n}}g^{−1}, X′_{α} = gX_{α}g^{−1}, α ∈ Σ}

which is a Chevalley basis related to the pair (T′, B′). Let {c′_{α}, α ∈ Σ} be the corresponding structure constants.

Lemma 4.8 For each α ∈ Σ one has c′_{α} = t_{1}^{−⟨α,α_{1}⟩} ⋯ t_{n}^{−⟨α,α_{n}⟩} · c_{α}.

Proof. Apply τ to the equality X′_{α} = gX_{α}g^{−1} and use relation (B) in 4.1.

The converse is also true.

Lemma 4.9 Let g ∈ G(K) be an element such that t = g^{−1+τ} ∈ T (K). Then T′ = gT g^{−1} is an F -defined maximal torus splitting over K and the restriction of the inner automorphism i_{g} to T is an F -defined isomorphism. The structure constants {c′_{α}} related to T′ are given by the formulas in Lemma 4.8.

Proof. This is clear.

Remark 4.10 The elements g ∈ G(K) such that g^{−1+τ} ∈ T (K) correspond to the elements of the kernel of μ : H^{1}(K/F, T (K)) → H^{1}(K/F, G(K)). The exact sequence in Galois cohomology attached to the exact sequence

1 ⟶ T (K) ⟶ G(K) ⟶ (G/T)(K) ⟶ 1

shows that they come from F -points of the homogeneous variety G/T. It was shown in [Cher89] that G/T is an F -rational variety, hence all such elements can be parametrized by points from an open subset of an affine space.

Remark 4.11 The restriction of i_{g} to T induces a bijection between H^{1}(K/F, T (K)) and H^{1}(K/F, T′(K)). If we identify these two sets (recall that T and T′ are isomorphic over F), then this bijection is the translation by the class of the cocycle ξ = (t_{τ}) ∈ Z^{1}(K/F, T (K)), where t_{τ} = g^{−1+τ} .

4.4 Type A_{n}

It follows from Lemma 4.6 that, for each root α ∈ Σ, the subgroup G_{α} of G is defined over F. Since it is a simple simply connected group of rank 1, we obtain that G_{α} ≃ SL(1, D_{α}), where D_{α} is a quaternion algebra over F.

Lemma 4.12 D_{α} ≃ (d, c_{α}).

Proof. Since T_{α} = R_{K/F}^{(1)} (G_{m,K}) ⊂ SL(1, D_{α}), the algebra D_{α} splits over K. Hence it is of the form D_{α} = (d, f), where f ∈ F^{×}, and we need to prove that

c_{α} ≡ f (mod N_{K/F} (K^{×})).

Let i, j ∈ D^{×} be elements such that i^{2} = d, j^{2} = f, ij = −ji. We may identify T_{α} with R_{K/F}^{(1)} (F (i)^{×}). A straightforward computation shows that the triple

H′_{α} = \frac{1}{\sqrt{d}} i, X′_{α} = \frac{−\sqrt{d} + i}{−2f\sqrt{d}} j, X′_{−α} = \frac{\sqrt{d} + i}{2\sqrt{d}} j

belonging to D⊗_{F} K is a Chevalley basis with respect to T_{α}. Note that under the standard identification D ⊗_{F} K ≃ M_{2}(K) the triple H′_{α}, X′_{α}, X′_{−α} corresponds to the matrices

( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ) , ( \begin{matrix}0 & 1 \\ 0 & 0 \end{matrix} ) , ( \begin{matrix}0 & 0 \\ 1 & 0 \end{matrix} ) .

From the above formulas for X_{α} and X′_{α} we easily get τ(X′_{α}) = fX′_{−α}.

Let us now come back to our former Chevalley basis H_{α}, X_{α}, X_{−α}. Recall that a Chevalley basis is unique up to an automorphism of the corresponding Lie algebra sl_{2}(K). It follows that there is an inner automorphism i_{d} of D_{K} which takes H′_{α}, X′_{α}, X′_{−α} into H_{α}, X_{α}, X_{−α}. Since H_{α} and H′_{α} are proportional (being elements of the Lie algebra of T_{α}), d lies in the Weyl group of T_{α}. Then X_{α} is proportional to either X′_{α} or X′_{−α}. Without loss of generality we may assume that X_{α} is proportional to X′_{α}. Then we have X_{α} = zX′_{α}, X_{−α} = z^{−1}X′_{−α} for some z ∈ K^{×} and hence

c_{α}X_{−α} = τ(X_{α}) = τ(z)τ(X′_{α}) = τ(z)fX′_{−α} = zτ(z)fX_{−α},

implying c_{α} = zτ(z)f, as required.

In conclusion of this subsection we consider outer forms of type ^{2}A_{n−1} splitting over a quadratic extension. Let

h = a_{1}x_{1}x_{1}^{τ} − a_{2}x_{2}x_{2}^{τ} + … + (−1)^{n−1}a_{n}x_{n}x_{n}^{τ}

be a hermitian form of dimension n relative to the quadratic extension K/F, where a_{1}, …, a_{n} are elements in F^{×}. Let T ⊂ SU(h) be a maximal F -torus consisting of all diagonal matrices in SU(h). Clearly, T is F -anisotropic and splits over K. As above, we denote the corresponding structure constants related to T and the standard Chevalley basis of SU(h)_{K} ≃ SL_{n,K} by {c_{α}, α ∈ Σ = Σ(SU(h), T)}.

Lemma 4.13 One has the following:

c_{α_{1}} ≡ a_{1}/a_{2}, c_{α_{2}} ≡ a_{2}/a_{3}, …, c_{α_{n−1}} ≡ a_{n−1}/a_{n} (mod N_{K/F} (K^{×})).

Proof. The subgroup G_{α_{i}} ⊂ SU(h) coincides with SU(h_{i}) ⊂ SU(h), where

h_{i} = a_{i}x_{i}x_{i}^{τ} − a_{i+1}x_{i+1}x_{i+1}^{τ}.

Since SU (h_{i}) ≃ SL (1, D_{α_{i}}) and D_{α_{i}} = (d, a_{α_{i}}/a_{α_{i+1}}), the result follows from Lemma 4.12.

4.5 A cohomological property

Proposition 4.14 The natural map H^{1}(K/F, T (K)) → H^{1}(K/F, G(K)) is surjective.

Proof. We are under the conditions of Lemma 6.28 in [PR94], p. 369, so the result follows from that lemma.

5 Groups E_{6}, E_{7} splitting over cubic extensions.

Throughout this section F denotes a field of characteristic ≠ 3, K/F a cyclic extension of degree 3 and τ a non-trivial automorphism of K/F. We also assume that Γ = Gal(F^{s}/F) is a pro-3-group. We let G_{0} denote a simple simply connected split group over F of type E_{6}, E_{7}.

Theorem 5.1 Any cocycle ξ ∈ Z^{1}(K/F, G_{0}) is equivalent to a cocycle with coefficients in a standard simple simply connected F -subgroup H < G_{0} of inner type A_{2}.

Proof. We split our proof into a sequence of simple observations. The main ingredients are the classification of Tits indices [T66], Steinberg theorem [St65] and the fact that Γ is a pro-3-group; in particular there are no simple F -anisotropic groups of types A_{1}, A_{3}, B_{n}, C_{n}, ^{1,2}D_{n}, G_{2}. We let G = ^{ξ}G_{0} denote the corresponding twisted group. Consider first the key case where G is an F -anisotropic group of type E_{6}.

5.1 Type E_{6}: anisotropic case

5.1.1 Construction of a special torus

Proposition 5.2 G contains a maximal F -defined torus S splitting over K.

Proof. We follow Harder’s arguments [H65, H66] (cf. [PR94], Chapter 6). In this part G is not necessary a strongly inner form. Let us start with the construction of a simple simply connected F -subgroup of G of type ^{3}D_{4} splitting over K. Since G splits over K, we choose a K-split maximal torus T. Let Σ = Σ(G, T) and let Π = {α_{1}, …, α_{6}} ⊂ Σ be a basis. Let 𝒫 ⊂ G be the standard parabolic subgroup over K corresponding to α_{6}.

Consider the connected component C of the intersection 𝒫 ∩ τ(𝒫) ∩ τ^{2}(𝒫). It is defined over F, hence reductive, since G is F -anisotropic. Denote the central torus of C by S_{1} and let C^{(1)} = [C, C] = C_{1} · ... · C_{s} be the decomposition of the semisimple part of C into an almost direct product of the simple components over F^{s}. It easily follows from dim C ≥ 30 that C^{(1)} contains a simple component, say C_{1}, of type not A_{n} (see Lemma 6.32 in [PR94], p. 380).

