RESTRICTING THE ROST INVARIANT TO THE CENTER

SKIP GARIBALDI AND ANNE QUÉGUINER-MATHIEU

Abstract. For simple simply connected algebraic groups of classical type, Merkurjev, Parimala, and Tignol gave a formula for the restriction of the Rost invariant to torsors induced from the center of the group. We complete their results by proving formulas for exceptional groups. Our method is somewhat different and recovers also their formula for classical groups.

1. Introduction

In the 1990’s, M. Rost proved that the group of degree 3 normalized invariants of an absolutely simple simply connected algebraic group G over k — that is, the group of natural transformations of the Galois cohomology functors

H^{1}(⋆, G) → H^{3}(⋆, ℚ/ℤ(2))

— is a cyclic group with a canonical generator. This canonical generator is known as the Rost invariant. Roughly speaking, it is the “first” nonzero invariant, in that there are no non-zero normalized invariants H^{1}(⋆, G) → H^{d}(⋆, ℚ/ℤ(d−1)) for d < 3 [KMRT, §31].

In general, there is no explicit formula for computing the Rost invariant. However, in [MPT], Merkurjev, Parimala and Tignol gave a nice description — for G of classical type — of the restriction of the Rost invariant to H^{1}(⋆, Z), where Z is the center of G. The purpose of this paper is to complete their results by providing an analogous description for exceptional groups (see Th. 2.3 for a precise statement for all types of groups).

Part of our proof is borrowed from [MPT]. Precisely, their Corollaries 1.2 to 1.6, which are stated for a general cycle module, show in our context that every group invariant

H^{1}(⋆, Z) → H^{3}(⋆, ℚ/ℤ(2))

— i.e., every invariant such that the map H^{1}(K, Z) → H^{3}(K, ℚ/ℤ(2)) is a group homomorphism for every extension K/k — is given by some cup-product with a class t ∈ H^{2}(k, Z), see Prop. 2.1 and §3 below. (Note that this class t does depend on a choice regarding the cup product, see Remark 2.5(iii).)

It remains to compute the class t_{R,G} ∈ H^{2}(k, Z) associated with composition

(1.1)  H^{1}(⋆, Z) ⟶ H^{1}(⋆, G) ⟶^{r_{G}} H^{3}(⋆, ℚ/ℤ(2)),

where the first map is induced by the inclusion of Z in G and r_{G} is the Rost invariant of G. Note that, even though it is not obvious from its definition, this composition actually is a group invariant, see [Ga 01b, 7.1] or [MPT, Cor. 1.8]. The Rost invariant is canonically determined; nevertheless, practically speaking, one only knows the group it generates. Consequently, we will determine the subgroup ⟨t_{R,G}⟩ of H^{2}(k, Z) generated by t_{R,G}.

Date: July 27, 2006.

For any semisimple group G, one has the Tits class t_{G}, an element of the group H^{2}(k, Z). It follows from [MPT] and from the present paper that either t_{R,G} is zero or t_{R,G} and t_{G} generate the same subgroup of H^{2}(k, Z), for a well-chosen cup product. Note that our argument for determining ⟨t_{R,G}⟩ is quite different from [MPT], in which they use concrete interpretations of the classical groups. Here, we first reduce to groups whose Tits index satisfies a certain condition (see (4.1)), using an injectivity result which follows from [MT, Th. B]. Second, we prove the result for those particular groups, by reducing to groups of inner type A for which the Rost invariant has a concrete description. This method applies to all absolutely simple groups except those of outer type A, and therefore can be used to recover the results of [MPT] on classical groups (except for their Th. 1.10). We give the details for types B, C, and D_{even}, as well as for the exceptional groups.

2. Statement of results

Throughout the paper, G denotes an absolutely simple simply connected algebraic group over a field k. We assume (except in §6) that the characteristic of k does not divide the exponent n of the center of G. That is, the characteristic is not 2 for G of type B, C, D, E_{7}; the characteristic is not 3 for G of type E_{6}; and the characteristic does not divide ℓ + 1 for G of type A_{ℓ}. This guarantees that the (scheme-theoretic) center Z of G is smooth. (See Remark 4.4 for comments on the characteristic hypothesis.)

The group H^{d}(k, ℚ/ℤ(d − 1)) is as defined in [M 03, App. A]; otherwise the notation H^{d}(k, A) stands for the Galois cohomology group H^{d}(Γ_{k}, A(k_{sep})), where Γ_{k} denotes the Galois group of a separable closure k_{sep} of k over k and A is a smooth algebraic group. Our hypothesis on the characteristic of k implies that the n-torsion of H^{d}(k, ℚ/ℤ(d − 1)) is naturally identified with H^{d}(k, μ_{n}^{⊗(d−1)}) for d = 2, 3. When G is not of type A_{ℓ} with ℓ ≥ 3 — e.g., in all cases studied in detail below — n is 2 or 3, hence μ_{n}^{⊗2} is isomorphic to ℤ/nℤ (see [KMRT, Ex. 11, p. 444]).

For any functor F from the category of field extensions of k to the category of groups, we let Inv^{3}(F) be the collection of group invariants of F with values in H^{3}(⋆, ℚ/ℤ(2)); it is an abelian group. If T is a quasi-trivial torus, i.e., T = R_{E/k}(𝔾_{m}) for some étale algebra E/k, it follows from Th. 1.1 of [MPT] that H^{2}(k, T) is isomorphic to Inv^{3}(T). Precisely, the invariant α^{E}(X) associated to a cohomology class X ∈ H^{2}(k, T) is defined by α^{E}(X)(y) = N_{E_{K}/K}(y.X_{K}) for every field extension K/k and y ∈ T (K) = E_{K}^{×}. The cup product appearing in this formula is given by the module structure on H^{⋆}(ℚ/ℤ(−1)) over the Milnor K-ring (see [M 03, Appendix A] for a definition). Using this, we may prove the following:

Proposition 2.1. The groups Inv^{3}(H^{1}(⋆, Z)) and H^{2}(k, Z) are (non-canonically) isomorphic.

While proving this proposition in section 3, we will exhibit particular such isomorphisms, which admit the following explicit description: for any cohomology class X ∈ H^{2}(k, Z), the corresponding invariant is given by

y ∈ H^{1}(K, Z) ↦ y.X_{K} ∈ H^{3}(K, ℚ/ℤ(2)).

Since it is a group invariant, its image is contained in the n-torsion, H^{3}(K, μ_{n}^{⊗2}), of H^{3}(K, ℚ/ℤ(2)). Further, the cup product is induced by a bilinear form Z(k_{sep}) × Z(k_{sep}) → μ_{n}^{⊗2}, which has to be specified.

We denote by μ_{n} the algebraic group of nth roots of unity and by μ_{n}_{[E]} the kernel of the norm map N_{E/k} : R_{E/k}(μ_{n}) → μ_{n}, for any quadratic étale algebra E/k. As recalled in [MPT], one deduces from the classification of absolutely simple simply connected groups that if G is classical, then Z is one of the following groups: μ_{n}, R_{E/k}(μ_{2}), and μ_{n}_{[E]}, where E is quadratic étale over k. If we wish to consider also exceptional groups, we need to add the centers of trialitarian D_{4} groups, which are isomorphic to the kernel, now denoted by R_{E/k}^{1}(μ_{2}), of the norm map N_{E/k} : R_{E/k}(μ_{2}) → μ_{2}, where E is cubic étale over k.

In most cases, namely when Z is μ_{n} or μ_{n}_{[E]}, there is a natural bilinear map

Z(k_{sep}) × Z(k_{sep}) → μ_{n}^{⊗2}.

