TOTARO’S QUESTION FOR G_{2}, F_{4}, AND E_{6}

SKIP GARIBALDI AND DETLEV W. HOFFMANN

Abstract. In a 2004 paper, Totaro asked whether a G-torsor X which has a zero-cycle of degree d > 0 will necessarily have a closed étale point of degree dividing d, where G is a connected algebraic group. This question is closely related to several conjectures regarding exceptional algebraic groups. Totaro gave a positive answer to his question in the following cases: G simple, split, and of type G_{2}, type F_{4}, or simply connected of type E_{6}. We extend the list of cases where the answer is “yes” to all groups of type G_{2} and some nonsplit groups of type F_{4} and E_{6}. No assumption on the characteristic of the base field is made. The key tool is a lemma regarding linkage of Pfister forms.

For certain linear algebraic groups G over a field k and certain homogeneous G-varieties X, Totaro asked in [To]:

(0.2)

If X has a zero-cycle of degree d > 0, does X necessarily have a closed étale point of degree dividing d?

(This question is closely related to earlier questions raised by Veisfeiler, Serre, and Colliot-Thélène.) This question can be rephrased as:

(0.2′)

If X has a (closed) point over finite extensions K_{1}, K_{2}, …, K_{n} of k, does X necessarily have a point over a separable extension of k of degree dividing every [K_{i} : k]?

The purpose of this note is to prove the following theorem.

Theorem 0.3. The answer to (0.2) is “yes” when X is a G-torsor and G is

• of type G_{2},

• reduced of type F_{4}, or

• simply connected of type ^{1}E_{6,6}^{0} (split) or of type ^{1}E_{6,2}^{28}.

A G-torsor is a principal G-bundle over Spec k. Torsors are often called principal homogeneous spaces, as in [Se GC]. It is possible that the answer to (0.2) is “yes” whenever G is semisimple and X is a G-torsor (and, in particular, is affine); no counterexamples are known. In contrast, for X projective, there are examples where the answer is “no” even when d = 1, see [Fl] and [Pa].

Date: December 7, 2004.

2000 Mathematics Subject Classification. 11E72 (20G15).

Every group of type F_{4} is the group of automorphisms of some uniquely determined Albert k-algebra J. We say that the group is reduced if the algebra J is reduced, i.e., if J is not a division algebra. (Recall that Albert algebras are 27-dimensional Jordan algebras. For a survey of the state of the art in 1992, see [PR 94]. Many useful and characteristic-free results about Albert algebras may be found in [Pe].)

The notation “^{1}E_{6,2}^{28}” is from Tits’ classification [Ti 66, p. 58]. Such groups have Tits index

[figure: (0.4)]

Totaro proved Theorem 0.3 under the additional assumption that G is split, see [To, 5.1]. The main tool that allows us to extend his result is our Proposition 1.6 and its analogue for cohomology, Cor. 1.7; they assert the existence of field extensions of small dimension that kill a difference of symbols in H^{p}(k, ℤ/2ℤ(p − 1)). The quadratic form theory that underlies Cor. 1.7 occupies half of this paper and concerns mainly the question of so-called linkage of Pfister forms (in all characteristics, including 2).

The last section gives some comments on extending Theorem 0.3 to include other exceptional groups. Except for this final section, we make no assumptions on the base field k.

Notation. Throughout, we write μ_{n} for the group scheme of n-th roots of unity, where n is a natural number. We write H^{p}(k, ℤ/nℤ(p − 1)) for the abelian group defined in [M 03, App. A]; when n is not divisible by the characteristic of k, it is the Galois cohomology group H^{p}(k, μ_{n}^{⊗(p−1)}).

1. Quadratic form preliminaries

For readers who are not very familiar with the algebraic theory of quadratic forms we will first present a few definitions and facts. Other basic results and terminologies that we use without further reference can all be found in Scharlau’s book [Sch] for characteristic ≠ 2, and in Baeza’s book [Ba 78] and the article [HL] by Laghribi and the second author for the characteristic 2 case. In particular, we will freely use certain properties of Pfister forms, Pfister neighbors, and function fields of quadratic forms.

If α and β are quadratic forms, then β is called a subform of α (denoted β ⊂ α), if there exists a form γ such that α ≅ β ⊥ γ, and α is said to dominate β (denoted β ≺ α) if β is isometric to the restriction of α to some subspace of the underlying vector space of α.

The Witt index i_{W} (ϕ) of a quadratic form ϕ is the maximal number m such that ϕ contains an orthogonal sum of m hyperbolic planes as a subform. Furthermore, if β is a bilinear form (in characteristic 2), then we denote by β_{q} the totally singular quadratic form defined by β_{q}(x) = β(x, x).

Every nonsingular quadratic form ϕ over k has attached to it an invariant e_{1}(ϕ) living in the Galois cohomology group H^{1}(k, ℤ/2ℤ). In characteristic not 2, we have H^{1}(k, ℤ/2ℤ) = k^{×}/k^{×2} and e_{1}(ϕ) = d_{±}(ϕ) is the signed discriminant. If char(k) = 2, then H^{1}(k, ℤ/2ℤ) = k/℘(k) and e_{1}(ϕ) = Δ(ϕ) is the Arf-invariant. Note that in all characteristics H^{1}(k, ℤ/2ℤ) is nothing else but the group H^{1}(k, ℤ/2ℤ(0)) from above.