Lemma 5.3 C_{1} is defined over F.

Proof. If C_{1} is defined over L/F, then there is a canonical embedding H_{1} = R_{L/F} (C_{1}) ↪ C, since C is F -defined. Consider a Levi subgroup ℒ ⊂ 𝒫 over K containing C. The parabolic subgroup 𝒫 corresponds to α_{6}, so the semisimple part [ℒ, ℒ] of ℒ has type D_{5}. It follows then from the inclusion [C, C] ⊂ [ℒ, ℒ] that

rank H_{1} = [L : F] · rank C_{1} ≤ 5.

But [L : F] is a power of 3 and C_{1} is not of type A_{n}, hence [L : F] = 1.

Lemma 5.4 C_{1} has type ^{3}D_{4}.

Proof. The conditions C_{1} is F -anisotropic and rank C_{1} ≤ 5 imply that C_{1} is of type either F_{4} or D_{4}. A group of type F_{4} being a group of dimension 52 can not be embedded into the group [ℒ, ℒ] of type D_{5}, since dim [ℒ, ℒ] = 45. So the result follows.

Lemma 5.5 C^{(1)} = C_{1}.

Proof. If C^{(1)} contains another simple component C_{2}, then its rank is 1, since C^{(1)} has rank at most 5. It follows that C_{2} is defined over F and hence F -isotropic being a group of type A_{1} — a contradiction.

Lemma 5.6 S_{1} ≃ R_{K/F}^{(1)} (G_{m,K}).

Proof. We have dim C ≥ 30. On the other hand, C^{(1)} = [C, C] is a group of dimension 28 and rank 4. Hence dim S_{1} = 2. According to the classification of 2-dimensional tori [V98] it suffices now to show that S_{1} is K-split or, what is the same, K-isotropic, since Γ is a pro-3-group.

Let ℒ be a Levi subgroup of 𝒫 over K containing C. Both groups ℒ and C are reductive and have the same rank 6. It follows that the central torus A of ℒ is contained in C. Since it commutes with [ℒ, ℒ] which contains [C, C] = C_{1}, we conclude that A ⊂ S_{1}. It remains to note that A is K-split.

Lemma 5.7 C_{1} contains a maximal torus S_{2} isomorphic to R_{K/F}^{(1)} (G_{m,K})×R_{K/F}^{(1)} (G_{m,K}).

Proof. Consider the following chain of inclusions:

C = S_{1} · C_{1} ⊂ C_{G}(S_{1}) ⊂ C_{G}(A) = ℒ.

It gives

[C, C] ⊂ [C_{G}(S_{1}), C_{G}(S_{1})] ⊂ [ℒ, ℒ].

Since [C, C] and [ℒ, ℒ] have type D_{4} and D_{5} respectively and [C_{G}(S_{1}), C_{G}(S_{1})] is a group of rank 4, we conclude that [C_{G}(S_{1}), C_{G}(S_{1})] = [C, C] = C_{1}.

Since S_{1} splits over K, so are C_{G}(S_{1}) and [C_{G}(S_{1}), C_{G}(S_{1})] = C_{1}. Thus C_{1} is an F -group of type D_{4} splitting over K. Then arguing just as in [PR94], p. 371–372, we obtain that C_{1} contains a torus of the required form.

Lemmas 5.6 and 5.7 show that S = S_{1} · S_{2} is an F -defined maximal torus of G splitting over K. Proposition 5.2 is proved.

Let S be the torus constructed in Proposition 5.2. According to Theorem 3.1 there exists an F -embedding S ↪ G_{0} such that

[ξ] ∈ Im [H^{1}(F, S) ⟶ H^{1}(F, G_{0})].

We fix this embedding and starting from this point we may forget about the way it was constructed. The only fact we need to know is that S is an almost direct product of three copies of R_{K/F}^{(1)} (G_{m,K}). Based on this fact we are going to show that there exists a decomposition

S ≃ R_{K/F}^{(1)} (G_{m,K}) × R_{K/F}^{(1)} (G_{m,K}) × R_{K/F}^{(1)} (G_{m,K})  (2)

and a new F -embedding η : S ↪ G_{0} with the following properties:

(a) the images of the first two components in decomposition (2) lie in an F -split standard subgroup of G_{0} isomorphic to SL_{3} × SL_{3},

(b) ξ is equivalent to a cocycle with coefficients in η(S).

5.1.2 The Weyl group W_{E_{6}}

Let Σ = Σ(G_{0}, S) be the root system of G_{0} with respect to S, W = W (Σ) the Weyl group of Σ and Π = {α_{1}, …, α_{6}} a fixed basis of Σ.

Lemma 5.8 Let W_{3} ⊂ W be a 3-Sylow subgroup. Then

W_{3} ≃ (ℤ/3 × ℤ/3 × ℤ/3) ⋊ ℤ/3,

where ℤ/3 acts by permuting the components of ℤ/3 × ℤ/3 × ℤ/3.

Proof. We have # (W_{3}) = 3^{4}. Let Σ_{1}, Σ_{2} and Σ_{3} be the three subroot systems of Σ of type A_{2} generated by roots ⟨α_{1}, α_{3}⟩, ⟨α_{5}, α_{6}⟩ and ⟨α_{2}, −β⟩ respectively, where β ∈ Σ is the positive root of maximal length with respect to the basis Π. Let w_{0}, w_{1} ∈ W be the elements of maximal length with respect to the basis {α_{1}, …, α_{6}} and {α_{1}, α_{3}, α_{4}, α_{2}, −β, α_{5}} respectively.

It is easy to see that w = w_{0} · w_{1} has order 3 and takes the roots α_{1}, α_{3}, α_{5}, α_{6}, α_{2}, −β into α_{6}, α_{5}, α_{2}, −β, α_{3}, α_{1} respectively. Therefore w permutes the components of the subsystem Σ′ = Σ_{1} × Σ_{2} × Σ_{3} ⊂ Σ and their Weyl groups. Let v_{1} ∈ W (Σ_{1}) be an arbitrary element of order 3. Then v_{2} = wv_{1}w^{−1} ∈ W (Σ_{2}) and v_{3} = wv_{2}w^{−1} ∈ W (Σ_{3}). It follows that v_{1}, v_{2}, v_{3} commutes and w permutes them; in particular, v_{1}, v_{2}, v_{3}, w generate a subgroup W_{3} ⊂ W of order 3^{4}.

Since S is an F -defined maximal torus splitting over K, the group Gal (K/F) acts on the root system Σ. Thus we have a natural embedding Gal (K/F) ↪ W. Choosing an appropriate basis of Σ we may assume that the image of Gal (K/F) lies in the 3-Sylow subgroup W_{3} constructed in Lemma 5.8.

Lemma 5.9 The image of Gal (K/F) lies in a subgroup ⟨v_{1}, v_{2}, v_{3}⟩ of W_{3}.

Proof. Let τ = v_{1}^{i_{1}}v_{2}^{i_{2}}v_{3}^{i_{4}}w^{i_{4}}, where i_{1}, …, i_{4} are integers. Since S is F -anisotropic, we have α_{1} + τ(α_{1}) + τ^{2}(α_{1}) = 0. On the other hand, assuming that i_{4} ≠ 0 we easily obtain that τ(α_{1}) ∈ Σ_{2} and τ^{2}(α_{1}) ∈ Σ_{3}. But then α_{1} + τ(α_{1}) + τ^{2}(α_{1}) ≠ 0 — a contradiction.

5.1.3 Special embedding S ↪ G_{0}

Lemma 5.9 shows that Σ_{1}, Σ_{2}, Σ_{3} are stable under the action of Gal (K/F), hence the subgroup G_{Σ_{1}} · G_{Σ_{2}} · G_{Σ_{3}} ⊂ G_{0} is F -defined and of type A_{2} × A_{2} × A_{2}. Note also that the three intersections S ∩G_{Σ_{i}} are 2-dimensional tori isomorphic to R_{K/F}^{(1)} (G_{m,K}). This follows from the facts that they are F -anisotropic, splitting over K and from the classification of 2-dimensional tori. Clearly, the tori S ∩ G_{Σ_{i}}, i = 1, 2, 3, generate S.