The only groups for which the cup-product has to be defined carefully are groups of type D_{ℓ} with ℓ even. Their center is—over k_{sep}—isomorphic to μ_{2} × μ_{2}. Rather than fix an identification between these two groups, we note that the fundamental weights of the root system give characters ω_{1}, ω_{ℓ−1}, ω_{ℓ} whose restriction to Z are the three nonzero homomorphisms Z → μ_{2}. We consider the cup product induced by the bilinear map with values in μ_{2}^{⊗2} = ℤ/2ℤ:

(2.2)  (x, y) ↦ \begin{cases}ω_{ℓ−1}(x) ⊗ ω_{ℓ}(y) + ω_{ℓ}(x) ⊗ ω_{ℓ−1}(y) & if ℓ ≡ 0 (mod 4), \\ ω_{ℓ−1}(x) ⊗ ω_{ℓ−1}(y) + ω_{ℓ}(x) ⊗ ω_{ℓ}(y) & if ℓ ≡ 2 (mod 4). \end{cases}

Since ω_{1} + ω_{ℓ−1} + ω_{ℓ} = 0 as characters, when ℓ ≡ 0 (mod 4), this cup product can be rewritten as

(x, y) ↦ ω_{1}(x) ⊗ ω_{1}(y) + ω_{ℓ−1}(x) ⊗ ω_{ℓ−1}(y) + ω_{ℓ}(x) ⊗ ω_{ℓ}(y).

With this in hand, we may summarize the results of [MPT] and the present paper in the following theorem:^{1}

Main Theorem 2.3. If G is of type A, C_{ℓ} (ℓ odd), D, E_{6} or E_{7}, then the composition (1.1) and the cup product with the Tits class of G generate the same subgroup of Inv^{3}(H^{1}(⋆, Z)). Otherwise the composition (1.1) is zero.

Example 2.4. For groups of inner type A — that is, when G is SL(A) for some central simple algebra A over k — the theorem asserts that (1.1) and the cup product with the Brauer class [A] of A generate the same subgroup of Inv^{3}(H^{1}(⋆, Z)). This follows from the explicit description of the Rost invariant given in [M 03, p. 138]. This example is a starting point for proving the theorem.

The new cases in the theorem are the groups of type E_{6}, E_{7}, and trialitarian D_{4}, treated below in sections 11, 9, and 12 respectively. The exceptional groups of type E_{8}, F_{4}, and G_{2} all have trivial center, so (1.1) is automatically zero for those groups.

Remark 2.5. (i) If the Tits class is zero, then a twisting argument shows directly that the composition (1.1) is zero, regardless of the type of G. Indeed, since the Tits class is zero, there is a simply connected quasi-split group G^{q} such that we may identify G with G^{q} twisted by a 1-cocycle α ∈ Z^{1}(k, G^{q}). We write Z^{q} for the center of G^{q}; the identification of G with G_{α}^{q} identifies Z with Z^{q}; we write i and i^{q} for the inclusions of Z and Z^{q} in G and G^{q}. The diagram

1There is a typo in Th. 1.13 of [MPT], which addresses the C_{ℓ} case: the words “odd” and “even” should be interchanged. For keeping the C_{odd} and C_{even} cases straight, we find it helpful to recall that B_{2} = C_{2}, cf. §10 below.

H^{1}(k, Z) ⟶^{i_{⋆}} H^{1}(k, G) ⟶^{r_{G}} H^{3}(k, ℚ/ℤ(2))
τ_{α}↓≅  ↓^{?+r_{G^{q}} (α)}
H^{1}(k, Z^{q}) ⟶^{i_{⋆}^{q}} H^{1}(k, G^{q}) ⟶^{r_{G^{q}}} H^{3}(k, ℚ/ℤ(2))

where τ_{α} is the twisting isomorphism, commutes by [Gi 00, p. 76, Lemma 7] or [MPT, Cor. 1.8]. Fix ζ ∈ Z^{1}(k, Z). The image of i_{⋆}ζ under τ_{α} is the class in H^{1}(k, G^{q}) of the 1-cocycle σ ↦ ζ_{σ}α_{σ} for σ ∈ Γ_{k}. But this is just ζ · α ∈ H^{1}(k, G^{q}), where the · represents the usual action of H^{1}(k, Z^{q}) on H^{1}(k, G^{q}). So the image of i_{⋆}ζ going counterclockwise around the diagram is

r_{G^{q}}(ζ · α) = r_{G^{q}}(i_{⋆}^{q}ζ) + r_{G^{q}}(α)

by [Ga 01b, 7.1]. But G^{q} is quasi-split, so it has a quasi-trivial maximal torus T^{q} that necessarily contains Z^{q}, hence i_{⋆}^{q} factors through the zero group H^{1}(k, T^{q}). We conclude that the image of i_{⋆}ζ in the lower right corner is r_{G^{q}}(α).

On the other hand, going clockwise around the diagram, we find that the image of i_{⋆}ζ in the lower right corner is r_{G}(i_{⋆}ζ) + r_{G^{q}}(α). The commutativity of the diagram proves the claim.

(ii) As opposed to (i), note: For groups of type B_{ℓ}, i.e., for G = Spin(q) for some (2ℓ + 1)-dimensional quadratic form q over k, the composition (1.1) is always zero, while the Tits class of G, which is the Brauer class of the even Clifford algebra of q, can be non-zero.

(iii) In the D_{ℓ} case with ℓ even, the bilinear form we use to compute the cup product does depend on whether ℓ/2 is even or odd. If we keep the same cup product in both cases, then composition (1.1) coincides alternately with the cup product with t_{G} and its conjugate.

(iv) The bilinear maps in (2.2) can be viewed as symmetric bilinear forms on a vector space of dimension 2 over the field with two elements. For ℓ ≡ 0 (mod 4), it is the wedge product, equivalently, a hyperbolic form. For ℓ ≡ 2 (mod 4), it is the unique (up to isomorphism) metabolic form that is not hyperbolic.

3. Invariants of H^{1}(⋆, Z)

The purpose of this section is to prove Proposition 2.1. In order to use the description of invariants of quasi-trivial tori recalled above, we need to embed Z in such a torus. More precisely, we are going to consider exact sequences

1 ⟶ Z ⟶^{j} T ⟶^{g} T ⟶ 1,

where T is a quasi-trivial torus, i.e., T = R_{E/k}(𝔾_{m}) for some étale k-algebra E. We call such an exact sequence admissible if the map g satisfies the following equation:

(3.1)  N_{E_{K}/K}(g_{K}(y).X_{K}) = N_{E_{K}/K}(y.H^{2}(g)(X)_{K}),

for every X ∈ H^{2}(k, T), field extension K/k, and y ∈ T (K) = E_{K}^{×}.

Given an admisible exact sequence, we claim it induces an isomorphism between Inv^{3}(H^{1}(⋆, Z)) and H^{2}(k, Z), as in Prop. 2.1. Indeed we can apply [MPT, Lemma 3.1]: Since the sequence

T (K) ⟶^{g_{K}} T (K) ⟶^{φ} H^{1}(K, Z) ⟶ 1.

is exact for every extension K/k, the top row of the diagram

(3.2)  \xymatrix{1 \ar[r] & Inv^{3}(H^{1}(⋆, Z)) \ar[r]^{φ^{⋆}} & Inv^{3}(T) \ar[r]^{g^{⋆}} & Inv^{3}(T) \\ 1 \ar[r] & H^{2}(k, Z) \ar@{-->}[u]^{β} \ar[r]^{H^{2}(j)} & H^{2}(k, T) \ar[u]^{α^{E}}_{≅} \ar[r]^{H^{2}(g)} & H^{2}(k, T) \ar[u]^{α^{E}}_{≅}}

is also exact, where α^{E} is the isomorphism described in §2. Since T is quasi-trivial, the group H^{1}(k, T) is zero and the bottom row of the diagram is also exact. An easy computation shows that under condition (3.1) the right-hand box is commutative. Since α^{E} is an isomorphism, this enable us to identify the kernels of both lines, producing an isomorphism β as in the diagram.

