ORTHOGONAL INVOLUTIONS ON ALGEBRAS OF DEGREE 16 AND THE KILLING FORM OF E_{8}

SKIP GARIBALDI

With an appendix by Kirill Zainoulline

Abstract. We exploit various inclusions of algebraic groups to give a new construction of groups of type E_{8}, determine the Killing forms of the resulting E_{8}’s, and define an invariant of central simple algebras of degree 16 with orthogonal involution “in I^{3}”, equivalently, groups of type D_{8} with a half-spin representation defined over the base field. The determination of the Killing form is done by restricting the adjoint representation to various twisted forms of PGL_{2} and requires very little computation.

An appendix by Kirill Zainoulline contains a type of “index reduction” result for groups of type D.

The first part of this paper (§§1–6) extends the Arason invariant e_{3} for quadratic forms in I^{3} to central simple algebras (A, σ) “in I^{3}” (this term is defined in §1) where A has degree 16 or has a hyperbolic involution. (The first case corresponds to simple linear algebraic groups of type D_{8} with a half-spin representation defined over the base field.) The invariant e_{3} detects whether (A, σ) is generically Pfister, see Cor. 2.7 below. We remark that the paper [BPQ] appears to rule out the existence of such an invariant by a counterexample. Our invariant exists exactly in the cases where their counterexample does not apply; surprisingly, this includes some interesting cases. The proofs in this part are not difficult, but we include this material to provide background and context for the later results. Proposition 1.4 generalizes the Arason-Pfister Hauptsatz for quadratic forms of dimension < 16, and depends on a result of Kirill Zainoulline presented in Appendix A.

The real work begins in the second part of the paper (§§7–10), where we use the inclusion PGL_{2} × PSp_{8} ⊂ PSO_{8} to give a formula for the Arason invariant in case (A, σ) can be written as a tensor product (Q,¯ )⊗(C, γ), where (Q,¯ ) is a quaternion algebra with its canonical symplectic involution.

We apply the preceding results in the third part of the paper (§§11–16) to studying algebraic groups of type E_{8}. We give a construction of groups of type E_{8} and compute the Rost invariant, Tits index (in some cases), and Killing form of the resulting E_{8}’s, see Th. 9.1, Prop. 12.1, and Th. 15.2. We compute the Killing form by branching to subgroups of type A_{1}, which is somewhat cleaner than computations of other Killing forms in the literature.

The E_{8}’s arising from our construction are an interesting class. On the one hand, they are uncomplicated enough to be tractable. On the other hand, up to odd-degree extensions of the ground field and assuming the validity of a pre-existing conjecture regarding groups of type D_{8}, they include all E_{8}’s whose Rost invariant is zero. Our construction produces all E_{8}’s over every number field.

2000 Mathematics Subject Classification. 20G15 (11E88, 17B25).

Contents

Part I. Extending the Arason invariant to orthogonal involutions 2 1. I^{n} 2 2. Extending the Arason invariant 4 3. Invariant e_{3}^{hyp}(A, σ) 5 4. I^{3} and D_{2n} 6 5. HSpin_{16} ⊂ E_{8} 7 6. Invariant e_{3}^{16}(A, σ) for algebras of degree 16 in I^{3} 8

Part II. The invariant e_{3}^{16} on decomposable involutions 9 7. An inclusion PGL_{2} × PSp_{8} ×μ_{2} ⊂ HSpin_{16} 9 8. Crux computation 11 9. Rost invariant 13 10. Value of e_{3}^{16} on decomposable involutions 14

Part III. Groups of type E_{8} constructed from 9 parameters 15 11. Construction of E_{8}’s 15 12. Tits index of groups of type E_{8} 16 13. Reduced Killing form up to Witt-equivalence 19 14. Calculation of the Killing form 20 15. The Killing form and E_{8}’s arising from (11.2) 22 16. A conjecture, and its consequences 25 Appendix A. Non-hyperbolicity of orthogonal involutions

By Kirill Zainoulline 25

References 27

Notation and conventions. We work over a field k of characteristic ≠ 2. Throughout the paper, (A, σ) denotes a central simple k-algebra with orthogonal involution.

We often write ¯ for the canonical symplectic involution on a quaternion algebra; it will be clear by context which quaternion algebra is intended. Similarly, we write hyp for a hyperbolic involution; context again will make it clear whether symplectic or orthogonal is intended.

For g in a group G, we write Int(g) for the automorphism x ↦ gxg^{−1}. General background on algebras with involution can be found in [KMRT]. For

the Rost invariant, see [Mer 03].

Part I. Extending the Arason invariant to orthogonal involutions

1. I^{n}

Definition 1.1. Let (A, σ) be a central simple algebra with orthogonal involution over a field k of characteristic ≠ 2. The function field k_{A} of the Severi-Brauer variety of A splits A, hence over k_{A} the involution σ is adjoint to a quadratic form q_{σ}. As an abbreviation, we say that (A, σ) is in I^{n}k (or simply “in I^{n}”) if q_{σ} belongs to I^{n}k_{A}, the n-th power of the fundamental ideal ideal in the Witt ring of k_{A}. Clearly, if (A, σ) is in I^{n}k, then (A ⊗ L, σ ⊗ Id) is in I^{n}L for every extension L/k.

ORTHOGONAL INVOLUTIONS AND E_{8} 3

We say that (A, σ) is generically Pfister if q_{σ} is a Pfister form, or more precisely is generically n-Pfister if q_{σ} is an n-Pfister form.

This first part of the paper is concerned with cohomological invariants of (A, σ) in case (A, σ) is in I^{3}, especially when A has degree 16. For context, we give some properties of algebras with involution in I^{3} or in I^{4} of small degree.

Example 1.2. (1) (A, σ) is in I if and only if deg A is even.

(2) (A, σ) is in I^{2} if and only if it is in I and the discriminant of σ is the identity in k^{×}/k^{×2}.

(3) (A, σ) is in I^{3} if and only if it is in I^{2} and the even Clifford algebra C_{0}(A, σ) is Brauer-equivalent to A × k.

(4) Suppose that (A, σ) is Witt-equivalent to (A′, σ′). Then (A, σ) is in I^{n} if and only if (A′, σ′) is in I^{n}.

(5) Suppose that deg A = 2^{n}. Then (A, σ) is in I^{n} if and only if (A, σ) is generically Pfister [Lam, X.5.6].

(6) Suppose that deg A = 2^{n} with n ≥ 2. If (A, σ) is completely decomposable (i.e., isomorphic to a tensor product of quaternion algebras with orthogonal involution), then (A, σ) is generically Pfister by [Bech], hence is in I^{n}.

Items (1) through (3) show that the property of an algebra being in I^{n} for n ≤ 3 can be detected by invariants defined over k, without going up to the generic splitting field k_{A}. Below we construct an invariant that detects whether (A, σ) belongs to I^{4} for A of degree 16.

Question 1.3. The converse to (6) holds for n = 1 (trivial), n = 2 [KPS], and n = 3 [KMRT, 42.11]. Does the converse also hold for n = 4? That is, does generically 4-Pfister imply completely decomposable? The answer is “yes” if A has index 1 (obvious) or 2 [Bech, Th. 2].

We return to this question in §16 below.

Proposition 1.4 (“Arason-Pfister”). Suppose that (A, σ) is in I^{n} for some n ≥ 1 and deg A < 2^{n}. If n ≤ 4, then σ is hyperbolic (and A is not a division algebra).

Proof. The case where A has index 1 is the Arason-Pfister Hauptsatz. Otherwise, the Hauptsatz implies that σ is hyperbolic over k_{A}. If A has index 2 we are done by [PSS, Prop. 3.3], and if deg A/ ind A is odd we are done by Prop. A.1.

The remaining case is where A has degree 8 and index 4. As (A, σ) is in I^{3}, one component of its even Clifford algebra is split and endowed with an orthogonal involution adjoint to an 8-dimensional quadratic form ϕ with trivial discriminant. The involution σ is hyperbolic over k_{A}, hence ϕ is also hyperbolic over k_{A}. As A is Brauer-equivalent to the full Clifford algebra of ϕ [KMRT, §42], ϕ is isotropic over the base field k by [Lag, Th. 4]. It follows that σ is hyperbolic over k [Ga 01a, 1.1]. □

The algebras of degree 8 in I^{3} are completely decomposable by [KMRT, 42.11]. For degree 10, we have the following nice observation pointed out to us by Jean-Pierre Tignol:

Lemma 1.5. If (A, σ) is in I^{3} and deg A ≡ 2 mod 4, then A is split.

In particular, if (A, σ) is of degree 10 and in I^{3}, then A is split (by the lemma), hence σ is isotropic by Pfister, see [Lam, XII.2.8] or [Ga, 17.8].

Proof of Lemma 1.5. Because the degree of A is congruent to 2 mod 4, the Brauer class γ of either component of the even Clifford algebra of (A, σ) satisfies 2γ = [A], see [KMRT, 9.15]. But (A, σ) belongs to I^{3}, so γ is 0 or [A]. Hence [A] = 0. □

Algebras (A, σ) in I^{3} of degree 12 are described in [GQ].

For (A, σ) in I^{3} of degree 14, the algebra A is split by Lemma 1.5, hence σ is adjoint to a quadratic form in I^{3}. These forms have been described by Rost, see [R] or [Ga, 17.8].

For (A, σ) in I^{3} and of degree ≥ 16, the main question to ask is: How to tell if (A, σ) is in I^{4}? We address that question in Cor. 2.6 below.

Remark 1.6 (I ⇒ H). In addition to the generically Pfister and completely decomposable algebras with involution, another interesting class of involution are the so-called I ⇒ H involutions. We say that a central simple k-algebra A with orthogonal involution σ has I ⇒ H if the degree of A is 2^{n} for some n ≥ 1, and for every extension K/k over which σ is isotropic, the involution σ is actually K-hyperbolic. If (A, σ) has I ⇒ H, then (A, σ) is generically Pfister, see [BPQ]. Conversely, if n ≤ 4 and (A, σ) is generically Pfister, then (A, σ) has I ⇒ H by the arguments in the proof of Prop. 1.4.

2. Extending the Arason invariant

In this section, (A, σ) denotes a central simple algebra in I^{3} over a field k. We use the notation k_{A} and q_{σ} from Def. 1.1.

In some cases, we can define an element

(2.1)  e_{3}(A, σ) ∈ H^{3}(k, ℤ/4ℤ)/E(A)

for E(A) := ker(H^{3}(k, ℤ/4ℤ) → H^{3}(k_{A}, ℤ/4ℤ)), such that

(2.2)

If K/k splits A, then e_{3}(A, σ) is the Arason invariant e_{3}(q_{σ⊗K}) in
H^{3}(k, ℤ/4ℤ).

and

(2.3)  res_{K/k} e_{3}(A, σ) = e_{3} [(A, σ) ⊗ K] for every extension K/k.

Clearly, properties (2.1) and (2.2) uniquely determine e_{3}(A, σ) if such an element exists. The existence is a triviality in case A is split. If A has index 2, then the element e_{3} [(A, σ) ⊗ k_{A}] ∈ H^{3}(k_{A}, ℤ/4ℤ) is unramified [Ber, Prop. 9] and so descends to define an element e_{3}(A, σ) as above by [KRS, Prop. A.1]. However, an element e_{3}(A, σ) need not exist if A has degree 8 and is division as Th. 3.9 in [BPQ] shows. We have:

Theorem 2.4. Suppose ind A ≤ 2 or 2 ind A divides deg A or deg A = 16. Then there exists an e_{3}(A, σ) as in (2.1) and (2.2).