We are now ready to construct a new embedding S ↪ G_{0} with the required properties (a) and (b). Let T ⊂ G_{0} be a maximal F -split torus and let {β_{1}, …, β_{6}} be a basis of the root system Σ(G_{0}, T). We consider the subsystems Σ′_{1}, Σ′_{2}, Σ′_{3} generated by {β_{1}, β_{3}}, {β_{5}, β_{6}}, {β_{2}, −β′} respectively. Here β′ is the root of maximal length with respect to the basis {β_{1}, …, β_{6}}. Since G_{Σ′_{1}}, G_{Σ′_{2}} and G_{Σ′_{3}} are isomorphic to SL_{3} over F,

there are F -embeddings ϕ_{i} : R_{K/F}^{(1)} (G_{m,K}) ↪ G_{Σ′_{i}}, i = 1, 2, 3. Then the torus

S′ = Im ϕ_{1} · Im ϕ_{2} · Im ϕ_{3}

is maximal in G_{0} isomorphic to S over F.

Proposition 5.10 [ξ] ∈ Im [H^{1}(F, S′) ⟶ H^{1}(F, G_{0})].

Proof. It follows from the constructions of S and S′ that there is an F -defined isomorphism S → S′ which takes Σ(G_{0}, S) into Σ(G_{0}, S′). Any such isomorphism can be extended to an inner automorphism i_{g} : G_{0} → G_{0} over F^{s}, by Theorem 32.1 in [Hum75]. Since the restriction i_{g}|_{S} is F -defined, we obtain that σ(g)^{−1} · g ∈ S for any σ ∈ Gal (F^{s}/F). Let ξ = (a_{σ}), where a_{σ} ∈ S. Then the equivalent cocycle

ξ′ = (g · a_{σ} · σ(g)^{−1}) = (g · a_{σ} · (σ(g)^{−1}g) · g^{−1})

takes values in S′ and we are done.

5.1.4 The direct product decomposition

In view of Proposition 5.10 the torus S′ satisfies condition (b). Let us prove that it satisfies condition (a) as well. We do not need the torus S any more and to ease notation we will denote S′ by S. We also denote Σ(G_{0}, S′) by Σ and let Π = {α_{1}, …, α_{6}} be the corresponding basis of Σ.

For any root α ∈ Σ we have α + τ(α) + τ^{2}(α) = 0, since S is F -anisotropic. It follows that the set {±α, ±τ(α)} generates an F -defined subsystem of Σ of rank at most 2. This subsystem has the automorphism τ, which is of order 3, hence it has type A_{2}. In particular, the subgroup G_{{α,τ(α)}} is an F -defined inner form of type A_{2} splitting over K. For α ∈ Σ we let

S_{{α,τ(α)}} = S ∩ G_{{α,τ(α)}}.

Since S_{{α,τ(α)}} splits over K, we obtain that S_{{α,τ(α)}} ≃ R_{K/F}^{(1)} (G_{m,K}).

Recall also that, by construction, the three subsystems of Σ generated by the roots {α_{1}, α_{3}}, {α_{5}, α_{6}} and {α_{2}, −β} are stable under the action of Gal (K/F) and have type A_{2}. Hence

S_{{α_{1},τ(α_{1})}} = S_{{α_{1},α_{3}}}, S_{{α_{5},τ(α_{5})}} = S_{{α_{5},α_{6}}}.

Proposition 5.11 The product morphism S_{{α_{1},α_{3}}} × S_{{α_{5},α_{6}}} × S_{{α_{4},τ(α_{4})}} ⟶ S is an F -isomorphism.

Proof. It suffices to show that it is an isomorphism over F^{s}. We use notation from Steinberg’s book [St67]. Recall that given a root α ∈ Σ, one can associate to it the oneparameter subgroup S_{α} = ⟨ h_{α}(t) | t ∈ \overline{F}^{×} ⟩ ≤ S. By 4.1 (A), S is the direct product of such subgroups corresponding to α_{1}, …, α_{6}. If α = m_{1}α_{1} + ⋯ + m_{6}α_{6}, then

h_{α}(t) = h_{α_{1}}(t)^{m_{1}} ⋯ h_{α_{6}}(t)^{m_{6}},  (3)

since all roots have the same length.

It is clear that τ(α_{4}) ≠ ±β, since otherwise in view of the fact that the root subsystem Σ_{3} = ⟨ α_{2}, −β ⟩ is F -defined we would have α_{4} ∈ Σ_{3}. Therefore τ(α_{4}) is of the form

τ(α_{4}) = ±(m_{1}α_{1} + m_{3}α_{3} + ⋯ + m_{6}α_{6} + α_{2}),

where m_{1}, m_{3}, …, m_{6} are positive integers. Then it follows from (3) that

S = S_{α_{1}} × S_{α_{3}} × ⋯ × S_{α_{6}} × S_{τ(α_{4})} = S_{{α_{1},α_{3}}} × S_{{α_{5},α_{6}}} × S_{{α_{4},τ(α_{4})}},

as required.

5.1.5 Concluding argument

Proof of Theorem 5.1 in anisotropic case. By Proposition 5.10 we may assume that ξ ∈ Z^{1}(F, S). Let ξ = (a_{σ}), where a_{σ} ∈ S. Applying Proposition 5.11 we split ξ into the product of three cocycles ξ = ξ_{1} ·ξ_{2} ·ξ_{3} with coefficients in S_{{α_{1},α_{3}}}, S_{{α_{5},α_{6}}} and S_{{α_{4},τ(α_{4})}} respectively. Since G_{Σ_{1}} and G_{Σ_{2}} are isomorphic to SL_{3}, ξ_{1} and ξ_{2} are trivial cocycles.

Let g_{1} ∈ G_{Σ_{1}} and g_{2} ∈ G_{Σ_{2}} be such that ξ_{1} = (g_{1}^{−1+σ}) and ξ_{2} = (g_{2}^{−1+σ}). Clearly, g_{1} and g_{2} commutes. It follows that the equivalent cocycle

ξ′ = ((g_{1}g_{2}) a_{σ} (g_{1}g_{2})^{−σ}) = (g_{1}g_{2})^{σ} [(g_{2}^{−σ}g_{2})(g_{1}^{−σ}g_{1})a_{σ}] (g_{1}g_{2})^{−σ}

has coefficients in

(g_{1}g_{2})^{σ} S_{{α_{4},τ(α_{4})}} (g_{1}g_{2})^{−σ} ≤ (g_{1}g_{2})^{σ} G_{{α_{4},τ(α_{4})}} (g_{1}g_{2})^{−σ}.

The condition (g_{1}g_{2})^{−1} (g_{1}g_{2})^{σ} ∈ S easily implies that

H = (g_{1}g_{2})^{σ} G_{{α_{4},τ(α_{4})}} (g_{1}g_{2})^{−σ}  (4)

is an F -defined subgroup of type A_{2} splitting over K, hence we are done.

Remark 5.12 Let S ↪ G_{0} be the embedding constructed in 5.1.3. Proposition 5.10 says that the canonical map H^{1}(K/F, S) → H^{1}(K/F, G_{0}) is surjective. By Proposition 5.11, any cocycle ξ ∈ Z^{1}(K/F, G_{0}) can be written as a triple

ξ = (a_{1}N_{K/F} (K^{×}), a_{2}N_{K/F} (K^{×}), a_{3}N_{K/F} (K^{×})),

where a_{1}, a_{2}, a_{3} ∈ F^{×}. One can show that the above subgroup (4) is isomorphic to SL(1, T), where T is a cubic central simple algebra of the form T = (K/F, a_{1}^{n_{1}}a_{2}^{n_{2}}c_{α_{4}}) for certain integers n_{1}, n_{2} and a constant c_{α_{4}} ∈ F^{×}. These integers and the constant can be computed explicitly, but we omit details. As a result, we obtain the following formula for the Rost invariant:

R_{G_{0}}([ξ]) = T ∪ (a_{3}).

5.1.6 Moving Lemma

The above argument will be used below several times. Let us formulate it for the future reference as follows.