To prove Prop. 2.1, it only remains to show that there exists an admissible exact sequence for the center Z of a given G. This is done in [MPT] in sections 3.1 to 3.4 when Z is isomorphic to μ_{n}, R_{E/k}(μ_{n}), or μ_{n}_{[E]} (see also sequences (3), (6), and (8)). The only remaining case is the center of a trialitarian D_{4}. For such a group, there exists a cubic field extension E/k and an embedding j : Z → R_{E/k}(𝔾_{m}) which identifies Z with the kernel R_{E/k}^{1}(𝔾_{m}) of the norm map. Over k_{sep}, it is given by j_{sep}(x) = (ω_{1}(x), ω_{ℓ−1}(x), ω_{ℓ}(x)) ∈ (k_{sep}^{×})^{3}.

Let us define g : R_{E/k}(𝔾_{m}) → R_{E/k}(𝔾_{m}) by g_{K}(x) = N_{E_{K}/K}(x) x^{−2}, for any field K/k and any x ∈ R_{E/k}(𝔾_{m})(K) = E_{K}^{×}, and consider the sequence

(3.3)  1 ⟶ Z ⟶^{j} R_{E/k}(𝔾_{m}) ⟶^{g} R_{E/k}(𝔾_{m}) ⟶ 1

Lemma 3.4. Sequence 3.3 is exact and admissible.

Proof. An element x in the kernel of g_{K} satisfies N_{E_{K}/K}(x) = x^{2}. Since E is cubic, taking the norm on both sides gives 1 = N_{E_{K}/K}(x) = x^{2}, and this proves x ∈ Z(K). The reader may easily check surjectivity; admissibility condition 3.1 follows from the projection formula, as in [MPT, 3.3]. □

Proposition 2.1 is now proved.

Note that, if you allow E/k to be a cubic étale algebra rather than a field extension, then the sequence (3.3) actually is an admissible exact sequence for any group of type D_{ℓ} with ℓ ≡ 0 mod 4. As we will see in the next paragraph, the corresponding isomorphism between Inv^{3}(H^{1}(⋆, Z)) and H^{2}(k, Z) is not the same as the one induced by sequence (3) of [MPT] (see also Remark 2.5(iii)).

Hence we have to specify which exact sequence we use: from now on, we will always consider the isomorphism induced by the following admissible exact sequence, depending on Z and G:

• If Z = μ_{n}, we take

1 ⟶ μ_{n} ⟶ 𝔾_{m} ⟶^{n} 𝔾_{m} ⟶ 1.

• If Z = μ_{n}_{[E]} with n odd (respectively even), we take sequence (6) (resp., (8)) of [MPT] • If G is of type D_{ℓ} with ℓ ≡ 2 mod 4, we take sequence (3) of [MPT].

• If G is of type D_{ℓ} with ℓ ≡ 0 mod 4, we take sequence (3.3).

In each of these cases, the isomorphism β from (3.2) is given by

(3.5)  β(t)(x) = x.t_{K},

where the cup product is induced by the bilinear map specified in § 2. (This is the description of the isomorphism β given just after the statement of Prop. 2.1.) This fact can be proved in each case by some explicit computation, going through the diagram. Let us do it for groups of type D_{ℓ} with ℓ ≡ 0 mod 4, and sequence (3.3).

For any t ∈ H^{2}(k, Z), the invariant β(t) is characterized by

(3.6)  β(t)(φ(x)) = α^{E}(H^{2}(j)(t))(x) = N_{E_{K}/K}(x.H^{2}(j)(t_{K}))

for any x ∈ T (K). We want to prove this is equal to φ(x).t_{K}, where the cup product is induced by the bilinear map

(x, y) ↦ ω_{1}(x) ⊗ ω_{1}(y) + ω_{ℓ−1}(x) ⊗ ω_{ℓ−1}(y) + ω_{ℓ}(x) ⊗ ω_{ℓ}(y).

Since Z has exponent 2, the map j factors as

Z ⟶^{j_{0}} R_{E/k}(μ_{2}) ⟶ R_{E/k}(𝔾_{m}),

and j_{0}_{sep}, as j_{sep}, coincides with the triple (ω_{1}, ω_{ℓ−1}, ω_{ℓ}). Hence, we have

(3.7)  φ(x).t_{K} = N_{E_{K}/K}(H^{1}(j_{0})(φ(x)).H^{2}(j_{0})(t_{K})).

To compare (3.6) and (3.7), we use the following commutative diagram

[figure: 1; Z; j_{0}; j; R_{E/k}(𝔾_{m}); g; R_{E/k}(𝔾_{m}); h; 1; 1; R_{E/k}(μ_{2}); R_{E/k}(𝔾_{m}); 2; R_{E/k}(𝔾_{m}); 1]

where the map h is defined by h_{K}(x) = N_{E_{K}/K}(x)x^{−1} = xg_{K}(x) for any x ∈ R_{E/k}(𝔾_{m})(K) = E_{K}^{×}. It induces a commutative square

\xymatrix{E_{K}^{×} \ar[r]^{φ} \ar[d]^{h_{K}} & H^{1}(K, Z) \ar[d]^{H^{1}(j_{0})} \\ E_{K}^{×} \ar[r] & H^{1}(K, R_{E/k}(μ_{2}))}

Hence H^{1}(j_{0})(φ(x)) = (h_{K}(x))_{2} = (x)_{2} + (g_{K}(x))_{2}, and

N_{E_{K}/K}(H^{1}(j_{0})(φ(x)).H^{2}(j_{0})(t_{K}))

= N_{E_{K}/K}((x)_{2} .H^{2}(j_{0})(t_{K})) + N_{E_{K}/K}((g_{K}(x))_{2} .H^{2}(j_{0})(t_{K}))
= N_{E_{K}/K}(x.H^{2}(j)(t_{K})) + N_{E_{K}/K}(g_{K}(x).H^{2}(j)(t_{K})),

where the cup product in the last line is again given by the module structure on H^{⋆}(ℚ/ℤ(−1)) over the Milnor K-ring (see [M 03, Appendix A]). By the admissibility condition (3.1), the second term in the sum is N_{E_{K}/K}(x.H^{2}(g)∘H^{2}(j)(t_{K})) = 0, so the expressions in (3.6) and (3.7) are equal. This finishes the proof of (3.5).

4. Reduction to groups having a particular Tits index

For each group G, we let t_{R,G} be the class in H^{2}(k, Z) corresponding to the composition (1.1) under the isomorphism specified in the previous section. The main theorem asserts that this class is zero or generates the same subgroup of H^{2}(k, Z) as the Tits class t_{G}, depending on the type of the group G we started with. A case by case proof will be given in sections 7 to 12. First, we prove some general facts, on which our strategy is based.