To illustrate the cases covered by the proposition, we note that for A of even degree between 8 and 16, the only omitted cases are where A has degree 8 and index 8 or A has degree 12 and index 4. These cases are genuinely forbidden by [BPQ] and the following example, which extends slightly the reasoning in [BPQ].

Example 2.5. Fix a field k_{0} and an algebra with orthogonal involution (A, σ) in I^{3} over k_{0}, where A has degree 12 and index 4. By extending scalars to various function fields of quadrics as in [Mer 92], we can construct an extension k/k_{0} such that H^{3}(k, ℤ/4ℤ) is zero and A ⊗ k still has index 4. We claim that there is no element e_{3}(A, σ) satisfying (2.1) and (2.2). Indeed, by (2.1), such an e_{3}(A, σ) would be zero. By (2.2), over the function field of the Severi-Brauer variety of A over k, the involution σ becomes hyperbolic. But this is impossible by Prop. A.1.

By adding hyperbolic planes, this example and the one from [BPQ] show that there exist (A, σ) ∈ I^{3}

• of index 8 and degree 8 + 16ℓ

• of index 4 and degree 12 + 8ℓ for all ℓ ≥ 0 such that no element e_{3}(A, σ) satisfies (2.1)–(2.2). Clearly, there are some difficulties for every degree congruent to 4, 8, or 12 mod 16.

As for the proof of Th. 2.4, the case of index ≤ 2 was treated in [Ber], as outlined above. In the remaining two cases, we define invariants e_{3}^{hyp} and e_{3}^{16} in §3 and §6 respectively that take values in H^{3}(k, ℤ/4ℤ)/[A] · H^{1}(k, μ_{2}). Clearly,

[A] · H^{1}(k, μ_{2}) is contained in E(A),^{a} and we define e_{3}(A, σ) to be the image of e_{3}^{hyp}(A, σ) or e_{3}^{16}(A, σ) in H^{3}(k, ℤ/4ℤ)/E(A). Property (2.2) is proved in Examples 3.5 and 6.3 below. The following corollaries are obvious:

Corollary 2.6. An algebra with involution (A, σ) as in Th. 2.4 belongs to I^{4} if and only if e_{3}(A, σ) is zero. □

Corollary 2.7. An algebra (A, σ) ∈ I^{3} of degree 16 is generically Pfister if and only if e_{3}(A, σ) is zero. □

3. Invariant e_{3}^{hyp}(A, σ)

Suppose that (A, σ) is in I^{3}, and 2 times the index of A divides the degree of A, i.e., there is a hyperbolic (orthogonal) involution “hyp” defined on A. We now define an element e_{3}^{hyp}(A, σ) ∈ H^{3}(k, ℤ/4ℤ)/[A] · H^{1}(k, μ_{2}) that agrees with the Arason invariant of (A, σ) in case A is split.

We may assume that A has degree ≥ 8; otherwise, σ is hyperbolic by Prop. 1.4 and we set e_{3}^{hyp}(A, σ) = 0. We assume further that 4 divides the degree of A; otherwise, A has index at most 2 and we set e_{3}^{hyp}(A, σ) to be the invariant defined by Berhuy. These two assumptions imply that Spin(A, hyp) is a simple algebraic group of type D_{ℓ} for ℓ even and ≥ 4.

Put Z for the center of Spin(A, hyp); it is isomorphic to μ_{2} × μ_{2}.

Lemma 3.1. The sequence

H^{1}(k, Z) ⟶ H^{1}(k, Spin(A, hyp)) ⟶^{q} H^{1}(k, Aut(Spin(A, hyp)))

is exact, and the fibers of q are the H^{1}(k, Z)-orbits in H^{1}(k, Spin(A, hyp)).

If one replaces Aut(Spin(A, hyp)) with its identity component, then the lemma is obviously true.

Sketch of proof of Lemma 3.1. Given a 1-cocycle with values in Spin(A, hyp), we write G for the group Spin(A, hyp) twisted by the 1-cocycle. The center of G is canonically identified with Z, and we want to show that the sequence

(3.2)  H^{1}(k, Z) ⟶ H^{1}(k, G) ⟶^{q} H^{1}(k, Aut(G))

is exact.

aThis inclusion is proper for some algebras A of index ≥ 8 by [Pey] and [Ka 98, 5.1].

Suppose that ĝ ∈ H^{1}(k, G) is killed by q, i.e., the twisted group G_{q(ĝ)} is isomorphic to G. By the exactness of the sequence

1 ⟶ Aut(G)^{∘} ⟶ Aut(G) ⟶ Aut(Δ) ⟶ 1

for Δ the Dynkin diagram of G [Sp, §16.3], the image γ of ĝ in H^{1}(k, Aut(G)^{∘}) is also the image of some π ∈ Aut(Δ)(k). The element π acts on Z, hence on H^{2}(k, Z), and since G_{q(ĝ)} is isomorphic to G,

(3.3)  The automorphism π fixes the Tits class of G in H^{2}(k, Z).

We now show that π is in the image of Aut(G)(k); we assume that π ≠ 1. We write G as Spin(A, τ) for some (A, τ) in I^{3}. The even Clifford algebra of (A, τ) is Brauer-equivalent to A × k. If ℓ ≠ 4, then (3.3) implies that A is isomorphic to M_{2ℓ}(k) [KMRT, p. 379], i.e., A is split, and a hyperplane reflection in Aut(G)(k) maps to π. If ℓ = 4, similar reasoning applies.

Since π is in the image of Aut(G)(k), the element γ ∈ H^{1}(k, Aut(G)^{∘}) is zero, hence ĝ comes from H^{1}(k, Z). This proves that (3.2) is exact. □

There is a class η′ ∈ H^{1}(k, Spin(A, hyp)) that maps to the class of Spin (A, σ) in H^{1}(k, Aut(G)), and Lemma 3.1 says that η′ is determined up to the action of H^{1}(k, Z).

We put:

(3.4)  e_{3}^{hyp}(A, σ) := r_{Spin(A,hyp)}(η′) ∈ \frac{H^{3}(k, ℤ/4ℤ)}{[A].H^{1}(k, μ_{2})} ,

where r denotes the Rost invariant. Note that e_{3}^{hyp}(A, σ) is well defined by the main result of [MPT].

Example 3.5. If A is split, then σ is adjoint to a quadratic form q_{σ} and e_{3}^{hyp}(A, σ) is the Arason invariant of q_{σ} by [KMRT, p. 432].

4. I^{3} and D_{2n}

Write Spin_{4n} for the split simply connected group of type D_{2n}. Its center is μ_{2} × μ_{2}. Up to isomorphism, Spin_{4n} has four quotients: itself, SO_{4n}, the adjoint group PSO_{4n}, and one other that we call a half-spin group^{b} and denote by HSpin_{4n}. We are interested in it because of the following result:

Lemma 4.1. The image of H^{1}(k, HSpin_{4n}) in H^{1}(k, PSO_{4n}) classifies pairs (A, σ) of degree 4n in I^{3}.

The algebras of (A, σ) in I^{3} with degree divisible by 4 are in some sense the most interesting ones. If the degree is not divisible by 4, then A is split by Lemma 1.5, and classifying such (A, σ) amounts to quadratic form theory.

Proof of Lemma 4.1. The proof can be summarized by writing: Combine pages 409 and 379 in [KMRT].

We identify the groups Spin_{4n} and PSO_{4n} with the corresponding groups for the split central simple algebra B of degree 4n with hyperbolic orthogonal involution τ. Fix a labeling C_{+} × C_{−} for the even Clifford algebra C_{0}(B, τ). Write π_{+} for the projection C_{0}(B, τ) → C_{+} and HSpin_{4n} for the image of Spin_{4n} in C_{+} under π_{+}.

bBourbaki writes “semi-spin” in [Bou].

Consider the following commutative diagram with exact rows:

(4.2)

1 ⟶ μ_{2} × μ_{2} ⟶ Spin_{4n} ⟶ PSO_{4n} ⟶ 1
↓^{π_{+}}  ↓^{π_{+}}  ∥
1 ⟶ μ_{2} ⟶ HSpin_{4n} ⟶ PSO_{4n} ⟶ 1

It induces a commutative diagram with exact rows:

(4.3)

H^{1}(k, Spin_{4n}) ⟶ H^{1}(k, PSO_{4n}) ⟶ H^{2}(k, μ_{2} × μ_{2})
↓^{π_{+}}  ∥  ↓^{π_{+}}
H^{1}(k, HSpin_{4n}) ⟶ H^{1}(k, PSO_{4n}) ⟶ H^{2}(k, μ_{2})

As in [KMRT, p. 409], the set H^{1}(k, PSO_{4n}) classifies triples (A, σ, ϕ) where A has degree 4n, the involution σ is orthogonal with trivial discriminant, and ϕ is a k-algebra isomorphism Z(C_{0}(A, σ)) ⟶^{∼} Z(C_{+} ×C_{−}). We view ϕ as a labeling of the components of the even Clifford algebra of (A, σ) as + and −. The image of such a triple (A, σ, ϕ) in H^{2}(k, μ_{2} × μ_{2}) is the Tits class of Spin(A, σ, ϕ), and it follows from [KMRT, p. 379] and the commutativity of (4.3) that the image of (A, σ, ϕ) in H^{2}(k, μ_{2}) is C_{+}(A, σ). So (A, σ, ϕ) is in the image of H^{1}(k, HSpin_{4n}) if and only if C_{+}(A, σ) is split.

We have proved that for every (A, σ) of degree 4n in I^{3}, there is some triple (A, σ, ϕ) in the image of H^{1}(k, HSpin_{4n}) → H^{1}(k, PSO_{4n}). Suppose now that (A, σ, ϕ) and (A, σ, ϕ′) are in the image of H^{1}(k, HSpin_{4n}); we will show they are equal; we may assume that ϕ ≠ ϕ′. If A is split, then a hyperplane reflection gives an isomorphism between (A, σ, ϕ) and (A, σ, ϕ′), and we are done. If A is nonsplit, then C_{−}(A, σ) (with numbering given by ϕ) is nonsplit and it is impossible that (A, σ, ϕ′) is in the image of H^{1}(k, HSpin_{4n}). This concludes the proof. □

5. HSpin_{16} ⊂ E_{8}

Write E_{8} for the split algebraic group of that type. We view it as generated by homomorphisms x_{ε} : 𝔾_{a} → E_{8} as ε varies over the root system E_{8} of that type, as in [St 68]. The root system D_{8} is contained in E_{8}, as can be seen from the completed Dynkin diagrams. Symbolically, we can see the inclusion as follows. Fix sets of simple roots δ_{1}, …, δ_{8} of D_{8} and ε_{1}, …, ε_{8} of E_{8}, numbered as in Figure 5A, where the unlabeled vertex denotes the negative −ε̃ of the highest root of E_{8}. The

[figure: δ_{1}; δ_{2}; δ_{3}; δ_{4}; δ_{5}; δ_{6}; δ_{7}; δ_{8}; ε_{1}; ε_{3}; ε_{4}; ε_{5}; ε_{6}; ε_{7}; ε_{8}; ε_{2}]

Figure 5A. Dynkin diagram of D_{8} and extended Dynkin diagram of E_{8}

inclusion of D_{8} in E_{8} is given by the following table:

(5.1)  D_{8} root δ_{1} δ_{2} δ_{3} δ_{4} δ_{5} δ_{6} δ_{7} δ_{8} ; E_{8} root −ε̃ ε_{8} ε_{7} ε_{6} ε_{5} ε_{4} ε_{2} ε_{3}

The subgroup of E_{8} generated by the x_{ε}’s for ε ∈ D_{8} is a subgroup of type D_{8}. This is standard, see e.g. [BdS]. Moreover, the subgroup of type D_{8} is HSpin_{16}. This can be seen by root system computations as in [Ti 90, §1.7] or with computations in the centers as in [GQ 08].