Recall that in characteristic ≠ 2 (resp. characteristic 2), a product of p binary quadratic (resp. bilinear) forms ⟨1, −a_{i}⟩ is called a p-fold Pfister form (resp. p-fold bilinear Pfister form): ⟨⟨a_{1}, ⋯, a_{p}⟩⟩ = ⟨1, −a_{1}⟩ ⊗ ⋯ ⊗ ⟨1, −a_{p}⟩. These forms generate additively the ideal I^{p}k, the p-th power of the fundamental ideal of even-dimensional forms in W k, the Witt ring of quadratic forms (resp. bilinear forms) over k.

In characteristic 2, a p-fold quadratic Pfister form is a product of a (p−1)- fold bilinear Pfister form ⟨⟨a_{1}, ⋯, a_{p−1}⟩⟩ and a nonsingular quadratic form [1, a_{p}] = X^{2} + XY + a_{p}Y^{2}, a_{p} ∈ k: ⟨⟨a_{1}, ⋯, a_{p−1}, a_{p}]] = ⟨⟨a_{1}, ⋯, a_{p−1}⟩⟩ ⊗ [1, a_{p}]. The p-fold quadratic Pfister forms generate the W k-submodule I_{q}^{p}k = (I^{p−1}k)W_{q}k of the Witt group W_{q}k of nonsingular quadratic forms over k.

Here, a p-Pfister form will always mean either a p-fold Pfister form in characteristic ≠ 2 or a p-fold quadratic Pfister form if the characteristic is 2, and we will only consider the case p ≥ 1. Also, we will use in characteristic ≠ 2 the notation I_{q}^{p}k instead of I^{p}k for uniformity’s sake, and we will denote the set of isometry classes of p-Pfister forms over k by P_{p}k.

A quadratic form ϕ is a Pfister neighbor of a Pfister form π if aϕ ⊂ π (char(k) ≠ 2) resp. aϕ ≺ π (char(k) = 2) for some a ∈ k^{×}, and 2 dim ϕ > dim π.

If ϕ is a Pfister neighbor of π and E/k is any field extension then i_{W} (ϕ_{E}) > 0 if and only if π_{E} is isotropic if and only if π_{E} is hyperbolic. If any of these equivalent conditions hold, then E(ϕ)/E and E(π)/E are purely transcendental, and any form ψ over k which is isotropic over E(ϕ) or E(π) is so already over E. Applied to E = k(ϕ) resp. E = k(π), this shows that a form ψ over k is isotropic over k(π) if and only if it is isotropic over k(ϕ).

In characteristic ≠ 2, it is well-known that if ϕ and π are Pfister forms with π ⊂ ϕ, then there exists a Pfister form ρ such that ϕ is isometric to ρ ⊗ π. The analogous result in characteristic 2 is the following.

Lemma 1.1. Suppose char(k) = 2. Let ϕ be a Pfister form over k and let x be in k^{×}.

(i) Let π be a Pfister form with xπ ⊂ ϕ. Then there exists a bilinear Pfister form ρ such that ϕ ≅ ρ ⊗ π.

(ii) Let ρ be a bilinear Pfister form such that xρ_{q} ≺ ϕ. Then there exists a Pfister form π such that ϕ ≅ ρ ⊗ π.

Proof. We assume that ϕ is anisotropic and leave the (much simpler) case of ϕ being isotropic (and hence hyperbolic) to the reader. We may also assume that x = 1 because of the roundness of Pfister forms. (Recall that a form is called round if the set of nonzero elements represented by the form is exactly the group of similarity factors of the form.) Indeed, since π resp. ρ_{q} represent 1, the hypotheses in (i) resp. (ii) imply that ϕ represents x, hence, the roundness of ϕ implies ϕ ≅ xϕ and thus π ⊂ ϕ resp. ρ_{q} ≺ ϕ.

(i) Let μ be a quadratic form such that ϕ ≅ μ ⊥ π. Now π = ⟨1⟩ ⊗ π and ⟨1⟩ is a 0-fold bilinear Pfister form. So let r ≥ 0 be maximal such that there exists an r-fold bilinear Pfister form ρ and a quadratic form η with ϕ ≅ η ⊥ (ρ ⊗ π).

Suppose dim η > 0, and let a ∈ k^{×} be represented by η. Then γ′ = ⟨a⟩_{q} ⊥ ρ ⊗ π is dominated by ϕ. In particular, ϕ becomes isotropic and hence hyperbolic over the function field k(γ′). But γ′ is a Pfister neighbor of the Pfister form γ = ⟨⟨a⟩⟩⊗ρ⊗π, hence ϕ becomes isotropic and hence hyperbolic over k(γ). Since γ and ϕ represent 1, it follows that there exists a form η′ such that ϕ ≅ η′ ⊥ γ (see, e.g., [HL, Th. 4.2]). We get a contradiction to the maximality of r as γ ≅ ρ′ ⊗ π with ρ′ = ⟨⟨a⟩⟩ ⊗ ρ a bilinear Pfister form of fold r + 1. Hence dim η = 0 and ϕ ≅ ρ ⊗ π.