Given G, we fix a maximal k-split torus S, a maximal k-torus T containing S, and a set of simple roots Δ of G with respect to T. Recall the Tits index of G as defined in [Ti 66]. It is the data of the Dynkin diagram of G together with the action of the Galois group Gal(k_{sep}/k) on Δ via the ∗-action, and the set Δ_{0} of those α ∈ Δ that vanish on S. Notations for Δ_{0} vary, but following [Ti 66], we circle a vertex if it belongs to Δ \ Δ_{0}, and circle together vertices that are in the same Galois orbit.

Let Δ_{r} be the subset of Δ consisting of those simple roots such that the corresponding fundamental weight belongs to the root lattice. We consider the following condition on G:

(4.1)  No α ∈ Δ_{r} vanishes on S.

In terms of Tits indices, this amounts to

(4.2)  Every vertex in Δ_{r} is circled in the Tits index of G.

Dynkin diagrams for E_{7}, E_{6}, and D_{even} with the vertices in Δ_{r} circled can be found in sections 9, 11, and 12 below. 4.3. We now give a method that reduces the proof of the main theorem 2.3 to the case where G satisfies condition (4.2). Let G be a simply connected absolutely almost simple group over k and further suppose that G has inner type, i.e., that the absolute Galois group acts trivially on the Dynkin diagram of G. Because G has inner type, the center Z of G is isomorphic to μ_{n} or μ_{2} × μ_{2}, and in §3 we fixed an injection j: Z → 𝔾_{m}^{×s} for s ≤ 3.

Consulting the tables in [Bou], we observe that every element of Δ_{r} is fixed by every automorphism of the Dynkin diagram, hence is fixed by the ∗-action. It follows that the variety of parabolic subgroups of G_{sep} of type Δ \ Δ_{r} (in the notation of [BT, 4.2, 5.12] or [MT, p. 33]) is defined over k, see [BT, 5.24] or [BS, 8.4]. This variety has a point over its function field F, hence G satisfies condition (4.2) over F. We have a commutative diagram with exact rows

1 ⟶ H^{2}(k, Z) ⟶^{H^{2}(j)} ⊕^{s}H^{2}(k, 𝔾_{m})
↓  ↓
1 ⟶ H^{2}(F, Z) ⟶^{H^{2}(j)} ⊕^{s}H^{2}(F, 𝔾_{m}).

The kernel of the restriction map H^{2}(k, 𝔾_{m}) → H^{2}(F, 𝔾_{m}) has been computed by Merkurjev and Tignol in [MT, Th. B]. Let Λ/Λ_{r} be the quotient of the weight lattice of G by its root lattice. The kernel of the restriction map is generated by the Tits algebras associated with the classes in Λ/Λ_{r} of the fundamental weights corresponding to those simple roots which belong to Δ_{r}. But we have precisely chosen Δ_{r} so that these classes are zero. Hence, the restriction map H^{2}(k, 𝔾_{m}) → H^{2}(F, 𝔾_{m}) is injective. By commutativity of the diagram, the map H^{2}(k, Z) → H^{2}(F, Z) is injective.

Let t_{R,G} and the Tits class t_{G} ∈ H^{2}(k, Z) be as defined in the introduction. The isomorphism β between H^{2}(k, Z) and Inv^{3}(H^{1}(⋆, Z)) from §3 is functorial in k, i.e., the image of t_{R,G} in H^{2}(F, Z) is the element corresponding to the composition (1.1) over F. Therefore, if the composition (1.1) is zero over F, then t_{R,G} is killed by F, and t_{R,G} is zero in H^{2}(k, Z), i.e., the composition (1.1) is zero over k. Alternatively, if the restrictions of t_{R,G} and t_{G} generate the same subgroup of H^{2}(F, Z), then t_{R,G} and t_{G} generate the same subgroup of H^{2}(k, Z).

In this way, we can reduce the proof of the main theorem for groups of inner type to the case of groups of inner type satisfying (4.2). This reduction can be generalized to treat also groups of outer type, but we will not use that, so we omit it.

Remark 4.4. (i) For groups of type A, Δ_{r} is the empty set, so (4.2) vacuously holds, the function field F is k itself, and this reduction is useless.

(ii) Our global hypothesis excluding fields of certain characteristics arises from this argument, i.e., from 4.3. The hypothesis on the characteristic is needed for the identification of H^{2}(k, Z) with Inv^{3}(H^{1}(⋆, Z)) from Prop. 2.1, because the proof of that proposition used [MPT, Th. 1.1] to provide the isomorphism α^{E}. In turn, the proof of the result from [MPT] needs that the n-torsion in H^{3}(k, ℚ/ℤ(2)) is a cycle module, in particular, one should be able to take residues relative to discrete valuations on k. This is only known for n not divisible by char k.

With small changes, our proof for groups of inner type satisfying (4.2) holds without any restriction on the characteristic of k.

5. A semisimple subgroup

In this section, we assume that the group G has Tits index satisfying condition (4.2). We produce a semisimple simply connected subgroup G′ of G that contains the center of G and describe how to compute the Rost invariant of some elements of G using G′. 5.1. Description of G by generators and relations. Recall from [St] that — over k_{sep} — G is generated by the images of homomorphisms

x_{α}: 𝔾_{a} → G

as α varies over the set of roots Φ of G with respect to T. Write Λ_{r} and Λ for the root and weight lattices of G with respect to T. Since G is simply connected, Λ is identified with the character group T^{⋆} = Hom_{k_{sep}}(T_{sep}, 𝔾_{m}) and Λ_{r} with the character group \overline{T}^{⋆} of the image \overline{T} of T in G/Z.

Write T_{⋆} for the Γ_{k}-module of loops or cocharacters, i.e., k_{sep}-homomorphisms 𝔾_{m} → T_{sep}. There is a natural pairing T^{⋆} × T_{⋆} → ℤ that enables us to identify T_{⋆} with the dual Hom(Λ, ℤ) of T^{⋆}, i.e., with the lattice of co-roots, denoted by Λ_{r}^{∨} (see [Bou, VI.1.1, Prop. 2]). We fix a set of simple roots Δ = {α_{1}, α_{2}, …, α_{ℓ}} of Φ; the coroots α̌_{1}, α̌_{2}, …, α̌_{ℓ} are a set of simple roots of the dual root system Φ^{∨} [Bou, VI.1.5, Rem. 5]. Given an element α̌ := ∑ c_{i}α̌_{i} in Λ_{r}^{∨}, the corresponding loop 𝔾_{m} → T_{sep} is

(5.2)  t ↦ ∏ h_{i}(t^{c_{i}}),

where h_{i}: 𝔾_{m} → T_{sep} is the loop corresponding to α̌_{i}. The map ∏ h_{i}: 𝔾_{m}^{×ℓ} → T_{sep} is an isomorphism [St, p. 44, Cor. (a)]. We have the identity:

(5.3)  hα̌(t)x_{β}(u)hα̌(t)^{−1} = x_{β}(t^{(β,α̌)}u)  (β ∈ Φ, α̌ ∈ Φ^{∨}).

5.4. Definition of G′. Assume that G satisfies (4.2). Write Λ^{∨} for the weight lattice of the dual root system; it has a basis, dual to the basis Δ of Λ_{r}, and we write ω̌_{j} for the “co-weight” such that (ω̌_{j}, α_{i}) = δ_{ij} (Kronecker delta). Write \overline{M} for the sublattice of Λ^{∨} generated by the ω̌_{j} for α_{j} ∈ Δ_{r}, and M for the intersection \overline{M} ∩ Λ_{r}^{∨}. It follows from [BT, 6.7, 6.9] that Γ_{k} acts trivially on \overline{M} — hence also on M — because Δ_{r} is pointwise fixed by the Galois action. (Compare [M 96, 5.2].) Consequently, the loops in M are k-homomorphisms 𝔾_{m} → T, and they generate a k-split torus S′ of rank |Δ_{r}| in G.