It follows from Table 5.1 (and is asserted in [D]), that:

(5.2)  The composition Spin_{16} → HSpin_{16} → E_{8} has Rost multiplier 1.

5.3. Let η ∈ H^{1}(k, HSpin_{16}) map to the class of (A, σ) in H^{1}(k, PSO_{16}), cf. Lemma 4.1. We write HSpin (A, σ) for the group HSpin_{16} twisted by η; it is the image of Spin (A, σ) in a split component of C_{0}(A, σ). We find homomorphisms

Spin (A, σ) → HSpin (A, σ) → (E_{8})_{η}.

The center of HSpin (A, σ) is a copy of μ_{2}, and we compute the image of an element c ∈ k^{×}/k^{×2} under the composition

k^{×}/k^{×2} = H^{1}(k, μ_{2}) → H^{1}(k, HSpin (A, σ)) → H^{1}(k, (E_{8})_{η}) ⟶^{r_{(E_{8})η}} H^{3}(k, ℚ/ℤ(2)).

The center of Spin (A, σ) is μ_{2} × μ_{2}, and it maps onto the center of HSpin (A, σ) via the map π_{+} from the proof of Lemma 4.1. The induced map H^{1}(k, μ_{2} × μ_{2}) → H^{1}(k, μ_{2}) is obviously surjective, so there is some γ ∈ H^{1}(k, μ_{2} × μ_{2}) such that π_{+}(γ) = (c). By (5.2), we have:

r_{(E_{8})_{η}}(c) = r_{Spin (A,σ)}(γ),

which is π_{+}(γ) · [A] by [MPT], i.e., (c) · [A].

6. Invariant e_{3}^{16}(A, σ) for algebras of degree 16 in I^{3}

Let (A, σ) be of degree 16 in I^{3}. Fix a class η ∈ H^{1}(k, HSpin_{16}) that maps to the class of (A, σ) in H^{1}(k, PSO_{16}). (Here we are using Lemma 4.1 to know that there is a uniquely determined element in H^{1}(k, PSO_{16}).) Consider the image r_{E_{8}}(η) of η under the map

H^{1}(k, HSpin_{16}) → H^{1}(k, E_{8}) ⟶^{r_{E_{8}}} H^{3}(k, ℚ/ℤ(2)).

Since (A, σ) is killed by an extension of k of degree a power of 2, the same is true for η, hence also for r_{E_{8}}(η). As r_{E_{8}}(η) is 60-torsion, we conclude that r_{E_{8}}(η) is 4-torsion, i.e., r_{E_{8}}(η) belongs to H^{3}(k, ℤ/4ℤ). We define

e_{3}^{16}(A, σ) := r_{E_{8}}(η) ∈ \frac{H^{3}(k, ℤ/4ℤ)}{[A].H^{1}(k, μ_{2})}

Theorem 6.1. The class e_{3}^{16}(A, σ) depends only on (A, σ) (and not on the choice of η).

Proof. Suppose that η, η′ ∈ H^{1}(k, HSpin_{16}) map to (A, σ) ∈ H^{1}(k, PSO_{16}). We consider the image τ(η′) of η′ in the twisted group H^{1}(k, (HSpin_{16})_{η}). Since τ(η′) maps to zero in H^{1}(k, (PSO_{16})_{η}), it is the image of some ζ ∈ H^{1}(k, μ_{2}), where μ_{2} denotes the center of (HSpin_{16})_{η}. In the diagram

(6.2)  \xymatrix{H^{1}(k, HSpin_{16}) \ar[r] \ar[d]_{≅}^{τ} & H^{1}(k, E_{8}) \ar[r]^{r_{E_{8}}} \ar[d]_{≅}^{τ} & H^{3}(k, ℚ/ℤ(2)) \ar[d]^{?−r_{E_{8}} (η)} \\ H^{1}(k, (HSpin_{16})_{η}) \ar[r] & H^{1}(k, (E_{8})_{η}) \ar[r]^{r_{(E_{8})η}} & H^{3}(k, ℚ/ℤ(2)),}

the left box obviously commutes and the right box commutes by [Gi, p. 76, Lemma 7]. Commutativity of the diagram and 5.3 give that

r_{E_{8}}(η′) = r_{E_{8}}(η) + ζ · [A],

as desired. □

Example 6.3. If A is split, then σ is adjoint to a quadratic form q_{σ}, and e_{3}^{16}(A, σ) equals the Arason invariant of q_{σ}. Indeed, if A is split, then there is a class γ ∈ H^{1}(k, Spin_{16}) that maps to η. Statement (5.2) gives:

r_{E_{8}}(η) = r_{Spin_{16}}(γ) = e_{3}(q_{σ}).

Part II. The invariant e_{3}^{16} on decomposable involutions

The purpose of this part is to compute e_{3}(A, σ) in case (A, σ) can be written as (Q,¯ ) ⊗ (C, γ) where (Q,¯ ) is a quaternion algebra endowed with its canonical involution and (C, γ) is a central simple algebra of degree 8 with symplectic involution. We do this by computing the value of the Rost invariant of E_{8} on a subgroup PGL_{2} × PSp_{8} ×μ_{2}; this finer computation will be used in §11.

7. An inclusion PGL_{2} × PSp_{8} ×μ_{2} ⊂ HSpin_{16}

7.1. Inclusions. We now describe concretely a homomorphism of groups PGL_{2} × PSp_{8} → HSpin_{16}. Write S_{n} for the n-by-n matrix whose only nonzero entries are 1s on the “second diagonal”, i.e., in the (j, n + 1 − j)-entries for various j. We identify Sp_{2n} with the symplectic group of M_{2n}(k) endowed with the involution γ_{2n} defined by

γ_{2n}(x) = Int ( \begin{matrix}0 & S_{n} \\ −S_{n} & 0 \end{matrix} )^{−1} x^{t}.

We identify Spin_{2n} with the spin group of M_{2n}(k) endowed with the involution σ_{2n} defined by

σ_{2n}(x) = Int(S_{2n}) x^{t}.

These are the realizations of the groups (stated on the level of Lie algebras) given in [Bou, §VIII.13].

We now define isomorphisms

(7.2)  (M_{2}(k), γ_{2}) ⊗ (M_{8}(k), γ_{8}) ⟶^{∼} (M_{16}(k), σ′_{16}) ⟶^{∼} (M_{16}(k), σ_{16}),

where

σ′_{16}(x) = Int ( \begin{matrix}0 & 0 & 0 & S_{4} \\ 0 & 0 & −S_{4} & 0 \\ 0 & −S_{4} & 0 & 0 \\ S_{4} & 0 & 0 & 0 \end{matrix} ) x^{t}.

We take the first isomorphism to be the usual Kronecker product defined on the standard basis elements by

E_{ij} ⊗ E_{qr} ↦ E_{8(i−1)+q,8(j−1)+r}.

The second isomorphism is conjugation by the matrix

( \begin{matrix}1_{4} & 0 & 0 & 0 \\ 0 & 0 & 1_{4} & 0 \\ 0 & −1_{4} & 0 & 0 \\ 0 & 0 & 0 & 1_{4} \end{matrix} )

where 1_{4} denotes the 4-by-4 identity matrix.

The homomorphism of groups induces a map on coroot lattices (= root lattices for the dual root systems) that describes the restriction of the group homomorphism to Cartan subalgebras on the level of Lie algebras. Using the concrete description of the group homomorphism above and the choice of Cartan, etc., from [Bou], we see that the map on coroots is given by Table 7B, where the simple roots of SL_{2}, Sp_{8}, Spin_{16}, and E_{8} are labelled α, γ, δ, and ε respectively and are numbered as in Figures 5A and 7A. For SL_{2}, Spin_{16}, and E_{8}, we fix the metric so that roots have

[figure: γ_{1}; γ_{2}; γ_{3}; γ_{4}; <]

Figure 7A. Dynkin diagram of C_{4}

length 2, which identifies the coroot and root lattices. The inclusion of HSpin_{16} in E_{8} is given by (5.1).

in D_{8}  in E_{8}
α_{1} ↦ δ_{1} + 2δ_{2} + 3δ_{3} + 4δ_{4} + 3δ_{5} + 2δ_{6} + δ_{7} ↦  −2ε_{1} − 2ε_{2} − 4ε_{3} − 4ε_{4} − 2ε_{5}
γ̌_{1} ↦  δ_{1} − δ_{7}  ↦ −2ε_{1} − 4ε_{2} − 4ε_{3} − 6ε_{4} − 5ε_{5} − 4ε_{6} − 3ε_{7} − 2ε_{8}
γ̌_{2} ↦  δ_{2} − δ_{6}  ↦  −ε_{4} + ε_{8}
γ̌_{3} ↦  δ_{3} − δ_{5}  ↦  −ε_{5} + ε_{7}
γ̌_{4} ↦  δ_{4} + 2δ_{5} + 2δ_{6} + δ_{7} + δ_{8}  ↦  ε_{2} + ε_{3} + 2ε_{4} + 2ε_{5} + ε_{6}

Table 7B. Homomorphisms SL_{2} ×Sp_{8} → Spin_{16} → E_{8} on the level of coroots

Either from the explicit tensor product in (7.2) or from the description of the center of Sp_{8} from [GQ 08, 8.5], we deduce inclusions:

(SL_{2} ×Sp_{8})/μ_{2} ⊂ Spin_{16} and PGL_{2} × PSp_{8} ⊂ HSpin_{16} .

Since the short coroot γ̌_{4} of Sp_{8} maps to a (co)root in D_{8}, the homomorphism Sp_{8} → Spin_{16} has Rost multiplier 1.

(The statements in the previous paragraph can also be deduced from the branching tables in [MP, p. 295], but of course those tables were constructed using data as in Table 7B. To get the statement on Rost multipliers, one uses [Mer 03, 7.9].)

We find a subgroup PGL_{2} × PSp_{8} ×μ_{2} of HSpin_{16} by taking the center of HSpin_{16} for the copy of μ_{2}.

7.3. The composition

H^{1}(k, PGL_{2} × PSp_{8} ×μ_{2}) → H^{1}(k, HSpin_{16}) → H^{1}(k, E_{8}) ⟶^{r_{E_{8}}} H^{3}(k, ℚ/ℤ(2))

defines an invariant of triples Q, (C, γ), c where Q is a quaternion algebra, (C, γ) is a central simple algebra of degree 8 with symplectic involution, and c is in k^{×}/k^{×2}. We abuse notation and write also r_{E_{8}} for this invariant.

For example, tracing through the proof of Th. 6.1, we find:

(7.4)  r_{E_{8}}(Q, (C, γ), c) = r_{E_{8}}(Q, (C, γ), 1) + (c) · [Q ⊗ C].

8. Crux computation

Lemma 8.1. The composition

H^{1}(k, PGL_{2}) × 1 ⊂ H^{1}(k, PGL_{2}) × H^{1}(k, PSp_{8} ×μ_{2}) → H^{1}(k, E_{8})

is identically zero.

We warm up by doing a toy version of a computation necessary for the proof of the lemma.