(ii) Write ρ_{q} ≅ ⟨1⟩_{q} ⊥ ρ′_{q} (with ρ′_{q} totally singular). Since ρ_{q} is dominated by ϕ, there exists b ∈ k such that γ′ = [1, b] ⊥ ρ′_{q} is also dominated by ϕ (see [HL, Cor. 3.3, Lemma 3.5]). Therefore, ϕ becomes isotropic and hence hyperbolic over k(γ′). But γ′ is a Pfister neighbor of the Pfister form γ = ρ ⊗ [1, b]. It follows that ϕ becomes hyperbolic over k(γ), and as in the proof of (i), we find a bilinear Pfister form ν such that ϕ ≅ ν ⊗ ρ ⊗ [1, b]. The claim follows by putting π = ν ⊗ [1, b]. □

Remark 1.2. Part (i) in the above lemma is nothing else but [Ba 78, Th. 4.4]. Our proof is quite different from Baeza’s original one in that it uses the machinery of function fields of quadratic forms. Part (ii) is basically the analogous result with the roles of bilinear and quadratic Pfister forms reversed.

Lemma 1.3. Let ϕ and ψ be anisotropic Pfister forms of fold p and q respectively. Then there exists an integer m ≥ 0 such that i_{W} (ϕ ⊥ −ψ) = 2^{m}.

Furthermore, if m > 0, then m is maximal for the property such that there exist an m-Pfister form π and Pfister forms (resp. bilinear Pfister forms if char(k) = 2) ρ and σ of fold p − m and q − m, respectively, with ϕ ≅ ρ ⊗ π and ψ ≅ σ ⊗ π.

Proof. If char(k) ≠ 2, this is essentially [EL, Prop.4.4]. In characteristic 2, the result is due to Faivre [Fa], but we include a proof of this case for the reader’s convenience.

So let us assume that char(k) = 2. Clearly, ϕ and ψ represent 1, so i_{W} (ϕ ⊥ −ψ) ≥ 1. Also, if π is an m-Pfister dividing both ϕ and ψ, then the roundness of Pfister forms implies that π is a subform of both ϕ and ψ, therefore

i_{W} (ϕ ⊥ −ψ) ≥ i_{W} (π ⊥ −π) = dim π = 2^{m}.

It thus suffices to consider the case i_{W} (ϕ ⊥ −ψ) ≥ 2. We have to show that the Witt index is equal to the maximal dimension of a Pfister form π which is a subform of both ϕ and ψ. The existence of the bilinear Pfister forms ρ and σ then follows from Lemma 1.1(i).

We first show that there exists a common 1-Pfister form as subform of both ϕ and ψ.

Now i_{W} (ϕ ⊥ −ψ) ≥ 2 implies that there is a binary form β (possibly singular) which is dominated both by ϕ and ψ (see [HL, Cor. 3.13]), and we may assume that β represents 1 as ϕ and ψ represent 1. If β is isometric to [1, b] for some b ∈ k, then we are done. Otherwise, β is isometric to ⟨1, b⟩_{q} for some b ∈ k^{×}. In this case, we necessarily have p, q ≥ 2 for dimension reasons and since ϕ and ψ are nonsingular. We then apply Lemma 1.1(ii) to find bilinear Pfister forms ρ′ and σ′, and c, d ∈ k such that ϕ ≅ ⟨⟨b⟩⟩⊗ρ′⊗[1, c] and ψ ≅ ⟨⟨b⟩⟩⊗σ′⊗[1, d]. But then there exist e, f, g ∈ k such that ⟨⟨b, c]] ≅ ⟨⟨e, g]] and ⟨⟨b, d]] ≅ ⟨⟨f, g]] (see, e.g., [Lam] for the equivalent “linkage” result for quaternion algebras). Clearly, [1, g] is the desired form.

Having found a 1-Pfister form as subform of both ϕ and ψ, let now π be an m-Pfister form which is a subform of both ϕ and ψ and with m maximal. Thus, we can write ϕ ≅ ϕ′ ⊥ π and ψ ≅ ψ′ ⊥ π, and we have

i_{W} (ϕ ⊥ −ψ) = i_{W} (ϕ′ ⊥ −ψ′) + 2^{m}.

