THE ROST INVARIANT HAS TRIVIAL KERNEL FOR QUASI-SPLIT GROUPS OF LOW RANK

R. SKIP GARIBALDI

Abstract. For G an almost simple simply connected algebraic group defined over a field F, Rost has shown that there exists a canonical map R_{G}: H^{1}(F, G) → H^{3}(F, ℚ/ℤ(2)). This includes the Arason invariant for quadratic forms and Rost’s mod 3 invariant for Albert algebras as special cases. We show that R_{G} has trivial kernel if G is quasi-split of type E_{6} or E_{7}. A case-by-case analysis shows that it has trivial kernel whenever G is quasi-split of low rank.

Contents

1. Quasi-split groups of type A, B, and D 5 2. Folded root systems 8 3. Small representations 10 4. ^{1}D_{4} ⊂ ^{2}E_{6} 14 5. ^{2}D_{5} ⊂ ^{2}E_{6} 18 6. Special cocycles 21 7. Quasi-split groups of type E_{6} and E_{7} 23 References 28

For G an almost simple simply connected algebraic group over a field F, the set of all natural transformations of functors

H^{1}(?, G) ⟶ H^{3}(?, ℚ/ℤ(2))

is a finite cyclic group [KMRT98, §31] with a canonical generator. (Here H^{i}(?, M) is the Galois cohomology functor which takes a field extension of your base field F and returns a group if M is abelian and a pointed set otherwise. When F has characteristic zero, ℚ/ℤ(2) is defined to be lim_{→} μ_{n}^{⊗2} for μ_{n} the algebraic groups of nth roots of unity; see [KMRT98, p. 431] or [Gilb, I.1(b)] for a more complete definition.)

Date: 30 October 2000.

1991 Mathematics Subject Classification. 20G10 (17B25).

This generator is called the Rost invariant of G and we denote it by R_{G}. In an abuse of notation, we also write R_{G} for the map H^{1}(F, G) ⟶ H^{1}(F, ℚ/ℤ(2)).

This map provides a useful invariant for algebraic structures classified by H^{1}(F, G), and an important and typically difficult question is to describe the kernel of R_{G}. For example, when G is split of type D_{n}, R_{G} is essentially the Arason invariant I^{3}F → H^{3}(F, μ_{2}^{⊗2}) for quadratic forms, where I^{n}F is as usual the nth power of the ideal IF of evendimensional quadratic forms in the Witt ring of F. That the kernel of the Arason invariant is precisely I^{4}F is a quite difficult result due independently to Merkurjev-Suslin [MS91] and Rost. (The proof of the main result of this paper somehow boils down to this one fact.) In general, one doesn’t even know if the kernel of R_{G} is trivial. On the other hand, the question becomes tractable if we assume that G is quasi-split. Generally R_{G} has nontrivial kernel; we give easy examples where G is split of type D_{8} (in 1.9) and B_{7} (in 1.6), and quasi-split of type ^{2}A_{6} (in 1.11). It should be mentioned that R_{G} can have nontrivial kernel when G is split of type E_{8} as well; Gille [Gil00] has produced an example by applying his results from [Gilb] to reduce the question to the same one for a split group of type D_{8}.

The principal result in this paper is to enlarge the list of quasi-split groups for which the Rost invariant is known to have trivial kernel.

Main Theorem 0.1. Suppose that G is a quasi-split simply connected group of type E_{6} or E_{7}. Then the Rost invariant R_{G} has trivial kernel.

0.2. There are some easy consequences of this theorem that may help the reader place it in context. The first is that as a vastly less powerful corollary, we obtain Serre’s “Conjecture II” for quasi-split groups of type E_{6} and E_{7}, in that if F has cohomological p-dimension ≤ 2 for p = 2, 3 (see [Ser94, I.3] for a definition), then the main theorem implies that H^{1}(F, G) is trivial. This conjecture appeared in print back in 1962 [Ser62], and remained open for such groups until the 1990s, when Chernousov (unpublished) and Gille [Gila] proved it (amongst other cases) independently and by different methods. Here we get it for free from the Main Theorem.

0.3. Another consequence is the following: Suppose that L is a field extension of F of degree relatively prime to 2 and 3 and that G is a group of type E_{6} or E_{7}. Serre asked in [Ser95, p. 233, Q. 1] if the natural map H^{1}(F, G) → H^{1}(L, G) is injective. Our Main Theorem gives the partial answer that it has trivial kernel in the case where G is quasi-split. This result was already known by experts in the area using arguments special to groups of type E_{6} and E_{7}, but as for Conjecture II we get it for free here.

0.4. There is also an application to finite-dimensional algebras. There is a large family of nonassociative algebras with involution called structurable algebras which includes central simple associative algebras with involution (as studied in [KMRT98]) and Jordan algebras (with involution the identity), see [All94] for a survey. The simple structurable algebras have all been classified, and they consist (roughly) of the two families already mentioned plus four others. I refer to one of these classes, which consists of 56-dimensional algebras all of which are isomorphic over a separably closed field and have automorphism group which is simply connected of type E_{6}, as Brown algebras. Now there is a natural equivalence relation defined on the set of structurable algebras called isotopy [AH81] which is weaker than isomorphism, and in the case of Jordan algebras is the same as the traditional notion of isotopy. For Albert algebras, it is known that any algebra isotopic to the split one is actually split. (This is equivalent to the cohomological statement that the map H^{1}(F, F_{4}) → H^{1}(F, E_{6}) induced by the embedding F_{4} → E_{6} described in 2.4 has trivial kernel.) The Main Theorem here combined with [Garb, 4.16(2), 5.12] shows that the same conclusion holds for Brown algebras, which was previously unknown. (This has the cohomological interpretation that the map H^{1}(F, E_{6}^{K}) → H^{1}(F, E_{7}) induced by the embedding E_{6}^{K} → E_{7} described in 3.5 has trivial kernel.)

The material in [KMRT98] is sufficient to show that the kernel of the Rost invariant is trivial for quasi-split groups of type G_{2}, D_{4} (including those of trialitarian type [KMRT98, 40.16]), and F_{4}, at least away from the “bad primes” 2 and 3. As easy corollaries to results needed for the E_{6} and E_{7} cases, we get analogous results for groups of type ^{2}A_{n}, B_{n}, and nontrialitarian groups of type D_{n} with small n in Section 1. So since H^{1}(F, G) is always trivial for G split of type A_{n} or C_{n}, we get the following:

Theorem 0.5. Suppose that G is an almost simple simply connected algebraic group. If G is

• quasi-split of (absolute) rank ≤ 5;

• quasi-split of type B_{6}, D_{6}, or E_{6}; or

• split of type D_{7} or E_{7}, then the Rost invariant R_{G} has trivial kernel.

The proofs of these theorems that we will give here and the material in [KMRT98] rely on the ground field having “good” characteristic, meaning for our purposes ≠ 2, 3. However, it is a consequence of Gille’s main theorem in [Gilb] that one only needs to prove that the Rost invariant has trivial kernel for fields of characteristic 0. Consequently, all fields considered here will be assumed to have characteristic ≠ 2, 3, but our two theorems will still hold for all characteristics. (Of course, in prime characteristic the group ℚ/ℤ(2) must be defined somewhat differently [Gilb], but this affects neither the statemtn of the theorems nor our proofs.)

Section 1 dispenses with the classical groups. (Some of that material is useful later.) Sections 2 and 3 contain the material necessary to reduce questions about the Rost invariant for a larger group to a subgroup. That material easily reduces the proof of the main theorem to considering the quasi-split ^{2}E_{6} case, which is treated in the remaining Sections 4 through 7.^{1}

Remark 0.6 (Noninjectivity for F_{4}). We caution the reader that even when the Rost invariant has trivial kernel, it may be far from injective. For example, for F_{4} the split group of type F_{4}, the set H^{1}(F, F_{4}) classifies Albert F -algebras. From known facts about Albert algebras, it is easy to show that two classes α_{1}, α_{2} corresponding to isotopic Jordan algebras J_{1}, J_{2} have the same Rost invariant. Since there are many isotopic Albert algebras which are not isomorphic (for example, over ℝ there are 3 isomorphism classes of Albert algebras and two of these are isotopic [Jac71, p. 119]), the Rost invariant for F_{4} has trivial kernel but is typically not injective.

Notations and conventions. We say that an algebraic group G is (absolutely) almost simple if it has finite center and no noncentral closed normal subgroups. When we say that a group is “of type T_{n}”, we implicitly mean that it is almost simple of that type. We will use the standard notations 𝔾_{m}, 𝔾_{a}, and μ_{n} for the algebraic groups with F -points F^{∗}, F, and the nth roots of unity in F, and G^{∘} will always denote the identity component of an algebraic group G. For a variety X we write X(F) for its F -points.

Our notation for quadratic forms will follow the standard reference [Lam73]. However, the reader should be warned of two quirks: We use the Pfister-approved notation for Pfister forms, so ≪a_{1}, …, a_{n} ≫ := ⟨1, −a_{1}⟩ ⊗ ⋯ ⊗ ⟨1, −a_{n}⟩, and we write H for the hyperbolic plane ⟨1, −1⟩.

1After this paper was released as a preprint, Chernousov sent to me a different proof of the^{2}E_{6} case [Che00], which uses a completely different argument. His proof will be published separately.

The standard reference for Galois cohomology is [Ser94, §I.5], and for algebras with involution (including the groups Spin (A, σ), O(A, σ), and SO(A, σ)) it is [KMRT98].

1. Quasi-split groups of type A, B, and D

As indicated in the introduction, the Rost invariant should have trivial kernel for quasi-split groups of small rank. To prove this for E_{6}, we will need a result on groups of type D, which also easily settles this question for groups of type A and B. (For the results in this section, our global hypothesis that our fields have characteristic ≠ 3 is not required; we need only assume characteristic ≠ 2.) For q a nondegenerate quadratic form over F, recall that there is a short exact sequence of algebraic groups

1 ⟶ C ⟶ Spin(q) ⟶ SO(q) ⟶ 1 (1.1)

with C ≅ μ_{2}.

Lemma 1.2. For q a d-dimensional nondegenerate quadratic form with anisotropic part of dimension d_{an} such that d ≥ 5 and d + d_{an} < 16, the kernel of the Rost invariant of Spin(q) is precisely the image of H^{1}(F, C) in H^{1}(F, Spin(q)).

