UNRAMIFIED COHOMOLOGY OF CLASSIFYING VARIETIES FOR EXCEPTIONAL SIMPLY CONNECTED GROUPS

R. SKIP GARIBALDI

Abstract. Let BG be a classifying variety for an exceptional simple algebraic group G. We compute the degree 3 unramified Galois cohomology of BG with values in (ℚ/ℤ)′(2) over a nearly arbitrary field F. Combined with a paper by Merkurjev, this completes the computation of these cohomology groups for G semisimple simply connected over (nearly) all fields.

Let G be an algebraic group over a field F with an embedding ρ: G ↪ SL_{n} over F. The isomorphism class of the variety X := SL_{n}/ρ(G) depends upon the embedding, but its stable birationality type does not. We call X a classifying space of G.

We will compute certain invariants of the stable birationality type of BG, specifically the unramified cohomology defined as follows. Let (ℚ/ℤ)′(d) be the module \underrightarrow{lim} μ_{n}^{⊗d} for n not divisible by char(F). For each d > 0, define H_{nr}^{d}(F (X)) to be the intersection of the kernels of the residue homorphisms

∂_{v}: H^{d}(F (X), (ℚ/ℤ)′(d − 1)) → H^{d−1}(F (v), (ℚ/ℤ)′(d − 2))

as v ranges over the discrete valuations of F (X) over F. If K is a purely transcendental extension of F, then the natural map H_{nr}^{d}(F (X)) → H_{nr}^{d}(K(X)) is an isomorphism, where the discrete valuation rings for the latter group are those containing K [Mer, 2.3]. That is, the group H_{nr}^{d}(F (X)) does not depend on ρ, but only upon the stable birationality type BG of X, so we write H_{nr}^{d}(BG) for H_{nr}^{d}(F (X)), or H_{nr}^{d}(B_{F} G) in order to emphasize the base field F.

The natural homomorphism H^{d}(F, (ℚ/ℤ)′(d − 1)) → H_{nr}^{d}(B_{F} G) is split by the evaluation at the distinguished point of BG, thus,

H_{nr}^{d}(B_{F} G) = H^{d}(F, (ℚ/ℤ)′(d − 1)) ⊕ H_{nr}^{d}(B_{F} G)_{norm},

where the latter group is the group of normalized classes. (For details about all of this, please see [Mer].)

The goal of this paper is to complete the computation of H_{nr}^{3}(BG)_{norm} for G simply connected semisimple and F (nearly) arbitrary. The computation of H_{nr}^{3}(BG)_{norm} for these groups G is quickly reduced to the case where G is simple simply connected [Mer, §4]. In [Mer], H_{nr}^{3}(BG)_{norm} was computed for G simple and classical. We compute it for the remaining cases, where G is exceptional, that is, where G is of type G_{2}, ^{3}D_{4}, ^{6}D_{4}, F_{4}, E_{6}, E_{7}, or E_{8}.

Date: 16 November 2001.

1991 Mathematics Subject Classification. 20G10 (14M17 14M20 17B25).

The author was partially supported by the NSF.

Main Theorem 0.1. Suppose that G is a simply connected simple exceptional algebraic group defined over a field F and that the characteristic of F is not one of the torsion primes for G. Then

H_{nr}^{3}(BG)_{norm} = \begin{cases}ℤ/2 & if G is of type ^{3}D_{4} and has a nonsplit Tits algebra \\ 0 & otherwise. \end{cases}

By torsion primes, we mean the primes associated with G in [Ser95, §2.2]:

type of G  torsion primes
G_{2}  2
3D_{4}, ^{6}D_{4}, F_{4}, E_{6}, E_{7}  2, 3
E_{8}  2, 3, 5

(The interested reader may apply Gille’s [Gil00, Th. 2] to transfer our Main Theorem to fields with characteristic a torsion prime of G.)

Since H_{nr}^{d} is homotopy invariant, we have: If BG is stably rational, then H_{nr}^{d}(BG)_{norm} = 0. (This is a general motivation for studying H_{nr}^{d}(F (X)), cf. [CT95, §4.2].) It was an open question whether BG is stably rational. The first counterexamples were provided in [Mer], where a simple group G was given with H_{nr}^{d}(BG)_{norm} ≠ 0. The results here give another such example.

The computations of H_{nr}^{3}(BG)_{norm} in this paper depend heavily on those in Merkurjev’s. The crux case for proving our Main Theorem is that of type ^{2}E_{6}, the work for which is done in Sections 4 and 5.

Remark 0.2. In contrast the generality of our base field F, computations of H_{nr}^{d}(F (X)) in the literature for X a smooth variety (e.g., a classifying space) typically assume that F is algebraically closed. The examples of nonrational classifying varieties BG provided here and in [Mer] require that F is not algebraically closed.

Notations and conventions. We say that an algebraic group is simple if it is ≠ 1, is connected, and has no nontrivial connected normal subgroups over an algebraic closure. (These groups are often called “absolutely almost simple”.) Simple groups are classified in, e.g., [KMRT98, Ch. VI]. We say that a group is of type T_{n} if it is simple with root system of type T_{n} and of type ^{t}T_{n} if additionally the absolute Galois group of F acts as a group of automorphisms of order t on the Dynkin diagram.