Example 8.2. Let O_{n} be the orthogonal group of the symmetric bilinear form f with matrix S_{n} as in 7.1. Fix a quadratic extension k(\sqrt{a})/k. For ι the nontrivial k-automorphism of k(\sqrt{a}) and c ∈ k^{×}, the 1-cocycle η ∈ Z^{1}(k(\sqrt{a})/k, O_{2}) defined by

η_{ι} = ( \begin{matrix}0 & c \\ c^{−1} & 0 \end{matrix} )

defines a bilinear form f_{η} over k. It is the restriction of f to the k-subspace of k(\sqrt{a})^{2} of elements fixed by η_{ι}ι. This subspace has basis

( \begin{matrix}c \\ 1 \end{matrix} ) and ( \begin{matrix}c\sqrt{a} \\ −\sqrt{a} \end{matrix} ) ,

so f_{η} is isomorphic to ⟨2c⟩⟨1, −a⟩.

Proof of Lemma 8.1. Fix a cocycle η ∈ Z^{1}(k, PGL_{2}). The rank 4 maximal torus in PSp_{8} (intersection with the torus in E_{8} specified by the pinning) is centralized by the image of η, so it gives a k-split torus S in the twisted group (E_{8})_{η}. A semisimple anisotropic kernel of (E_{8})_{η} is contained in the derived subgroup D of the centralizer of S. The root system of D (over an algebraic closure of k) consists of the roots of E_{8} orthogonal to the elements of the coroot lattice with image lying in S, which are given in Table 7B. The roots of D form a system of rank 4 with simple roots ϕ_{1}, ϕ_{2}, ϕ_{3}, ϕ_{4} as in Table 8A; they span a system of type D_{4} with Dynkin diagram

\begin{matrix}D_{4} & ϕ_{1} & ϕ_{2} & ϕ_{3} & ϕ_{4} \\ E_{8} & ε_{6} & ε_{1} + ε_{2} + 2ε_{3} + 2ε_{4} + ε_{5} & ε_{5} + ε_{6} + ε_{7} & ε_{4} + ε_{5} + ε_{6} + ε_{7} + ε_{8} \end{matrix}

Table 8A. Simple roots in the centralizer of the C_{4}-torus in E_{8}

as in Figure 8B, where the unlabeled vertex is the negative −ϕ̃ of the highest root ϕ_{1} + 2ϕ_{2} + ϕ_{3} + ϕ_{4} = ε̃.

[figure: ϕ_{1}; ϕ_{2}; ϕ_{3}; ϕ_{4}]

Figure 8B. Extended Dynkin diagram of centralizer of the C_{4}- torus in E_{8}

We now compute the map SL_{2} → D. On the level of tori, it is given by Table 7B. On the level of Lie algebras, we compute using the explicit map (7.2) that the element ( \begin{matrix}0 & 1 \\ 0 & 0 \end{matrix} ) of 𝔰𝔩_{2} maps to

( \begin{matrix}0 & 1 \\ 0 & 0 \end{matrix} ) ↦ ( \begin{matrix}0 & 0 & 1_{4} & 0 \\ 0 & 0 & 0 & 1_{4} \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix} ) ↦ ( \begin{matrix}0 & 1_{4} & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & −1_{4} \\ 0 & 0 & 0 & 0 \end{matrix} )

in M_{16}(k). In terms of the Chevalley basis of the Lie algebra of E_{8}, ( \begin{matrix}0 & 1 \\ 0 & 0 \end{matrix} ) maps to

X_{ϕ_{1}} + X_{ϕ_{3}} + X_{ϕ_{4}} + X_{−ϕ̃}.

The cocycle η represents a quaternion algebra (a, b) over k. If Q is split, then η is zero and we are done. Otherwise, a is not a square. Replacing η with an equivalent cocycle, we may assume that η belongs to Z^{1}(k(\sqrt{a})/k, PGL_{2}) and takes the value

η_{ι} = Int ( \begin{matrix}0 & b \\ 1 & 0 \end{matrix} ) = Int ( \begin{matrix}0 & −\sqrt{−b} \\ 1/\sqrt{−b} & 0 \end{matrix} )

on the non-identity k-automorphism ι of k(\sqrt{a}). That is, η_{ι} is conjugation by the element w_{α_{1}}(−\sqrt{−b}) in Steinberg’s notation for generators of a Chevalley group from [St 68]. The image of η in E_{8} is the cocycle η̂ with

(8.3)  η̂_{ι} := ∏_{ϕ} w_{ϕ}(−\sqrt{−b})

where ϕ ranges over the set Σ := {−ϕ̃, ϕ_{1}, ϕ_{3}, ϕ_{4}}. (The order of terms in the product does not matter, as the roots in Σ are pairwise orthogonal.) We compute the action of this element on each x_{ϕ}: 𝔾_{a} → D_{4} for ϕ ∈ Σ. Using the orthogonality of the roots in Σ, we have:

(8.4)  Int(η̂_{ι})x_{ϕ}(u) = Int(w_{ϕ}(−\sqrt{−b}))x_{ϕ}(u) = x_{ϕ}(u/b),

where the second equality is by the identities in [St 68, p. 66].

We now identify D (over an algebraic closure) with Spin_{8} using the pinning of Spin_{8} from [Bou] and project η̂ to a 1-cocycle with values in SO_{8}. This cocycle defines a quadratic form q that we claim is hyperbolic. Indeed, from equation (8.4), we deduce that the image of η̂ in SO_{8} is the matrix

( \begin{matrix}& & & & & & & −1 \\ & & & & & & 1/b \\ & & & & & −b \\ & & & & 1 \\ & & & 1 \\ & & −1/b \\ & b \\ −1 \end{matrix} )

This preserves the hyperbolic planes in k^{8} spanned by the 1st and 8th, 2nd and 7th, etc., standard basis vectors, so we can compute the quadratic form by restricting to each of these planes as in Example 8.2. One finds that q is isomorphic to

⟨2⟩ ⊗ ⟨−1, b^{−1}, −b, 1⟩ ⊗ ⟨1, −a⟩,

which is hyperbolic because the middle term is. In particular, the twisted group (SO_{8})η̂ is split, and the same is true for D. We conclude that (E_{8})_{η} is split and the image of η in H^{1}(k, E_{8}) is zero. □

Remark 8.5. In the case where k contains a square root of every element of the prime field F, one can given an easier proof of Lemma 8.1 as follows. Repeat the first paragraph. The composition

H^{1}(∗, PGL_{2}) → H^{1}(∗, E_{8}) ⟶^{r_{E_{8}}} H^{3}(∗, ℚ/ℤ(2))

gives a normalized invariant of PGL_{2}, which is necessarily of the form Q ↦ [Q]·x for some fixed x ∈ H^{1}(F, ℤ/2ℤ) by [Se 03, 18.1, §23]. Thus every element of H^{1}(k, E_{8}) coming from H^{1}(k, PGL_{2}) has Rost invariant zero (because x is killed by k) and is isotropic (obviously), hence is zero by Prop. 12.1(1) below.

9. Rost invariant

Theorem 9.1. The composition

H^{1}(k, PGL_{2}) × H^{1}(k, PSp_{8}) × H^{1}(k, μ_{2}) → H^{1}(k, E_{8}) ⟶^{r_{E_{8}}} H^{3}(k, ℚ/ℤ(2))

is given by

Q, (C, γ), c ↦ Δ(C, γ) + (c) · [Q ⊗ C] ∈ H^{3}(k, ℤ/2ℤ).

Here Δ refers to the discriminant of symplectic involutions on algebras of degree 8 defined in [GPT].

Proof. By (7.4), it suffices to prove the case where c = 1. \underline{Step 1.} We first verify the proposition in case C has index at most 2 and γ is hyperbolic; we write (C, γ) = (Q′ ⊗ M_{4}(k),¯ ⊗ hyp) for some quaternion algebra Q′. In this way, we restrict r_{E_{8}} to an invariant H^{1}(k, PGL_{2}) × H^{1}(k, PGL_{2}) → H^{3}(k, ℚ/ℤ(2)), which we claim is zero.

We argue as in [Se 03, §17]. We view H^{1}(k, PGL_{2}) as the image of H^{1}(k, ℤ/2ℤ × ℤ/2ℤ) via the map that sends elements a, b ∈ k^{×}/k^{×2} to the quaternion algebra (a, b). By restriction, r_{E_{8}} can be viewed as an invariant of (ℤ/2ℤ)^{×4}; its image consists of elements killed by a quadratic extension (by Lemma 8.1), so belong to the 2-torsion in H^{3}(k, ℤ/4ℤ), which is H^{3}(k, ℤ/2ℤ) by [MS]. Because the image of H^{1}(k, (ℤ/2ℤ)^{×4}) lies in H^{3}(k, ℤ/2ℤ) and the value of r_{E_{8}} on an element is unaltered if we interchange the first two coordinates (corresponding to the quaternion algebra Q) or the third and fourth coordinates (corresponding to Q′), we deduce that r_{E_{8}} is of the form

Q, Q′ ↦ λ_{0} + λ_{Q} · [Q] + λ_{Q′} · [Q′]

for uniquely determined elements λ_{0}, λ_{Q}, λ_{Q′} in H^{•}(k, ℤ/2ℤ). (There is no term involving [Q] · [Q′] because such a term would have degree at least 4.)

The element λ_{0} is zero, because the Rost invariant r_{E_{8}} is normalized. The coefficient λ_{Q} is zero by Lemma 8.1.

Write K for the field obtained by adjoining indeterminates a, b to k and Q^{gen} for the generic quaternion algebra (a, b) over K. On the one hand, the value of r_{E_{8}} on the triple

Q^{gen}, (Q^{gen} ⊗ M_{4}(k),¯ ⊗ hyp), 1 is λ_{Q′} · [Q^{gen}].

On the other hand, the algebra with involution

(A, σ) := (Q^{gen},¯ ) ⊗ (Q^{gen} ⊗ M_{4}(k),¯ ⊗ hyp)

has A split and σ hyperbolic, so e_{3}^{16}(A, σ) is zero as an element of H^{3}(k, ℤ/4ℤ). Thus λ_{Q′} = 0 and r_{E_{8}} sends Q, Q′ to 0. This verifies the proposition for C of index at most 2 and γ hyperbolic.

\underline{Step 2.} For a fixed quaternion algebra Q, the map

(C, γ) ↦ r_{E_{8}}(Q, (C, γ), 1) ∈ H^{3}(K, ℤ/2ℤ)

defines an invariant of H^{1}(k, PSp_{8}) that is zero on the trivial class by Step 1, hence by [GPT, 4.1] is of the form

(9.2)  (C, γ) ↦ λ_{1} · [C] + λ_{0} · Δ(C, γ)

for uniquely determined elements λ_{i} ∈ H^{i}(k, ℤ/2ℤ) for i = 0, 1 (which may depend on Q).

In case C has index 2 and γ is hyperbolic, (C, γ) maps to zero by Step 1 and Δ(C, γ) = 0, so λ_{1} = 0.

We are left with deciding whether the element λ_{0} is 0 or 1 in H^{0}(k, ℤ/2ℤ). Suppose that C has index 2 and put η for the image of Q, (C, hyp), 1 in H^{1}(k, E_{8}). The homomorphism

Sp(C, hyp) = (Sp_{8})_{η} → (E_{8})_{η}

has Rost multiplier 1 by 7.1, i.e., the induced map

H^{1}(k, Sp(C, hyp)) → H^{1}(k, (E_{8})_{η}) ⟶^{r_{E_{8}}} H^{3}(k, ℚ/ℤ(2))

is the Rost invariant of Sp(C, hyp). In particular it is not zero, so λ_{0} is not zero, i.e., λ_{0} = 1. This proves the proposition. □

10. Value of e_{3}^{16} on decomposable involutions

Let Q be a quaternion algebra and let (C, γ) be a central simple algebra of degree 8 with symplectic involution. The tensor product (Q,¯ ) ⊗ (C, γ) has degree 16 (clearly), trivial discriminant [KMRT, 7.3(5)], and one component of the even Clifford algebra is split [Tao, 4.15, 4.16], so the tensor product belongs to I^{3}.