Moving Lemma Let G be a simple group over F and let T be a maximal torus of G over F. Let ξ ∈ Z^{1}(F, T) be a cocycle. Assume that ξ = ξ_{1} · ξ_{2} is the product of two cocycles with coefficients in T such that ξ_{1} takes values in T ∩ H, where H is a proper F -subgroup of G normalizing by T, and ξ_{2} viewed as a cocycle with coefficients in G is trivial. Then ξ is equivalent to a cocycle with coefficients in a proper F -subgroup of G of the same type as H. Proof. Let ξ_{1} = (a_{σ}), ξ_{2} = (b_{σ}), where σ ∈ Gal (F^{s}/F), a_{σ} ∈ T (F^{s}) ∩ H(F^{s}) and b_{σ} ∈ T (F^{s}). Since ξ_{2} is trivial, there is g ∈ G(F^{s}) such that b_{σ} = g^{−1+σ} for all σ ∈ Gal (F^{s}/F). Then the equivalent cocycle ξ′ = (c_{σ}), where

c_{σ} = g(a_{σ}b_{σ})g^{−σ} = ga_{σ}g^{−1},

takes values in a proper F -subgroup gHg^{−1} which has the same type as H.

5.2 Type E_{6}: isotropic case

This case is much easier.

Lemma 5.13 Let the twisted group G = ^{ξ}G_{0} be F -isotropic. Then it splits over F; in particular ξ is trivial.

Proof. Assume the contrary. According to [T66] all admissible Tits’ indices of type ^{1}E_{6} corresponding to isotropic groups are as follows:

[figure: (i); α_{1}; α_{3}; α_{4}; α_{5}; α_{6}; α_{2}; (ii); α_{1}; α_{3}; α_{4}; α_{5}; α_{6}; α_{2}]  (5)

In case (i) the semisimple anisotropic kernel of G is a classical group of type D_{4}. Since Γ is a pro-3-group, any such a group is F -split.

Consider case (ii). Applying Theorem 3.2 we obtain that ξ is equivalent to a cocycle with coefficients in SL_{3} × SL_{3}. Hence ξ is trivial and G is F -split.

5.3 Type E_{7}

Proposition 5.14 There exists no F -anisotropic groups of type E_{7} splitting over K.

Proof. Assume the contrary. Let G be an F -anisotropic simple group of type E_{7} splitting over K. To get a contradiction we can proceed as in case E_{6}. Namely, let 𝒫 ⊂ G be the parabolic subgroup over K corresponding to the root α_{7}. Then we consider the connected component C of the intersection 𝒫 ∩ τ(𝒫) ∩ τ^{2}(𝒫). As in case E_{6}, it is easy to conclude that C is a reductive group of dimension at least 52 whose semisimple part [C, C] contains an F -defined simple component C_{1} of type not A_{n}.

Let ℒ ⊂ 𝒫 be a Levi subgroup containing C_{1}. Its semisimple part has type E_{6}, hence C_{1} is a group of type either ^{3}D_{4}, F_{4}, or E_{6}. By a dimension argument, C_{1} cannot be of type ^{3}D_{4}. If C_{1} is of type E_{6}, then C_{1} is the semisimple part of ℒ. It follows that C_{G}(C_{1}) is defined over F. But it coincides with the central torus A of ℒ whose dimension is 1. Hence A is F -defined and F -isotropic being a torus of dimension 1.

Assume now that C_{1} is a group of type F_{4} and let us consider C_{G}(C_{1}). Since it contains A, C_{G}(C_{1}) is a nontrivial reductive group over F whose rank is either 1, 2 or 3.

Let S ⊂ C_{G}(C_{1}) be a maximal torus over F. It commutes with C_{1}, by construction, hence C_{1} ⊂ [C_{G}(S), C_{G}(S)].

If dim S = 1, then S is F -isotropic. If dim S = 3, then C_{1} = [C_{G}(S), C_{G}(S)] would be a standard subgroup of G of type F_{4}. But a root system of type E_{7} does not contain a subsystem of type F_{4}.

Thus dim S = 2 and C_{G}(S) is a reductive group whose semisimple part [C_{G}(S), C_{G}(S)] has rank 5 and contains the subgroup C_{1} of type F_{4}. It follows that [C_{G}(S), C_{G}(S)] contains another simple component over F of rank 1. But any such a group is F -isotropic.

We are now ready to complete the proof of Theorem 5.1 in the case of E_{7} by reducing it to E_{6}. By Proposition 5.14, G is F -isotropic. Looking at Tits tables [T66] and taking into consideration the fact that Γ is a pro-3- group we obtain that the only possibility for the semisimple F -anisotropic kernel of G is to be a group of type E_{6}. Applying Theorem 3.2 we get that ξ is equivalent to a cocycle with coefficients in an F -split standard subgroup of G_{0} of type E_{6}. So the result follows.

6 The triviality of the kernel of R_{G} for E_{6}, E_{7}

Theorem 6.1 Let G_{0} be a quasi-split simple simply connectedgroup of type E_{6}, E_{7} defined over a field F of characteristic ≠ 2, 3. Then Ker R_{G_{0}} = 1.

We split the proof into three parts considering inner forms of type E_{6}, outer forms of type E_{6} and type E_{7} separately. Let G_{0} be the given quasi-split group and let [ξ] be an element of Ker R_{G_{0}}. Let G = ^{ξ}G_{0} be the corresponding twisted group. The triviality of ξ is clearly equivalent to saying that G is quasi-split over F.

6.1 Inner forms of type E_{6}

Our strategy is to reduce to the case where G is an F -isotropic group and then to apply Theorem 3.2 and Proposition 2.4.

6.1.1 Reduction to a pro-p-case

It is entirely based on the following theorem which is due to P. Gille [G93], [G97].

Theorem 6.2 Let G be a simple simply connected group of type E_{6} or E_{7} over F. Assume that there exist finite field extensions E_{1}, …, E_{s} of F splitting G and such that the

g.c.d.{ [E_{1} : F], …, [E_{s} : F] } = 1.

Then G splits over F.

Let Γ_{p} ⊂ Γ be a Sylow p-subgroup and let F_{p} = (F^{s})^{Γ_{p}} be fixed points of Γ_{p}. Assuming that Theorem 6.1 is proved over F_{p} we can find for each p a finite extension E_{p}/F contained in F_{p} and splitting G. Since the g.c.d of all degrees [E_{p} : F] is clearly 1, we obtain from Gille’s theorem that G splits over F.

6.1.2 Reduction to the case p = 2, 3

By Theorem 3.1, we may assume that ξ takes values in an F -defined maximal torus T ⊂ G_{0}.

Lemma 6.3 Any element of the group H^{1}(F, T) has order of the form 2^{n}3^{m}.

Proof. See Proposition 6.21 in [PR94], p. 375.

Lemma 6.3 implies that in the case p ≠ 2, 3 any cocycle from Z^{1}(F, T) is trivial.

6.1.3 The case p = 3

Theorem 5.1 says that ξ is equivalent to a cocycle with coefficients in a standard simple simply connected F -subgroup of G_{0} of type A_{2}. Then, by Theorem 2.1 and Proposition 2.4, ξ is trivial.

6.1.4 The case p = 2

Assume first that G be F -isotropic. According to [T66], we have two possibilities for an F -anisotropic kernel of G:

(i) an inner form of type A_{2} × A_{2}. But Γ is a pro-2-group, hence there are no anisotropic inner forms of type A_{2} over F; (ii) an inner form of type D_{4}. In this case, by Theorem 3.2, ξ is equivalent to a cocycle with coefficients in a standard simple simply connected F -split subgroup of G_{0} of type D_{4}, hence it is trivial, by Propositions 2.2 and 2.4.

Let G be now F -anisotropic. Let E/F be a finite Galois extension which splits G and is minimal with this property. Since Γ is a pro-2-group, there is a chain of subfields

F = E_{0} ⊂ E_{1} ⊂ … ⊂ E_{s} = E

such that [E_{i+1} : E_{i}] = 2 and s ≥ 1. Since G is not split over E_{s−1} and the F -isotropic case has been already studied, G is anisotropic over that field. Thus it suffices to treat the case where G splits over a quadratic extension K/F. Then G has a maximal torus T defined over F and splitting over K (see § 4). Since G is a group of inner type, Gal (K/F) has the natural embedding into the Weyl group. We also know that the image of the nontrivial element of Gal (K/F) acts on the corresponding root system Σ(G, T) by multiplication by −1. But the Weyl group of type E_{6} does not contain −1: a contradiction.

6.2 Outer forms of type E_{6}

Let G_{0} be a quasi-split group of type E_{6} and let K = F (\sqrt{d}) be the quadratic extension over which G_{0} becomes split. Since the inner case has been already treated, we have Res_{K}([ξ]) = 1, hence G splits over K.

6.2.1 Isotropic case

Proposition 6.4 If G is F -isotropic, then [ξ] = 1.