We use the standard notation μ_{n} for the algebraic groups with F -points the nth roots of unity in F. If G is defined over F and L is an extension of F, we set G_{L} := G ×_{F} L.

The cohomology here is all Galois cohomology, for which the standard reference is [Ser94, §I.5].

1. Preliminaries

1.1. Tits algebras. Let V be an irreducible representation of a simple group G over F. The F -algebra End_{G}(V) is a skew field by Schur’s Lemma, and it is finite-dimensional over F; it is called a Tits algebra for G. If it is a (commutative) field, we say that it is split.

1.2. For G simple simply connected, there is a canonical and nontrivial morphism of functors

r_{G}: H^{1}(∗, G) ⟶ H^{3}(∗, (ℚ/ℤ)′(2))

called the Rost invariant [KMRT98, 31.40]. It has finite order n_{G}, and this order depends only upon the type of G and the (Schur) indices of its Tits algebras. We will repeatedly make us of the fact that the value of n_{G} is known and can be found by looking in [KMRT98, pp. 437–442] (or see [Mer01] for proofs). The value of n_{G} for G exceptional is:

3,6D_{4}, all Tits ^{3,6}D_{4}, some Tits

type of G

G_{2}

alg’s split alg’s nonsplit F_{4}

1E_{6}

2E_{6}

E_{7} E_{8}
n_{G}  2  6  12  6 6 12 12 60

2. Ramification

We say that a scalar multiple mr_{G} is ramified (resp. unramified) if there is some (resp. no) field extension E of F such that the image of the composition

H^{1}(E((t)), G) ⟶^{mr_{G}} H^{3}(E((t)), (ℚ/ℤ)′(2)) ⟶^{∂} H^{2}(E, (ℚ/ℤ)′(1))

is nontrivial for ∂ the residue with respect to the canonical discrete valuation of E((t)) over E.

Lemma 2.1. [Mer] H_{nr}^{3}(BG)_{norm} is isomorphic to the set of unramified multiples of the Rost invariant.

Since the Rost invariant is torsion, H_{nr}^{3}(BG)_{norm} is necessarily finite.

Strongly Inner Lemma 2.2. Let G be a simply connected simple group over F. Fix α ∈ H^{1}(F, G) and m ∈ ℤ. Then mr_{G} is ramified if and only if mr_{G_{α}} is. Moreover,

H_{nr}^{d}(BG)_{norm} = H_{nr}^{d}(BG_{α})_{norm}.

The notation G_{α} means an algebraic group with the same set of points as G over a separable closure F_{sep} of F (i.e., G_{α}(F_{sep}) = G(F_{sep})), but with a different Galois action: For γ ∈ Gal(F_{sep}/F) and g ∈ G_{α}(F_{sep}), we set

γg = a_{γ}γ(g)a_{γ}^{−1},

where juxtaposition denotes the usual action on G(F_{sep}) and a is a 1-cocycle representing the class α. Such a group is known as a strongly inner form of G (continuing the assumption that G is simply connected), and its isomorphism class is independent of the choice of a.

Proof: Let E be an extension of F, and consider the diagram

H^{1}(E((t)), G) ⟶^{mr_{G}} H^{3}(E((t)), (ℚ/ℤ)′(2)) ⟶^{∂} H^{2}(E, (ℚ/ℤ)′(1))
τ_{α}↓≅  ^{·−r_{G}(α)}↓  ∥
H^{1}(E((t)), G_{α}) ⟶^{mr_{Gα}} H^{3}(E((t)), (ℚ/ℤ)′(2)) ⟶^{∂} H^{2}(E, (ℚ/ℤ)′(1)),

where τ_{α} is the isomorphism arising from a choice of some 1-cocycle a representing α. The left box commutes by [Gil00, p. 76, Lem. 7]. The right box commutes because ∂ is a group homomorphism and ∂(r_{G}(α)) = 0.

Hence mr_{G} is ramified if and only if mr_{G_{α}} is. Since G and G_{α} are strongly inner forms of each other, they have the same Rost numbers; the displayed equation follows.

3. A_{2} ⊂ D_{4}

A simple group is said to be trialitarian if it is of type ^{3}D_{4} or ^{6}D_{4}.

Lemma 3.1. (char F ≠ 2, 3) Let L = F (ℓ^{1/3}) be a cubic field extension of F. Let G^{q} be the quasi-split trialitarian group over F associated with L. Then G^{q} contains a subgroup isomorphic to P GL_{3} such that the induced diagram

H^{1}(F, P GL_{3}) ⟶ H^{3}(F, μ_{3}^{⊗2})
↓  ↓
H^{1}(F, G^{q}) ⟶^{r_{G^{q}}} H^{3}(F, (ℚ/ℤ)′(2))

commutes up to sign, where the arrow on top is given by composing the connecting homomorphism H^{1}(F, P GL_{3}) → H^{2}(F, μ_{3}) with the map

· ∪ (ℓ): H^{2}(F, μ_{3}) → H^{3}(F, μ_{3}^{⊗2}).