If i_{W} (ϕ ⊥ −ψ) > 2^{m} then ϕ′ ⊥ −ψ′ is isotropic, hence ϕ′ and ψ′ represent a common element a ∈ k^{×}, and an argument as in the proof of Lemma 1.1(i) then shows that the (m + 1)-Pfister ⟨⟨a⟩⟩ ⊗ π is a subform of both ϕ and ψ, a contradiction to the maximality of m. This completes the proof. □

In the above lemma, we restricted ourselves to the case of anisotropic Pfister forms. It is obvious that if, say, ϕ is isotropic and hence hyperbolic, then i_{W} (ϕ ⊥ −ψ) = 2^{p−1} or 2^{p−1} + 2^{q−1}, depending on whether ψ is anisotropic or not. In this case, ϕ divides ψ if ψ is hyperbolic and q ≥ p, and ψ divides ϕ if p > q or p = q and ψ is hyperbolic. If ψ is anisotropic and q ≥ p, then any (p − 1)-Pfister dividing ψ will also divide the hyperbolic form ϕ, but there is obviously no p-Pfister dividing both ϕ and ψ.

Lemma 1.4. Let ϕ, ψ be p-Pfister forms over k, with p ≥ 2. Suppose that there is an odd-degree extension L of k such that ϕ and ψ are isometric over a quadratic extension of L. Then there exists a 1-Pfister α dividing both ϕ and ψ. In particular, ϕ and ψ are both hyperbolic over k or some separable quadratic extension of k.

If p = 1, let c = e_{1}(ϕ ⊥ −ψ) ∈ H^{1}(k, ℤ/2ℤ). We then have that ϕ and ψ are isometric if and only if c ∈ H^{1}(k, ℤ/2ℤ) is trivial. Hence ϕ and ψ are isometric over k(\sqrt{c}) (char(k) ≠ 2) resp. k(℘^{−1}(c)) (char(k) = 2). But ϕ and ψ need not be divisible by the same 1-Pfister.

The statement of the lemma might also be expressed by saying that the two Pfister forms have a common right slot. In other words, there exist c ∈ k and a_{1}, …, a_{p−1}, b_{1}, …, b_{p−1} ∈ k^{×} such that the following holds:

If char(k) ≠ 2, we have ϕ ≅ ⟨⟨a_{1}, …, a_{p−1}, c⟩⟩, ψ ≅ ⟨⟨b_{1}, …, b_{p−1}, c⟩⟩, and the extension is k(\sqrt{c})/k.

If char(k) = 2, we have ϕ ≅ ⟨⟨a_{1}, …, a_{p−1}, c]], ψ ≅ ⟨⟨b_{1}, …, b_{p−1}, c]], and the extension is k(℘^{−1}(c))/k.

Of course, the order of the slots only matters in characteristic 2.

Proof. Suppose first that ϕ is isotropic and hence hyperbolic, and let b_{1}, …, b_{p−1} ∈ k^{×}, c ∈ k be such that ψ ≅ ⟨⟨b_{1}, …, b_{p−1}, c⟩⟩ (char(k) ≠ 2) resp. ψ ≅ ⟨⟨b_{1}, …, b_{p−1}, c]] (char(k) = 2). Since ϕ is hyperbolic, we have ϕ ≅ ⟨⟨−1, …, −1, c⟩⟩ resp. ϕ ≅ ⟨⟨−1, …, −1, c]], and the result is obvious.

So let us assume that both ϕ and ψ are anisotropic. Set q to be the anisotropic part of ϕ ⊥ −ψ. Let m be maximal such that ϕ and ψ contain a common m-Pfister as subform. By Lemma 1.3, we have

dim q = 2^{p+1} − 2i_{W} (ϕ ⊥ −ψ) = 2^{p+1} − 2^{m+1},

and it suffices to show that m is not zero; clearly we may assume that m < p, i.e., that dim q > 0.

By Springer’s Theorem for odd-degree extensions (see, e.g., [Sch, 2.5.3]), q is L-anisotropic. Now q is hyperbolic over a quadratic extension E of L. We consider three cases:

(i) char(k) = 2 and E = L(\sqrt{b}) for some b ∈ L^{×} (E/L is inseparable); (ii) char(k) ≠ 2 and E = L(\sqrt{b}) for some b ∈ L^{×}; (iii) char(k) = 2 and E = L(℘^{−1}(b)) for some b ∈ L^{×} (E/L is separable);

In cases (i) and (ii), there exists a nonsingular quadratic form η over L such that

q_{L} ≅ ⟨⟨b⟩⟩ ⊗ η

(see [Sch, 2.5.2] in case (i) and [Ah] in case (ii)). In case (iii), there exists a bilinear form η over L such that

q_{L} ≅ η ⊗ [1, b]

(see [Ba 74]).

In case (i), the nonsingularity of η implies that dim η ≡ 0 mod 2, hence dim q ≡ 0 mod 4 and thus m ≥ 1.

Suppose we are in case (ii) or (iii), and put β = ⟨⟨b⟩⟩ or β = [1, b], respectively. Note that β is obviously anisotropic, and that q_{L} ∈ I_{q}^{2}L as ϕ and ψ are p-Pfisters with p ≥ 2. If dim η is odd, then there exists c ∈ L^{×} such that

0 ≡ q_{L} ≡ cβ mod I_{q}^{2}L ,

but this contradicts the Arason-Pfister Hauptsatz which states that anisotropic forms in I_{q}^{n}L are of dimension ≥ 2^{n}. Hence, again dim η ≡ 0 mod 2 and therefore m ≥ 1. □

The following field-theoretic lemma is folklore but we include a proof for the reader’s convenience.