Proof. According to Tits’ classification [T66] the F -index of G is one of the following:

[figure: (a); α_{2}; α_{4}; α_{3}; α_{1}; α_{6}; α_{5}; (b); α_{2}; α_{4}; α_{3}; α_{1}; α_{6}; α_{5}; (c); α_{2}; α_{4}; α_{3}; α_{1}; α_{6}; α_{5}; (d); α_{2}; α_{4}; α_{3}; α_{1}; α_{6}; α_{5}]

Lemma 6.5 The Tits index of G is not of the form (a), (b) or (c).

Proof. Let T ⊂ G be a maximal F -torus containing a maximal F -split torus of G. We fix a basis Π = {α_{1}, …, α_{6}} of Σ(G, T) corresponding to the above pictures. According to [T66] the maximal F -split subtorus of T is the identity component of the subgroup of T defined by the following system of equations:

\{ \begin{matrix}γ(t) & = & σ^{∗}(γ)(t) & for all γ ∈ Π and all σ ∈ Gal(F^{s}/F), \\ α_{i}(t) & = & 1 & for all non-distinguished vertices α_{i}, \end{matrix}

where t ∈ T and σ^{∗} denotes the ∗-action of Γ on Π (see [T66]). It follows from these equations and the above pictures that in all cases (a), (b) and (c) the maximal F -split subtorus of T contains the F -split one-dimensional subtorus S_{1} = ⟨h_{β}(u) | u ∈ (F^{s})^{×}⟩ corresponding to the root β of maximal length. Then the semisimple part of the centralizer C_{G}(S_{1}) is F -defined and coincides with the subgroup G_{Σ_{1}} ⊂ G generated by roots Σ_{1} = {α_{1}, α_{3}, …, α_{6}}. Let S_{2} ⊂ G_{Σ_{1}} be an arbitrary maximal F -defined torus and let S = S_{1} · S_{2}.

Arguing as in Theorem 3.2 it is easy to see that there is an F -embedding S ↪ G_{0} such that [ξ] ∈ Im [H^{1}(F, S) → H^{1}(F, G_{0})] and the image of S_{2} lies in a standard quasi-split simple simply connected F -subgroup H of type ^{2}A_{5}. Since S/S_{2} is an F -split torus, any cocycle in Z^{1}(F, S) is equivalent to a cocycle with coefficients in S_{2}. Thus ξ is equivalent to a cocycle with coefficients in H, hence ξ is trivial, by Propositions 2.3 and 2.4.

Lemma 6.6 A group with the Tits index (d) can not split over a quadratic extension. In particular, the Tits index of G is not of the form (d).

Proof. Let H be an F -isotropic group with the Tits index of the form (d). If H splits over a quadratic extension K/F, then so is its semisimple F -anisotropic kernel L. In case (d) L is an almost simple simply connected F -anisotropic group of type ^{2}D_{4}. Since L splits over a quadratic extension of the ground field, there are two possibilities.

Case I. L ≃ Spin (f), where f is an F -anisotropic quadratic form of dimension 8. Since L has type ^{2}D_{4}, the discriminant of f is nontrivial. On the other hand, L is F -anisotropic and splits over K, hence f is of the form

f ≃ a_{1}(x_{1}^{2} − dx_{2}^{2}) + a_{2}(x_{1}^{3} − dx_{4}^{2}) + a_{3}(x_{5}^{2} − dx_{6}^{2}) + a_{4}(x_{7}^{2} − dx_{8}^{2}),

where a_{1}, …, a_{4} ∈ F^{×}. It follows that f has trivial discriminant — a contradiction.

Case II. L ≃ Spin (D, f), where D is a quaternion algebra over F and f is a skewhermitian form over D (with respect to the standard involution on D). Let E be the function field of the Severi-Brauer variety corresponding to D. By a result of Parimala, Sridharan and Suresh [PSS99], L is still anisotropic over E and clearly has type ^{2}D_{4} over E. Then we can proceed to Case I to get a contradiction.

Lemmas 6.5 and 6.6 complete the proof of Proposition 6.4.

6.2.2 Anisotropic case

By Theorem 3.1, there is a maximal F -anisotropic torus S ⊂ G_{0} splitting over K and such that

[ξ] ∈ Im [H^{1}(F, S) → H^{1}(F, G_{0})].  (6)

Then according to Proposition 4.14 for any maximal F -anisotropic torus of G_{0} splitting over K, condition (6) still holds. We keep the notation introduced in Section 4. In particular, given a Chevalley basis of the Lie algebra of G_{0} with respect to S we denote the corresponding structure constants by {c_{α}, α ∈ Σ = Σ(G_{0}, S)}.

Lemma 6.7 There exists a maximal F -anisotropic torus S ⊂ G_{0} splitting over K and such that the structure constants are of the form

c_{α_{1}} ≡ 1, c_{α_{2}} ≡ 1, …, c_{α_{6}} ≡ 1 (mod N_{K/F} (K^{×})).

Proof. Let T ⊂ G_{0} be a centralizer of a maximal F -split torus in G_{0}. The Tits F -index of G_{0} is of the form

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

Here {α_{1}, …, α_{6}} is a basis of Σ(G_{0}, T). Let β ∈ Σ(G_{0}, T) be the root of maximal length with respect to the basis {α_{1}, …, α_{6}}. It follows from (7) that the subgroup G_{β} ⊂ G_{0} corresponding to β is an F -split group of rank 1, hence G_{β} ≃ SL_{2}. Let S_{1} ⊂ G_{β} be a maximal F -anisotropic torus of dimension 1 which is isomorphic to R_{K/F}^{(1)} (G_{m,K}).

The subgroup H = G_{Σ_{1}} ⊂ G_{0} corresponding to the set Σ_{1} = {α_{1}, α_{3}, … α_{6}} is a quasi-split F -group of type ^{2}A_{5}, hence it is F -isomorphic to SU (h), where

h = x_{1}x_{1}^{τ} − x_{2}x_{2}^{τ} + x_{3}x_{3}^{τ} − x_{4}x_{4}^{τ} + x_{5}x_{5}^{τ} − x_{6}x_{6}^{τ}.  (8)

We choose the F -anisotropic torus S_{2} ⊂ H = SU(h) consisting of all diagonal isometries of h and we let S = S_{1} · S_{2}. Since G_{β} commutes with H, S is a maximal F -anisotropic torus in G_{0} splitting over K.

Let Σ be the root system of G_{0} with respect to S. We may now forget about T and we work with the root system Σ related to S. In order to ease notation we denote a basis in Σ by the same letters {α_{1}, …, α_{6}}. We choose a Chevalley basis (with respect to S) of the Lie algebra of G_{0} and a basis {α_{1}, …, α_{6}} of Σ in such a way that its subset {α_{1}, α_{3}, …, α_{6}} is a basis of the root system of SU (h) with respect to S_{2}. Then by Lemma 4.13,

c_{α_{1}} ≡ 1, c_{α_{3}} ≡ 1, …, c_{α_{6}} ≡ 1 (mod N_{K/F} (K^{×})).

We now want to modify S (if necessary) in such a way that c_{α_{2}} ≡ 1 (mod N_{K/F} (K^{×})) as well.

Assume that c_{α_{2}} ≢ 1. Consider the cocycle ζ = (b_{τ}) ∈ Z^{1}(K/F, S_{1}(K)), where b_{τ} = h_{β}(c_{α_{2}}) and β is the root of Σ of maximal length with respect to the ordering determined by the basis {α_{1}, …, α_{6}}. Since G_{β} ≃ SL_{2}, ζ viewed as a cocycle with coefficients in G_{β} is trivial.

Let b_{τ} = g^{−1+τ} , where g ∈ G_{β}(K). We claim that the torus S′ = gSg^{−1} is as required. Indeed, by Lemma 4.9, S′ is defined over F and splits over K. Since g commutes with H, Lemma 4.8 shows that the constants c_{α_{1}}, c_{α_{3}}, …, c_{α_{6}} related to the tori S and S′ coincide, hence are as required. Since ⟨β, α_{2}⟩ = 1, Lemma 4.8 again shows that the constant c_{α_{2}} related to S′ equals 1.

Applying the Moving Lemma 5.1.6 we want next to modify our cocycle ξ, i.e. take an equivalent one, in such a way that the new one has coefficients in a simple simply connected F -group of type D_{5}.