The reason that we may only claim that the diagram commutes up to sign is because the Rost invariant is only known up to sign. (The Rost invariant is canonically determined, but in some cases the Rost invariant of a particular cocycle is only known up to a scalar multiple which is relatively prime to the order of the Rost invariant.)

For ρ: G → H a map between simple simply connected algebraic groups over F, there is a positive integer n_{ρ} called the Dynkin index of ρ, see [Mer] or [Gar, 2.1] (where it is called the “Rost multiplier” of ρ). It was defined by Dynkin — without use of the Rost invariant, of course — in the case G = SL_{n}, see [Dyn57, p. 130]. One of the properties of n_{ρ} is that for E any extension of F, the composition

H^{1}(E, H) ⟶^{ρ} H^{1}(E, G) ⟶^{r_{G}} H^{3}(E, (ℚ/ℤ)′(2))

is n_{ρ}r_{H}.

We will compute the Dynkin index in relatively easy situations. Let Λ_{c,H} and Λ_{c,G} denote the coroot lattices for H and G. The map H → G induces a map Λ_{c,H} → Λ_{c,G}, and these lattices are endowed with unique minimal positive definite Weyl-invariant quadratic forms, which are 1 on short coroots. The restriction of the form q on Λ_{c,G} to the image of a short coroot of H is n_{ρ}.

Proof of Lemma 3.1: The map x ↦ Tr(x^{2}) defines a quadratic form on M_{3}(F) and the group P GL_{3} preserves the 8-dimensional subspace of trace zero elements. This gives rise to an embedding P GL_{3} → SO_{8} which lifts to a map P GL_{3} → Spin_{8} [KMRT98, pp. 504, 505]. We can twist Spin_{8} to obtain G^{q}. The subgroup P GL_{3} of Spin_{8} is preserved by this twist, and we obtain an embedding of P GL_{3} in G^{q}.

The Springer construction [KMRT98, §38.A] gives an embedding of G^{q} in F_{4} the split group of type F_{4}. By inspecting this embedding over L, we see that it has Dynkin index 1. Consequently, the Rost invariant r_{G^{q}} factors through F_{4}, so we can rewrite our desired composition as

H^{1}(F, P GL_{3}) ⟶ H^{1}(F, G^{q}) ⟶ H^{1}(F, F_{4}) ⟶^{r_{F_{4}}} H^{3}(F, μ_{6}^{⊗2}).

The set H^{1}(F, F_{4}) classifies Albert F -algebras, and by [KMRT98, 39.9] the image of [A] ∈ H^{1}(F, P GL_{3}) in H^{1}(F, F_{4}) is the class of a certain kind of Albert algebra called a first Tits construction and denoted by J(A, ℓ). Then the Rost invariant is [A] ∪ (ℓ) as desired by [Ros91] or [PR96].

4. ^{3}D_{4},^{6}D_{4} ⊂ ^{2}E_{6}

The main result of this section is to describe an embedding of isotropic but not quasi-split trialitarian groups into groups of type ^{2}E_{6}. In this section, we assume that F has characteristic ≠ 2, 3. 4.1. Related triples [KMRT98], [Gar98], [Gar01]. Write ℭ for the split Cayley algebra over F which has a canonical involution and a quadratic norm form 𝔫. Let ⋆ denote the (nonassociative, non-unital) multiplication on ℭ defined by x ⋆ y := \overline{x} \overline{y}, where juxtaposition denotes the usual multiplication. We say that a similarity t ∈ GL(ℭ) of 𝔫 with similarity factor μ(t) is proper if det t = μ(t)^{4} (as opposed to −μ(t)^{4}). The subgroup of GL(ℭ) consisting of such similarities is denoted GO(ℭ)^{∘}.

A triple \underline{t} = (t_{0}, t_{1}, t_{2}) in (GO(ℭ)^{∘})^{×3} is said to be related if

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

for all i = 0, 1, 2 and x, y ∈ ℭ, with subscripts taken modulo 3. This defines a closed subgroup of (GO(ℭ)^{∘})^{×3} which we call Rel (ℭ). For a related triple \underline{t}, it is always the case that μ(t_{0})μ(t_{1})μ(t_{2}) = 1.

The simply connected split group Spin_{8} of type D_{4} is isomorphic to the intersection of Rel (ℭ) and O(ℭ)^{×3} (where O(ℭ) is the group of isometries of 𝔫) by [KMRT98, 35.7], and we identify the two groups by this isomorphism.

4.2. Albert algebras [KMRT98, Ch. IX]. The split Albert algebra J has as its underlying vector space the matrices in M_{3}(ℭ) fixed by the conjugate transpose. With that in mind, we may write a general element of J as

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

where ε_{i} ∈ F, c_{i} ∈ ℭ, and the entries given as · are forced by symmetry. The algebra J is strictly power associative and hence endowed with a canonically determined norm form (normalized so that the identity has norm 1) N. For \underline{t} a related triple, the linear transformation g\underline{t} of J 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}e_{2} \end{matrix} )

is an isometry of the norm on J. This defines an injection of Spin_{8} into the group Inv (J) of isometries of the norm.