Proof: The set H^{1}(F, SO(q)) classifies quadratic forms of the same dimension and discriminant as q [KMRT98, 29.29]. For α ∈ H^{1}(F, Spin(q)) we set q_{α} to be the quadratic form corresponding to the image of α in H^{1}(F, SO(q)). Then q_{α}−q is not only even-dimensional with trivial discriminant (i.e., q_{α} − q ∈ I^{2}F), but since q_{α} comes from H^{1}(F, Spin(q)), it has the same Clifford invariant as q [KMRT98, 31.11] and so q_{α} −q ∈ I^{3}F by Merkurjev’s Theorem. As described in [KMRT98, p. 437], the Rost invariant of α is the Arason invariant e_{3}(q_{α} − q) ∈ H^{3}(F, ℤ/2). (Since ℤ/2 = μ_{2}^{⊗2}, we can consider ℤ/2 to be a subgroup of ℚ/ℤ(2) and hence H^{3}(F, ℤ/2) is a subgroup of H^{3}(F, ℚ/ℤ(2)).)

Suppose first that α is in the image of H^{1}(F, C). Sequence (1.1) induces an exact sequence

SO(q)(F) ⟶ H^{1}(F, C) ⟶ H^{1}(F, Spin(q)) ⟶ H^{1}(F, SO(q)), (1.3)

and since the Rost invariant R_{Spin(q)} “factors through” H^{1}(F, SO(q)), certainly R_{Spin(q)}(α) is trivial.

Conversely, suppose that α is in the kernel of the Rost invariant. Then e_{3}(q_{α} − q) is trivial, but as mentioned in the introduction the kernel of e_{3} is precisely I^{4}F. Since dim q_{α} = dim q = d, the hypotheses on q ensure that the dimension of the anisotropic part of q_{α}−q is strictly less than 16. Since q_{α} − q ∈ I^{4}F, it is hyperbolic by the Arason-Pfister Hauptsatz [Lam73, X.3.1]. Thus q_{α} ≃ q and α is in the kernel of the map H^{1}(F, Spin(q)) → H^{1}(F, SO(q)), which is just the image of H^{1}(F, C).

The first map in (1.3) is the spinor norm, which immediately produces the following lemma.

Corollary 1.4. Suppose that q is as in Lemma 1.2. Then the kernel of the Rost invariant is isomorphic to F^{∗}/SN(q)F^{∗2}, where SN(q) is the image of the spinor norm map SO(q)(F) → F^{∗}/F^{∗2}.

1.5. Quasi-split simply connected groups of type B_{n} are actually split, so of the form Spin(q) for q = nH ⊥ ⟨1⟩. In terms of the lemma, d = 2n + 1 and d_{an} = 1. So q satisfies the hypotheses for 2 ≤ n ≤ 6. Since q is isotropic, it has surjective spinor norm, so the Rost invariant for a split group of type B_{n} has trivial kernel for 2 ≤ n ≤ 6.

Example 1.6 (B_{7}). As just mentioned, the split simply connected group of type B_{7} is isomorphic to Spin(q) for q = 7H ⊥ ⟨1⟩. We will show that the Rost invariant R_{Spin(q)} can have nontrivial kernel. Sequence (1.1) induces an exact sequence

H^{1}(F, Spin(q)) ⟶ H^{1}(F, SO(q)) ⟶^{∂} H^{2}(F, μ_{2}) (1.7)

where the set H^{1}(F, SO(q)) classifies nondegenerate quadratic forms with the same dimension (15) and discriminant (1 · F^{∗2}) as q.

Fix a base field F and a nonhyperbolic 4-fold Pfister form φ over F (e.g. F = ℝ, φ = ≪−1, −1, −1, −1≫). Set q_{α} = −φ′ for φ′ such that φ = ⟨1⟩ ⊥ φ′. Then disc q_{α} = (−1)^{\binom{15}{2}} det(−φ′) = 1 · F^{∗2}, so there is a unique element of H^{1}(F, SO(q)) corresponding to q_{α}. The image of q_{α} under the connecting homomorphism ∂ is [C_{0}(q_{α} − q)], which by [Lam73, V.2.10] is the same as [C(q_{α} − q)] which is trivial since q_{α} − q = −φ ∈ I^{3}F. Thus q_{α} is the image of some α in H^{1}(F, Spin(q)). But then R_{Spin(q)}(α) = e_{3}(q_{α} − q) = e_{3}(−φ), which is trivial since φ ∈ I^{4}F.

1.8. An analysis for groups of type D_{n} similar to the one in 1.5 for B_{n} shows that the Rost invariant for a simply connected group is trivial for groups of type ^{1}D_{n} with 3 ≤ n ≤ 7 and for groups of type ^{2}D_{n} with 3 ≤ n ≤ 6. As in the B case, we show that one of these bounds is sharp.

Example 1.9 (^{1}D_{8}). The situation here is quite similar to the one in Example 1.6, except that q = 8H. We use the same base field F and nonsplit 4-fold Pfister form φ from before. Then there is a unique element of H^{1}(F, SO(q)) corresponding to φ and since φ = φ − q ∈ I^{4}F, the same reasoning shows that there is a nontrivial class in H^{1}(F, Spin(q)) which is the inverse image of φ and which has trivial Rost invariant.

Lemma 1.2 easily deals with quasi-split groups of type ^{2}A_{n} of low rank.

Corollary 1.10. If G is a quasi-split simply connected group of type 2A_{n} with n ≤ 5, the kernel of the Rost invariant R_{G} is trivial.

Proof: Set K to be the quadratic field extension of F which splits G and take (V, h^{d}) to be a “maximally split” (n + 1)-dimensional hermitian form over K. (See below for a more explicit description.) Then G is none other than SU(V, h^{d}), the algebraic group with F -points

SU(V, h^{d})(F) = {g ∈ GL(V)(K) | h(gv, gv′) = h(v, v′) for all v, v′ ∈ V and det g = 1}.

The trace form of h^{d} is defined to be the quadratic form q^{d} on V considered as a 2(n + 1)-dimensional vector space over F given by q^{d}(v) = h^{d}(v, v). Then

h^{d} = \begin{cases}mH & if n + 1 = 2m, \\ mH ⊥ ⟨1⟩ & if n + 1 = 2m + 1 \end{cases} and q^{d} = \begin{cases}2mH & if n + 1 = 2m, \\ 2mH ⊥ ≪d≫ & if n + 1 = 2m + 1, \end{cases}

where K = F (\sqrt{d}) if n = 2m for some integer m, and the H occurring in the description of h^{d} is the usual unitary hyperbolic plane as described in [Sch85, 7.7.3].

The set H^{1}(F, G) classifies nonsingular hermitian forms h on V which have the same dimension and discriminant as h^{d} [KMRT98, p. 403]. Now G ↪ SO(V, q^{d}) and the corresponding map H^{1}(F, G) → H^{1}(F, SO(V, q^{d})) sends h to its trace form q, and this map is an injection by [Sch85, 10.1.1(ii)]. Moreover, the Rost invariant R_{G}(h) is just e_{3}(q − q^{d}) by [KMRT98, 31.44]. Since dim q^{d} = 2n + 2 < 14 and the anisotropic part of q^{d} has dimension 0 (if n + 1 is even) and 2 (if n + 1 is odd), as in the proof of Lemma 1.2, if R_{G}(h) is trivial, q ≃ q^{d} and so h ≃ h^{d}.

Example 1.11 (^{2}A_{6}). Take F = ℝ, K = ℂ, and consider G = SU(V, h^{d}) for h^{d} the hermitian form 3H ⊥ ⟨1⟩ over K, so that G is simply connected quasi-split of type^{2}A_{6}. Then the hermitian form h = ⟨−1, −1, −1, −1, −1, −1, −1⟩ has trace form q = −7≪−1≫ which is not hyperbolic, so h corresponds to a (unique) nontrivial class in H^{1}(F, G). However,

q − q^{d} = −7≪−1≫ − ≪−1≫ = −≪−1, −1, −1, −1≫ ∈ I^{4}F ,

so R_{G}(h) is trivial.

2. Folded root systems

2.1. The Rost multiplier. In general, for an arbitrary algebraic group G we define the set G_{∗} of loops in G to be the homomorphisms 𝔾_{m} → G. Then as in [KMRT98, p. 432], we set Q(G) to be the abelian group of all integer-valued functions on G_{∗} such that

(1) for ^{g}f the loop given by ^{g}f(x) = gf(x)g^{−1}, q(^{g}f) = q(f) for all g ∈ G and f ∈ G_{∗}; and

(2) for any two loops f and h with commuting images, the function ℤ × ℤ ⟶ ℤ given by (k, m) ↦ q(f^{k}h^{m}) is a quadratic form.

When G is an almost simple group, Q(G) is cyclic with a canonical generator which is positive definite [KMRT98, 31.27], hence is identified with ℤ. Now suppose that we have two almost simple simply connected groups H ↪ G. This map clearly induces a map H_{∗} → G_{∗}, so we in turn have a map ℤ = Q(G) → Q(H) = ℤ. Because the canonical generators are positive definite, this map must be multiplication by a positive integer n, which we define to be the Rost multiplier of the embedding.

The naturality of the Rost invariant implies that we have a commutative diagram

H^{1}(F, H) ⟶_{R_{H}} H^{3}(F, ℚ/ℤ(2))
↓  ^{n·}↓
H^{1}(F, G) ⟶^{R_{G}} H^{3}(F, ℚ/ℤ(2)),

where n is the Rost multiplier of the embedding [KMRT98, 31.34]. This is the motivation for our study of this invariant.

2.2. Luckily, it can be quite easy to compute such a “Rost multiplier”. Suppose that G and H are split and contain split maximal tori S and T respectively such that the image of T lies in S. Since G and H are simply connected, the character groups X(T) and X(S) are identified with the weight lattices, but the character groups are dual to the loop groups S_{∗} and T_{∗} [Bor91, 8.6] and the weight lattices are dual to the lattices generated by the coroots, which we denote by Λ_{c,G} and Λ_{c,H}, respectively. (By a coroot, we mean the roots of the dual root system, which are denoted by α̌ in [Bou68, VI.1] for α a root.) Putting these dualities together, we obtain identifications S_{∗} = Λ_{c,G} and T_{∗} = Λ_{c,H}, so the embedding T → S induces a map Λ_{c,H} → Λ_{c,G}. Now the dual root systems (whose roots are the coroots) are indeed root systems [Bou68, VI.1.1, Prop. 2] and so they each have a unique minimal Weylgroup invariant positive-definite integer-valued quadratic form [Bou68, VI.1.2, Prop. 7], say q and r (for the forms for G and H respectively). Hence q induces such a form on Λ_{c,H}, which must be of the form nr for some natural number n. This n is the Rost multiplier of the inclusion.