Corollary 10.1 (of Th. 9.1). We have:

e_{3}^{16} [(Q,¯ ) ⊗ (C, γ)] = Δ(C, γ) ∈ H^{3}(k, ℤ/4ℤ)/[Q ⊗ C].H^{1}(k, μ_{2}). □

10.2. We compare the invariants e_{3}^{16} (from §6) and e_{3}^{hyp} (from §3). The invariant e_{3}^{16} is only defined on (A, σ) in I^{3} where A has degree 16. For such (A, σ), the invariant e_{3}^{hyp} is only defined if A is isomorphic to M_{2}(C) for a central simple algebra C of degree 8. It turns out that the two invariants agree if the algebra C is decomposable.

Corollary 10.3. If (A, σ) is in I^{3} and A is isomorphic to M_{2}(Q_{1} ⊗ Q_{2} ⊗ Q_{3}) for quaternion algebras Q_{1}, Q_{2}, Q_{3} (e.g., this holds if the index of A is ≤ 4), then e_{3}^{16}(A, σ) = e_{3}^{hyp}(A, σ).

Proof. The hypothesis on A implies that A supports a hyperbolic involution hyp:

(A, hyp) ≅ (M_{2}(k),¯ ) ⊗ (C, γ) for (C, γ) = ⊗_{i=1}^{3}(Q_{i},¯ ).

Let η, η′ ∈ H^{1}(k, HSpin_{16}) have image (A, hyp), (A, σ) in H^{1}(k, PSO_{16}) respectively. The bottom row of (6.2) can be rewritten as

H^{1}(k, HSpin(A, hyp)) → H^{1}(k, (E_{8})_{η}) ⟶^{r_{(E_{8})η}} H^{3}(k, ℚ/ℤ(2)).

For ν ∈ H^{1}(k, Spin(A, hyp)) mapping to τ(η′) ∈ H^{1}(k, HSpin(A, hyp)), we have:

r_{(E_{8})_{η}}(τ(η′)) = r_{Spin(A,hyp)}(ν) by (5.2).

The equality

e_{3}^{hyp}(A, σ) = e_{3}^{16}(A, σ) − e_{3}^{16}(A, hyp),

follows by commutativity of (6.2). Since (C, γ) is completely decomposable, Δ(C, γ) is zero and e_{3}^{16}(A, hyp) = 0 by Cor. 10.1, which proves the corollary. □

Part III. Groups of type E_{8} constructed from 9 parameters

11. Construction of E_{8}’s

In 7.1, we gave concrete descriptions of embeddings PGL_{2} × PSp_{8} ×μ_{2} ⊂ HSpin_{16} ⊂ E_{8}. Similarly, we can give an explicit embedding PGL_{2}^{×3} ⊂ PSp_{8} as in [D, Table 9]. On the level of coroot lattices the total inclusion

(11.1)  PGL_{2}^{×4} ×μ_{2} ⊂ PGL_{2} × PSp_{8} ×μ_{2} ⊂ HSpin_{16} ⊂ E_{8}

is described in Table 11A. We remark that the four copies of PGL_{2} are not normalized by a maximal torus of E_{8}.

simple (co)root in copy of PGL_{2}

in C_{4} in E_{8}
α_{1}  −(2ε_{1} + 2ε_{2} + 4ε_{3} + 4ε_{4} + 2ε_{5})
α_{2}  γ̌_{1} − γ̌_{3}  −(2ε_{1} + 4ε_{2} + 4ε_{3} + 6ε_{4} + 4ε_{5} + 4ε_{6} + 4ε_{7} + 2ε_{8})
α_{3}  γ̌_{1} + γ̌_{3}  −(2ε_{1} + 4ε_{2} + 4ε_{3} + 6ε_{4} + 6ε_{5} + 4ε_{6} + 2ε_{7} + 2ε_{8})
α_{4}  γ̌_{1} + 2γ̌_{2} + γ̌_{3}  −(2ε_{1} + 4ε_{2} + 4ε_{3} + 8ε_{4} + 6ε_{5} + 4ε_{6} + 2ε_{7})

Table 11A. Inclusion PGL_{2}^{×4} ⊂ PGL_{2} × PSp_{8} ⊂ E_{8} on the level of coroots

α_{i} is a simple (co)root in the i-th copy of PGL_{2}. That copy is inside PSp_{8} for i ≠ 1.

Applying Galois cohomology to (11.1) gives a function

(11.2)  H^{1}(k, PGL_{2}^{×4} ×μ_{2}) → H^{1}(k, E_{8}).

The first set classifies quadruples (Q_{1}, Q_{2}, Q_{3}, Q_{4}) of quaternion k-algebras together with an element c ∈ k^{×}/k^{×2}, and the second set classifies groups of type E_{8} over k. Therefore we may view the function (11.2) as a construction of groups of type E_{8} via Galois descent.

Corollary 11.3 (of Th. 9.1). The Rost invariant of a group of type E_{8} constructed from (Q_{1}, Q_{2}, Q_{3}, Q_{4}, c) is (c). ∑[Q_{i}].

Proof. As (C, γ) is the tensor product ⊗_{i=2}^{4}(Q_{i},¯ ), it is decomposable, and so has discriminant zero [GPT]. Th. 9.1 gives the claim. □

11.4. How much can we vary the data (Q_{1}, Q_{2}, Q_{3}, Q_{4}, c) without changing the resulting group of type E_{8}? For example, let Q′_{2}, Q′_{3}, Q′_{4} be quaternion algebras such that the tensor products ⊗_{i=2}^{4}(Q′_{i},¯ ) and ⊗_{i=2}^{4}(Q_{i},¯ ) are isomorphic as algebras with involution. Then the images of (Q_{1}, Q_{2}, Q_{3}, Q_{4}, c) and (Q_{1}, Q′_{2}, Q′_{3}, Q′_{4}, c) in H^{1}(k, PGL_{2}) × H^{1}(k, PSp_{8}) × H^{1}(k, μ_{2}) agree, hence one obtains the same group of type E_{8} from the two inputs.

We also have:

Proposition 11.5. For every permutation π, the group of type E_{8} constructed from (Q_{π1}, Q_{π2}, Q_{π3}, Q_{π4}, c) is the same.

Proof. We compare the images η and η_{π} of (Q_{1}, Q_{2}, Q_{3}, Q_{4}, c) and (Q_{π1}, Q_{π2}, Q_{π3}, Q_{π4}, c) respectively in H^{1}(k, HSpin_{16}). As both η and η_{π} map to the class of ⊗(Q_{i},¯ ) in H^{1}(k, PSO_{16}), the class of η_{π} is ζ_{π} · η for some ζ_{π} ∈ H^{1}(k, μ_{2}). The element ζ_{π} is uniquely determined as an element of the abelian group Γ := H^{1}(k, μ_{2})/ im(PSO_{16})_{η}(k), where (PSO_{16})_{η}(k) maps into H^{1}(k, μ_{2}) via the connecting homomorphism arising from the exact sequence at the bottom of diagram (4.2), see [Se 02, §I.5.5, Cor. 2]. This defines a homomorphism ζ from the symmetric group on 4 letters, S_{4}, to Γ.

As Γ is abelian, the homomorphism ζ factors through the commutator subgroup of S_{4}, the alternating group. But ζ vanishes on the odd permutation (3 4) by 11.4, so ζ is the zero homomorphism. This proves the proposition. □

11.6. Tits’s construction. In [Ti 66a], Tits gave a construction of algebraic groups of type E_{8} with inputs an octonion algebra and an Albert algebra. In terms of algebraic groups, there is an (essentially unique) inclusion of G_{2} × F_{4} in E_{8} [D, p. 226], and Tits’s construction is the resulting map in Galois cohomology:

H^{1}(k, G_{2}) × H^{1}(k, F_{4}) → H^{1}(k, E_{8}).

His construction and ours from (11.2) overlap, but they are distinct.

We compute the Rost invariant of a group G of type E_{8} constructed by Tits’s recipe from an octonion algebra with 3-Pfister norm form γ_{3} and an Albert algebra A. Because the inclusions G_{2} ⊂ E_{8} and F_{4} ⊂ E_{8} both have Rost multiplier 1 [D, p. 192], we have:

r_{E_{8}}(G) = e_{3}(γ_{3}) + r_{F_{4}}(A),

where r_{F_{4}} denotes the Rost invariant relative to the split group of type F_{4}. Associated with A are Pfister forms ϕ_{3} and ϕ_{5}, where ϕ_{i} has dimension 2^{i} and ϕ_{3} divides ϕ_{5}, see [Se 03, 22.5]. We find:

15r_{E_{8}}(G) = e_{3}(γ_{3} + ϕ_{3}) ∈ H^{3}(k, ℤ/2ℤ).

12. Tits index of groups of type E_{8}

In this section, we note some relationships between the Tits index of a group G of type E_{8} over k and its Rost invariant r_{E_{8}}(G).

Recall that if r_{E_{8}}(G) is killed by a quadratic extension or is 2-torsion, then it belongs to H^{3}(k, ℤ/2ℤ). A symbol is an element of the image of the cup product map H^{1}(k, ℤ/2ℤ)^{×3} → H^{3}(k, ℤ/2ℤ). The symbol length of an element x ∈ H^{3}(k, ℤ/2ℤ) is the smallest integer n such that x is equal to a sum of n symbols in H^{3}(k, ℤ/2ℤ). Zero is the unique element with symbol length 0.

Proposition 12.1. Let G be an isotropic group of type E_{8}. Then:

(1) If r_{E_{8}}(G) is zero, then G is split.

(2) If r_{E_{8}}(G) is split by a quadratic extension of k, then r_{E_{8}}(G) has symbol length ≤ 3 in H^{3}(k, ℤ/2ℤ).

(3) If r_{E_{8}}(G) is 2-torsion and G has k-rank ≥ 2, then the Tits index is given by Table 12A.

12.2. Before proving the proposition, we give some context for it. We consider a group G constructed via (11.2) from quaternion algebras Q_{1}, Q_{2}, Q_{3}, Q_{4} and some c ∈ k^{×}/k^{×2}. If at least one of the Q_{i} is split, then G contains a subgroup isomorphic to PGL_{2} and so is isotropic. If at least two of the Q_{i} are split or a tensor product of some three of them is split, then G contains a subgroup isomorphic to PGL_{2} × PGL_{2} or PSp_{8} respectively, and so has k-rank ≥ 2. In any case, the Rost invariant of G is zero over k(\sqrt{c}), which is either k or a quadratic extension of k.

index  symbol length of r_{E_{8}}(G)
split  0

[figure: 1]

[figure: 2]

Table 12A. Tits index versus symbol length for isotropic groups of type E_{8} such that r_{E_{8}}(G) is 2-torsion

In particular, if one wishes to use (11.2) to construct groups of type E_{8} that are non-split but in the kernel of the Rost invariant, then none of the Q_{i} can be split, nor can any tensor product of three of them.

Example 12.3. If the quaternion algebras Q_{2}, Q_{3}, Q_{4} are split and (c) · [Q_{1}] ≠ 0. Then (c) · [Q_{1}] is a symbol in H^{3}(k, ℤ/2ℤ) corresponding to a 3-Pfister form q. By Proposition 12.1(3), G has semisimple anisotropic kernel Spin(q).