Lemma 1.5. Let K/k be a finite separable extension of degree p^{n}m where p is a prime not dividing m. Then there exist finite separable extensions E/L/k such that K ⊂ E, [E : L] = p^{n}, and p does not divide [L : k].

Proof. Let N/k be any finite Galois extension containing K, and let S′ be any p-Sylow of Gal(N/K). Since Gal(N/K) is a subgroup of Gal(N/k), there is a p-Sylow S of Gal(N/k) containing S′. Let L = N^{S} and E = N^{S′}. Clearly, E contains K, p does not divide [L : k], and by comparing degrees we have [E : L] = p^{n}. □

Proposition 1.6. Let ϕ and ψ be p-Pfister forms over a field k. Suppose that there exist finite extensions K_{1}, …, K_{n} of k such that ϕ and ψ are isometric over K_{i} for all i. Then there is a Galois extension K of k such that ϕ and ψ are K-isometric and [K : k] divides

g := gcd{4, [K_{1} : k], [K_{2} : k], …, [K_{n} : k]}.

If p ≥ 2 and g is even, then K may be chosen to kill both ϕ and ψ. If g = 1, then there exists a Galois extension of degree dividing 2 that kills both ϕ and ψ.

Proof. If ϕ = ⟨⟨⋯, a⟩⟩, ψ = ⟨⟨⋯, b⟩⟩ (char(k) ≠ 2) resp. ϕ = ⟨⟨⋯, a]], ψ = ⟨⟨⋯, b]] (char(k) = 2), then k(\sqrt{a}, \sqrt{b}) resp. k(℘^{−1}(a), ℘^{−1}(b)) is a Galois extension of degree dividing 4 which obviously kills ϕ and ψ simultaneously.

So we may assume that the gcd g in the statement is 1 or 2. If g = 1, then [K_{i} : k] is odd for some i, hence ϕ and ψ are k-isomorphic by Springer’s theorem. In this case, we may choose a = b above and thus get the quadratic Galois extension which kills both forms.

Finally, suppose that g = 2. In this case, we have [K_{i} : k] ≡ 2 mod 4 for some i which we fix.

Let us first consider the case char(k) ≠ 2. We then can write K_{i} as a separable extension K′_{i} followed by a purely inseparable one which necessarily is of odd degree. Then [K′_{i} : k] ≡ 2 mod 4, and using Springer’s theorem, we may then assume that K_{i} is separable over k by replacing K_{i} by K′_{i} if necessary. By Lemma 1.5, there are separable extensions E/L/k such that K_{i} ⊂ E, [E : L] = 2, and [L : k] is odd. Since K_{i} ⊂ E, we have that ϕ and ψ are isometric over E, and we conclude by applying Lemma 1.4.

Now suppose char(k) = 2. If K_{i}/k is separable, the same reasoning as above applies. If it is inseparable, then this is only possible for reasons of degree and characteristic if there is an intermediate field L such that L/k is of odd degree (and thus separable) and K_{i}/L is inseparable of degree 2. Again, Lemma 1.4 yields the desired conclusion. □

The above proposition allows an interpretation in terms of symbols in the group H^{p}(k, ℤ/2ℤ(p − 1)) mentioned in the introduction. Recall that in characteristic ≠ 2, H^{p}(k, ℤ/2ℤ(p − 1)) is just the usual Galois cohomology group with mod 2 coefficients, H^{p}(k, ℤ/2ℤ), and that there is a well-defined injective map P_{p}k → H^{p}(k, ℤ/2ℤ) sending ⟨⟨a_{1}, ⋯, a_{p}⟩⟩ to the symbol (a_{1})∪ ⋯∪(a_{p}). This map is well-defined, see Elman-Lam [EL, 3.2] or Arason [Ar, 1.6], and it is injective as a direct consequence of Voevodsky’s proof of the Milnor conjecture.

As already mentioned, in characteristic 2, we have H^{1}(k, ℤ/2ℤ(0)) = k/℘(k). One can then show that H^{p}(k, ℤ/2ℤ(p − 1)) is nothing else but the group h_{p} defined in [AB], namely the quotient of the group (K_{p−1}k/2K_{p−1}k)⊗ H^{1}(k, ℤ/2ℤ(0)) by the subgroup R_{p} generated by elements of type

{a_{1}, ⋯, a_{p−1}} ⊗ (b) ∈ (K_{p−1}k/2K_{p−1}k) ⊗ H^{1}(k, ℤ/2ℤ(0))

such that a_{i} is a norm of the extension k(℘^{−1}(b))/k for some i (or, equivalently, that a_{i} is represented by the quadratic form [1, b]). Again, we have a well-defined and injective map P_{p}k → h_{p} sending ⟨⟨a_{1}, ⋯, a_{p−1}, b]] to the “symbol” {a_{1}, ⋯, a_{p−1}} ⊗ (b) mod R_{p}, see [AB] (or [Ba 01] for a short exposition of these facts).