Criterion (2) in the definition of Q(G) implies that its canonical generator is identified with the positive-definite Weyl-group invariant quadratic form on the dual root system which takes the value 1 on short coroots. (Short roots correspond to long roots, where we adopt the convention that short = long in the event that all roots have the same length. In that case, the quadratic form is very easy to identify, in that its Gram matrix is simply the Cartan matrix of the root system with all entries divided by 2.) So one can simply compute the image of a short coroot from H in the dual root system for G to find the Rost multiplier of the embedding.

Example 2.3 (SL_{n} → SL_{2n}). The block diagonal embedding SL_{n} ↪ SL_{2n} via x ↦ ( \begin{matrix}x \\ & x \end{matrix}) has Rost multiplier 2. The embedding given by x ↦ ( \begin{matrix}x \\ & 1 \end{matrix} ) has Rost multiplier 1.

Example 2.4 (Folding). The split simply connected group of type E_{6} can be realized as the group Inv (J) of invertible linear maps of the split Albert algebra J which preserve the cubic norm form. The algebra J has a nondegenerate symmetric bilinear trace form T given by setting T (x, y) to be the trace of the product x · y [Jac68, p. 240, Thm. 5], and for φ ∈ Inv (J)(F) we define φ^{†} ∈ GL(J)(F) to be the unique map satisfying T (φ(j), φ^{†}(j′)) = T (j, j′) for all j, j′ ∈ J. This defines an outer automorphism of E_{6} = Inv (J) [Jac61, p. 76, Prop. 3] and the subgroup of elements fixed by this automorphism is the split group F_{4} of F -algebra automorphisms of J.

We would like to compute the Rost multiplier of the embedding F_{4} ⊂ E_{6}. We fix an F -split maximal torus S in G := E_{6} which is preserved by the automorphism (such as the one denoted by “S_{6}” in [Garb, pf. of 7.2]) and fix a set of simple roots Δ of G with respect to S. We would like our outer automorphism to leave Δ invariant, although it probably does not do so. However, two things are apparent from the definition of the Rost multiplier: it is not changed if we extend scalars nor if modify the automorphism φ ↦ φ^{†} by an inner automorphism of E_{6}. So we may assume that the base field is separably closed and so that the F -points of the Weyl group of G with respect to S (i.e., the F -points of N_{G}(S)/S) is the full Weyl group of the root system of G with respect to S. Then we may modify our outer automorphism by an element of the Weyl group so that F_{4} is described as the subgroup of E_{6} fixed by the automorphism f induced by the automorphism of Δ which is given by the unique nontrivial automorphism of the Dynkin diagram. That is, we set H := F_{4} = G^{f} (= the subgroup of G of elements fixed by f), and T := (S^{f})^{∘} (= the identity component of T ∩ G^{f}) is a maximal torus in H. Then the restrictions of elements of Δ to T give a root system of H = F_{4} with respect to T [Sch69, p. 108] and the fibers of this restriction map are the orbits of f in Δ [Sch69, 3.5].

Now Λ_{c,G} is a free ℤ-module with basis Δ̌ = {δ̌ | δ ∈ Δ} which is permuted by f and Λ_{c,H} is the fixed sublattice. So Λ_{c,H} has a basis consisting of one element for each orbit of f in Δ̌, and this element is given by the sum of the elements in the orbit in Δ̌. Since no two roots in Δ lying in the same orbit under f are connected in the Dynkin diagram, the description of the roots of H with respect to T above shows that the spanning set in Λ_{c,H} just described is in fact a set of simple coroots of H with respect to T, which is dual to the set of simple roots given by restrictions of elements of Δ.

So we would like to compute the value of q on a particular simple coroot α̌ of H with respect to T under the composition Λ_{c,H} → Λ_{c,G} ⟶^{q} ℤ. Since each orbit of f in Δ̌ consists of totally disconnected sets of vertices of the Dynkin diagram and all coroots δ̌ in Δ̌ have q(δ̌) = 1 as a consequence of hypothesis (2) in the definition of Q(G), the value of q on α̌ is precisely the size of the orbit in Δ̌ which gave rise to α̌. Since there is some element of Δ̌ is fixed by f, q takes the value 1 on some α̌, so by the discussion in 2.2 the Rost multiplier of the embedding H ⊂ G is one.

Remark 2.5. Presumably this same argument also works in the other instances where one obtains a root system by “folding up” another root system all of whose roots have the same length, i.e., C_{ℓ+1} ⊂ A_{2ℓ+1}, B_{n−1} ⊂ D_{n}, and G_{2} ⊂ D_{4}. The other root system consisting of roots of the same length, A_{2ℓ}, folds up to give the smaller root system BC_{ℓ}, see [Hec84, Table I].

3. Small representations

We say a representation V of an algebraic group G is small if G has an open orbit in ℙ(V). We are interested in small representations in the case where G is simple, which have all been classified as a consequence of the (more general) classification of prehomogeneous vector spaces, see [Kim88] for a survey. These small representations also provide “standard relative sections” in the language of [Pop94, 1.7], and in that sense were classified in [Èla72, Table 1]. Our motivation for studying these representations comes from the following easy lemma, which was pointed out to me by Rost.

Lemma 3.1. Suppose that G is an algebraic group over a field F such that G has a small representation V, and that F is infinite or G is connected. Let H be the subgroup of G consisting of elements which stabilize some F -point in the open orbit in ℙ(V). Then the natural map

H^{1}(F, H) → H^{1}(F, G)

is surjective.

Proof: If the base field F is finite, then by hypothesis G is connected, and by Lang’s Theorem H^{1}(F, G) is trivial so the lemma holds. So we may assume that F is infinite.

Fix a 1-cocycle z ∈ Z^{1}(F, G). For U the open orbit in ℙ(V), z defines a new open subset U_{z} in ℙ(V) which is the same as U over the separable closure F_{s} of F but has a different Galois action: For u ∈ U_{z}(F_{s}) and σ ∈ Gaℓ (F_{s}/F), σ acts by

σ ∗ u = z_{σ}σu

where juxtaposition denotes the usual action.

Since the representation gives a map G → GL(V), ℙ(V)_{z} ≅ ℙ(V). Thus since F is infinite, ℙ(V)_{z}(F) is dense in ℙ(V)_{z}(F_{s}). Since U_{z}(F_{s}) is open in ℙ(V)_{z}(F_{s}), the two sets U_{z}(F_{s}) and ℙ(V)_{z}(F) must meet nontrivially, i.e., U_{z} has some F -point which we will denote by x_{z}.

Now let x ∈ U(F) be the point with stabilizer subgroup H and fix some g ∈ G(F_{s}) such that gx = x_{z}. Then for all σ ∈ Gaℓ (F_{s}/F), the element g^{−1}z_{σ}σgσ^{−1} fixes x and so lies in H(F_{s}). Thus z is cohomologous to something in the image of Z^{1}(F, H).

Example 3.2 (O_{n−1} ⊂ O_{n}). Write O_{n} for the orthogonal group of the dot product on F^{n}. Then the subgroup of O_{n} which stabilizes [v] ∈ ℙ(F^{n}) where v has nonzero length is just O_{n−1} × μ_{2}, where O_{n−1} is the orthogonal group for the (n − 1)-dimensional space of vectors in F^{n} which are orthogonal to v. Iterating this process recovers the fact that all nondegenerate quadratic forms are diagonalizable, a.k.a. the Spectral Theorem.

Example 3.3 (Spin_{n} [Igu70], [GV78], [Pop80]). For Spin_{n} the spin group for an n-dimensional maximally split quadratic form, the spin representation (if n is odd) or the half-spin representation (if n is even) is small for n ≤ 12 and n = 14. In the n = 14 case, the stabilizer subgroup is isomorphic to (G_{2} × G_{2}) ⋊ μ_{8}, and this leads to structural statements about 14-dimensional forms in I^{3}F, see [Ros99].

Example 3.4 (F_{4} × μ_{3} ⊂ E_{6}). We write E_{6} for the split group of type E_{6} which can be realized as Inv (J) as described in 2.4. By [Jac61, p. 71, Thm. 7], E_{6} acts transitively on the subset of J consisting of elements of norm 1, so certainly this is a small representation.

We take H to be the subgroup of E_{6} which fixes the identity element 1_{J} of J projectively. Then since the norm form is cubic, μ_{3} is contained in H and is central (since it consists of scalar endomorphisms), and any element of H differs by an element of μ_{3} from something which fixes 1_{J} absolutely. This subgroup of elements fixing 1_{J} is well-known — it is none other than the automorphism group F_{4} of J [Jac59, p. 186, Thm. 4], which is split of type F_{4}. So H ≅ F_{4} × μ_{3}, and the resulting surjective map H^{1}(F, F_{4} × μ_{3}) → H^{1}(F, E_{6}) is the statement that H^{1}(F, E_{6}) classifies cubic forms of the form λN for N the norm form on some Albert F -algebra and λ ∈ F^{∗}, see [Spr62]. This can also be interpreted in terms of structurable algebras, see [Garb, 2.8(1)].

Example 3.5 (E_{6} ⋊ μ_{4} ⊂ E_{7}). Write E_{7} for the split simply connected group of type E_{7} over F. It is the group of vector space automorphisms of V = ( \begin{matrix}F & J \\ J & F \end{matrix} ) which preserve a quartic form q as given in [Bro69, p. 87]. Then E_{7} acts transitively on the open subset of ℙ(V) consisting of points [v] such that q(v) ≠ 0 by [Fer72, 7.7].

We set H to be the subgroup of E_{7} which stabilizes the vector v = ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ) projectively. This vector has q(v) ≠ 0, and so by [Fer72, 3.7] there are two uniquely determined (up to scalar multiples) “strictly regular” elements e_{1} and e_{2} such that v lies in their span. These are e_{1} = ( \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} ) and e_{2} = ( \begin{matrix}0 & 0 \\ 0 & 1 \end{matrix} ). Since E_{7} preserves the property of being strictly regular, every element of H must projectively stabilize e_{1} and e_{2} as well, and perhaps interchange them.