Example 12.4. For “generic” c and quaternion algebras Q_{i}, construction (11.2) gives a group G of type E_{8} whose Rost invariant is killed by a quadratic extension of k and has symbol length 4. The group G is anisotropic by Prop. 12.1.

We prepare the proof of Prop. 12.1 with lemmas on groups of type D_{6} and E_{7}.

Lemma 12.5. The Witt index of a 12-dimensional quadratic form q ∈ I^{3} is given by the table:

Witt index of q  0 2 6
symbol length of e_{3}(q) in H^{3}(k, ℤ/2ℤ) 2 1 0

Proof. The Witt index of q cannot be 1 because 10-dimensional forms in I^{3} are isotropic. Similarly, it cannot be 3, 4, or 5 by the Arason-Pfister Hauptsatz. This shows that 0, 2, and 6 are the only possibilities.

The form is hyperbolic if and only if it belongs to I^{4}, i.e., e_{3}(q) is zero; this proves the last column of the table. If the Witt index is 2, then q = ⟨c⟩γ ⊕ 2H for some c ∈ k^{×} and anisotropic 3-Pfister form γ, so e_{3}(q) is a symbol. Finally, suppose that e_{3}(q) has symbol length 1, i.e., q − γ is in I^{4} for some anisotropic 3-Pfister form γ. Over the function field K of γ, the form q is hyperbolic by Arason-Pfister, so q = ⟨c⟩γ ⊕ 2H for some c ∈ k^{×} by [Lam, X.4.11]. □

In the next lemma, we write E_{7} for the split simply connected group of that type, and E_{6}^{K} for the quasi-split simply connected group of type E_{6} associated with a quadratic étale k-algebra K.

Lemma 12.6. There is an inclusion of E_{6}^{K} in E_{7} with Rost multiplier 1 such that the induced map H^{1}(K/k, E_{6}^{K}) → H^{1}(K/k, E_{7}) is surjective.

A class η ∈ H^{1}(k, E_{7}) is split by K if and only if K kills the Rost invariant r_{E_{7}}(η) by [Ga 01b]. It follows from [Ga 01b, 3.6] that there is some quadratic étale k-algebra L such that η is in the image of H^{1}(K/k, E_{6}^{L}) → H^{1}(K/k, E_{7}). The point of the lemma is to arrange that L = K.

Proof of Lemma 12.6. We view E_{7} as the identity component of the group preserving a quartic form on the 56-dimensional vector space ( \begin{matrix}k & J \\ J & k \end{matrix} ) for J the split Albert algebra, cf. [Ga 01c]. Write S for the subgroup of E_{7} that stabilizes the subspaces ( \begin{matrix}k & 0 \\ 0 & 0 \end{matrix} ) and ( \begin{matrix}0 & 0 \\ 0 & k \end{matrix}); it is reductive and its derived subgroup is the split simply connected group of type E_{6}.

As in [Ga 01b, 3.5], for i a square root of −1, the map

( \begin{matrix}α & x \\ y & β \end{matrix}) ↦ ( \begin{matrix}iβ & iy \\ ix & iα \end{matrix} )

is a k(i)-point of E_{7}, and this gives an inclusion μ_{4} ↪ E_{7}. Twisting E_{7} by a 1cocycle ν ∈ Z^{1}(k, μ_{4}) that maps to the class of K in H^{1}(k, μ_{2}), we find inclusions

E_{6}^{K} = (E_{6})_{ν} ⊂ S_{ν} ⊂ (E_{7})_{ν} ≅ E_{7}.

Write ι for the nontrivial k-automorphism of K. The group S_{ν} is the intersection P ∩ ι(P) for a maximal parabolic K-subgroup P of (E_{7})_{ν}, hence the map H^{1}(K/k, S_{ν}) → H^{1}(K/k, E_{7}) is surjective [PR, pp. 369, 383].

We have an exact sequence

1 ⟶ E_{6}^{K} ⟶ S_{ν} ⟶^{π} R_{K/k}^{1}(𝔾_{m}) ⟶ 1

where π is the map that sends the endomorphism

( \begin{matrix}α & x \\ y & β \end{matrix}) ↦ ( \begin{matrix}μα & f(x) \\ f^{†}(y) & μ^{−1}β \end{matrix} )

to μ. Formula [Ga 01c, 1.6] gives an explicit splitting s of π defined over k such the image of s is contained in the parabolic subgroup Q of (E_{6})_{ν} stabilizing the subspace W from [Ga 01c, 6.8].

Fix an element η′ ∈ H^{1}(K/k, S_{ν}) that maps to η ∈ H^{1}(K/k, E_{7}). We twist S_{ν} by sπ(η′). The image of η′′ of η′ in H^{1}(K/k, (S_{ν})_{sπ(η′)}) under the twisting map takes values in the semisimple part D := ((E_{6})_{ν})_{sπ(η′)}. But D contains the k-parabolic Q, hence D is quasi-split or has semisimple anisotropic kernel a transfer R_{K/k}(H) where H is anisotropic of type ^{1}A_{2}. But this is impossible because D is split by a quartic extension of k, so D is the quasi-split group E_{6}^{K}. □

Proof of Prop. 12.1. Statement (1) is standard, so we only sketch the proof. The semisimple anisotropic kernel of an isotropic but non-split group of type E_{8} is a simply connected group of type E_{7}, D_{7}, E_{6}, D_{5}, or D_{4} by [Ti 66b, p. 60], but the Rost invariant has zero kernel for a split group of that type [Ga 01b]. Statement (1) now follows by Tits’s Witt-type theorem [Ti 66b, 2.7.2(d)].

For (3), we may assume that G is not split, equivalently that r_{E_{8}}(G) has positive symbol length. Because the k-rank of G is at least 2, Tits’s table in [Ti 66b, p. 60] shows that the semisimple anisotropic kernel of G is a strongly inner simply connected group of type E_{6}, D_{6}, or D_{4}. The first case is impossible because r_{E_{8}}(G) is 2-torsion. Statement (3) now follows from Tits’s Witt-type theorem and Lemma 12.5.

To prove (2), by (3) we may assume that G has k-rank 1, hence that the semisimple anisotropic kernel of G is a strongly inner simply connected group of type D_{7} or E_{7}. In the first case, r_{E_{8}}(G) is the Arason invariant of a 14-dimensional form in I^{3}, hence has symbol length ≤ 3 by [HT, Prop. 2.3]. For the second case, by Lemma 12.6 it suffices to prove that the Rost invariant of every element of H^{1}(K/k, E_{6}^{K}) has symbol length at most 3, which is [C, p. 321]. □

Presumably the methods of [C] can be used to give an alternative proof of Prop. 12.1(2) that avoids Lemma 12.6.

13. Reduced Killing form up to Witt-equivalence

Recall that the reduced Killing form of G — which we denote by redKill_{G} — is equal to the usual Killing form divided by twice the dual Coxeter number [GN, §5]. For a group G of type E_{8}, all the roots of G have the same length and the dual Coxeter number equals the (usual) Coxeter number, which is 30. Hence the usual Killing form Kill_{G} satisfies Kill_{G} = 60 redKill_{G} and Kill_{G} is zero in characteristics 2, 3, 5.

We identify the bilinear form redKill_{G} with the quadratic form (and element of the Witt ring) x ↦ redKill_{G}(x, x).

Example 13.1 (the split group). The reduced Killing form for the split group E_{8} of that type is Witt-equivalent to the 8-dimensional form ⟨1, 1, …, 1⟩, which we denote simply by 8. To see this, note that the positive root subalgebras span a totally isotropic subspace parallel to the isotropic subspace spanned by the negative root subalgebras, so redKill_{E_{8}} is Witt-equivalent to its restriction to the Lie algebra of a split maximal torus. By [GN], this restriction is isomorphic to the quadratic form x ↦ xCx^{t} for C the Cartan matrix of the root system, and it is easy to check that this quadratic form is 8.

Example 13.2 (Tits’s groups). For a group G of type E_{8} obtained from Tits’s construction as in 11.6, we have

redKill_{G} = ⟨2⟩ [8 − (4γ_{3} + 4ϕ_{3} + ⟨2⟩γ_{3}(ϕ_{5} − ϕ_{3}))]

by [J, p. 117, (144)], where the Killing forms for the subalgebras of type F_{4} and G_{2} are given in [Se 03, 27.20].

13.3. The map G ↦ ⟨2⟩(redKill_{E_{8}} −redKill_{G}) defines a Witt-invariant of H^{1}(∗, E_{8}) in the sense of [Se 03, §27], i.e., a collection of maps

κ: groups of type E_{8} over K → W (K)

for every extension K/k (together with some compatibility condition), where W (K) denotes the Witt ring of K.

Example 13.4 (groups over ℝ). For each of the three groups of type E_{8} over the real numbers, we list the Rost invariant, the (signature of the) Killing form, and the value of κ. All three groups are obtained by Tits’s construction [J, p, 121], so the Killing form and Rost invariant are provided by the formulas in 13.2 and 11.6. The Rost invariant r_{E_{8}} takes values in H^{3}(ℝ, ℚ/ℤ(2)) = H^{3}(ℝ, ℤ/2ℤ) = ℤ/2ℤ.

G  r_{E_{8}}(G) Kill_{G} κ(G)
split  0  8  0
other  1  −24 32 ∈ I^{5}
anisotropic/compact  0  −248 256 ∈ I^{8}

The rest of this section is concerned with the question over the field k: In what power of I does κ(G) lie?

Lemma 13.5. For every group G of type E_{8}, κ(G) belongs to I^{3}.

Proof. Because E_{8} is simply connected, the adjoint representation E_{8} → O(redKill_{E_{8}}) lifts to a homomorphism E_{8} → Spin(redKill_{E_{8}}). For G a group of type E_{8} over an extension K/k, the reduced Killing form of G is the image of G under the composition

H^{1}(K, E_{8}) → H^{1}(K, Spin(redKill_{E_{8}})) → H^{1}(K, O(redKill_{E_{8}})),

so it belongs to I^{3}K, see e.g. [KMRT, p. 437]. □

Lemma 13.6. For every group G of type E_{8}, we have 30r_{E_{8}}(G) = e_{3}(κ(G)). The form κ(G) is in I^{4} if and only if 30r_{E_{8}}(G) = 0.

Proof. We have a commutative diagram, where 𝔢_{8} denotes the Lie algebra of E_{8} and arrows are labeled with Rost multipliers:

\xymatrix{E_{8} \ar[r] \ar[dr]^{60} & Spin(𝔢_{8}) \ar[d]^{2} \\  & SL(𝔢_{8})}

Therefore, the Rost multiplier of the top arrow is 60/2 = 30 [Mer 03, p. 122]. The first claim follows. The second claim amounts to the observation that the kernel of e_{3} is I^{4}. □

Because the Rost invariant r_{E_{8}} has order 60 [Mer 03, 16.8], the value of r_{E_{8}} on a versal E_{8}-torsor G has order 60 [Se 03, 12.3]. In particular, κ(G) does not belong to I^{4}, so Lemma 13.5 cannot be directly strengthened.

We return to the question “In what power of I does κ(G) lie?” in §16 below.

14. Calculation of the Killing form

We now compute the Killing form of a group G of type E_{8} constructed via (11.2).

Theorem 14.1 (char k ≠ 2, 3). Let G be a group of type E_{8} constructed from quaternion algebras Q_{1}, Q_{2}, Q_{3}, Q_{4} and c ∈ k^{×}/k^{×2} as in (11.2). Then the reduced Killing form of G is isomorphic to

8⟨2c⟩ − ⟨2⟩⟨1, −c⟩ (∑_{i} Q′_{i} + ∑_{i<j<ℓ} Q′_{i}Q′_{j}Q′_{ℓ}) .