4.3. Construction of a quasi-split ^{2}E_{6}. The algebra J is also endowed with a linear trace map Tr: J → F and the bilinear form T defined by T (x, y) = Tr(xy) is symmetric and nondegenerate. For each f ∈ GL(J), there is a unique f^{†} ∈ GL(J) such that T (f(x), f^{†}(y)) = T (x, y) for all x, y ∈ J. The map f ↦ f^{†} restricts to be an outer automorphism on Inv (J) defined over F. For ι the nontrivial F - automorphism of K, we define the group E_{6}^{K} to be Inv (J) with a twisted ι-action: For f ∈ E_{6}^{K}(K), we have ^{ι}f = ιf^{†}ι where the action on the left is the action in E_{6}^{K} and juxtaposition denotes the usual action in Inv (J). This group is quasi-split of type ^{2}E_{6} [Gar01, 2.9(2)].

The norm 𝔫 induces an involution σ on GL(ℭ) such that 𝔫(f(x), y) = 𝔫(x, σ(f)y) for all x, y ∈ ℭ and f ∈ GL(ℭ). We define σ(\underline{t}) to mean (σ(t_{0}), σ(t_{1}), σ(t_{2})). Then g† ; \underline{t} = g_{σ(\underline{t})^{−1}}. A related triple \underline{t} lies in Spin_{8} if and only if σ(\underline{t}) = \underline{t}^{−1}, so the embedding of Spin_{8} in E_{6}^{K} given by g is defined not just over K, but also over F.

4.4. Let L be a cubic extension of F endowed with a quaternion algebra Q whose corestriction down to F is split. By [Gar98, 4.7], there is a simply connected isotropic trialitarian group T over F whose Tits algebra is M_{4}(Q). By [KMRT98, 43.9], there is a b ∈ F^{∗} such that Q ⊗_{L} L(\sqrt{b}) is split.

Lemma 4.5. (char F ≠ 2, 3) Let L, Q, T, and b be as in 4.4. There exists a simply connected group ^{2}E_{6} of that type over F such that:

1. ^{2}E_{6} has split Tits algebras.

2. ^{2}E_{6} is of type ^{1}E_{6} over F (\sqrt{b}).

3. T is a subgroup of ^{2}E_{6} with Dynkin index 1.

By [Gar98], every isotropic simply connected trialitarian group occurs as in 4.4.

Proof: If Q is split, then T is the quasi-split trialitarian group associated to the cubic extension L/F. So T is a subgroup with Dynkin index 1 of the group Aut(J) of automorphisms of J, which is the split group of type F_{4} from the proof of Lemma 3.1. Since Aut(J) is the subgroup of Inv (J) consisting of the elements fixed by the outer automorphism †, it is also a subgroup of the quasi-split group E_{6}^{F (\sqrt{b})}. This embedding also has Dynkin index 1 [Gar, 2.4], so we are done in this case.

We may assume that Q is not L-split. Since Q becomes split over L(\sqrt{b}), it is isomorphic to (a, b)_{L} for some a ∈ L^{∗}. Since cor_{L/F} [Q] is split by hypothesis and Brauer-equivalent to (N_{L/F} (a), b)_{F}, we have that α := N_{L/F} (a) is a norm from F (\sqrt{b}). For a′ := a^{3}α^{−1}, the algebra (a, b)_{L} is isomorphic to (a′, b)_{L} and N_{L/F} (a′) = 1. Replacing a with a′, we may assume that N_{L/F} (a) = 1.

Set P = L^{c}(\sqrt{b}) for L^{c} the normal closure of L/F. Since Q is not split over L, it is not split over L^{c} by [Gar98, 3.2]. In particular, L^{c} does not contain a square root of b, so P is a quadratic extension of L^{c}.

To simplify our argument, we assume that L is not Galois over F, so Gal(L^{c}/F) is isomorphic to S_{3}. (The case where L is Galois over F is only easier.) Then, the group Gal(P/F) is isomorphic to S_{3} × μ_{2} ≅ ℤ/6 ⋊ μ_{2}, where the factor of ℤ/6 corresponds to the subgroup Gal(P/Δ) for Δ the unique quadratic extension of F in L^{c}. We fix generators ζ := ((1 2 3), −1) (which generates the copy of ℤ/6) and τ = ((2 3), 1) (which generates the copy of μ_{2} in ℤ/6 ⋊ μ_{2} corresponding to Gal(P/L(\sqrt{b}))).

We construct the group ^{2}E_{6} by giving a 1-cocycle z ∈ Z^{1}(P/F, E_{6}^{K}) for K = F (\sqrt{b}). Define a related triple \underline{t} = (t_{0}, t_{1}, t_{2}) by setting t_{i} = m_{i}P, for

m_{i} = diag (1, ρ^{i}(a), −ρ^{i}(a), ρ^{i+2}(a)^{−1}, ρ^{i+1}(a)^{−1}, −1, 1, ρ^{i}(a))

with ρ := ζ^{2}, and P the matrix permuting the basis vectors as (1 2)(3 6)(4 5)(7 8), for the basis of ℭ used in [Gar98] and [Gar]. This is indeed a related triple since N_{L/F} (a) = 1, see [Gar98, 1.5(3), 1.6, 1.8]. Also, define maps r, π ∈ Inv (J)(F) by

r ( \begin{matrix}ε_{0} & c_{2} & · \\ · & ε_{1} & c_{0} \\ c_{1} & · & ε_{2} \end{matrix} ) = ( \begin{matrix}ε_{2} & c_{1} & · \\ · & ε_{0} & c_{2} \\ c_{0} & · & ε_{1} \end{matrix} ) and π ( \begin{matrix}ε_{0} & c_{2} & · \\ · & ε_{1} & c_{0} \\ c_{1} & · & ε_{2} \end{matrix} ) = ( \begin{matrix}ε_{0} & \overline{c_{1}} & · \\ · & ε_{2} & \overline{c_{0}} \\ \overline{c_{2}} & · & ε_{1} \end{matrix} ) .