To do so, let S be the torus constructed in Lemma 6.7. Let L ⊂ G_{0} be a subgroup generated by the simple roots α_{2}, …, α_{6} from the root system Σ(G_{0}, S). Clearly, L is an F -defined group of type D_{5} splitting over K. Since S splits over K, we have H^{1}(F, S) = H^{1}(K/F, S(K)). Hence ξ can be written in the form ξ = (s_{τ}), where

s_{τ} = h_{α_{1}}(u_{1}) ⋯ h_{α_{6}}(u_{6}),

and τ is the nontrivial automorphism of K/F. It easily follows from s_{τ} τ(s_{τ}) = 1 that u_{1}, …, u_{6} ∈ F^{×}.

Since c_{α_{1}} ≡ 1, we have G_{α_{1}} ≃ SL_{2} over F, by Lemma 4.12. Hence there is g ∈ G_{α_{1}}(K) such that g^{−τ+1} = h_{α_{1}}(u_{1}). Consider the equivalent cocycle ξ̃ = (s̃_{τ}), where

s̃_{τ} = g^{τ} h_{α_{1}}(u_{1}) ⋯ h_{α_{6}}(u_{6})g^{−1} = g h_{α_{2}}(u_{2}) ⋯ h_{α_{6}}(u_{6}) g^{−1}.

It has coefficients in L̃ = g L g^{−1}.

The group L̃ is a simple simply connected group of type D_{5} defined over F. It contains the maximal torus S̃_{2} = g S_{2} g^{−1} splitting over K. In order to stress that L̃ sits inside G_{0}, we number simple roots in Σ(L̃, S̃_{2}) as follows.

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

Lemma 6.8 The structure constants of L̃ with respect to the torus S̃_{2} are as follows:

c_{α_{2}} ≡ c_{α_{4}} ≡ c_{α_{5}} ≡ c_{α_{6}} ≡ 1 (mod N_{K/F} (K^{×})), c_{α_{3}} ≡ u_{1} (mod N_{K/F} (K^{×}));

in particular, L̃ has F -rank at least 2.

Proof. The first statement follows from Lemma 4.8, the second one from the fact that the subgroups G_{α_{2}} and G_{α_{5}} of L̃ corresponding to α_{2} and α_{5} split over F, by Lemma 4.12, and commutes.

Since L̃ splits over the quadratic extension K/F, there are two possibilities for L̃.

Lemma 6.9 Assume that L̃ ≃ Spin (f), where f is a quadratic form over F of dimension 10. Then L̃ is quasi-split.

Proof. We have to exclude the possibilities for the Witt index i of f to be i = 2, 3.

Assume that i = 2. Then f is the direct sum of two hyperbolic planes and a certain F -anisotropic quadratic form of dimension 6. It follows that the dimension of a maximal F -split torus in L̃ equals 2 and the semisimple part of its centralizer has type A_{3} = D_{3}.

On the other hand, all maximal F -split tori in L̃ are conjugate over F. Since G_{α_{2}} and G_{α_{5}} are split over F, they contain F -split tori of dimension 1, say T_{2} and T_{5}. Then simple calculations with roots show that the semisimple part of the centralizer CL̃(T_{2} · T_{5}) has type A_{1} × A_{1} — a contradiction.

Assume that i = 3. Let f_{an} be the anisotropic part of f. It has dimension 4 and splits over K, in particular f_{an} has trivial discriminant. It follows that the function field E of the projective quadric f_{an} = 0 splits f. Hence G_{0} has rank at least 5 over E which is impossible, since E does not contain K.

Since L̃ is quasi-split and R_{G_{0}}([ξ]) = RL̃([ξ̃]) = 1, Proposition 2.2 gives [ξ̃] = 1.

It remains to consider the case L̃ ≃ Spin (D, f), where D is a quaternion algebra over F and f is a skew-hermitian form over D (with respect to the standard involution on D). Let E be the function field of the Severi-Brauer variety of D. The extension E/F splits D and so we can reduce to the previous case if we show that the twisted group G = ^{ξ}G_{0} is still anisotropic over E.

Lemma 6.10 G is anisotropic over E.

Proof. Assume the contrary, i.e. G is E-isotropic. Then G is quasi-split over E, since the isotropic case has been already treated. Let U be the unipotent radical of a Borel subgroup B ⊂ G over E. The twisted group ξ̃(L̃) being F -anisotropic is still anisotropic over E, by [PSS99]. Hence the intersection U ∩ ξ̃(L̃) is trivial. On the other hand, we have dim ξ̃(L̃) = 45, dim U = 36, so dim U ∩ ξ̃(L̃) is at least 3.

6.2.3 A formula for the Rost invariant

We keep the notation from the previous subsection. Using the above material one can easily produce a formula for the Rost invariant for any cocycle ξ ∈ Z^{1}(K/F, G_{0}(K)). Namely, let S be the torus from Lemma 6.7. By Proposition 4.14, we have the surjection

H^{1}(K/F, S(K)) → H^{1}(K/F, G_{0}(K)).

Hence we may assume that ξ has coefficients in S.

Let ξ = (a_{τ}) and let a_{τ} = h_{α_{1}}(t_{1}) ⋯ h_{α_{6}}(t_{6}), where t_{1}, …, t_{6} ∈ F^{×}. We write ξ as the product ξ = ξ_{1} · ξ_{2}, where

ξ_{1} = (h_{α_{1}}(t_{1})h_{α_{4}}(t_{4})h_{α_{6}}(t_{6})), ξ_{2} = (h_{α_{3}}(t_{3})h_{α_{2}}(t_{2})h_{α_{5}}(t_{5})).

Since, by Lemma 4.12, the subgroup of G_{0} generated by the roots α_{2}, α_{3}, α_{5} splits over F, ξ_{2} is a trivial cocycle. Then, by the Moving Lemma 5.1.6, ξ is equivalent to a cocycle with coefficients in an F -subgroup H ⊂ G_{0} of type A_{1} × A_{1} × A_{1}.

According to Lemma 4.8 the three components of H correspond to the quaternion algebras D_{1} = (d, t_{3}), D_{4} = (d, t_{2}t_{3}t_{5}), D_{6} = (d, t_{5}). Then, by Proposition 2.4 and from the formula for the Rost invariant for groups of inner type A_{n}, we obtain that the cocycle R_{G_{0}}(ξ) is given by

R_{G_{0}}(ξ) = (d) ∪ (t_{3}) ∪ (t_{1}) + (d) ∪ (t_{2}t_{3}t_{5}) ∪ (t_{4}) + (d) ∪ (t_{5}) ∪ (t_{6}).

6.3 Type E_{7}

This case is entirely similar to case E_{6}. By Gille’s Theorem 6.2 and Lemma 6.3, we may assume that Gal (F^{s}/F) is a pro-p-group, where p = 2, 3. If p = 3, then in view of Theorem 5.1 ξ is equivalent to a cocycle with coefficients in a subgroup of G_{0} of type A_{2}. Therefore ξ is trivial, by Theorem 2.1.

Let now p = 2. Then we may assume that G is split over a quadratic extension K/F. It is easy to see that G, and hence G_{0}, contains a maximal F -anisotropic torus S splitting over K. By Theorem 3.1, there is an embedding S ↪ G_{0} such that ξ is equivalent to a cocycle with coefficients in the image of S. Arguing as in E_{6} we may assume that all structure constants of S corresponding to all simple roots are 1 modulo norms N_{K/F} (K^{×}). After that we can modify ξ (as in Lemma 6.8) in such a way that the new cocycle lies in a simple simply connected F -subgroup H of G_{0} of type E_{6} with the following structure constants:

c_{α_{1}} ≡ c_{α_{2}} ≡ c_{α_{3}} ≡ c_{α_{4}} ≡ c_{α_{5}} ≡ 1 (mod N_{K/F} (K^{×})), c_{α_{6}} ≡ u_{1} (mod N_{K/F} (K^{×})).

It follows that F -rank of H is at least 3. Looking at the tables [T66], we see that H is either quasi-split or split over F. Hence the result follows from Proposition 2.4, since H is standard.

6.4 Type ^{3,6}D_{4}

This case is known (see [KMRT98]). However in order to get a self-contained proof of Serre’s Conjecture II for groups of exceptional types over (C_{2})–fields (except E_{8}) let us show how this case follows from the above.