Corollary 1.7. Let x and y be symbols in H^{p}(k, ℤ/2ℤ(p−1)). Suppose that there exist finite extensions K_{1}, …, K_{n} of k such that x and y agree over every K_{i}. Then there exists a Galois extension K of k such that x and y agree over K and [K : k] divides

g := gcd{4, [K_{1} : k], [K_{2} : k], …, [K_{n} : k]}.

If p ≥ 2 and g is even, the extension K may be chosen to kill both x and y. If g = 1, then there exists a Galois extension of degree dividing 2 that kills both x and y.

2. Proof of Theorem 0.3: type G_{2}

Every group G of type G_{2} and every G-torsor X may be (in a compatible manner) identified with an octonion k-algebra [KMRT, 33.24]. Such algebras are determined by their norm form, which is a 3-Pfister form [SV, §1.7].

Suppose that the G-torsor X is trivial (= has a closed point) over finite extensions K_{1}, K_{2}, …, K_{n} of k. Let ϕ_{G} and ϕ_{X} be 3-Pfister forms corresponding to G and X respectively. The quadratic forms are isomorphic over K_{i} for all i by hypothesis, and Prop. 1.6 gives a separable extension K of k such that ϕ_{G} and ϕ_{X} are K-isomorphic and [K : k] divides [K_{i} : k] for all i. That is, X has a K-point, and the answer to (0.2′)—equivalently, (0.2)—is “yes” for every group G of type G_{2} and every G-torsor X.

3. Proof of Theorem 0.3: type F_{4}

An Albert k-algebra J has p-Pfister form invariants f_{p}(J) for p = 3, 5. Every Albert algebra also has a “mod 3” invariant g_{3}(J) living in the group H^{3}(k, ℤ/3ℤ(2)). (The correspondence between 3-Pfister forms and elements of H^{3}(k, ℤ/2ℤ(2)) identifies the pair (f_{3}(J), g_{3}(J)) with an element of H^{3}(k, ℤ/6ℤ(2)). This element is the Rost invariant of J with respect to the split group of type F_{4}, up to sign.) The algebra J is reduced if and only if g_{3}(J) is zero [Ro 91], [PR 96]. There is a unique Albert algebra — the split Albert algebra — with f_{3} hyperbolic and g_{3} equal to zero.

Let G be a reduced group of F_{4}, i.e., G is of the form Aut(J) for a reduced Albert k-algebra J. We are given a G-torsor X and finite extensions K_{1}, K_{2}, …, K_{n} of k such that X is trivial over K_{i} for all i. The torsor X is the collection Iso(J′, J) of isomorphisms J′ ⟶^{∼} J for some Albert k-algebra J′, and by hypothesis J′ and J are K_{i}-isomorphic for all i. We will construct a separable extension K of k such that J′ and J are K-isomorphic and the degree [K : k] divides the gcd g of the degrees [K_{i} : k]. For ℓ = 2, 3, we first construct an ℓ-primary separable extension E_{ℓ} of k which makes the mod ℓ invariants of J and J′ equal.

\underline{ℓ = 2:} If g is odd, then [K_{i} : k] is odd for some i. Hence f_{3} and f_{5} agree for J and J′, and we take E_{2} = k. If g is even, let E_{2} be the Galois extension provided by Prop. 1.6 that kills f_{3}(J) and f_{3}(J′). Since f_{3} divides f_{5}, the extension E_{2} kills f_{5}(J) and f_{5}(J′) also.

\underline{ℓ = 3:} If g_{3}(J′) is zero, we may take E_{3} = k. If g is not divisible by 3, then [K_{i} : k] is not divisible by 3 for some i and g_{3}(J′) is zero. Otherwise, J′ is a division algebra and so contains a separable cubic subfield, which we take to be E_{3}. Then J′ ⊗_{k} E_{3} has zero divisors because E_{3} ⊗_{k} E_{3} does, whence g_{3}(J′) is killed by E_{3}.

Set K to be the compositum of E_{2} and E_{3} in some separable closure of k. Since E_{2} is Galois over k, the degree [K : k] divides the product [E_{2} : k][E_{3} : k], hence K has degree dividing g. By construction, K kills g_{3}(J′), so J′ is reduced over K. Since f_{i}(J) agrees with f_{i}(J′) over K for i = 3, 5, the algebras J and J′ are isomorphic over K by [SV, 5.8.1], [Pe, 4.1].

4. The connecting homomorphism for E_{6}

The purpose of this section is to prove a technical result we will need to prove Theorem 0.3 for groups of type E_{6}.

We say that a group G has trivial Tits algebras if the k-algebra End_{G}(V) is a field for every irreducible representation V of G. From here until the end of the paper, H^{1} denotes flat cohomology as in [DG] or [Wa]. For smooth groups—in particular, for all the groups we consider here except for central subgroups of semisimple groups in bad characteristic—it agrees with the Galois cohomology defined in [Se GC].

Lemma 4.1. Let C be a central subgroup of an algebraic group G of type E_{6}. If G is isotropic and has trivial Tits algebras, then the natural map

∂: (G/C)(k) → H^{1}(k, C)

is surjective.