Note that

r^{†} = r, π^{†} = π, and r^{−1}g\underline{t}r = g_{ρ(\underline{t})}.

We define z by setting

z_{ζ} = g\underline{t}r and z_{τ} = π.

Observing that

ζg\underline{t}ζ^{−1} = g_{ρ(σ(\underline{t})^{−1})} = g_{ρ(\underline{t})^{−1}},

we have the formulas

(z_{ζ}ζ)^{2} = r^{2}ζ^{2} and (z_{ζ}ζ)^{3} = g\underline{t}ζ^{3}.

Since πτ and g\underline{t} commute, it is easy to verify that z is a 1-cocycle in Z^{1}(P/F, E_{6}^{K}). Set ^{2}E_{6} to be the twisted group (E_{6}^{K})_{z}; it automatically satisfies (1) and (2).

Since the values of z normalize Spin_{8} ⊂ E_{6}^{K}, the group (Spin_{8})_{z} is a subgroup of ^{2}E_{6} with Dynkin index 1. The restriction of z to Spin_{8} is the descent given in [Gar98, 4.7] to construct T, i.e., (Spin_{8})_{z} is isomorphic to T, hence (3).

Remark 4.6. The isotropic group^{2}E_{6} occurring in the preceding proposition is typically not quasi-split, even over L. This can be seen by examining the Rost invariant for (z) ∈ H^{1}(P/L, E_{6}^{K}), which is typically nontrivial by [Gar, 6.7].

5. A construction

The purpose of this section is to construct a suitable cubic field extension so that we may apply 4.5. We suppose that the characteristic of F is ≠ 2, 3. The following arguments are somewhat simpler than previously, thanks to suggestions by Adrian Wadsworth.

Lemma 5.1. (char F ≠ 2, 3) For p, q ∈ F^{∗}, the ring L = F (t)[x]/(x^{3} +px+qt) is a cubic field extension of F (t) which is not Galois over F (t). There is a prolongation of the t-adic valuation on F (t) to L which is unramified with residue degree 1 and with respect to which x has value 1.

Proof: If L is not a field, then there is some a ∈ F (t) such that a^{3} + pa + qt = 0. Since a is integral over the UFD F [t], it belongs to F [t], so it makes sense to speak of the degree of a. In particular, at least two of the terms a^{3}, pa, and qt must have the same degree, which is also the maximum of the degrees. This implies that a cannot have positive degree. But then qt, with degree 1, is the unique term of maximal degree, which is a contradiction.

A similar argument shows that the discriminant −4p^{3} − 27q^{2}t^{2} of L is not a square in F (t): Any square root b ∈ F (t) of the discriminant would belong to F [t] and have degree 1. Then the coefficient of t in b^{2} would be nonzero. Thus L is not Galois over F (t).

Hensel’s Lemma gives that x^{3} + px + tq has a linear factor of the form x − π in F ((t))[x], where π has value 1. The map x ↦ π gives an isomorphism of L with the subfield F (t)(π) of F ((t)), and the t-adic valuation obviously extends to L so that x has value 1. Since F ((t)) is the completion of F (t) with respect to the t-adic valuation and hence is unramified with residue degree 1, the claims about ramification and residue degree of our prolongation to L follow.

Lemma 5.2. (char F ≠ 2, 3) Let p, b ∈ F^{∗} be such that the quaternion algebra (p, b)_{F} is nonsplit. Let L be as in Lemma 5.1. Then the quaternion algebra (x, b)_{L} is nonsplit and is not defined over F (t).

Proof: Since N_{L/F (t)}(x) = −qt, the corestriction of (x, b)_{L} down to F (t) is Brauerequivalent to (−qt, b)_{F (t)}. This algebra is split if and only if the quadratic form ⟨1, −b, qt⟩ is isotropic over F (t). Over the completion F ((t)), this form has residue forms ⟨1, −b⟩ and ⟨q⟩. Since the algebra (p, b)_{F} is nonsplit, the first form is anisotropic, hence ⟨1, −b, qt⟩ is anisotropic over F ((t)) by Springer’s Theorem [Lam73, VI.1.9]. Thus (−qt, b)_{F (t)} is nonsplit, and hence so is (x, b)_{L}.

For the sake of contradiction, we suppose that (x, b)_{L} is defined over F. Since cor_{L/F (t)} res_{L/F (t)} is multiplication by 3 on the Brauer group, (x, b)_{L} is isomorphic to (−qt, b)_{L}. This implies that the algebra (−xqt, b)_{L} is split. Since

−x(qt) = −x(−x^{3} − px) = x^{4} + px^{2} ≡ x^{2} + p mod L^{∗2},

the algebra (x^{2} + p, b)_{L} is split.