We define G′ to be the derived subgroup of the centralizer in G of S′.

Over k_{sep}, we can describe G′ concretely. It follows from (5.3) that G′ is generated by the images of the x_{α}’s, where α varies over the roots in Φ whose support does not meet Δ_{r}. In particular, G′ is semisimple and even simply connected by [SS, 5.4b]. The Dynkin diagram of G′ is obtained by deleting the vertices Δ_{r} from the Dynkin diagram of G. The intersection of G′ with the maximal torus T is, again over k_{sep}, the image of ∏_{α_{i}∉Δ_{r}} h_{i}.

We are going to compute t_{R,G} by reducing to this subgroup G′. One reason why this is possible is the following:

Proposition 5.5. The center Z of G is contained in G′.

Proof. It suffices to check this over an algebraic closure of k, so we may assume that G′ and G are split. It follows from equation (5.3) that the center of G is the subgroup of T cut out by the roots, i.e., it is the intersection

Z(G) = ∩_{α∈Φ} ker α.

Of course, we may replace the condition “α ∈ Φ” with “α is in the root lattice”. In particular, the fundamental weight ω_{j} corresponding to α_{j} ∈ Δ_{r} belongs to the root lattice, so Z(G) is contained in the kernel of ω_{j}. Recall that ω_{j} is given by the formula

ω_{j} (∏_{i} h_{i}(t_{i})) = t_{j}.

Therefore, Z(G) is a subgroup of ∏_{α_{i}∉Δ_{r}} h_{i}(𝔾_{m}). But this is the maximal torus of G′. □

5.6. As G′ is simply connected, it is of the form G′_{1} × G′_{2} × ⋯ × G′_{s}, where each G′_{i} is simply connected and absolutely simple with Dynkin diagram a connected component of Δ \ Δ_{r}.

Lemma. An element a ∈ Z^{1}(k, ∏_{i} G′_{i}) can be written as ∏_{i} a_{i} for a_{i} ∈ Z^{1}(k, G′_{i}), and the Rost invariant r_{G}(a) equals ∑_{i} m_{i}r_{G′_{i}}(a_{i}), where m_{i} is the Rost multiplier of the inclusion G′_{i} ⊂ G.

Proof. We prove this by induction on the number r of a_{i}’s such that a_{i} is not the zero cocycle in H^{1}(k, G′_{i}). The case r = 1 is trivial. We treat the general case. Let i be such that a_{i} is not zero. Consider the group G_{a_{i}} obtained by twisting G by a_{i}; since G′_{i} commutes with G′_{j} for j ≠ i, the twisted group contains ∏_{j≠i} G′_{j} in an obvious way. We find a diagram

H^{1}(k, ∏_{j≠i} G_{j}) ⟶ H^{1}(k, G_{a_{i}}) ⟶^{r_{Ga_{i}}} H^{3}(k, ℚ/ℤ(2))
τ_{a_{i}} ↓≅  ↓^{?+r_{G}(a_{i})}
H^{1}(k, G) ⟶^{r_{G}} H^{3}(k, ℚ/ℤ(2))

where τ_{a_{i}} denotes the twisting map. The diagram commutes as in Remark 2.5. Starting with ∏_{j≠i} a_{j} in the upper left, we find ∑_{j≠i} m_{j}r_{G′_{j}}(a_{j}) in the upper right by the induction hypothesis and a in H^{1}(k, G) in the lower left. The commutativity of the diagram gives the desired equation. □

5.7. Continue the notation of 5.6. We are mainly interested in the case where G is “simply laced”, i.e., where all roots have the same length. In that case, the inclusions G′_{i} ⊂ G all have Rost multiplier one—see e.g. [Ga 01b, 2.2] or [M 03, 7.9.2]—and the formula from Lemma 5.6 simply says:

(5.8)  r_{G}(a) = ∑_{i} r_{G′_{i}}(a_{i}).

6. The center and the torus

For the duration of this section, the field k is arbitrary, with no restriction on the characteristic. We compute how the scheme-theoretic center Z of a semisimple simply connected group G sits inside a fixed maximal torus, in terms of the root system. We do this using an exact sequence from [M 96], which we first recall. 6.1. Isogenies of tori. Let T → \overline{T} be a k-isogeny of tori with kernel Z, i.e., Z is finite and the sequence

1 ⟶ Z ⟶ T ⟶^{π} \overline{T} ⟶ 1

is exact. As before, T_{⋆} (resp. \overline{T}_{⋆}) is the Γ_{k}-module of loops or cocharacters, i.e., k_{sep} homomorphisms 𝔾_{m} → T_{sep} (resp. 𝔾_{m} → \overline{T}_{sep}).

Proposition 6.2. There is a map z such that

1 ⟶ T_{⋆} ⟶^{π_{⋆}} \overline{T}_{⋆} ⟶^{z} \underrightarrow{lim}_{n} Hom_{k_{sep}}(μ_{n}, Z_{sep}) ⟶ 1

is an exact sequence of Γ_{k}-modules.

This is sketched in §1 of Merkurjev’s paper (loc.cit.). We give an explicit proof, including a precise description of z which will be used later.

We first prove the following lemma.

Lemma 6.3. The image of every k-homomorphism from μ_{n} to a split torus S is contained in a rank 1 subtorus of S.

Proof. We may assume that μ_{n} injects into S. Dualizing, such an injection corresponds to a surjection π: ℤ^{r} → ℤ/nℤ, where r is the rank of S. By the fundamental theorem for finitely generated abelian groups, there is a basis b_{1}, …, b_{r} of ℤ^{r} such that the kernel of π has basis b_{1}, b_{2}, …, b_{r−1}, nb_{r}. Projection on the b_{r}-coordinate defines a map π_{r} : ℤ^{r} → ℤ, and π factors through π_{r}. Dualizing again gives the lemma. □

Proof of Prop. 6.2. Since π has finite kernel, the induced map π_{⋆} is injective. We use it to identify T_{⋆} with a submodule of \overline{T}_{⋆}. Since the tori have the same rank, the quotient \overline{T}_{⋆}/T_{⋆} is finite.

We now define z. For ω ∈ \overline{T}_{⋆}, let n be the smallest positive integer such that nω belongs to T_{⋆}. We define z_{ω} : μ_{n} → T_{sep} to be the restriction of the loop nω: 𝔾_{m} → T_{sep} to μ_{n}. Every character χ of \overline{T} maps \overline{T}_{⋆} → ℤ, hence χ(nω) is in nℤ and the composition

μ_{n} ⟶^{z_{ω}} T ⟶ \overline{T} ⟶^{χ} 𝔾_{m}

is 1. Since subgroups of T are cut out by characters, the image of z_{ω} belongs to Z. It is an exercise to verify that the sequence displayed in the lemma is exact at \overline{T}_{⋆}. Note that the map z is Γ_{k}-equivariant.