This lemma is an E_{6}-analogue of a well-known fact about classical groups. Specifically, let G be the spin group of a nonsingular quadratic from q on a vector space V. Let C be the kernel of the “vector representation” G → GL(V); it is isomorphic to μ_{2}. The group H^{1}(k, C) is identified with k^{×}/k^{×2} by the Kummer exact sequence, and the connecting homomorphism ∂ is the spinor norm. When q is isotropic, ∂ is surjective, as is well-known in quadratic form theory [Ba 78, p. 78]. The proof of Lemma 4.1 below is easily adapted to give an alternative proof that the spinor norm is surjective.

The crux case of Lemma 4.1 is where G is of type ^{1}E_{6,2}^{28}, precisely the case we need in the rest of the paper. When k has good characteristic (i.e., not 2 or 3), the result in the crux case is an easy consequence of the theory of Albert algebras. We give an algebraic-group-theoretic proof that is valid in all characteristics.

We will use without comment the fact that ∂ fits naturally into an exact sequence

(G/C)(k) ⟶^{∂} H^{1}(k, C) ⟶^{ι} H^{1}(k, G).

In particular, ∂ is surjective if and only if ι is the zero map.

Proof of Lemma 4.1. Clearly we may assume that C is not the trivial group, hence that G is not adjoint, i.e., G is simply connected. Since the center of G is a twisted form of μ_{3}, the subgroup C must be the entire center.

The case where G is split is standard. Indeed, G contains a maximal k-torus T that is k-split. The center C is contained in every maximal torus, hence the map ι above factors through H^{1}(k, T). The group H^{1}(k, T) is zero by Hilbert 90, since T is split.

Suppose now that G is of type ^{1}E_{6,2}^{28}, i.e., has Tits index (0.4). Fix a maximal k-torus T in G containing a maximal k-split torus. Fix a set of simple roots Δ = {α_{1}, …, α_{6}} of G with respect to T, numbered as in [Bo]. Let Λ (resp. Λ_{r}) denote the weight (resp. root) lattice. Let ω_{i} be the fundamental weight corresponding to α_{i}. The absolute Galois group Γ of k acts naturally on Λ and the fixed sublattice Λ^{Γ} consists of the weights ω such that (α_{i}, ω) = 0 for all i ≠ 1, 6 by [BT, p. 108, Cor. 6.9]. In particular, the weight ω_{1} lies in Λ^{Γ}.

The weight (resp. root) lattice is the group of cocharacters 𝔾_{m} → T/C (resp. 𝔾_{m} → T), and the weight ω_{1} corresponds to a k-defined cocharacter \overline{ℓ}. Consulting the tables in the back of [Bo], we find that ω_{1} is not in the root lattice, but 3ω_{1} is. Let ℓ: 𝔾_{m} → T be the k-cocharacter corresponding to 3ω_{1}. We have a commutative diagram with exact rows

1 ⟶ μ_{3} ⟶ 𝔾_{m} ⟶^{3} 𝔾_{m} ⟶ 1
↓  _{ℓ}↓  \sqrt{ℓ}
1 ⟶ C ⟶ T ⟶ T/C ⟶ 1

Here, the arrow μ_{3} → C arises because the composition μ_{3} → 𝔾_{m} → T/C is the zero map. Since ω_{1} is not in the root lattice, ℓ is an injection, ergo the map μ_{3} → C is an injection. That is, the center C of G is contained in a rank 1 k-split torus, namely the image of ℓ. The map ι is zero by the same argument as in the case where G is split. (This paragraph and the previous one are an adaptation of the arguments behind the applications of the Gille-Merkurjev Norm Principle in the last section of [M 96]. The map μ_{3} → C constructed here is h(\overline{ℓ}) in the notation of [M 96, 1.3].)

If G is of type ^{1}E_{6}, the remaining possibility is that it has Tits index

[figure]

We claim that this is impossible. Indeed, the semisimple anisotropic kernel G_{an} of G is of type ^{1}A_{2} ×^{1}A_{2}. Since the Tits algebras of G are trivial, so are the Tits algebras of G_{an} by [Ti 71, 5.5.5], hence G_{an} is isotropic. This is a contradiction.

Now suppose that G is of type ^{2}E_{6} and let K be the separable quadratic extension of k over which G is of type ^{1}E_{6}. The hypothesis on the Tits algebras ensures that G is split or of type ^{1}E_{6,2}^{28} over K, hence that C is contained in a K-defined rank 1 split torus S in G. Not only is the map

∂_{K}: (G/C)(K) → H^{1}(K, C)

surjective, it is even surjective when one restricts to the rational subgroup S/C of G/C. The Gille-Merkurjev Norm Principle from [Gi], [M 96] shows that the image of ∂ contains the corestriction cor_{K/k} H^{1}(K, C). Since K/k is quadratic and C is 3-torsion, a restriction-corestriction argument shows that the corestriction H^{1}(K, C) → H^{1}(k, C) is surjective. □

We remark that the Gille-Merkurjev Norm Principle is proved for Galois cohomology and here we are using flat cohomology, and the two cohomology theories may give different values for H^{1}(k, C) in characteristic 3. However, the proof of the norm principle in [M 96] goes through with no changes since flat and Galois cohomology agree for reductive groups. The only point requiring checking is Merkurjev’s Lemma 3.11, which is easy to translate.