Now, the map ω defined by

ω ( \begin{matrix}α & j \\ j′ & β \end{matrix} ) = ( \begin{matrix}iβ & ij′ \\ ij & iα \end{matrix} )

lies in H, where i is some fixed square root of −1 in the separable closure of F. We would like to describe an arbitrary h ∈ H, which after modification by ω we may assume projectively stabilizes each of e_{1} and e_{2}. Then by [Bro69, p. 96, Lem. 12], h must be of the form

h ( \begin{matrix}α & j \\ j′ & β \end{matrix} ) = ( \begin{matrix}μ^{−1}α & φ(j) \\ φ^{†}(j′) & μβ \end{matrix} )

where φ is a similarity of the norm form on J with multiplier μ and φ^{†} is as defined in 2.4. Since h also stabilizes v, we must have that μ = ±1. In particular, after modifying h by ω^{2} = −1, we may assume that h has the form

h ( \begin{matrix}α & j \\ j′ & β \end{matrix} ) = ( \begin{matrix}α & φ(j) \\ φ^{†}(j′) & β \end{matrix} )

where φ preserves the cubic norm on J and so lies in E_{6}. So we have shown that H ≅ E_{6} ⋊ μ_{4}.

The surjection on Galois cohomology coming from this example will be more useful if we can replace E_{6} ⋊ μ_{4} with an almost simple group. For K a quadratic étale F -algebra, we write E_{6}^{K} for the simply connected quasi-split group of type E_{6} over F which is split by an extension L of F if and only if L ⊗_{F} K ≅ L × L.

Proposition 3.6. (Cf. [Garb, 4.14]) Suppose that F has characteristic ≠ 2, 3. For each α ∈ H^{1}(F, E_{7}) there is some quadratic étale F -algebra K such that E_{6}^{K} embeds in E_{7} with Rost multiplier 1 and α is in the image of the induced map H^{1}(F, E_{6}^{K}) → H^{1}(F, E_{7}).

Proof: Fix some a ∈ Z^{1}(F, E_{6} ⋊ μ_{4}) representing α. The natural projection E_{6} ⋊ μ_{4} → μ_{4} has an obvious section given by sending i ↦ ω, and we set b to be the image of a given by the map induced by the composition E_{6} ⋊ μ_{4} → μ_{4} → E_{6} ⋊ μ_{4}. We twist E_{6} ⋊ μ_{4} by b to obtain a new group (E_{6} ⋊ μ_{4})_{b}, with a twisted Galois action ∗ so that

σ ∗ g = b_{σ}(^{σ}g)b_{σ}^{−1},

where ^{σ}g denotes the usual action. Then we have an isomorphism

H^{1}(F, (E_{6} ⋊ μ_{4})_{b}) ⟶_{τ_{b}}^{∼} H^{1}(F, E_{6} ⋊ μ_{4})

where τ_{b}^{−1}(α) is the class of a 1-cocycle given by σ ↦ a_{σ}b_{σ}^{−1} with values in the identity component of the twisted group (E_{6}⋊μ_{4})_{b}. This identity component is just E_{6} twisted by b, and we would like to show that it is isomorphic to E_{6}^{K} for some quadratic étale F -algebra K. We observe that if σ in the absolute Galois group of F has b_{σ} = ±1, then σ acts in the usual manner upon the twisted E_{6}. On the other hand, if b_{σ} = ±ω, then the twisted action is given by

(σ ∗ h) ( \begin{matrix}α & j \\ j′ & β \end{matrix} ) = (±ω)σhσ^{−1}(±ω)^{−1} ( \begin{matrix}α & j \\ j′ & β \end{matrix} ) = ( \begin{matrix}α & σφ^{†}σ^{−1}(j′) \\ σφσ^{−1}(j) & β \end{matrix} ) .

Since this is precisely the description of the Galois action on E_{6}^{K} given in [Garb, 2.4] for K determined by the image of b under the composition H^{1}(F, E_{6} ⋊ μ_{4}) → H^{1}(F, μ_{4}) → H^{1}(F, μ_{2}) = F^{∗}/F^{∗2}, we have (E_{6})_{b} ≅ E_{6}^{K}. To see that E_{6}^{K} embeds in E_{7}, we observe that the 1-cocycle b is trivial in H^{1}(F, E_{7}) by [Garb, 4.10, 5.10], so we have a map

E_{6}^{K} ⊂ (E_{6} ⋊ μ_{4})_{b} ↪ (E_{7})_{b} ⟶_{f}^{∼} E_{7}

where (by a simple computation having nothing to do with E_{7}) H^{1}(f) = τ_{b}. This proves the proposition aside from the claim about the Rost multiplier.

But that claim is easy in the split case (where K = F × F), since the embedding of E_{6} in E_{7} comes from the obvious embedding of root systems. Then since the Rost multiplier is invariant under scalar extensions, the embeddings of quasi-split groups of type E_{6}^{K} in E_{7} given above all have Rost multiplier 1 as well.

4. ^{1}D_{4} ⊂ ^{2}E_{6}

For the remainder of the paper we will study the quasi-split group E_{6}^{K} of type ^{2}E_{6} defined in 3.5. In this section we introduce a particular subgroup G of E_{6}^{K} which is reductive of semisimple type ^{1}D_{4}. Defining G will necessitate digging more deeply in to the structure of Albert and Cayley algebras.

Definition 4.1. Fix ℭ to be the split Cayley algebra endowed with hyperbolic norm form 𝔫 and canonical involution denoted by ¯ . We define some algebraic groups related to ℭ. (For more information about Cayley algebras, see [KMRT98, §33.C] or [Sch66, Ch. III, §4].) First, if t ∈ GL(ℭ) satisfies 𝔫(t(c)) = m𝔫(c) for some m ∈ F^{∗} and all c ∈ ℭ, we say that m is a similarity of 𝔫 with multiplier μ(t) := m. (Note that if σ_{𝔫} is the involution on End_{F} (ℭ) which is adjoint for 𝔫 so that 𝔫(tc, c′) = 𝔫(c, σ_{𝔫}(t)c′) for all c, c′ ∈ ℭ, then μ(t) = σ_{𝔫}(t)t.) Then we set GO^{∘}(ℭ, 𝔫) to be the algebraic group with F -points

GO^{∘}(ℭ, 𝔫)(F) := {t ∈ GL(ℭ) |t is a similarity of 𝔫 with multiplier μ(t) such that det(t) = μ(t)^{4}} .

We can also define a new, seemingly uglier multiplication ⋆ on ℭ by setting x ⋆ y := x̄ȳ as in [KMRT98, §34.A]. Then a related triple is a triple (t_{0}, t_{1}, t_{2}) in GO^{∘}(ℭ, 𝔫)^{×3} such that

μ(t_{i})^{−1}t_{i}(x ⋆ y) = t_{i+2}(x) ⋆ t_{i+1}(y)

for all x, y ∈ ℭ and i = 0, 1, 2 with subscripts taken modulo 3. We write Rel (ℭ, 𝔫) for the algebraic subgroup of GO^{∘}(ℭ, 𝔫)^{×3} consisting of related triples and Spin(𝔫) for the subgroup of Rel (ℭ, 𝔫) consisting of triples with multiplier one (i.e., those triples such that μ(t_{i}) = 1 for all i).

4.2. The vector space underlying the split Albert F -algebra J is the subspace of M_{3}(ℭ) consisting of elements fixed by the conjugate transpose ∗ which applies ¯ to each entry and takes the transpose. It is the algebra denoted by ℌ(ℭ_{3}) in the notation of [Jac68, §I.5] and has multiplication a · b := (ab + ba)/2, where juxtaposition denotes the usual multiplication on M_{3}(ℭ). When writing down explicit elements of J, we will use a “·” to indicate entries whose values are forced by this symmetry condition. Then Rel (ℭ, 𝔫) embeds in the group Inv (J) of norm isometries of J via the map \underline{t} ↦ g\underline{t} given by

g\underline{t} ( \begin{matrix}ε_{0} & c_{2} & · \\ · & ε_{1} & c_{0} \\ c_{1} & · & ε_{2} \end{matrix} ) = ( \begin{matrix}μ(t_{0})^{−1}ε_{0} & t_{2}(c_{2}) & · \\ · & μ(t_{1})^{−1}ε_{1} & t_{0}(c_{0}) \\ t_{1}(c_{1}) & · & μ(t_{2})^{−1}ε_{2} \end{matrix} ) . (4.3)

4.4. Definition of G. Since Rel (ℭ, 𝔫) embeds in Inv (J) over F, it embeds in E_{6}^{K} over K. However, we can identify E_{6}^{K} with Inv (J) with a different ι-action where ^{ι}f := ιf^{†}ι, where ι is the nontrivial F -automorphism of K and juxtaposition denotes the usual action, and we fix this identification for the rest of the paper. Then the map Rel (ℭ, 𝔫) → E_{6}^{K} is not defined over F, because for \underline{t} = (t_{0}, t_{1}, t_{2}) ∈ Rel (ℭ, 𝔫)(F_{s}) and g\underline{t} ∈ E_{6}^{K}, we have ^{ι}g\underline{t} = g_{ισ_{𝔫}(\underline{t})^{−1}ι} which is typically not the same as g_{ι\underline{t}ι} where σ_{𝔫}(\underline{t}) means to apply σ_{𝔫} to each component of \underline{t}. So we define G to be the algebraic group over F which is the same as Rel (ℭ, 𝔫) over K but with a different ι-action: for \underline{t} ∈ G(F_{s}), we set ^{ι}\underline{t} := ισ_{𝔫}(\underline{t})^{−1}ι. Then G injects into E_{6}^{K} over F via the map g from (4.3).

This group G is reductive with absolute rank 6 and semisimple part Spin(𝔫) of type ^{1}D_{4}. Its importance is given by the following lemma, excavated from a paper by Ferrar:

Ferrar’s Lemma 4.5. [Fer69, p. 65, Lem. 3] The natural map H^{1}(K/F, G) → H^{1}(K/F, E_{6}^{K}) is surjective.

Comments: Ferrar proved this by explicit computations in the Jordan algebra. However, this can also be seen with more algebraic grouptheoretic methods, as was pointed out to me by Gille: We must assume that our base field has characteristic zero, which as we observed in the introduction doesn’t impair our main results in any way. Let T be a maximal torus in G defined over F. Then [PR94, p. 329, Lem. 6.17] combined with the fact that all maximal tori in E_{6}^{K} are conjugate over F says that there is a Borel subgroup B of E_{6}^{K} defined over K such that T = B ∩ ^{ι}B. Consequently, the natural map H^{1}(K/F, T) → H^{1}(K/F, E_{6}^{K}) is a surjection by [PR94, p. 369, Lem. 6.28].