Let L̂ be a completion of L with respect to the prolongation of the t-adic valuation on F (t) given by Lemma 5.1. The norm of (x^{2} + p, b)_{L} is the form ⟨1, −(x^{2} + p), −b, b(x^{2} + p)⟩ over L. Since x has value 1, over L̂ this form has one residue form ⟨1, −p, −b, bp⟩ over the residue field F. This is the norm of the algebra (p, b)_{F}, which is anisotropic because the algebra is nonsplit. By Springer’s Theorem, the norm of (x^{2} + p, b)_{L} is anisotropic over L̂, hence the algebra is nonsplit, which contradicts our assumption that (x, b)_{L} is defined over F.

5.3. Application. We will make use of this construction as follows: Let G^{q} be a quasi-split simply connected group of type ^{2}E_{6} over F which becomes of type ^{1}E_{6} over K = F (\sqrt{b}).

Let F_{0} := F (p) for p an indeterminate. Since b is not a square in F, the quaternion algebra (b, p)_{F_{0}} is nonsplit. Set F_{1} := F_{0}(t) and L_{1} := F_{0}(t)[x]/(x^{3} + px + t). Then L_{1} is a cubic non-Galois extension of F_{1} (by 5.1) and the algebra (x, b)_{L_{1}} is nonsplit and not defined over F_{1} (by 5.2). Set F′ to be the function field of the Severi-Brauer variety of (−t, b)_{F_{1}}. Since F_{1} is algebraically closed in F′, L′ := L_{1} ⊗_{F_{1}} F′ is a field which is not Galois over F′ and b is still a nonsquare in F′.

Since (−t, b)_{F_{1}} is Brauer-equivalent to the corestriction of (x, b)_{L_{1}}, and (−t, b)_{L′} is split, the corestriction of (x, b)_{L′} down to F′ is split. Thus there is a simply connected isotropic trialitarian group T over F′ with Tits algebra (x, b)_{L′}. It is of type ^{6}D_{4} since L′ is not Galois over F′. Since (x, b)_{L_{1}} is not defined over F_{1}, it is not isomorphic to (−t, b)_{L_{1}}, hence (x, b)_{L′} is not split. By 4.5, there is a simple simply connected group G′ over L′ of type ^{2}E_{6} which is a strongly inner form of G_{L′}^{q} such that T is a subgroup of G′ with Dynkin index 1.

6. Proof of the main theorem

Let G be a simple simply connected algebraic group over F such that char F is not a torsion prime for G.

6.1. Let α : H → G be a homomorphism of simple simply connected algebraic groups with Dynkin index 1. If mr_{H} is ramified, then so is mr_{G}. If n_{G} = n_{H} and H_{nr}^{3}(BH)_{norm} = 0, then H_{nr}^{3}(BG)_{norm} = 0.

Lemma 6.2. [Mer, 4.4] If there exists an extension K of F such that n_{G_{K}} = n_{G} and H_{nr}^{3}(B_{K}G)_{norm} = 0, then H_{nr}^{3}(BG)_{norm} = 0.

Proof: If mr_{G} is unramified over F, then mr_{G_{K}} = 0. Since n_{G_{K}} = n_{G}, we have mr_{G} = 0.

As a corollary, we obtain the following reduction:

Corollary 6.3. Let G be a simple simply connected group over F and let G^{q} be the unique quasi-split inner form of G. Suppose that n_{G} = n_{G^{q}}. If there exists an extension F′ of F such that n_{G_{F′}^{q}} = n_{G^{q}} and H_{nr}^{3}(B_{F′}G^{q})_{norm} = 0, then H_{nr}^{3}(B_{F} G)_{norm} = 0.

Proof: The hypotheses on F′ give H_{nr}^{3}(B_{F} G^{q})_{norm} = 0 by the lemma. Let K be a generic quasi-splitting field for G as in [KR94]. Since n_{G} = n_{G^{q}} = n_{G_{K}^{q}}, we have H_{nr}^{3}(B_{F} G)_{norm} = 0 by the lemma.

The remainder of this section is the proof of the Main Theorem 0.1, broken up by the type of the simple exceptional group G. We must show in all cases except for 6.6 below that every nontrivial multiple of the Rost invariant is ramified. By hypothesis, char F is not a torsion prime for G, and inspection of the table in 1.2 shows that the order n_{G} of the Rost invariant r_{G} is relatively prime to char F. Consequently [MS83] the n_{G}-torsion of H^{3}(F, (ℚ/ℤ)′(2)) (which contains the image of r_{G}) is identified with the subgroup H^{3}(F, μ_{n_{G}}^{⊗2}).

6.4. Types F_{4} and G_{2}. Let G^{d} be the split group of the same type as G, where G has type F_{4} or G_{2}. The Rost invariant (up to some invertible scalar multiple) has been explicitly computed for the split groups of type G_{2} and F_{4} [KMRT98, pp. 441, 533–537], and all nontrivial multiples of it are clearly ramified. Since the groups G^{d} and G have the same Rost number, H_{nr}^{d}(BG)_{norm} = H_{nr}^{d}(BG^{d})_{norm} = 0. 6.5. Type ^{3,6}D_{4} with split Tits algebras. Let G be simply connected trialitarian with associated cubic extension L/F and quasi-split inner form G^{q}.