It remains to prove that z is surjective. Let ϕ: μ_{n} → Z_{sep} be a homomorphism defined over k_{sep}. By Lemma 6.3, there is a rank 1 subtorus S_{sep} of T_{sep} such that the image of ϕ is contained in S_{sep}; let ρ be a loop generating S_{⋆}. By hypothesis, χ(πρ) is in nℤ for every character χ ∈ \overline{T}^{⋆}, so there is some ω ∈ \overline{T}_{⋆} such that πρ = nω. Then ϕ and z_{ω} agree in \underrightarrow{lim} Hom(μ_{n}, Z_{sep}). □

In our situation (with the notations from 5.1 and 5.4), the center Z of G is contained in the maximal torus T and the sequence

1 ⟶ Z ⟶ T ⟶ \overline{T} ⟶ 1

is exact. For the same reason that T_{⋆} is identified with the coroot lattice Λ_{r}^{∨}, the lattice \overline{T}_{⋆} is identified with Λ^{∨}. Hence the exact sequence given by Prop. 6.2 can be re-written as

(6.4)  1 ⟶ Λ_{r}^{∨} ⟶ Λ^{∨} ⟶ \underrightarrow{lim}_{n} Hom_{k_{sep}}(μ_{n}, Z_{sep}) ⟶ 1

We close this section with an application of this exact sequence. We will only use it for groups of type B and C, but we include it because of independent interest. Write Δ_{c} for the set of α_{j} ∈ Δ such that ω̌_{j} (as defined in 5.4) is a minuscule weight for the dual root system Φ^{∨}. Recall [Bou, §VI.1, Exercise 24] that the minuscule weights are the minimal nonzero dominant weights.

Corollary 6.5. If G is of inner type and Δ_{c} does not meet Δ_{0}, then the center Z of G is contained in the maximal k-split subtorus S of T.

The condition on Δ_{c} is the same as saying: every element of Δ_{c} is circled in the index of G.

Remark 6.6. In the situation of the corollary, H^{1}(k, S) is zero, so the natural map H_{fppf}^{1}(k, Z) → H^{1}(k, G) is zero. (If Z is smooth—as in the rest of this paper—the group H_{fppf}^{1}(k, Z) agrees with the Galois cohomology group H^{1}(k, Z).) This has two useful consequences:

(i) The composition (1.1) is zero.

(ii) The connecting homomorphism (G/Z)(k) → H_{fppf}^{1}(k, Z) is surjective.

Proof of Cor. 6.5. Since the minuscule weights generate Λ^{∨}/Λ_{r}^{∨} [Bou, §VI.2, Exercise 5], Z is generated by the images of zω̌_{j} for α_{j} ∈ Δ_{c}. Using the same argument as in the proof of Prop. 6.2, we see that Z is contained in the torus Q (defined over k_{sep}) corresponding to the n_{j}ω̌_{j} for α_{j} ∈ Δ_{c}, where n_{j} is the smallest natural number such that n_{j}ω̌_{j} is in Λ_{r}^{∨}.

Since G has inner type, the maximal k-split torus S in T is the intersection of the kernels of α ∈ Δ_{0} by [BT, 6.7, 6.9]. For such an α and for α_{j} ∈ Δ_{c}, we have α ≠ α_{j} by hypothesis, so the inner product (α, ω̌_{j}) is zero. We conclude that the torus Q is contained in S. □

7. Type B

Before proving the main theorem for exceptional groups, we show that our method gives a new and very short proof for groups of type B_{ℓ} (ℓ ≥ 2), i.e., that composition (1.1) is zero. By 4.3, it is enough to prove this for groups having Tits index satisfying (4.2). Consulting the tables in [Bou], we find that

Δ_{r} = {α_{1}, α_{2}, …, α_{ℓ−1}}.

The dual root system is C_{ℓ}, and its only minuscule weight is ω̌_{1}, i.e.,

Δ_{c} = {α_{1}}.

By (4.2), the vertex α_{1} is circled in the index of G, so by Remark 6.6(i) the composition (1.1) is zero.

Note that groups of type B_{ℓ} satisfying the hypotheses of Cor. 6.5 are those of the form Spin(q) where q is a (2ℓ + 1)-dimensional isotropic quadratic form. For these groups, Remark 6.6(ii) says: The spinor norm map SO(q)(k) → k^{×}/k^{×2} is surjective.

8. Concrete description of the center

We now give a concrete description of the center Z inside a simply connected group G in terms of the generators and relations for G as in 5.1; consequently we work over the separably closed field k_{sep}. (But see 8.10 below.) Note that, by exactness of sequence (6.4), it is enough to compute zω̌ for ω̌ belonging to a set of representatives in Λ^{∨} of generators of the quotient group Λ^{∨}/Λ_{r}^{∨}.

Example 8.1 (E_{7}). The center of a simply connected group G of type E_{7} is isomorphic to μ_{2}. As all roots have the same length, we can normalize the Weylinvariant inner product so that all roots have length 2; this identifies Φ with the inverse system Φ^{∨}. The fundamental weight

ω_{7} = α_{1} + \frac{3}{2}α_{2} + 2α_{3} + 3α_{4} + \frac{5}{2}α_{5} + 2α_{6} + \frac{3}{2}α_{7}

is not in the root lattice and so maps to the nonidentity element in Λ/Λ_{r}. (We systematically number roots as in Bourbaki; for a diagram see §9.) The corresponding map z_{ω_{7}} : μ_{2} → Z_{sep} is given by

z_{ω_{7}}(−1) = h_{2ω_{7}}(−1) = h_{2}(−1)h_{5}(−1)h_{7}(−1).

Here we can see Prop. 5.5 explicitly: Δ \ Δ_{r} is {α_{2}, α_{5}, α_{7}}, and the image of this morphism is contained in G′.

Example 8.2 (E_{6}). Let now G be of type E_{6}. The center of G is isomorphic to μ_{3}. Again we can identify Φ and Φ^{∨}. The fundamental weight

ω_{1} = \frac{4}{3}α_{1} + α_{2} + \frac{5}{3}α_{3} + 2α_{4} + \frac{4}{3}α_{5} + \frac{2}{3}α_{6}

is not in in the root lattice; so its class generates Λ/Λ_{r}. The corresponding isomorphism z_{ω_{1}} : μ_{3} → Z_{sep} is defined by

(8.3)  z_{ω_{1}}(ζ) = h_{3ω_{1}}(ζ) = h_{1}(ζ) h_{3}(ζ^{2}) h_{5}(ζ) h_{6}(ζ^{2}).

Example 8.4 (C_{ℓ}). Let now G be of type C_{ℓ}; the center Z is isomorphic to μ_{2}. In this case, we must be somewhat more careful because the root system has roots of different lengths. The inverse system Φ^{∨} is B_{ℓ}. We write ω̌_{ℓ} for the fundamental weight corresponding to the root α̌_{ℓ} in the inverse system, i.e.,

ω̌_{ℓ} = \frac{1}{2}(α̌_{1} + 2α̌_{2} + ⋯ + ℓα̌_{ℓ}).

The corresponding isomorphism μ_{2} → Z_{sep} is given by

(8.5)  zω̌_{ℓ}(−1) = \begin{cases}h_{2ω̌_{ℓ}}(−1) = h_{1}(−1)h_{3}(−1) ⋯ h_{ℓ−1}(−1) & if ℓ is even, \\ h_{2ω̌_{ℓ}}(−1) = h_{1}(−1)h_{3}(−1) ⋯ h_{ℓ}(−1) & if ℓ is odd. \end{cases}

Example 8.6 (D_{even}). For G of inner type D_{ℓ} with ℓ even, the center Z is isomorphic to μ_{2} × μ_{2}. As for E_{7}, all roots have the same length and we identify the root system with the inverse system. The fundamental weights

ω_{ℓ−1} = \frac{1}{2}α_{1} + α_{2} + \frac{3}{2}α_{3} + ⋯ + \frac{ℓ − 2}{2}α_{ℓ−2} + \frac{ℓ}{4}α_{ℓ−1} + \frac{ℓ − 2}{4}α_{ℓ}

ω_{ℓ} = \frac{1}{2}α_{1} + α_{2} + \frac{3}{2}α_{3} + ⋯ + \frac{ℓ − 2}{2}α_{ℓ−2} + \frac{ℓ − 2}{4}α_{ℓ−1} + \frac{ℓ}{4}α_{ℓ}

have two different residues in Λ/Λ_{r}. As this is the Klein four-group, they generate it. It follows that the images of the corresponding maps z_{ω_{ℓ−1}}, z_{ω_{ℓ}} : μ_{2} → Z_{sep} generate Z_{sep}. Applying 6.1, we find that Z_{sep} is generated by the images of the homomorphisms z_{0}, z_{1} defined by

z_{0}(−1) := h_{1}(−1)h_{3}(−1) ⋯ h_{ℓ−3}(−1) h_{ℓ}(−1)
(8.7)  z_{1}(−1) := h_{1}(−1)h_{3}(−1) ⋯ h_{ℓ−3}(−1) h_{ℓ−1}(−1)

Note that we have abandoned tying the weights with their corresponding homomorphism, because this depends on the parity of ℓ/2.