Now imagine how the argument for proving the main theorem in the ^{2}E_{6} case must proceed: We apply some simple argumentation and Ferrar’s Lemma to show that any class in H^{1}(F, E_{6}^{K}) with trivial Rost invariant must come from H^{1}(K/F, G). Then we apply some facts about Rost invariants on this smaller group to obtain the theorem. However, G is reductive, so we want to put our class with trivial Rost invariant into a semisimple subgroup if we hope to apply our results from Section 1. So we need to do something about the center of G. 4.6. The center P of G. For the moment, set N_{1} to be the algebraic group with F -points the elements of K^{∗} with norm 1 in F. This group is the same as 𝔾_{m} over K, but has a different ι-action given by ^{ι}λ = ι(λ)^{−1}. It is often denoted by something like R_{K/F}^{(1)} (𝔾_{m,K}).

Now, since 𝔾_{m} is the center of GO^{∘}(ℭ, 𝔫), the center of Rel (ℭ, 𝔫) is the subgroup of 𝔾_{m}^{×3} consisting of triples whose product is one. But we are concerned with G, which has a different ι-action; its center P is then isomorphic to the subgroup of N_{1}^{×3} consisting of triples whose product is 1.

4.7. We have a map G → P given by sending each t_{i} to its multiplier μ(t_{i}) = σ_{𝔫}(t_{i})t_{i} ∈ N_{1}. The semisimple part Spin(𝔫) of G is the kernel of this map, and we have a short exact sequence 1 → Spin(𝔫) → G → P → 1. This sequence is even exact over K (instead of just over a separable closure of F) because the map G → P is surjective over K by [KMRT98, 35.4].

So consider the map H^{1}(K/F, G) → H^{1}(K/F, P). We need to describe this second group. We have a short exact sequence over K

1 ⟶ P ⟶ N_{1}^{×3} ⟶^{π} N_{1} ⟶ 1,

where π is the product map, which induces an exact sequence

1 ⟶ H^{1}(K/F, P) ⟶ H^{1}(K/F, N_{1}^{×3}) ⟶^{H^{1}(π)} H^{1}(K/F, N_{1}).

(The second map is an injection because the product map π is a surjection on F -points.) Now any 1-cocycle in Z^{1}(K/F, N_{1}) is determined by its value at ι, and the condition that it is a 1-cocycle forces that this value lies in F^{∗}. Then the obvious check shows that two such are cohomologous if and only if they differ by a norm from K^{∗}. So H^{1}(K/F, P) is isomorphic to the subgroup of (F^{∗}/N_{K/F} (K^{∗}))^{×3} with product one.

So if we have a 1-cocycle γ ∈ Z^{1}(K/F, G), it is determined by its value at ι, and the image of γ in H^{1}(K/F, P) is the multiplier of its value there.

One natural question arising from this discussion is the following: If you have a 1-cocycle in H^{1}(K/F, E_{6}^{K}), then it comes from H^{1}(K/F, G) by Ferrar’s Lemma and so has an image in H^{1}(K/F, P). Is that image an invariant of the original class in H^{1}(K/F, E_{6}^{K})? The answer is a very definite no, as is shown in the following lemma. (We will give explicit situations where the hypotheses are satisfied nontrivially in 6.6 and 7.10.) Let e_{i} ∈ J be the element whose only nonzero entry is a 1 in the (i + 1, i + 1)-position.

Moving Lemma 4.8. Let η be a 1-cocycle in Z^{1}(K/F, G) whose image in Z^{1}(K/F, P) takes the value \underline{a} at ι. Suppose that there is some j ∈ e_{0} × J_{K} such that

j^{#} = 0  and  T (j, η_{ι}ιj) = r ∈ F^{∗}.

Then η is cohomologous in H^{1}(K/F, E_{6}^{K}) to a 1-cocycle coming from Z^{1}(K/F, G) whose image in Z^{1}(K/F, P) takes the value (r, a_{0}, (a_{0}r)^{−1}) at ι.

The hypotheses in the lemma make use of the Freudenthal cross product ×: J × J → J, which is a commutative bilinear map defined by the relation 6N(j) = T (j, j × j) for all j ∈ J. The map #: J → J is defined by 2j^{#} := j × j.

Proof: The proof is an adaptation of an argument in [Fer80, p. 277]. We first observe that the three elements j, e_{0}, and e_{0} × j′ for j′ := η_{ι}ιj all have “rank one”, i.e., are sent to zero by the map x ↦ x^{#}. Furthermore, since e_{0} × J is preserved by ι and G(K) [Garb, 1.7], j′ = e_{0} × y for some y and applying a linearization of [McC69, (19)] gives us

e_{0} × (e_{0} × j′) = T (e_{0}, e_{0})(e_{0} × y) = j′.

Thus this triple is “in general position”, meaning that for N trilinearized so that N(x, x, x) = N(x), we have

6N(e_{0}, j, e_{0} × j′) = T (e_{0} × j, e_{0} × j′) = T (j, e_{0} × (e_{0} × j′)) = T (j, j′) = r ≠ 0.

Thus by [SV68, 3.11], there is some f ∈ Inv (J)(K) such that

f(j) = ρ_{0}e_{0}, f(e_{0}) = ρ_{1}e_{1}, and f(e_{0} × j′) = ρ_{2}e_{2}

for ρ_{i} ∈ K^{∗}. Since 𝔫 is hyperbolic, G(K) is split over K and the map G(K) → P (K) is surjective. So there is some g = g\underline{t} ∈ Inv (J)(K) such that g(e_{0}) = ρ_{0}^{−1}e_{0}, g(e_{1}) = ρ_{1}^{−1}e_{1}, and g(e_{2}) = ρ_{0}ρ_{1}e_{2}. By replacing f with gf, we may assume that ρ_{0} = ρ_{1} = 1. Moreover, f preserves N, and so r = 6N(e_{0}, e_{1}, ρ_{2}e_{2}) = ρ_{2}.

So we set η′ ∈ Z^{1}(K/F, E_{6}^{K}) to be the cocycle cohomologous to η given by η′_{ι} = f^{†}η_{ι}ιf^{−1}ι. Keeping in mind the facts that e_{i}×e_{i+1} = e_{i+2}; f(u × v) = f^{†}(u) × f^{†}(v) for all u, v ∈ J_{K}; and j × (e_{0} × j′) = re_{0} (as can be verified by examining the explicit formula for × given in [Jac68, p. 358, (4)], although the reader should be cautioned that our definition of × — which agrees with the one in [KMRT98] and [McC69] — differs from Jacobson’s by a factor of 2), one can now easily calculate that f^{†}(e_{0}) = e_{1} and f^{†}(j′) = re_{0}. Then it follows that

η′_{ι}(e_{0}) = re_{0}, and η′_{ι}(e_{1}) = a_{0}^{−1}e_{1}.

Finally, since η_{ι}ι(e_{0} × j′) = a_{0}e_{0} × j, we have η′_{ι}(e_{2}) = (a_{0}/r)e_{2}, and we are done.

5. ^{2}D_{5} ⊂ ^{2}E_{6}

For the purpose of making computations, we will need to make use of another subgroup of E_{6}^{K}, which we define to be the subgroup consisting of elements h such that h and h^{†} both fix the element e_{0} ∈ J. Since the map h → h^{†} is a group homomorphism on Inv (J), it is clear that H is indeed a subgroup of E_{6}^{K} over K, and it is certainly preserved by the ι-action so it is even defined over F. Our first task is to describe it explicitly. 5.1. Fix a particular basis u_{1}, u_{2}, …, u_{8} for the split Cayley algebra ℭ as given in [Gar98, p. 388]. One important thing for us to know about this basis is that when we bilinearize the norm form 𝔫 so that 𝔫(x, x) = 𝔫(x), we have

𝔫(u_{i}, u_{j}) = \begin{cases}1 & if i + j = 9 \\ 0 & otherwise, \end{cases}

so that the Gram matrix of 𝔫 with respect to this basis is a matrix we will denote by S_{8}, meaning that it is the 8 × 8 matrix which has zeroes everywhere except for a line of ones connecting the (1, 8) and the (8, 1) entries. Also, the canonical involution ¯ is given by

\overline{u_{i}} = \begin{cases}−u_{i} & if i ≠ 4, 5 \\ u_{5} & if i = 4 \\ u_{4} & if i = 5. \end{cases}

5.2. Over K, H is isomorphic to Spin_{10} ⋊ℤ/2. Let A denote the 10-dimensional subspace e_{0} × J of J, which is A = ( \begin{matrix}0 & 0 & · \\ · & F & ℭ \\ 0 & · & F \end{matrix} ). We first observe that if f ∈ H, then f(e_{0} × j) = f^{†}(e_{0}) × f^{†}(j), so f(A) = A. The multiplication on J restricts to give A the structure of a central simple Jordan algebra as well, albeit with a different unit element. It has norm form N_{A} given by

N_{A} ( \begin{matrix}0 & 0 & · \\ · & α & c \\ 0 & · & β \end{matrix} ) = αβ − 𝔫(c).

Next we extend scalars to F (t) and consider N(te_{0} + j) = N(f(te_{0} + j)) = N(te_{0}+f(j)). The coefficient of t in this expression is T (e_{0}, j^{#}) = T (e_{0}, f(j)^{#}). For j actually lying in A, T (e_{0}, j^{#}) = N_{A}(j), so f must restrict to preserve the norm on A. We write O(A) for the algebraic subgroup of GL(A) consisting of maps which preserve the norm N_{A}, and we have proven that restriction provides a map H → O(A) which is defined over K. (It may not be defined over F because of the strange ι-action on H inherited from E_{6}^{K}.)

Clearly, O(A) is an orthogonal group for the 10-dimensional quadratic form H ⊥ −𝔫. Moreover, the map H → O(A) has kernel of order 2: Anything in H which maps to the identity in O(A) fixes all of the diagonal idempotents, and so must be of the form g\underline{t} for some \underline{t} ∈ Spin(𝔫) [Sod66, p. 155, Thm. 1]. However, t_{0} must also be the identity, so \underline{t} = (1, 1, 1) or (1, −1, −1) by [Gar98, 1.5(2)].