Let G_{0} be a simple simply connected quasi-split trialitarian group over F. Denote the minimal extension of F over which G_{0} becomes split by L and let Γ = Gal (L/F). If G_{0} is a group of type ^{3}D_{4}, then [L : F] = 3 and Γ ≃ ℤ/3; otherwise we have [L : F] = 6 and Γ ≃ S_{3}. We start with the following

Proposition 6.11 Let ξ ∈ Z^{1}(F, G_{0}) be a cocycle such that the twisted group G = ^{ξ}G_{0} is quasi-split over a quadratic extension K/F. Assume that K and L are linearly disjoint over F. Then ξ is equivalent to a cocycle with coefficients in a standard simple simply connected F -subgroup of G_{0} of type A_{1}.

Proof. Let σ ∈ Γ be an arbitrary element of order 3. We fix a subfield P ⊆ L of degree 3 over F. If [L : F] = 3, the field P coincides with L. If [L : F] = 6, let λ be the non-trivial automorphism L/P. We have clearly λσλ^{−1} = σ^{2}. Let τ be the nontrivial automorphism of K/F. Since K and L are linearly disjoint over F, we can extend the automorphisms

σ, λ ∈ Gal (L/F), τ ∈ Gal (K/F)

in a natural way to the extension E/F, where E = K ·L. For simplicity we denote these extensions by the same letters σ, λ, τ.

Let B ⊂ G be a Borel subgroup over K such that S = B ∩ τ(B) is a maximal torus. S is split over E. Let Σ = Σ(G, S) and let Π = {α_{1}, …, α_{4}} ⊂ Σ be the basis corresponding to the Borel subgroup B. The action of Γ on root subgroups was described in [ChT99] and is given by the formulas

\begin{matrix}σ[x_{±α_{1}}(u)] = x_{±α_{3}}[σ(u)], & σ[x_{±α_{3}}(u)] = x_{±α_{4}}[σ(u)], \\ σ[x_{±α_{4}}(u)] = x_{±α_{1}}[σ(u)], & σ[x_{±α_{2}}(u)] = x_{±α_{2}}[σ(u)], \\ λ[x_{±α_{1}}(u)] = x_{±α_{1}}[λ(u)], & λ[x_{±α_{2}}(u)] = x_{±α_{2}}[λ(u)], \\ λ[x_{±α_{3}}(u)] = x_{±α_{4}}[λ(u)], & λ[x_{±α_{4}}(u)] = x_{±α_{3}}[λ(u)], \\ τ([x_{α_{i}}(u)] = x_{−α_{i}}(τ(u)). \end{matrix}  (9)

Formulas (9) show that S ≃ S_{1} × S_{2}, where

S_{1} = S ∩ G_{α_{2}} = ⟨ h_{α_{2}}(u) ⟩ ≃ R_{K/F}^{(1)} (G_{m,K}),

S_{2} = S ∩ G_{{α_{1},α_{3},α_{4}}} = ⟨ h_{α_{1}}(u_{1})h_{α_{3}}(u_{3})h_{α_{4}}(u_{4}) ⟩ ≃ R_{P/F} [R_{P ·K/P}^{(1)} (G_{m,P ·K})].

The rest of the proof is the same as in Section 5.1.3. Namely, we construct the following embedding ψ : S ↪ G_{0}. Let T ⊂ G_{0} be the centralizer of a maximal F -split torus of G_{0}. The Tits index of G_{0} over F is of the form:

[figure: α_{2}; α_{3}; α_{1}; α_{4}]  (10)

This picture shows that the subgroup H_{1} ⊂ G_{0} (respectively H_{2}) generated by α_{2} (respectively α_{1}, α_{3}, α_{4}) from the root system Σ(G_{0}, T) is isomorphic to SL_{2,F} (respectively R_{P/F} (SL_{2,P})). Hence there are natural embeddings S_{1} ↪ H_{1}, S_{2} ↪ H_{2} which can be extended to ψ : S ↪ G_{0}. Arguing as in Proposition 5.10 we obtain that ξ is equivalent to a cocycle with coefficients in ψ(S). Then it can be written as a product of two cocycles ξ = ξ_{1} · ξ_{2} with coefficients in ψ(S_{1}) and ψ(S_{2}) respectively. Since ξ_{2} has coefficients in H_{2} which has trivial Galois cohomology, the Moving Lemma 5.1.6 completes the proof.

Theorem 6.12 If ξ ∈ Z^{1}(F, G_{0}) is a cocycle such that R_{G_{0}}([ξ]) = 1, then ξ is trivial.

Proof. The triviality of ξ is equivalent to saying that the twisted group G = ^{ξ}G_{0} is quasi-split over F. As usual we distinguish two cases: isotropic and anisotropic.

Let first G be F -isotropic. We claim that then G is quasi-split over F. Assume the contrary. Then the F -index of G is of the form

[figure: (11)]

Let S_{1} be a maximal F -split torus in G. It follows from the picture that the semisimple part of the centralizer C_{G}(S_{1}) coincides with the subgroup G_{Σ_{2}} ⊂ G generated by the nondistinguished roots Σ_{2} = {α_{1}, α_{3}, α_{4}}. Let S_{2} ⊂ G_{Σ_{2}} be an arbitrary maximal F -defined torus and let S = S_{1} · S_{2}.

Arguing as in Theorem 3.2 we obtain that there is an F -embedding S ↪ G_{0} such that [ξ] ∈ Im [H^{1}(F, S) → H^{1}(F, G_{0})] and the image of S_{2} lies in a standard quasi-split simple simply connected F -subgroup H ⊂ G_{0} of type A_{1} × A_{1} × A_{1}. Since S/S_{2} is a split torus over F, any cocycle in Z^{1}(F, S) is equivalent to a cocycle with coefficients in S_{2}. It follows that ξ is equivalent to a cocycle with coefficients in H and H has trivial Galois cohomology.

Let G be F -anisotropic. Consider first the case [L : F] = 3. Since Ker R_{G_{0}}^{L} = 1, we obtain that [ξ] ∈ H^{1}(L/F, G_{0}(L)). Then the argument on p. 372–373 in [PR94] shows that ξ up to equivalence lies in a standard simple simply connected subgroup of G_{0} of inner type A_{2}. Hence ξ is trivial, by Theorem 2.1 and Proposition 2.4.

The last case is [L : F] = 6. Let K_{1} = F (\sqrt{d}) be the quadratic extension contained in L. Since we have already proved that Ker R_{G_{0}}^{K_{1}} = 1, we obtain that ξ ∈ Z^{1}(K_{1}/F, G_{0}(K_{1})), in particular G is quasi-split over K_{1}. Then we can easily find as above (see also [PR94], p. 354) an F -subgroup H of G of type A_{1}. Let K/F be a quadratic extension splitting H and linearly disjoint with L. Then G is isotropic over K and hence quasi-split over K, for the isotropic case has been already treated. Since L and K are linearly disjoint, Proposition 6.11 and Theorem 2.1 show that ξ is trivial.

7 Serre’s Conjecture II and the Hasse principle Conjecture II

Conjecture II (Serre [S94]) Let F be a perfect field with cd (F) ≤ 2 and let G be a simply connected semisimple linear algebraic group over F. Then H^{1}(F, G) = 1.

The Hasse Principle Conjecture II ( Colliot-Thélène, Scheiderer [CT96, Sch96]) Let F be a field with vcd (F) ≤ 2 and let G be a simply connected semisimple linear algebraic group over F. Then the canonical map

θ : H^{1}(F, G) ⟶ ∏_{ξ∈Ω_{F}} H^{1}(F_{ξ}, G)

induced by the restriction maps has trivial kernel. Here Ω_{F} denotes the set of all orderings of F and F_{ξ} is a real closure of F at the ordering ξ.

Theorem 7.1 Let G_{0} be a quasi-split simple simply connected group over a perfect field F which is of type ^{3,6}D_{4}, E_{6}, E_{7}. Assume that cd_{p} (F) ≤ 2, where p = 2, 3. Then H^{1}(F, G_{0}) = 1.

Proof. Let ζ ∈ Z^{1}(F, G_{0}). Since cd_{p} (F) ≤ 2, p = 2, 3, we get R_{G_{0}}([ζ]) = 1. It follows then from Theorem 6.1 that [ζ] = 1.

Theorem 7.2 Let G_{0} be a quasi-split simple simply connected algebraic group over F which is of type ^{3,6}D_{4}, E_{6}, E_{7}. Assume that vcd (F) ≤ 2. Then

θ : H^{1}(F, G_{0}) ⟶ ∏_{ξ∈Ω_{F}} H^{1}(F_{ξ}, G_{0})

has trivial kernel.