Here we have written Q′_{i} for the unique 3-dimensional quadratic form such that ⟨1⟩ ⊕ Q′_{i} is the norm on Q_{i}.

We prove the theorem by restricting the adjoint representation 𝔤 of G to the subgroup HSpin (A, σ) for (A, σ) = ⊗(Q_{i},¯ ). We can compute the restriction of G to HSpin (A, σ) over an algebraic closure, where we find that 𝔤 is a direct sum of the adjoint representation of HSpin_{16} and the natural half-spin representation [MP, p. 305]. Both representations of HSpin_{16} are irreducible: the half-spin representation because it is minuscule; for the adjoint representation see [St 61, 2.6]. As the reduced Killing form redKill_{G} is invariant under HSpin_{16}, it follows that these two irreps of HSpin_{16} in 𝔤 are orthogonal relative to redKill_{G}. We compute the restriction of the reduced Killing form to each summand separately.

(We use the reduced Killing form instead of the usual one, in order to get a nontrivial result in characteristic 5. The statement of the theorem makes sense in characteristic 3, but our proof does not work in that case.)

14.2. Killing form of HSpin (A, σ). The Lie algebra of HSpin (A, σ) can be identified with the space of σ-skew-symmetric elements in A. Because (A, σ) is a tensor product of quaternion algebras, this space can be described inductively as a tensor product of the subspaces of the Q_{i} that are symmetric or skew-symmetric under ¯ . (It is clear that such tensor products can be formed that belong to the skew elements in A, and dimension count shows that all skew elements of A are obtained in this way.) The trace quadratic form q ↦ tr_{Q_{i}}(q^{2}) restricts to be ⟨2⟩ on the ¯ -symmetric elements and ⟨−2⟩Q′_{i} on the ¯ -skew-symmetric elements. We conclude that the form a ↦ tr_{A}(a^{2}) on A restricts to

(14.3)  ⟨−1⟩ (∑_{i} Q′_{i} + ∑_{i<j<ℓ} Q′_{i}Q′_{j}Q′_{ℓ})

on the σ-skew-symmetric elements.

The form (14.3) is invariant under HSpin (A, σ), so it is a scalar multiple of the Killing form. By [Bou, chap. VIII, §13, Exercise 12], the Killing form of HSpin (A, σ) is ⟨h_{D_{8}}⟩ times the form (14.3), where h_{D_{8}} denotes the Coxeter number, which is 14.

Note that for X_{α}, X_{−α} belonging to our fixed pinning of E_{8} and spanning the highest and lowest root subalgebras of HSpin_{16}, we have Kill_{HSpin_{16}}(X_{α}, X_{−α}) = 2h_{D_{8}}, but Kill_{E_{8}}(X_{α}, X_{−α}) = 2h_{E_{8}} by [SS, pp. E-14, E-15], where h_{E_{8}} is the Coxeter number of E_{8}, i.e., 30. We conclude that the restriction of the Killing form of E_{8} to the adjoint representation of HSpin (A, σ) is ⟨h_{E_{8}}⟩ times the form (14.3), i.e., the reduced Killing form of G restricts to be ⟨2⟩ times (14.3) on the Lie algebra of HSpin_{16}.

14.4. Half-spin representation. We restrict the half-spin representation of HSpin (A, σ) to the product of the PGL(Q_{i})’s. Putting ω_{i} for the unique fundamental dominant weight of PGL(Q_{i}), the half-spin representation decomposes as a direct sum of the four 5-dimensional representations with highest weight 4ω_{i} and the four 27-dimensional representations with highest weight 2ω_{i} + 2ω_{j} + 2ω_{ℓ} with i < j < ℓ.

Now the representations of PGL(Q_{i}) with highest weights 2ω_{i} and 4ω_{i} support PGL(Q_{i})-invariant quadratic forms isomorphic to

⟨2⟩Q′_{i} and ⟨2⟩Q_{i} + ⟨6⟩

respectively by [Ga 07]. (The cited result concerns Weyl modules, but the modules with these highest weights are irreducible in characteristic ≠ 2, 3.) Because the Q_{i} are interchangeable (Prop. 11.5), the half-spin representation contributes

(14.5)  ⟨2⟩⟨c m_{2}⟩ ∑_{i<j<ℓ} Q′_{i}Q′_{j}Q′_{ℓ} ⊕ ⟨c m_{4}⟩ ∑_{i}(⟨2⟩Q_{i} + ⟨6⟩)

to the reduced Killing form of G, where m_{2}, m_{4} are elements of k^{×} that are not yet determined.

We first compute m_{4} up to sign, for which it suffices to consider the case c = 1. As the Q_{i}’s are interchangeable, we focus on Q_{1}. The only root of E_{8} that restricts to 4ω_{1} is −(ε_{1} + ε_{2} + 2ε_{3} + 2ε_{4} + ε_{5}). We write X_{1} for the element of the Chevalley basis of E_{8} corresponding to that root and X_{−1} for the element corresponding to the negation of the root; X_{1} and X_{−1} are highest and lowest weight vectors respectively for the irrep of PGL_{2}^{×4} with highest weight 4ω_{1}. We have

( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ) X_{1} = ±X_{−1}

because all the roots in E_{8} have the same length. Therefore,

redKill_{E_{8}}(X_{1}, ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ) X_{1}) = ±redKill_{E_{8}}(X_{1}, X_{−1}) = ±1.

On the other hand, for f the symmetric bilinear form on the representation of PGL(Q_{1}) with highest weight 4ω_{1} such that f is isomorphic to ⟨2⟩Q_{1} + ⟨6⟩, we have

f(X_{1}, ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ) X_{1}) = 1,

see [Ga 07, 2.4]. Consequently, m_{4} = ±1. Similarly, m_{2} is also 1 or −1, possibly with a different sign from m_{4}.

14.6. Combining the results of 14.2 and 14.4, we find that the reduced Killing form of G is

(14.7)  4⟨2c m_{4}⟩⟨1, 3⟩ + ⟨2⟩⟨−1, c m_{4}⟩ ∑ Q′_{i} + ⟨2⟩⟨−1, c m_{2}⟩ ∑ Q′_{i}Q′_{j}Q′_{ℓ}.

As m_{2} and m_{4} are ±1 and are defined over ℤ, to compute their values in k^{×}/k^{×2} it suffices to compute their values in ℝ^{×}/ℝ^{×2}. Consider the case k = ℝ and c = 1 and where exactly one of the Q_{i} is nonsplit. Then

∑ Q′_{i} = 0 and ∑ Q′_{i}Q′_{j}Q′_{ℓ} = 8.

Plugging this data into (14.7), we find that the reduced Killing form is 8⟨m_{2}, m_{4}⟩−8. On the other hand, the group is isotropic and has Rost invariant zero, so it is split and the reduced Killing form is 8. It follows that ⟨m_{2}, m_{4}⟩ is 2, i.e., m_{2} = m_{4} = 1.

Returning to the general case, 4⟨3⟩ is isomorphic to 4⟨1⟩. This completes the proof of Th. 14.1. □.

Example 14.8. As a corollary of the proof, we can see that in case q is a 4-Pfister quadratic form, there is a Spin(q)-invariant quadratic form on each half-spin representation that is isomorphic to 8q. Indeed, we can write q as a product q_{1}q_{2} of 2-Pfister forms, and let Q_{i} be the quaternion algebra with norm q_{i}. The tensor product (Q_{i},¯ ) ⊗ (Q_{i},¯ ) is M_{4}(k) endowed with an orthogonal involution adjoint to q_{i}. Setting Q_{3} = Q_{1} and Q_{4} = Q_{2} and evaluating (14.5) with c = μ_{2} = μ_{4} = 1 gives the claim.

15. The Killing form and E_{8}’s arising from (11.2)

This section gives some properties of the groups of type E_{8} constructed via (11.2), proved using the calculation of the Killing form in the preceding section.

Example 15.1. The classification of groups of type E_{8} over the real numbers was recalled in Example 13.4. Such groups are distinguished (up to isomorphism) by their Killing forms. We now observe that construction (11.2) produces all three groups. There are only two possibilities (split or not) for each of the four quaternion algebras as well as for c.

c # of nonsplit quaternion algebras signature of Killing form

description of group

1 0 through 4  8  split
−1 0 or 2  8  split
−1 1 or 3  −24  isotropic
−1 4  −248  anisotropic/compact

Chernousov’s Hasse Principle [PR, p. 286, Th. 6.6] for E_{8} implies that construction 11.2 produces every group of type E_{8} over a number field.

Recall the definition of κ(G) from 13.3; it measures the difference of the reduced Killing form of G from the same form for the split E_{8}.

Proposition 15.2 (char k ≠ 2, 3). For a group G of type E_{8} constructed from quaternion algebras Q_{1}, Q_{2}, Q_{3}, Q_{4} and c ∈ k^{×}/k^{×2}, we have

κ(G) = ⟨⟨c⟩⟩ [4 ∑_{i} Q_{i} − 2 ∑_{i<j} Q_{i}Q_{j} + ∑_{i<j<ℓ} Q_{i}Q_{j}Q_{ℓ}] ∈ I^{5}.

We use the notation ⟨⟨x_{1}, …, x_{n}⟩⟩ for the tensor product ⟨1, −x_{1}⟩⊗⋯⊗⟨1, −x_{n}⟩.

Proof of Prop. 15.2. Example 13.1 and Th. 14.1 give

κ(G) = ⟨⟨c⟩⟩ (8 + ∑_{i} Q′_{i} + ∑_{i<j<ℓ} Q′_{i}Q′_{j}Q′_{ℓ}) .

Replacing each Q′_{i} with Q_{i} − 1 and expanding gives the displayed formula. The form is obviously in I^{5}. □

15.3. It is easy to see that I^{5} in Prop. 15.2 “cannot be improved”, i.e., that κ(G) need not lie in I^{6}. One can take k to be ℝ with indeterminates x, y, c adjoined, and put Q_{1} := (x, y) and Q_{2}, Q_{3}, Q_{4} split. The resulting G has κ(G) = 4⟨⟨c, x, y⟩⟩.

Similarly, if −1 is a square in k, then 2 = 0 in the Witt ring and κ(G) = ⟨⟨c⟩⟩ ∑ Q_{i}Q_{j}Q_{ℓ} belongs to I^{7}. Again, this cannot be improved, as can be seen by taking Q_{1}, Q_{2}, Q_{3} to be “generic” quaternion algebras, and Q_{4} to be split.

However, the Killing forms of the groups G constructed from (11.2) are mainly of interest in case the Rost invariant r(G) is zero. That case is treated by the following theorem provided by Detlev Hoffmann.

Theorem 15.4 (Hoffmann). Let G be a group of type E_{8} constructed via (11.2) from quaternion algebras Q_{1}, Q_{2}, Q_{3}, Q_{4} and c ∈ k^{×}. If r(G) = 0, then

κ(G) = 2⟨⟨c⟩⟩Q_{1}Q_{2}Q_{4} ∈ I^{8}.

Proof. The hypothesis implies that ⟨⟨c⟩⟩ ∑ Q_{i} belongs to I^{4}, hence that ⟨⟨c⟩⟩(Q′_{1} − Q′_{2}) and ⟨⟨c⟩⟩(Q′_{3} − Q′_{4}) are congruent mod I^{4}, so [H, Cor.]^{c} gives that

⟨⟨c⟩⟩(Q_{1} − Q_{2}) = ⟨m⟩⟨⟨c⟩⟩(Q_{3} − Q_{4}) for some m ∈ k^{×}.

cThe statement of this result includes the hypothesis that the 12-dimensional form ⟨⟨c⟩⟩(Q′_{1} − Q′_{2}) is anisotropic, but it is unnecessary.