We would like to show that the map H → O(A) is surjective. Note that O(A) is generated by

• the special orthogonal group SO(B) for B the codimension 1 subspace of A spanned by ℭ and e_{1} − e_{2} (where e_{i} denotes the element of J whose only nonzero entry is a 1 in the (i + 1, i + 1)- position) endowed with the quadratic form given by restricting N_{A};

• anything in O(A) with determinant −1; and

• anything in O(A) which doesn’t leave B invariant. Since for f ∈ Aut (J), f^{†} = f, the subgroup Aut (J/e_{0}) of elements of Aut (J) which fix e_{0} is a subgroup of H. As described in [Jac68, p. 376, Thm. 4], Aut (J/e_{0}) ≅ Spin(B) and the restriction to B gives the surjection onto SO(B). The map

( \begin{matrix}ε_{0} & c_{2} & · \\ · & ε_{1} & c_{0} \\ c_{1} & · & ε_{2} \end{matrix} ) ↦ ( \begin{matrix}ε_{0} & c_{1} & · \\ · & ε_{2} & c_{0} \\ c_{2} & · & ε_{1} \end{matrix} )

lies in H(K) and restricts to have determinant −1 on A. Finally, we consider Freudenthal’s maps from [Jac61, p. 74]. For E_{ij} ∈ M_{3}(ℭ) the matrix whose only nonzero entry is a 1 in the (i, j)-position, 1_{3} is the 3 × 3 identity matrix, x ∈ ℭ, and a ∈ J, he defines a map ψ_{ij}(x) ∈ Inv (J) given by

ψ_{ij}(x)(a) = (1_{3} + xE_{ij})a(1_{3} + xE_{ij})^{∗},

where juxtaposition denotes the usual multiplication in M_{3}(ℭ), not the Jordan multiplication. So ψ_{ij}(x) ∈ H(K) if i, j ≠ 1. In particular, ψ_{32}(u_{5})|_{A} is given by

ψ_{32}(u_{5})|_{A} ( \begin{matrix}ε_{1} & c_{0} \\ · & ε_{2} \end{matrix} ) = ( \begin{matrix}ε_{1} & c_{0}−ε_{1}u_{4} \\ · & ε_{2}−𝔫(c_{0},u_{4}) \end{matrix}) ,

which doesn’t leave B invariant.

Finally, we observe that H^{∘} is isomorphic to Spin(A). The inverse image, call it H′, of SO(A) maps onto SO(A) with a kernel which is central and of order 2. Consequently, H′ is absolutely almost simple and hence must be isomorphic to Spin(A). Since H′ is connected and [G : H′] = 2, H^{∘} = H′.

5.3. Over F, H is isomorphic to Spin(4H ⊥ ⟨−2, 2k⟩) ⋊ ℤ/2. To compute the isomorphism class of H over F, we observe that the map h ↦ h^{†} restricts to the identity on the kernel of the K-map H → O(A), so the ι-action on H induces one on O(A), which we will calculate explicitly.

Fix the basis (u_{1}, u_{2}, u_{3}, u_{4}, e_{1}, e_{2}, u_{5}, …, u_{8}) for A so that the Gram matrix for N_{A} becomes

( \begin{matrix}& & −S_{4} \\ & S_{2} \\ −S_{4} \end{matrix} ) ,

for S_{2} and S_{4} defined analogously to how S_{8} was in 5.1. Then SO(A) is generated by

• a torus T consisting of diagonal matrices with diagonal entries (d_{1}, d_{2}, …, d_{5}, d_{5}^{−1}, d_{4}^{−1}, …, d_{1}^{−1});

• root groups U_{ij}: 𝔾_{a} → SO(A) given by

U_{ij}(r) = 1_{10} + rE_{ij} − rE_{j^{∗}i^{∗}}

for 1_{10} the 10 × 10 identity matrix, i^{∗} := 11 − i, and (i, j) = (i, i + 1) for i = 1, 2, 3, and their transposes; and

• root groups V_{ij}: 𝔾_{a} → SO(A) given by

V_{ij}(r) = 1_{10} + r(E_{ij} + E_{j^{∗}i^{∗}})

for (i, j) = (4, 5) and (4, 6), and their transposes. (Note that V_{45}(r) = ψ_{32}(ru_{5})|_{A} and V_{46}(r) = ψ_{23}(ru_{4})|_{A} for r ∈ F = 𝔾_{a}(F).)

Since the torus lies in the image of Rel (ℭ, 𝔫) and g† ; \underline{t} = g_{σ_{𝔫}(\underline{t})^{−1}}, the action on T and on the first kind of root groups is the usual ι-action. However,

V_{45}(r)^{†} = ψ_{32}(ru_{5})^{†} = ψ_{23}(−ru_{4}) = V_{46}(−r).

So the map h ↦ h^{†} induces on SO(A) the map f ↦ MfM^{−1} for

M = ( \begin{matrix}1_{4} \\ & −S_{2} \\ & & 1_{4} \end{matrix} ) .

Write η for the 1-cocycle in Z^{1}(K/F, O(A)) given by η_{ι} = M. The K-map H → O(A) descends to a map over F from H onto the twisted group O(A)_{η}, so we wish to describe the group O(A)_{η}.

But this is now just a problem of explicitly computing a quadratic form given by descending down a quadratic extension. So we need to find a K-basis of A ⊗ K consisting of elements fixed by the map a ⊗ κ ↦ M(a) ⊗ ι(κ). Then O(A)_{η} ≅ O(q), where q is the restriction of N_{A} to the F -span of those fixed vectors. Such a K-basis is given by u_{i} for 1 ≤ i ≤ 8, e_{1} − e_{2}, and \sqrt{k}e_{1} + \sqrt{k}e_{2}. These vectors give an orthogonal basis for a quadratic form 4H ⊥ ⟨−2, 2k⟩, which proves the claim.

We close this section by proving a little lemma which foreshadows the way we will prove the Main Theorem for quasi-split groups of type 2E_{6}.

Lemma 5.4. The restriction of the Rost invariant on H^{1}(F, E_{6}^{K}) to the image of H^{1}(F, H^{∘}) has trivial kernel.

Proof: We first observe that the embedding of H^{∘} in E_{6}^{K} has Rost multiplier one. Since the Rost multiplier is invariant under scalar extension, we may work over K, where this embedding is described in 5.2. Then some of the coroots (identified with copies of 𝔾_{m} lying in the maximal torus T from 5.3) for H^{∘} are the same as those for Spin(𝔫) considered as a subgroup of Inv (J) via the map g. Since the embedding Spin(𝔫) ↪ Inv (J) has Rost multiplier one, so does H^{∘} ↪ E_{6}^{K}.

We are left with showing that anything in the kernel of the Rost invariant on H^{1}(F, H^{∘}) maps to be trivial in H^{1}(F, E_{6}^{K}). By Lemma 1.2, the kernel of the Rost invariant is precisely the image of H^{1}(F, C), where C ≅ μ_{2} has for its unique nontrivial element the map g_{(1,−1,−1)} for g as in (4.3). Now Spin(𝔫) is a subgroup of H^{∘} over F and it contains this subgroup C so we have a composition

H^{1}(F, C) → H^{1}(F, Spin(𝔫)) → H^{1}(F, H^{∘}) → H^{1}(F, E_{6}^{K}). (5.5)

Since 𝔫 is hyperbolic, the spinor norm SO(𝔫)(F) → F^{∗}/F^{∗2} = H^{1}(F, C) is surjective, so by the exactness of (1.3) the first map in (5.5) is trivial.

6. Special cocycles

Definition 6.1. For \underline{a} = (a_{0}, a_{1}, a_{2}) ∈ (F^{∗})^{×3} with product 1, we define a “special” cocycle z := z_{K,\underline{a}} in H^{1}(K/F, G). Set z_{ι} = (z_{0}, z_{1}, z_{2}) where z_{j} = m_{j}(\underline{a})dP for P the permutation matrix giving the map u_{k} ↦ u_{π(k)} for π the permutation (1 2)(3 6)(4 5)(7 8), m_{j}(\underline{a}) the diagonal matrix

m_{j}(\underline{a}) := diag(1, a_{j}, a_{j}, a_{j+2}^{−1}, a_{j+1}^{−1}, 1, 1, a_{j}) (6.2) with subscripts taken modulo 3, and

d := diag(1, 1, −1, 1, 1, −1, 1, 1).

Then the z_{j} form a related triple by [Gar98, 1.6, 1.7, 1.5(3)], so z_{ι} ∈ G(K). Since P is an isometry of 𝔫, σ_{𝔫}(P) = P^{−1} = P. Then

ιz_{j} = σ_{𝔫}(m_{j}(\underline{a})dP)^{−1} = diag (a_{j}^{−1}, 1, 1, a_{j+1}, a_{j+2}, a_{j}^{−1}, a_{j}^{−1}, 1) d P

and so z is indeed in Z^{1}(K/F, G).

6.3. Freedom in the definition. Of course, some of these special cocycles are cohomologically equivalent in H^{1}(K/F, G). If \underline{a} and \underline{a′} are two triples in (F^{∗})^{×3} such that a_{j}^{−1}a′_{j} ∈ N_{K/F} (K^{∗}) for all j, fix λ_{j} ∈ K^{∗} such that a_{j}^{−1}a′_{j} = λ_{j}ι(λ_{j}). Then for \underline{ℓ} = (ℓ_{0}, ℓ_{1}, ℓ_{2}) with ℓ_{j} = P m_{j}(\underline{λ})P, \underline{ℓ} is a related triple by [Gar98], so \underline{ℓ} ∈ G(K). Then ^{ι}\underline{ℓ} (z_{K,\underline{a′}})_{ι} \underline{ℓ}^{−1} = (z_{K,\underline{a}})_{ι}, i.e., the two cocycles z_{K,\underline{a}} and z_{K,\underline{a′}} are cohomologous.