Let F′ be the extension obtained from F by adjoining an indeterminate x and, if they are not already in F, a primitive cube root of unity ζ and a square root of the discriminant of the cubic extension L of F. By Corollary 6.3, it is sufficient to show that mr_{G_{F′}^{q}} is ramified or trivial for all m. Note that L′ := L ⊗_{F} F′ is a cubic Galois extension of F′, hence by Kummer theory is of the form L′ = F′(ℓ^{1/3}) for some ℓ ∈ (F′)^{∗}

If mr_{G_{F′}^{q}} is nontrivial, it has order 2, 3, or 6. If it has order 2, it must equal 3r_{G_{F′}^{q}}. Since n_{G_{L′}^{q}} = 2, we have 3r_{G_{L′}^{q}} = r_{G_{L′}^{q}}, which is ramified [Mer, 8.7]. Hence 3r_{G_{F′}^{q}} is ramified. If mr_{G_{F′}^{q}} has order 6, then 3mr_{G_{F′}^{q}} has order 2 and so is ramified, hence mr_{G_{F′}^{q}} is ramified.

The remaining cases are mr_{G_{F′}^{q}} of order 3, i.e., m ≡ 2, 4 mod 6. As described in 3.1, P GL_{3} is a subgroup G_{F′}^{q}. Let D the cyclic central simple algebra over F′((t)) determined by the extension F′((t))(\sqrt[3]{t}) and x ∈ F′((t))^{∗}/F′((t))^{∗3}. (See [KMRT98, §30.A] for information on cyclic algebras.) The image of [D] ∈ H^{1}(F′((t)), P GL_{3}) under mr_{G_{F′}^{q}} is up to sign m[D] ∪ (ℓ) = m(t) ∪ (x) ∪ (ℓ). This has residue m(x) ∪ (ℓ). Since ℓ is not a cube in F′ and m ≡ 2, 4 mod 6, this class is nontrivial. Hence mr_{G_{F′}^{q}} is ramified. This completes the case where G is trialitarian with split Tits algebras.

6.6. Type ^{3,6}D_{4} with nonsplit Tits algebra. Suppose that G is of type ^{3}D_{4} or 6D_{4} with nonsplit Tits algebra, and let K be a generic quasi-splitting variety for G. Since mr_{G_{K}} is ramified if it is nontrivial for m ≢ 6 mod 12, the same is true of mr_{G} for such m.

We now focus on 6r_{G}, which is nontrivial and whose image lies in H^{3}(F, μ_{2}^{⊗2}). Let L be a cubic extension of F associated with G. If G is of type ^{6}D_{4} (over F), then G_{L} is of type ^{2}D_{4}, and 6r_{G_{L}} = 2r_{G_{L}} is ramified by [Mer, 8.3]. So we may assume that G is of type ^{3}D_{4}. We will have proved that H_{nr}^{3}(BG)_{norm} = ℤ/2 if we show that for every extension E of F, the composition

H^{1}(E((t)), G_{E((t))}) ⟶^{6r_{G_{E((t))}}} H^{3}(E((t)), μ_{2}^{⊗2}) ⟶^{∂} H^{2}(E, μ_{2}) (6.7)

is trivial.

If G is of type ^{1}D_{4} over E, then G_{E} is isomorphic to Spin(A, σ) for some central simple E-algebra A of degree 8 with orthogonal involution σ such that the even Clifford algebra C_{0}(A, σ) is isomorphic to a direct sum C_{+} × C_{−} of two central simple E-algebras of degree 8. If at least one of the algebras A, C_{+}, C_{−} is split, then n_{G_{E}} = 2, so 6r_{G_{E}} = 0. If none of the three is split, then 6r_{G_{E}} = 2r_{G_{E}}, and the composition 6.7 is 0 by [Mer, 8.5(2)].

Otherwise, G_{E} is of type ^{3}D_{4}, i.e., L_{E} := L ⊗_{F} E is a field. We have a commutative diagram

H^{1}(E((t)), G_{E((t))}) ⟶^{6r_{G_{E((t))}}} H^{3}(E((t)), μ_{2}^{⊗2}) ⟶^{∂} H^{2}(E, μ_{2})
res↓  ^{res}↓  ^{res}^{L_{E} /E} ↓
H^{1}(L_{E}((t)), G_{L_{E}((t))}) ⟶^{6r_{G_{L_{E} ((t))}}} H^{3}(L_{E}((t)), μ_{2}^{⊗2}) ⟶^{∂} H^{2}(L_{E}, μ_{2}).

Since G_{L_{E}((t))} is of type ^{1}D_{4}, the composition of the two bottom arrows is 0, hence the image of H^{1}(E((t)), G_{E((t))}) in H^{3}(L_{E}, μ_{2}) is 0. Since μ_{2} is 2-torsion and [L_{E} : E] = 3, the map res_{L_{E}/E} is invertible, and so the composition (6.7), which is the top row of the diagram, is 0. This proves the Main Theorem for G trialitarian with nonsplit Tits algebras.