Example 8.8 (A_{2}). Suppose now that G has type A_{2}, i.e., G is isomorphic to SL_{3} and its center is isomorphic to μ_{3}. Since the fundamental weight ω_{1} = \frac{2}{3}α_{1} + \frac{1}{3}α_{2} does not belong to the root lattice, we get an isomorphism z_{ω_{1}} : μ_{3} → Z_{sep} defined by

(8.9)  z_{ω_{1}}(ζ) = h_{1}(ζ^{2})h_{2}(ζ).

8.10. Let G be a simply connected semisimple group of inner type over k. Each element ω̌ of Λ^{∨} defines an isomorphism

zω̌: μ_{n} → μ_{n} ⊂ G

over k_{sep}. But the only morphisms μ_{n} → μ_{n} are raising to a power, so zω̌ is Γ_{k}-equivariant, and hence is defined over k by [Borel, AG.14.3].

9. Type E_{7}

This section proves the main theorem 2.3 for a group G of type E_{7}. In this case, the center Z is identified with μ_{2}. We prove that t_{R,G} coincides with the Tits class t_{G}, i.e. the composition (1.1) is the cup product with t_{G}, where the cup product is induced by the obvious bilinear map Z(k_{sep}) × Z(k_{sep}) = μ_{2}(k_{sep}) × μ_{2}(k_{sep}) → μ_{2}(k_{sep})^{⊗2}. (The group H^{2}(k, Z) is 2-torsion, so t_{R,G} and t_{G} generate the same subgroup of H^{2}(k, Z) if and only if they are equal.)

By 4.3, it is enough to prove this for groups of type E_{7} with Tits index satisfying (4.2). Let G be such a group, i.e. assume the index of G is

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

or possibly has more circled vertices. (If the index has more circles, then G is split, see [Ti 66, p. 57].)

Write G′_{i} with i = 2, 5, 7 for the components of G′, where G′_{i} corresponds to the vertex α_{i} in the Dynkin diagram. Each is isomorphic to SL_{1}(Q_{i}) for some quaternion algebra Q_{i}. Since the weights ω_{2}, ω_{5}, ω_{7} all have the same image in Λ/Λ_{r}, the Q_{i} are all isomorphic by [Ti 71, p. 211] and we simply write Q. By [Ti 71, loc. cit.], [Q] ∈ H^{2}(k, μ_{2}) is also the Tits class of G. The center Z is contained in the product Z_{2} × Z_{5} × Z_{7} of the centers of G′_{2}, G′_{5}, G′_{7} by Prop. 5.5. By the description of the center given in example 8.1, the map

μ_{2} = Z → Z_{2} × Z_{5} × Z_{7} = μ_{2} × μ_{2} × μ_{2}

is −1 ↦ (−1, −1, −1). (Even though Example 8.1 treats G over the separable closure, it still applies to our nonsplit G over k by 8.10.) Applying equation 5.8 and the type A_{1} case of the main theorem (see Remark 2.4), we get that the class t_{R,G} is 3[Q] = [Q], which is the Tits class of G.

This concludes the proof of the main theorem for groups of type E_{7}.

10. Type C

The argument of the previous section can be adapted to recover the theorem for groups of type C_{ℓ}. (Alternatively, in the case where ℓ even, the same argument as for type B in §7 shows that composition (1.1) is zero.) Since the result is not new for type C, we only briefly sketch the proof. We want to show that t_{R,G} is 0 if ℓ is even and t_{G} if ℓ is odd. Recall that a group of type C_{ℓ} has Dynkin diagram

[figure: 1; 2; 3; ℓ − 2; ℓ − 1; ℓ; <]

As before, it is enough to prove it for groups whose Tits index satisfy condition (4.2), i.e., such that α_{i} is circled for i even.

As before, we write G′_{i} for the component of G′ corresponding to the vertex α_{i}, for 1 ≤ i ≤ n, i odd. We may identify all those with SL(Q) for some quaternion algebra Q. The description of the center given in Example 8.4 shows that Z maps to the product of the centers of the G′_{i} by (−1) ↦ (−1, …, −1).

It remains to apply Lemma 5.6, which asserts r_{G}(a) = ∑_{i} m_{i}r_{G′_{i}}(a_{i}). Note that we do have to care about Rost multipliers here, since groups of type C are not simply laced. If 1 ≤ i ≤ n − 1, i odd, then the root α_{i} is short in C_{ℓ}. So the corresponding co-root α̌_{i} is long and the inclusion G′_{i} ⊂ G has Rost multiplier 2, see e.g. [Ga 01b, 2.2] or [M 03, 7.9.2]. Since the Rost invariant has values in H^{3}(k, μ_{2}^{⊗2}) which is 2-torsion, we get r_{G}(a) = 0 if ℓ is even, and r_{G}(a) = r_{G′_{ℓ}}(a_{ℓ}) if ℓ is odd. In this last case, applying the formula for groups of type A, we get that t_{R,G} = [Q], which is the Tits class of G.

11. Type E_{6}

In this section, we prove the theorem for a group G of type E_{6}. 11.1. Groups of inner type E_{6}. Let us first assume G is of inner type, that is Γ_{k} acts trivially on the Dynkin diagram. The center Z is identified with μ_{3} and we want to prove that t_{R,G} and t_{G} generate the same subgroup of H^{2}(k, Z). Again by 4.3, it suffices to consider groups whose Tits index satisfies condition (4.2). Let G be such a group, i.e. we assume the Tits index of G is

[figure: 1; 3; 4; 5; 6; 2]

or has more circled vertices. (As in the type E_{7} case in §9, if more vertices are circled, then G is split.)

Write G′_{1} and G′_{5} for the components of G′ corresponding to the subdiagrams with vertices 1, 3 and 5, 6 respectively. They are of type A_{2}. Moreover, since the fundamental weights ω_{1} and ω_{5} have the same class in Λ/Λ_{r}, you may identify G′_{1} and G′_{5} with SL(D) for some central simple algebra D of degree 3 over k. The center of SL(D) is naturally identified with μ_{3}, and the composition

H^{1}(k, μ_{3}) → H^{1}(k, SL(D)) = H^{1}(k, G′_{i}) ⟶^{r_{G′_{i}}} H^{3}(k, ℚ/ℤ(2))

is the cup product with m[D] for m = 1 or 2 by Example 2.4.