6.4. We will twist by these cocycles to move a cocycle in H^{1}(F, G) so that it takes values in a semisimple group. For now, we just observe that the semisimple group we get from one of them, Spin(𝔫)_{z}, is described in [Gar98, pp. 403, 404]: Let k ∈ F^{∗} be such that K = F (\sqrt{k}) and let Q_{i} denote the quaternion algebra (k, a_{i})_{F} generated by elements x, y such that x^{2} = k, y^{2} = a_{i}, and xy = −yx. Then the group Spin(𝔫)_{z} is isomorphic to Spin(A_{i}, σ_{i}) where A_{i} is isomorphic to M_{4}(Q_{i}), σ_{i} is an isotropic orthogonal involution with trivial discriminant, and

(C_{0}(A_{i}, σ_{i}), \underline{σ_{i}}) ≅ (A_{i+1}, σ_{i+1}) × (A_{i+2}, σ_{i+2}), (6.5)

where the subscripts are taken modulo 3. (These properties specify the σ_{i} up to isomorphism [Gara, 2.3].)

The Moving Lemma lets us say something useful about the Rost invariant of our special cocycles.

Corollary 6.6. The Rost invariant of z_{K,\underline{a}} is trivial if and only if the 1-cocycle is cohomologically trivial in H^{1}(F, E_{6}^{K}).

Proof: Consider the element j = ( \begin{matrix}0 & 0 & · \\ · & 0 & c \\ 0 & · & 0 \end{matrix} ) in e_{0} ×J_{K} for c = (u_{2} +u_{8})/2. Then 𝔫(c) = 0 and so consulting the explicit formula for j^{#} in [Jac68, p. 358], we see that j^{#} = 0. Moreover, for z := z_{K,\underline{a}}, we have z_{ι}ιj =

( \begin{matrix}0 & 0 & · \\ · & 0 & c′ \\ 0 & · & 0 \end{matrix} ) for c′ = (u_{1} + u_{7})/2. Then

T (j, j′) = c\overline{c′} + \overline{c}c′ = 2𝔫(c, c′) = 1.

Applying the Moving Lemma shows that z is equivalent in H^{1}(K/F, E_{6}^{K}) to some z′ ∈ Z^{1}(K/F, G) whose image in H^{1}(K/F, P) is (1, a_{0}^{−1}, a_{0}). So in particular we may assume that σ_{𝔫}(z′_{ι})z′_{ι} = (1, a_{0}^{−1}, a_{0}). Since the restriction of z′_{ι} to the 10-dimensional subalgebra A defined in 5.2 has determinant 1, it must lie in H^{∘}. The statement about the triviality is now a consequence of Lemma 5.4

In a special case we can calculate the value of the Rost invariant of our special cocycles quite explicitly.

Lemma 6.7. For a, k ∈ F^{∗} such that K = F (\sqrt{k}), the Rost invariant of the 1-cocycle z_{K,(1,a,a^{−1})} is (a) ∪ (k) ∪ (−1).

Proof: The cocycle z := z_{K,(1,a,a^{−1})} takes values in H and restricts to have determinant one on the subalgebra A defined in 5.2, so z ∈ Z^{1}(K/F, H^{∘}). Since the embedding H^{∘} ⊂ E_{6}^{K} has Rost multiplier 1, to compute the Rost invariant of z, we may compute the Rost invariant of z in H^{1}(F, H^{∘}). But recall that H^{∘} ≅ Spin(q) for q = 4H ⊥ ⟨−2, 2k⟩ and that H^{1}(F, SO(q)) classifies nondegenerate quadratic forms of the same dimension and discriminant as q. So we can compute the Rost invariant of z by computing the quadratic form q_{z} corresponding to the image of z in H^{1}(F, SO(q)), which is just the restriction of q ⊗ K to the vector subspace fixed by the action a ⊗ κ ↦ z_{ι}M(a) ⊗ ι(κ) for M as in 5.3.

We will perform the Galois descent calculation by decomposing A⊗K into 2-dimensional subspaces and calculating the Galois action on those subspaces.

subspace restriction of  F -basis for  contribution
basis  z_{ι}M  fixed subspace  to q_{z}
(u_{1}, u_{2})  S_{2}  totally
(u_{7}, u_{8})  S_{2}  isotropic
(u_{3}, u_{6})  −S_{2}  u_{3} − u_{6}, \sqrt{k}u_{3} + \sqrt{k}u_{6}  ⟨2, −2k⟩
(u_{4}, u_{5})  ( \begin{matrix}& a \\ a^{−1} \end{matrix} )  au_{4} + u_{5}, −a\sqrt{k}u_{4} + \sqrt{k}u_{5} ⟨−2a, 2ak⟩
(e_{1}, e_{2}) ( \begin{matrix}& −a^{−1} \\ −a \end{matrix} ) −e_{1} + ae_{2}, \sqrt{k}e_{1} + a\sqrt{k}e_{2} ⟨−2a, 2ak⟩

The first two subspaces form a complementary pair of totally isotropic subspaces, so they contribute two hyperbolic planes to q_{z}. Thus the image of z is q_{z} = 2H ⊥ ⟨2, −2k, −2a, 2ak, −2a, 2ak⟩ and the Rost invariant of z is the Arason invariant of q_{z} − q = ⟨2⟩≪a, k, −1≫.

7. Quasi-split groups of type E_{6} and E_{7}

This section consists solely of a proof of the main theorem, beginning with a nearly trivial lemma.

Lemma 7.1. Suppose that C is a central subgroup in an almost simple simply connected group Γ. Then H^{1}(F, C) acts on H^{1}(F, Γ) and for ζ ∈ H^{1}(F, C) and γ ∈ H^{1}(F, Γ), we have

R_{Γ}(ζ · γ) = R_{Γ}(ζ) + R_{Γ}(γ),

where R_{Γ}(ζ) denotes the image of ζ under the composition H^{1}(F, C) ⟶

H^{1}(F, Γ) ⟶^{R_{Γ}} H^{3}(F, ℚ/ℤ(2)).

Proof: We pick a 1-cocycle z ∈ Z^{1}(F, C) which represents ζ. Then we have a diagram

H^{1}(F, Γ)  H^{1}(F, Γ_{z}) ⟶_{τ_{z}}^{∼} H^{1}(F, Γ)
R_{Γ}↓  ^{R_{Γz}} ↓  ↓^{R_{Γ}}
H^{3}(F, ℚ/ℤ(2))  H^{3}(F, ℚ/ℤ(2)) ⟶ H^{3}(F, ℚ/ℤ(2))

where the group Γ_{z} is the usual twist of Γ by the cocycle z, so that it is just the group Γ with a different Galois action so that a member σ of the Galois group maps g ↦ z_{σ} ^{σ}gz_{σ}^{−1}. In our case, z_{σ} is always central, so in fact Γ_{z} = Γ. The map τ_{z} is the usual twisting map [Ser94, I.5.5], defined by sending a ∈ Z^{1}(F, Γ_{z}) to the 1-cocycle σ ↦ a_{σ}z_{σ}. The composition of the two maps on the top row is then the action of ζ.

The left-hand box commutes because the Rost invariant is canonical. The right-hand box commutes, where the bottom map is given by η ↦ η + R_{Γ}(ζ) by [Gilb, p. 16, Lem. 8].

This result has the obvious corollary that the induced map H^{1}(F, C) ⟶ H^{3}(F, ℚ/ℤ(2)) is a group homomorphism. 7.2. Groups of type ^{1}E_{6}. Suppose first that our simply connected quasi-split group of type E_{6} is split and denote it simply by E_{6}. From Example 3.4, we have an embedding F_{4} × μ_{3} ↪ E_{6} which induces a surjection on H^{1} terms. So for ε ∈ H^{1}(F, E_{6}), we can find a ϕ ∈ H^{1}(F, F_{4}) and ζ ∈ H^{1}(F, μ_{3}) such that ϕ ⊕ ζ ↦ ε. Now we simply note that since E_{6} is split, the image of H^{1}(F, μ_{3}) → H^{1}(F, E_{6}) is trivial. So if ε is in the kernel of the Rost invariant R_{E_{6}}, by Lemma 7.1 ϕ must be killed by the composition

H^{1}(F, F_{4}) → H^{1}(F, E_{6}) ⟶^{R_{E_{6}}} H^{3}(F, ℚ/ℤ(2)).

As described in 2.4, the Rost multiplier of the embedding F_{4} ⊂ E_{6} is 1, so ϕ lies in the kernel of the Rost invariant R_{F_{4}}, which is known to be trivial. So ε is the image of ζ, which we have already observed is trivial.

Remark 7.3 (Noninjectivity for ^{1}E_{6}). We want to provide an example to show that the Rost invariant is noninjective for the group E_{6}. We can not simply apply Remark 0.6 and the fact that the embedding F_{4} ↪ E_{6} has Rost multiplier one, since two isotopic Albert algebras have the same image in H^{1}(F, E_{6}).

So instead fix a ground field F which supports a division (= nonreduced) Albert F -algebra J. Then over the field F (t), the norm N of J doesn’t represent t as can be seen by some elementary valuation theory [Jac68, p. 417, Lem. 1]. Consequently, N is not isomorphic to tN over F (t), so the images of the two classes (J) ⊕ (1) and (J) ⊕ (t) under the map H^{1}(F, F_{4}) × H^{1}(F, μ_{3}) → H^{1}(F, E_{6}) are distinct by [Garb, 2.8(2)]. However, since the image of H^{1}(F, μ_{3}) → H^{1}(F, E_{6}) is trivial, by Lemma 7.1 the two classes in H^{1}(F, E_{6}) have the same Rost invariant.

7.4. Groups of type ^{2}E_{6}. Suppose now that our quasi-split simply connected group of type E_{6} is not actually split, so that it only becomes split over some quadratic field extension K of F. We write E_{6}^{K} for this group, as we have since Section 4. Then by the split case, any α ∈ H^{1}(F, E_{6}^{K}) which is in the kernel of the Rost invariant must become trivial over K and so must come from H^{1}(K/F, E_{6}^{K}). Applying Ferrar’s Lemma 4.5, we have that α is the image of some β ∈ H^{1}(K/F, G).

7.5. Twisting. We fix a triple \underline{a} = (a_{0}, a_{1}, a_{2}) ∈ (F^{∗})^{×3} such that a_{0}a_{1}a_{2} = 1 which represents the image of β in H^{1}(K/F, P). (This makes sense thanks to the description of H^{1}(K/F, P) in 4.7.) Then we set z := z_{K,\underline{a}} as defined in 6.1, and we can twist E_{6}^{K} by z to obtain a diagram

H^{1}(F, G_{z}) ⟶ H^{1}(F, (E_{6}^{K})_{z}) ⟶^{R_{(E^{K}_{6} )z}} H^{3}(F, ℚ/ℤ(2))
τ_{z}↓  ^{τ_{z}}↓  ↓^{·+R_{E^{K}_{6}} (z)}
H^{1}(F, G) ⟶ H^{1}(F, E_{6}^{K}) ⟶^{R_{E^{K}_{6}}} H^{3}(F, ℚ/ℤ(2)),

where the right vertical arrow has the specified value by [Gilb, p. 16, Lem. 8].