Proof. Let ζ ∈ Z^{1}(F, G_{0}) be a locally trivial cocycle. Let K = F (\sqrt{−1}). By Theorem 7.1, the restriction of ζ at K is trivial. It follows that R_{G_{0}}([ζ]) has coefficients in ℤ/2. Hence the result follows from Theorem 6.1 and the following theorem which is due to Arason.

Theorem 7.3 The Hasse principle holds for H^{3}(F, ℤ/2).

Proof. Let η ∈ Z^{3}(F, ℤ/2) be a locally trivial cocycle. Then by Arason’s theorem [A75], Satz 3, there is an integer r such that η ∪ (−1)^{r} = 0. On the other hand from the exact sequence

H^{i}(K, ℤ/2) ⟶^{cor} H^{i}(F, ℤ/2) ⟶^{∪(−1)} H^{i+1}(F, ℤ/2) ⟶^{res} H^{i+1}(K, ℤ/2)

( [Ar75], Corollary 4.6 ) and from the equalities

H^{i}(K, ℤ/2) = H^{i+1}(K, ℤ/2) = 0, i ≥ 3

we conclude that the product ∪(−1) is an isomorphism. Therefore, η = 0.

8 (C_{2})-fields

A field F has (C_{2})-property if every homogeneous equation f(x_{1}, …, x_{n}) = 0 of degree d with coefficients in F has a nontrivial solution in F^{n} if n > d^{2}. Property (C_{2}) implies cd(F) ≤ 2, but the converse is not true in general case. The main property of (C_{2})-fields we are going to use is

Theorem 8.1 ([Art82]) Every central simple algebra A over a (C_{2})-field F of exponent p has index p if p = 2, 3.

Theorem 8.2 Let G be a simple simply connected algebraic group of type ^{3,6}D_{4}, E_{6}, E_{7} defined over a (C_{2})-field F. Then H^{1}(F, G) = 1.

Proof. Denote a simple simply connected ( resp. adjoint ) F -quasi-split group of the same type as G by G_{0} ( resp. \overline{G}_{0} ). In the case ^{3,6}D_{4}, as in 6.4, let L/F be the minimal Galois extension over which G_{0} becomes split and P/F a cubic extension contained in L. We have the natural map φ : H^{1}(F, \overline{G}_{0}) ⟶ H^{2}(F, Z), where Z is the center of G_{0}. Let ξ ∈ Z^{1}(F, \overline{G}_{0}) be such that G = ^{ξ}G_{0}. Then there is the natural bijection (see [S94])

φ^{−1}(φ([ξ])) ⟶ H^{1}(F, G)/ ∼,

where ∼ is the equivalence relation given by multiplication on elements from H^{1}(F, Z). It follows that we need only to show that the fiber φ^{−1}(φ([ξ])) consists of one element, namely [ξ], and any cocycle with coefficients in the centre of G is trivial, viewed as a cocycle in G.

8.1 Type ^{3,6}D_{4}

Let [ξ_{1}] ∈ φ^{−1}(φ([ξ])) and let G_{1} = ^{ξ_{1}}G_{0} be the corresponding twisted group. It is known that

H^{2}(F, Z) ≃ Ker [_{2}Br P ⟶ _{2}BrF ].

Since exponent of an algebra over P coincides with its index, φ([ξ]) can be represented by a quaternion algebra over P. Then, by Proposition 43.9 in [KMRT98], there is a quadratic extension K/F which kills φ([ξ]) = φ([ξ_{1}]). This implies, by Theorem 7.1, that G and G_{1} are quasi-split over K. Arguing as in Proposition 6.11 (see also [PR94], p. 354) we can easily see that each of G and G_{1} contains a simple simply connected subgroup of outer type A_{2} splitting over K. Since cd_{2}(F) ≤ 2, any such subgroup is isotropic over F. Hence both groups G and G_{1} are F -isotropic.

If one of G, G_{1} is quasi-split over F, there is nothing to prove. Otherwise, both of them have the same Tits index (11). Denote by H_{0} the subgroup in G_{0} generated by the root subgroups corresponding to the roots α_{1}, α_{3}, α_{4} from diagram (10). The group H_{0} is a semisimple simply connected group of type A_{1} × A_{1} × A_{1} containing the center Z. Let \overline{H}_{0} ⊂ \overline{G}_{0} be its image in the adjoint group. Taking into consideration the diagram (11) and arguing as in Theorem 3.2 we get immediately that up to equivalence the cocycles ξ and ξ_{1} lies in Z^{1}(F, \overline{H}_{0}). Hence [ξ] = [ξ_{1}], since for groups of classical types Conjecture II holds by [BP95].

To show that

Im [ H^{1}(F, Z) ⟶ H^{1}(F, G) ] = 1

it suffices to note that the center Z lies in the semisimple F -anisotropic kernel of G which has type A_{1} × A_{1} × A_{1} (see diagram (11)) and hence has trivial Galois cohomology.

The above argument shows that we can apply the same proof for the other types E_{6}, E_{7} if we find an F -split torus S ⊂ G_{0} satisfying the following conditions:

(a) [ξ], [ξ_{1}] ∈ Im [ H^{1}(F, C_{\overline{G}_{0}}(\overline{S})) ⟶ H^{1}(F, \overline{G}_{0}) ], where \overline{S} is the image of S in \overline{G}_{0},

(b) H_{0} = [ C_{G_{0}}(S), C_{G_{0}}(S) ] is a semisimple simply connected group of classical type containing the center Z of G_{0},

(c) the centralizer C_{G_{0}}(S) is an almost direct product of S and H_{0}.

8.2 Type ^{1}E_{6}

We have Z ≃ μ_{3}, hence there is a cubic cyclic extension E/F killing φ([ξ]) = φ([ξ_{1}]).

Lemma 8.3 G and G_{1} are F -isotropic.

Proof. Assume the contrary. Let C_{1} be a subgroup constructed in Lemma 5.3. It has type D_{4}, hence is F -isotropic by the argument above — a contradiction.

All admissible F -isotropic Tits indices are given by diagrams (5). Since G, G_{1} are split over a cubic extension E/F we conclude that the only case which can occur is represented by diagram (5), index (ii). Then one can easily see that the torus S_{1} constructed in Theorem 3.2 satisfies properties (a), (b) and (c).

8.3 Type ^{2}E_{6}

Let K/F be the quadratic extension over which G_{0} becomes a group of inner type and let τ be the nontrivial automorphism K/F.

Lemma 8.4 G and G_{1} are F -isotropic.

Proof. Assume the contrary. We have already proved that G is K-isotropic. If it splits over K, then it contains an F -defined subgroup of type A_{2} splitting over the quadratic extension K/F. Since F has (C_{2})-property, this subgroup is F -isotropic.

Thus G is not K-split, hence its K-Tits index is of the form (ii) in diagram (5) (note that the index (i) can not occur since its anisotropic kernel is a classical group of type D_{4} and any such group is F -isotropic over a (C_{2})-field). Let P be a minimal parabolic subgroup of G over K in generic position and let C be the connected component of P ∩τ(P). Since, by our assumption, G is F -anisotropic, C is a reductive F -group. By dimension argument, C_{K} is a Levi subgroup of P. This implies that [C, C] is a semisimple Kanisotropic kernel of G. It follows then from picture (ii) in diagram (5) that the centraliser C_{G}([C, C]) is an F -defined subgroup of G of type A_{2} splitting over K. But any such group is F -isotropic.

All admissible F -isotropic Tits indices of type ^{2}E_{6} are given by the diagrams in Proposition 6.4. Since the K-Tits indices of G, G_{1} are of the form (ii) in (5), we conclude that over F they are of the form either (a) or (c) from the diagrams in Proposition 6.4. In both cases the subgroups in G, G_{1} corresponding to the root of maximal length are isomorphic to SL_{2} over F, hence contains a 1-dimensional split torus S. The rest of the proof is the same as for inner forms of type E_{6}.

8.4 Type E_{7}

As above one can easily see that both G, G_{1} are F -isotropic. Looking at the Tits tables [T66] and taking into the consideration the fact (which has been already proved above) that all simple groups of type not A_{n} are F -isotropic we obtain that G, G_{1} have the same Tits indices of the form

[figure]

The rest of the proof is the same as for inner forms of type E_{6}. Thus we complete the proof of Theorem 8.2.

Remark 8.5 Note that under the proof of Theorem 8.2 we have showed that any simple group of type ^{3,6}D_{4}, E_{6}, E_{7} is isotropic and also described all admissible Tits indices of such groups.

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