Comparing the description of the center of G and the G′_{i} given by formulas 8.3 and 8.9, one sees that the map Z → G′_{1}×G′_{5} is ζ ↦ (ζ^{2}, ζ^{2}). Hence the induced map H^{1}(k, Z) → H^{1}(k, Z′_{1}) × H^{2}(k, Z′_{5}) is given by a ↦ (2a, 2a). Applying Equation (5.8), we see that the composition (1.1) is

a ↦ 4a.m[D] ∈ H^{3}(k, μ_{3}^{⊗2}).

But 4a equals a because the exponent of Z is 3. Therefore, t_{R,G} equals m[D], i.e., generates the same subgroup of H^{2}(k, Z) as [D], which is t_{G} [Ti 71, p. 211]. 11.2. Groups of outer E_{6} type. Suppose now that the group G is of outer type, i.e., the absolute Galois group acts nontrivially on the Dynkin diagram of G. The kernel of the homomorphism Γ_{k} → Aut(Δ) = ℤ/2 defines a quadratic extension L of k, over which the group becomes of inner type. Its center Z is isomorphic to μ_{3}_{[L]}.

By the inner type case, we know that t_{R,G} − m t_{G} is killed by L. Since it lives in the group H^{2}(k, Z) which is of exponent 3, while L is quadratic over k, a restriction/corestriction argument shows that t_{R,G} − m t_{G} is zero in H^{2}(k, Z). This concludes the proof for groups of type E_{6}.

12. Type D_{ℓ} for ℓ even

We now prove the main theorem for groups of type D_{ℓ} for ℓ even, including trialitarian D_{4}. 12.1. Groups of inner type D_{ℓ}. As before, we first assume that G is of inner type, so that Z is identified with μ_{2} × μ_{2}. We want to prove that the composition (1.1) is the cup product with the Tits class, where the cup product is induced by the bilinear map given in (2.2). By 4.3, it is enough to consider groups with Tits index satisfying condition (4.2), i.e., with Dynkin diagram

[figure: 1; 2; 3; 4; ℓ − 3; ℓ − 2; ℓ − 1; ℓ]

or having more circled vertices.

We write G′_{i}, for i = 1, 3, …, ℓ − 3, ℓ − 1 and ℓ for the component of G′ corresponding to the vertex α_{i}. They are of type A_{1}, and may be identified with SL(Q_{i}), for some quaternion algebras Q_{i}. Each has center Z′_{i} isomorphic to μ_{2}. Moreover, since ω_{1}, …, ω_{ℓ−3} and ω_{ℓ−1} + ω_{ℓ} have the same class in Λ/Λ_{r}, we may assume Q_{1} = ⋯ = Q_{ℓ−3} = Q, for some quaternion algebra Q satisfying [Q] = [Q_{ℓ−1}]+[Q_{ℓ}].

Since G is of inner type, the homomorphisms z_{0}, z_{1} from Example 8.6 are kdefined even though G need not be split. Hence, we get a map

(z_{0}, z_{1}): μ_{2} × μ_{2} → Z ⊂ Z′_{1} × ⋯ × Z′_{ℓ−3} × Z′_{ℓ−1} × Z′_{ℓ}.

The induced map H^{1}(k, Z) → ∏_{i} H^{1}(k, Z′_{i}) is given by

(a_{0}, a_{1}) ↦ (a_{0} + a_{1}, …, a_{0} + a_{1}, a_{1}, a_{0}).

Applying Equation (5.8), and the formula for groups of type A (see 2.4), we get r_{G}(a_{0}, a_{1}) = \frac{ℓ−2}{2}(a_{0} + a_{1}).[Q] + a_{1} .[Q_{ℓ−1}] + a_{0} .[Q_{ℓ}], that is

(12.2)  r_{G}(a_{0}, a_{1}) = \begin{cases}a_{0} .[Q_{ℓ−1}] + a_{1} .[Q_{ℓ}] & if ℓ ≡ 0 (mod 4) \\ a_{0} .[Q_{ℓ}] + a_{1} .[Q_{ℓ−1}] & if ℓ ≡ 2 (mod 4) \end{cases}

On the other hand, we have:

(12.3)  ω_{ℓ−i}(a_{0}, a_{1}) = a_{i} and ω_{ℓ−i}(t_{G}) = [Q_{ℓ−i}]

for i = 0, 1. Combining (12.2) and (12.3) with the definition of the cup product in (2.2), we conclude that composition (1.1) equals the cup product with the Tits class t_{G}. 12.4. Groups of outer D_{ℓ} type. For the groups of type ^{2}D_{ℓ}, one can reduce to the ^{1}D_{ℓ} case as in [MPT, p. 817]. Alternatively, one can apply the same method as for groups of inner type. In this case, G′ is a direct product of (ℓ − 2)/2 copies of SL(Q) and a transfer R_{L/k}SL(Q′), where Q′ is a quaternion algebra over a quadratic extension L of k. To apply (5.8), one needs to know the Rost invariant of R_{L/k}SL(Q′), which is specified in [M 03, 9.8].

It remains only to treat the trialitarian groups, i.e., the groups G of type ^{3}D_{4} or 6D_{4}. Recall what this means. The automorphism group of the Dynkin diagram Δ of G is the symmetric group on three letters, and the superscript 3 or 6 denotes the size of the image of homomorphism Γ_{k} → Aut(Δ). The kernel of this homomorphism fixes at least one separable cubic field extension of k; we pick one and call it L. The group G is of type ^{1}D_{4} or ^{2}D_{4} over L. As the exponent of Z (i.e., 2) is relatively prime to the dimension of L/k (i.e., 3), the proof may be completed using some restriction/corestriction argument as for groups of outer type E_{6}.

13. Application to groups of type E_{7}

We now apply the main theorem to prove a result about groups of type E_{7}. A simply connected group of type E_{7} can be described in terms of “gifts” as in [Ga 01a]. A gift is a triple (A, σ, π) where A is a central simple k-algebra of degree 56, σ is a symplectic involution on A, and π: A → A is k-linear and satisfies certain axioms, see [Ga 01a, Def. 3.2] for a precise statement. The Brauer class of A is the Tits class t_{G}. The k-points of G are the elements a ∈ A^{×} such that σ(a)a = 1 and Int(a) π = π Int(a), where Int(a) denotes the automorphism x ↦ axa^{−1} of A.

The associated adjoint group \overline{G} is a subgroup of P GL(A); it has k-points Int(a) for a ∈ A^{×} such that σ(a)a is in k^{×} and Int(a) π = π Int(a). The exact sequence

1 ⟶ μ_{2} ⟶ G ⟶ \overline{G} ⟶ 1

gives a connecting homomorphism

δ: \overline{G}(k) → H^{1}(k, μ_{2}) = k^{×}/k^{×2}.

As in [KMRT, §31], one finds

δ(Int(a)) = σ(a)a ∈ k^{×}/k^{×2}.

If λk^{×2} is in the image of δ, then its image in H^{1}(k, G) is zero and the composition (1.1) sends λ to zero. The main theorem gives:

Corollary 13.1. If λk^{×2} is in the image of δ, then (λ).[A] = 0.

This result was previously observed in the case k = ℝ in [Ga 01a, 6.2].

Acknowledgements. The first author thanks Université Paris 13 for its hospitality while some of the work on this paper was performed.

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Department of Mathematics & Computer Science, Emory University, Atlanta, GA 30322, USA

   E-mail address: skip@member.ams.org
   URL: http://www.mathcs.emory.edu/~skip/

Laboratoire Analyse, Géométrie & Applications, UMR CNRS 7539 – Institut Galilée, Université Paris 13, 93430 Villetaneuse, France

   E-mail address: queguin@math.univ-paris13.fr
   URL: http://www-math.univ-paris13.fr/~queguin/