7.6. The image of τ_{z}^{−1}(β) in H^{1}(F, SO(A, σ)). We want to say something about what kind of class β′ := τ_{z}^{−1}(β) can be. In particular, its image in H^{1}(K/F, P_{z}) is trivial, so β′ comes from the semisimple part of G_{z}, which is isomorphic to Spin (A, σ) for (A, σ) one of the three algebras A_{i} described in 6.4.

So we can think of β′ as lying in H^{1}(K/F, Spin (A, σ)) and consider its image in H^{1}(K/F, SO(A, σ)). Let L be a generic splitting field of A (e.g., a function field of its Severi-Brauer variety) and consider the image of β′ in H^{1}(L, SO(A, σ)). Since A is split by L, σ becomes adjoint to the quadratic form ≪k, a_{i+1} ≫ ⊥ 2H [Gara, 2.3]. The image of β′ determines an 8-dimensional quadratic form q over L, and the Rost invariant of β′ is just the class of q − ≪k, a_{i+1} ≫ in I^{3}L/I^{4}L. However, by the twisting argument above, the Rost invariant of β′ over F is −R_{E_{6}^{K}}(z). Since A is split over L, a_{i} ∈ L^{∗} is a norm from KL, so by 6.3 and Lemma 6.7 the Rost invariant becomes (k) ∪ (a_{i+1}) ∪ (−1) over L.

So we have that for ϕ = ≪k, a_{i+1} ≫, q − ϕ ∈ I^{3}L and q − ϕ ≡ ϕ≪−1≫ mod I^{4}L. But then

q + ϕ = (q − ϕ) + 2ϕ ≡ 4ϕ ≡ 0 mod I^{4}L.

So q + ϕ ∈ I^{4}L. However dim(q ⊥ ϕ) = 12 < 16, so by the Arason-Pfister Hauptsatz, q ⊥ ϕ is hyperbolic and q ≅ ⟨−1⟩ϕ ⊥ 2H.

The consequence of all that is that the image of β′ in H^{1}(L, SO(A, σ)) is the same as the image of −1 ∈ F^{∗}/F^{∗2} = H^{1}(F, Z(SO(A, σ))). Since A is Brauer-equivalent to a quaternion algebra, it follows from the material in [Sch85, Ch. 10] that the canonical map H^{1}(F, SO(A, σ)) → H^{1}(L, SO(A, σ)) is injective. (This was shown independently in [PSS] and [Dej99, 5.3.1].) So the image of β′ in H^{1}(F, SO(A, σ)) must also be −1.

7.7. More generally, any simply connected group Γ of type ^{1}D_{4} is isomorphic to Spin(A_{i}, σ_{i}) for three central simple algebras A_{i} of degree 8 with i = 0, 1, 2 endowed with an orthogonal involution σ_{i} with trivial discriminant and related as in (6.5).

Each of the three descriptions of Γ comes paired with natural maps Γ → SO(A_{i}, σ_{i}) → P Γ for P Γ the adjoint group associated to Γ. The kernel of the second map is Z(SO(A_{i}, σ_{i})) ≅ μ_{2}, and the kernel of the composition is Z(Γ), which is isomorphic to the subgroup of μ_{2}^{×3} with product one. Then the group H^{1}(F, Z(Γ)) can be identified with the set of triples \underline{b} = (b_{0}, b_{1}, b_{2}) ∈ F^{∗}/F^{∗2} with product 1 [KMRT98, 44.14] and where the map H^{1}(F, Z(Γ)) → H^{1}(F, Z(SO(A_{i}, σ_{i}))) is given by \underline{b} ↦ b_{i}.

Lemma 7.8. (Notation as in the preceding paragraph.) Suppose an element η ∈ H^{1}(F, Γ) has the same image in H^{1}(F, SO(A_{i}, σ_{i})) as c_{i} ∈ F^{∗}/F^{∗2} = H^{1}(F, Z(SO(A_{i}, σ_{i}))) for i = 1, 2. Then η is the image of ((c_{1}c_{2})^{−1}, c_{1}, c_{2}) coming from H^{1}(F, Z(Γ)).

Proof: We have a short exact sequence 1 → Z(SO(A_{i}, σ_{i})) → SO(A_{i}, σ_{i}) → P Γ → 1, so η is killed by the composition H^{1}(F, Γ) → H^{1}(F, SO(A_{i}, σ_{i})) → H^{1}(F, P Γ) for i = 1. Thus η lies in the image of H^{1}(F, Z(Γ)).

For general Galois-cohomological reasons, the map H^{1}(F, Z(Γ)) → H^{1}(F, Γ) is a group homomorphism. (Although the second set doesn’t have a group structure, the image of the first set does.) The kernel of this map can be described fully by suitably applying [KMRT98, 35.4], but for our purposes it is enough to observe that it contains all elements of the form (s, s^{−1}, 1) for s a spinor norm of an element in SO(A_{2}, σ_{2})(F) and symmetrically. Let G(A_{i}, σ_{i})^{∘} be the algebraic group of proper similarity factors, i.e., the group with F -points

G(A_{i}, σ_{i})^{∘}(F) = {m ∈ F^{∗} | ∃ f ∈ A_{i}^{∗} such that m = σ_{i}(f)f and Nrd_{A_{i}}(f) = m^{4}} .

Then for every m_{0} ∈ G(A_{0}, σ_{0})^{∘}(F), the kernel contains an element of the form (m_{0}, m_{1}, m_{2}) and symmetrically. Conversely, if (b_{0}, b_{1}, b_{2}) is in the kernel, then b_{i} ∈ G(A_{i}, σ_{i})^{∘}(F) for all i.

It is also the case that the natural map F^{∗}/F^{∗2} = H^{1}(F, Z(SO(A_{i}, σ_{i}))) → H^{1}(F, SO(A_{i}, σ_{i})) is a group homomorphism, and its kernel is precisely G(A_{i}, σ_{i})^{∘}(F). So the fact that η maps to the image of c_{2} in H^{1}(F, SO(A_{2}, σ_{2})) means that η is the image of some class (n_{0}, n_{1}, n_{2}) in H^{1}(F, Z(Γ)) such that n_{2} = c_{2}.

Now consider the middle component of this triple. By hypothesis, n_{1} = m_{1}c_{1} for some m_{1} ∈ G(A_{1}, σ_{1})^{∘}(F). By [Mer96, p. 262, Prop.], the group SN(A_{2}, σ_{2})(F) of spinor norms from SO(A_{2}, σ_{2})(F) is F^{∗2} times the norms from field extensions E which split A_{2} and make σ_{2} isotropic. By [Mer96, p. 263, Prop.], G(A_{1}, σ_{1})^{∘}(F) is equal to the group generated by the norms from every extension field E which splits A_{1} and makes σ_{1} hyperbolic. Since the (A_{i}, σ_{i}) are related by (6.5), any extension which splits A_{1} and makes σ_{1} hyperbolic certainly splits A_{2} and makes σ_{2} isotropic, so SN(A_{2}, σ_{2})(F) ⊇ G(A_{1}, σ_{1})^{∘}(F). Consequently, (n_{0}, n_{1}, n_{2}) has the same image in H^{1}(F, Γ) as ((c_{1}c_{2})^{−1}, c_{1}, c_{2}).

7.9. β′ is in the image of H^{1}(K/F, Z(Spin (A, σ))). Let (A, σ) = (A_{0}, σ_{0}) for (A_{i}, σ_{i}) as in 6.4. Combining the result from 7.6 with Lemma 7.8, we have that β′ ∈ H^{1}(F, Spin (A, σ)) is the image of (1, −1, −1) ∈ H^{1}(F, Z(Spin (A, σ))). However, for k ∈ F^{∗} such that K = F (\sqrt{k}), since K certainly splits A and makes σ hyperbolic and −k = N_{K/F} (\sqrt{k}), by Merkurjev’s norm principle [Mer96, p. 262, Prop.] there is some element of SO(A, σ)(F) with spinor norm −k. Then as described in the proof of Lemma 7.8, β′ is also the image of (1, k, k^{−1}) ∈ H^{1}(F, Z(Spin (A, σ))), which itself is in the image of H^{1}(K/F, Z(Spin (A, σ))).

7.10. Consider the 1-cocycle b = τ_{z}(b′) ∈ Z^{1}(K/F, G) for b′ the image of (1, k, k^{−1}) as above. (Note that b represents the class of β and is the 1-cocycle which takes the value g_{(1,−1,−1)}z_{K,\underline{a}} at ι.) For j and c as in the proof of 6.6, we set j′ := b_{ι}ιj, so that j′ = ( \begin{matrix}0 & 0 & · \\ · & 0 & c′ \\ 0 & · & 0 \end{matrix} ) for c′ = (u_{1} + u_{7})/2. So T (j, j′) = 2𝔫(c, c′) = 1. Then by the Moving Lemma 4.8, we may replace β by a different inverse image of α in H^{1}(K/F, G) and so assume that \underline{a} = (1, a_{0}, a_{0}^{−1}).

Any element of G with multiplier (1, ·, ·) lies in H, and since such an element restricts to have determinant one on the subspace A defined in 5.2, it in fact lies in H^{∘}. Thus α is in the image of H^{1}(F, H^{∘}). Since the Rost invariant of α is trivial, α must be the trivial class by Lemma 5.4.

7.11. Groups of type E_{7}. We are left with proving that the Rost invariant has trivial kernel for G split of type E_{7}, but this follows directly from the same conclusion for quasi-split groups of type E_{6}, thanks to Proposition 3.6.

Acknowledgements. I thank ETH in Zürich for its hospitality while I performed part of the research contained herein; Ph. Gille and Zinovy Reichstein for helpful discussions; Vladimir Chernousov for catching an error in an earlier version of this paper; and absolutely most of all Markus Rost for explaining so much about his invariants and for posing this problem to me.

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R. Skip Garibaldi

e-mail: skip@member.ams.org
web: http://www.math.ucla.edu/~skip/

UCLA Dept. of Mathematics Los Angeles, CA 90095-1555