6.8. Type ^{1}E_{6}. The split F_{4} injects into the split group E_{6}^{d} of type E_{6} with Rost number 1 [Gar, 3.4], and we have n_{F_{4}} = n_{E_{6}^{d}} = 6. Hence H_{nr}^{3}(BE_{6}^{d})_{norm} = 0.

Since all groups of type^{1}E_{6} have the same Rost number, we have H_{nr}^{3}(BG)_{norm} = 0 by 6.3.

6.9. Type ^{2}E_{6}. Let G^{q} be the quasi-split inner form of the given group G of type 2E_{6}. Write K = F (\sqrt{b}) for the quadratic extension over which the groups are of type ^{1}E_{6}. Let T, L′, and F′ be as in 5.3. We have n_{T} = n_{G′} = 12. By 6.6 and 6.1, we have H_{nr}^{3}(B_{F′}G′)_{norm} = 0. Since G_{F′}^{q} and G′ are strongly inner forms over F′, we have H_{nr}^{3}(B_{F′}G^{q})_{norm} = 0 by 2.2. Hence H_{nr}^{3}(B_{F} G)_{norm} = 0 by Corollary 6.3.

6.10. Type E_{7}. Let G^{q} be the unique (quasi-)split inner form of our given group G of type E_{7}. Set F′ = F (x) and K = F′(\sqrt{x}). There is a quasi-split simply connected group E_{6}^{K} over F of type ^{2}E_{6} associated with the extension K/F′. This group injects into G_{F′}^{q} with Dynkin index 1 [Gar, §3]. Since n_{E_{6}^{K}} = n_{G_{F′}^{q}} = 12, H_{nr}^{3}(B_{F′}G^{q})_{norm} = 0 by 6.1 and 6.9. Then H_{nr}^{3}(B_{F} G)_{norm} = 0 by 6.3.

6.11. Type E_{8}. Again, let G^{q} be the unique (quasi-)split inner form of our given group G, which in this case is of type E_{8}.

For m an integer such that 1 ≤ m < 60 = n_{G^{q}}, we will show that mr_{G^{q}} is ramified.

Suppose first that 5mr_{G^{q}} is nonzero. The map x ↦ 5x sends the 60-torsion in H^{3}(F, (ℚ/ℤ)′(2)) to the 12-torsion, i.e., it is a map H^{3}(F, μ_{60}^{⊗2}) → H^{3}(F, μ_{12}^{⊗2}). Since the split group of type E_{7} obviously lies in G^{q} with Dynkin index 1, 5r_{G^{q}} restricts to be 5r_{E_{7}}, which also has order 12 and all of whose nontrivial multiples are ramified. Thus 5mr_{G^{q}} is ramified, hence mr_{G^{q}} is also.

Otherwise, 5mr_{G^{q}} = 0, so m = 12, 24, 36, or 48. Let E be the extension of F generated by two indeterminates x and y and, if it is not already in F, a primitive 5th root of unity ζ. The cyclic algebra D given by the cyclic extension E(x^{1/5}) and the class of y ∈ E^{∗}/E^{∗5} is an E-central division algebra.

If we remove the correct vertex from the extended Dynkin diagram of the split group E_{8} of type E_{8}, we are left with a diagram of type A_{4} × A_{4}. By [Tit90, §1], this reflects the existence of an exact sequence

1 → μ_{5} → SL_{5} × SL_{5} → H → 1 (6.12)

where the map μ_{5} → SL_{5} × SL_{5} is given by ζ ↦ (ζ Id, ζ^{2} Id) and H is a subgroup of E_{8}. Note that the first copy of SL_{5} injects into H, hence E_{8}, and this injection has Dynkin index 1. As described in the [Gil01, §1], there is a class z ∈ H^{1}(E, H) such that twisting (6.12) by z gives

1 → μ_{5} → SL_{1}(D) × SL_{1}(D′) → H_{z} → 1,

where D′ is Brauer-equivalent to D^{⊗3}. Thus SL_{1}(D) is a subgroup of H_{z} ⊂ (G^{q})_{z} with Dynkin index 1.

Since D is nonsplit, n_{SL_{1}(D)} = 5 (note that characteristic ≠ 5 by hypothesis), so mr_{G_{z}^{q}} restricts to mr_{SL_{1}(D)} = mr_{SL_{1}(D)}. Since SL_{1}(D) has Rost number 5, this is nonzero, and it is ramified by [Mer, 5.1]. Since all group of type E_{8} are strongly inner forms of each other, the same conclusion holds for mr_{G_{E}^{q}} by the Strongly Inner Lemma 2.2, and mr_{G^{q}} by 6.2.

Since H_{nr}^{3}(B_{F} G^{q})_{norm} = 0, we have H_{nr}^{3}(B_{F} G)_{norm} = 0 for G of type E_{8} by 6.3.

This completes the proof of the Main Theorem 0.1.

Acknowledgements. Thanks are due to A.S. Merkurjev for helpful conversations on this topic.

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UCLA Department of Mathematics, Los Angeles, CA 90095-1555

   E-mail address: skip@member.ams.org