This implies that

(15.5)  ⟨⟨c⟩⟩(Q_{1} − Q_{2})^{2} = ⟨⟨c⟩⟩(Q_{3} − Q_{4})^{2}

in the Witt ring. Further,

(15.6)  ⟨⟨c⟩⟩Q_{1}Q_{2}(Q_{3} − Q_{4}) = ⟨m⟩⟨⟨c⟩⟩Q_{1}Q_{2}(Q_{1} − Q_{2}) = 0

and similarly

(15.7)  ⟨⟨c⟩⟩Q_{1}(Q_{3} − Q_{4})^{2} = ⟨⟨c⟩⟩Q_{1}(Q_{1} − Q_{2})^{2} = 4⟨⟨c⟩⟩Q_{1}(Q_{1} − Q_{2}).

Of course, the roles of Q_{1} and Q_{2} are not special, and the same identities hold for every permutation of the subscripts.

We now compute:

⟨⟨c⟩⟩ (4 ∑ Q_{i} − 2 ∑ Q_{i}Q_{j}) = ⟨⟨c⟩⟩(∑ Q_{i}^{2} − 2 ∑ Q_{i}Q_{j})

= ⟨⟨c⟩⟩ (∑_{i<j}(Q_{i} − Q_{j})^{2} − 2 ∑ Q_{i}^{2}) .

Applying (15.5), we find:

⟨⟨c⟩⟩ (4 ∑ Q_{i} − 2 ∑ Q_{i}Q_{j}) = ⟨⟨c⟩⟩ (2(Q_{1} − Q_{2})^{2} + 2(Q_{1} − Q_{3})^{2} + 2(Q_{1} − Q_{4})^{2} − 2 ∑ Q_{i}^{2})
= ⟨⟨c⟩⟩ [4Q_{1}^{2} − Q_{1}(4Q_{2} + 4Q_{3} + 4Q_{4})] .

We can replace the 4Q_{3} + 4Q_{4} with (Q_{3} − Q_{4})^{2} + 2Q_{3}Q_{4}, and further replace that with 4Q_{1} − 4Q_{2} + 2Q_{3}Q_{4} by (15.7), i.e.,

⟨⟨c⟩⟩ (4 ∑ Q_{i} − 2 ∑ Q_{i}Q_{j}) = ⟨⟨c⟩⟩ [4Q_{1}^{2} − 4Q_{1}Q_{2} − 4Q_{1}^{2} + 4Q_{1}Q_{2} − 2Q_{1}Q_{3}Q_{4}]
= −2⟨⟨c⟩⟩Q_{1}Q_{3}Q_{4}.

Evaluating the full Killing form, we have:

κ(G) = ⟨⟨c⟩⟩(Q_{1}Q_{2}Q_{3} − Q_{1}Q_{3}Q_{4} + Q_{2}Q_{3}Q_{4} + Q_{1}Q_{2}Q_{4})
= ⟨⟨c⟩⟩ [Q_{1}Q_{2}(Q_{3} − Q_{4}) + 2Q_{1}Q_{2}Q_{4} − (Q_{1} − Q_{2})Q_{3}Q_{4}] ,

and the claim follows by applying (15.6) twice. □

Of course, the roles of Q_{1}, Q_{2}, Q_{4} in the theorem are not special, and one can take any three of the four quaternion algebras by (15.6).

Remark 15.8. One can view Th. 15.4 as giving a relationship amongst four symbols in H^{d}(k, ℤ/2ℤ) that sum to zero. For three symbols, one has the Elman-Lam Linkage Thoerem [Lam, X.6.22].

Example 15.9. Suppose that k is a field with 2^{3} · I^{5} ≠ 0 in the Witt ring; for example, this happens if k is formally real. Then there is a 5-Pfister form ϕ such that 2^{3}ϕ ≠ 0. We write ϕ = ⟨⟨c⟩⟩q_{1}q_{2} for 2-Pfister forms q_{1}, q_{2}, and c ∈ k^{×}. Taking

• construction (11.2) where Q_{1}, Q_{2}, Q_{3}, Q_{4} have norms q_{1}, q_{2}, q_{1}, q_{2} respectively, or

• Tits’s construction 11.6 with a reduced Albert algebra such that γ_{3} = ϕ_{3} =

⟨⟨c⟩⟩q_{1} and ϕ_{5} = ϕ, one obtains a group G of type E_{8} over k that has zero Rost invariant, has κ(G) = 8ϕ nonzero, and is anisotropic (by Prop. 12.1(1)).

16. A conjecture, and its consequences

Consider the following statement:

(16.1)

For every odd-degree separable extension K/k and every central simple K-algebra of degree 16 with orthogonal involution (A, σ) in I^{3}K, we have: If e_{3}(A, σ) is zero, then there is an odd-degree separable extension L/K such that (A, σ) ⊗ L is completely decomposable.

By Prop. 2.7, if e_{3}(A, σ) is zero, then e_{3}(A, σ) is generically Pfister. That is, (16.1) is somewhat weaker than a “yes” answer to Question 1.3. It is natural to hope that (16.1) holds for every field k of characteristic different from 2.

We have:

Theorem 16.2. Suppose that k is a field of characteristic ≠ 2, 3 for which (16.1) holds. Then the map G ↦ redKill_{E_{8}} − redKill_{G} defines a function

Groups of type E_{8} over k whose Rost invariant has odd order → I^{8}(k).

Proof. We first observe that there is a separable extension K/k of odd degree such that res_{K/k} G ∈ H^{1}(K, E_{8}) is the image of some η ∈ H^{1}(K, HSpin_{16}). This follows from the fact that HSpin_{16} contains a maximal torus of E_{8} and that the 2-Sylow subgroups of the Weyl groups have order 2^{14} in both cases, see e.g. the proof of [Ga, 13.7]. (For HSpin_{16}, one checks that the 2-primary part of 2^{7} 8! is 2^{14}.)

We enlarge K so that it also kills r(G). By (16.1), there is an odd-degree extension L/K such that the image of η in H^{1}(L, PSO_{16}) is the class of a tensor product ⊗_{i=1}^{4}(Q_{i},¯ ), and it follows that res_{L/k} G is in the image of (11.2).

By Lemma 13.6, κ(G) is in I^{4}k. But over the odd-degree extension L/k, κ(G) is in I^{8}L by Th. 15.4. Write e_{n} for the invariant I^{n}(∗) → H^{n}(∗, ℤ/2ℤ) defined in [OVV] such that for every extension F/k, the map (e_{n})_{F} is an additive homomorphism with kernel I^{n+1}F. Here, e_{4}(κ(G)) is killed by L and so is zero, hence κ(G) is in I^{5}k. Repeating this with e_{5}, e_{6}, and e_{7} shows that κ(G) is in I^{8}k. □

Acknowledgments. I thank D.W. Hoffmann and K. Zainoulline for providing Theorem 15.4 and Prop. A.1 respectively, and V. Chernousov, R. Parimala, A. Quéguiner-Mathieu, and J.-P. Tignol for their comments. I gratefully acknowledge the support by NSF grant DMS-0654502. Part of the research for this paper was done during a visit at IHES, and I thank them for providing an excellent working environment.

Appendix A. Non-hyperbolicity of orthogonal involutions By Kirill Zainoulline

The purpose of the following notes is to prove the following

Proposition A.1. Let (A, σ) be a central simple algebra with orthogonal involution over a field k of characteristic ≠ 2. If deg A/ ind A is odd, then the involution σ is not hyperbolic over the function field of the Severi-Brauer variety of A.

This result can also be deduced from the divisibility of the Witt index of (A, σ) proved recently by N. Karpenko (see [Ka 08, Th. 3.3]). Our arguments use the notion of the J-invariant instead.

Proof of Prop. A.1. The case where A has index 1 is clear and the index 2 case is [PSS, Prop. 3.3], so we may assume that A has index at least 4 and hence degree is divisible by 4. Further, we may assume that (A, σ) is in I^{3} as in Example 1.2(3), otherwise the conclusion is obvious.

Consider the groups G = HSpin(A, σ) and G′ = PGL(A). Let X and X′ be the respective varieties of Borel subgroups.

(A.2)  Assume that σ is hyperbolic over the function field k(SB(A)).

Then the group G is split over k(SB(A)) and, hence, over k(X′). Since the group G is split over k(X), the algebra A and the group G′ are split over k(X).

By the main result of the paper [PSZ] (Theorems 4.9 and 5.1) to any simple linear algebraic group of inner type over k and its torsion prime p one may associate an indecomposable Chow motive R_{p} such that over the algebraic closure k̄ of k the generating function of R_{p} is given by the product of r cyclotomic polynomials

∏_{i=1}^{r} \frac{1 − t^{d_{i}2^{j_{i}}}}{1 − t^{d_{i}}} , where 0 ≤ j_{i} ≤ k_{i} and d_{i} > 0

and the explicit values of the parameters d_{i} and bounds k_{i} are provided in [PSZ, Table 6.3]. The r-tuple of integers (j_{1}, j_{2}, …, j_{r}) is called the J-invariant.

Let R_{2}(G) and R_{2}(G′) be the respective motives for the groups G and G′ and for p = 2. By [PSZ, Prop. 5.3] applied to G over k(X′) and G′ over k(X) we obtain the following motivic reformulation of the assumption (A.2):

(A.3)  R_{2}(G) ≃ R_{2}(G′).

Since the group G′ is a twisted form of the group PGL_{deg A}, by the first line of [PSZ, Table 6.3] the J-invariant of G′ has only one entry (r = 1), and the parameter d_{1} is 1. Then by the proof of [PSZ, Lemma 7.3], we obtain that the J-invariant is the list consisting of the single element s, where 2^{s} is the index of A. Hence the generating function of R_{2}(G′) is (1 − t^{2^{s}})/(1 − t).

Similarly, since the group G is a twisted form of the group HSpin_{deg A}, by [PSZ, Table 6.3] the J-invariant of G has \frac{1}{4} deg A entries with d_{i} = 2i−1 and the following inequality holds

j_{1} ≤ k_{1} = v_{2}(\frac{1}{2} deg A) = v_{2}(2^{s−1} · \frac{deg A}{ind A}) = s − 1 < s,

where v_{2} is the 2-adic valuation. (Here we essentially use that \frac{deg A}{ind A} is odd.)

The isomorphism (A.3) implies the equality of the respective generating functions, namely

\frac{1 − t^{2^{s}}}{1 − t} = ∏_{i=1}^{s/2} \frac{1 − t^{(2i−1)2^{j_{i}}}}{1 − t^{2i−1}} , where j_{1} < s.

We claim that it never holds. Indeed, comparing the coefficients at t^{2} and t^{3} of the polynomials at the left and the right hand side, we conclude that j_{1} ≥ 2 and j_{2} = 0. Then comparing them consequently at powers t^{2i−2} and t^{2i−1}, i ≥ 3 we conclude that 2^{j_{1}} ≥ 2i − 1 and j_{i} = 0. Therefore, j_{3} = ⋯ = j_{s/2} = 0 and j_{1} must coincide with s, which is not the case, since j_{1} < s.

Hence, the assumption (A.2) fails and the lemma is proven. □

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(Garibaldi) 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/

(Zainoulline) Mathematisches Institut der LMU München, Theresienstr. 39, 80333 München, Germany

   E-mail address: kirill@mathematik.uni-muenchen.de