Remark 4.2. The same sort of proof gives an E_{7} version of Lemma 4.1: Let C be a central subgroup of an algebraic group G of type E_{7}. If G is isotropic, has trivial Tits algebras, and is not of type E_{7,1}^{66}, then the natural map (G/C)(k) → H^{1}(k, C) is surjective.

5. Proof of Theorem 0.3: type E_{6}

In this entire section, we assume that G is a simply connected group of type E_{6} that is split or of type ^{1}E_{6,2}^{28}. We write Z for the center of G; it is isomorphic to the group scheme μ_{3} of cube roots of unity.

Lemma 5.1. The group G contains a subgroup H that is reduced of type F_{4}, and the natural map

H^{1}(k, H) × H^{1}(k, Z) → H^{1}(k, G)

is surjective.

Proof. Write G_{sp} for the split simply connected group of type E_{6}. The split group H_{sp} of type F_{4} is contained in G_{sp}, and the induced map

(5.2)  H^{1}(k, H_{sp}) × H^{1}(k, Z) → H^{1}(k, G_{sp})

is surjective. This has been shown in [Ga, 3.4] under the assumption that k has characteristic different from 2, 3, but this restriction is unnecessary by the following reasoning. In all characteristics G_{sp} is the group of isometries of a cubic form—a norm for an Albert algebra J—and the elements of norm 1 in J form an open orbit in ℙ(J) [As, 3.16(3)]. The stabilizer of the identity element in J is of type F_{4} in all characteristics by [Sp, 4.6], hence the stabilizer of the identity element in ℙ(J) is F_{4} × μ_{3}. The conclusion now follows by [Ga, Lemma 3.1], where the last paragraph of the proof is replaced with an appeal to [DG, p. 373, Prop. III.4.4.6b], a flat cohomology analogue of [Se GC, I.5.4, Prop. 37]. This completes the proof when G is split.

Now suppose that G is of type ^{1}E_{6,2}^{28}. The semisimple anisotropic kernel G_{an} is simply connected of type D_{4}. It has trivial Tits algebras [Ti 71, 5.5.5], so is obtained by twisting the split simply connected group Spin_{8} of type D_{4} by a 1-cocycle α with values in Spin_{8}. The group Spin_{8} is even a subgroup of H_{sp} in G_{sp}. Twisting the split group G_{sp} by α produces a group of type 1E_{6} with the same semisimple anisotropic kernel as G, hence the group is isomorphic to G by Tits’ Witt-type theorem. Twisting H_{sp} by α, we find a group H of type F_{4} that is reduced. Twisting everything in (5.2) by α gives the lemma. □

Let r_{G} denote the Rost invariant of G as defined in [M 03]. The composition

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

is equal to an integer multiple nr_{H} of the Rost invariant r_{H} : H^{1}(k, H) → H^{3}(k, ℚ/ℤ(2)). Over a separable closure of k, the subgroup H from Lemma 5.1 is similar to the standard inclusion of the split F_{4} into the split E_{6}, and the integer multiple for this inclusion is 1 by [Ga, 2.4]. Since this multiplying factor does not change under scalar extensions [M 03, Prop. 7.9(4)], the composition (5.3) is equal to r_{H}.

By Lemma 4.1, the image of the map H^{1}(k, Z) → H^{1}(k, G) is zero. Since the Rost invariant is compatible with twisting, for (α, λ) ∈ H^{1}(k, H) × H^{1}(k, Z) we have:

(5.4)  r_{G}(α, λ) = r_{H}(α).

Proposition 5.5. The Rost invariant H^{1}(k, G) → H^{3}(k, ℚ/ℤ(2)) has kernel zero.

Proof. Suppose β is in the kernel. It is the image of some pair (α, λ) in H^{1}(k, H) × H^{1}(k, Z) such that r_{H}(α) is zero by 5.1 and (5.4).

The group H is the automorphism group of a reduced Albert algebra J. Let J′ be the Albert algebra corresponding to α. Since r_{H}(α) is zero, the Rost invariants of J and J′ relative to the split group of type F_{4} agree, i.e., f_{3}(J) = f_{3}(J′) and g_{3}(J) = g_{3}(J′). In particular, J′ is also reduced. An explicit formula for the cubic norm form on a reduced Albert algebra is well-known, and since the 3-Pfister forms f_{3}(J) and f_{3}(J′) are the same, it is easy to cook up an isometry between the norms on J and J′. Since G may be viewed as the group of isometries of the cubic norm on J, the image of α in H^{1}(k, G) is zero by descent.

The group H^{1}(k, Z) acts on H^{1}(k, G), and β is λ · (im α). Hence β is in the image of the map H^{1}(k, Z) → H^{1}(k, G). But this map has image zero by Lemma 4.1, hence β is zero. □

Now we may prove Theorem 0.3 for our G. By hypothesis, we are given a G-torsor X and finite extensions K_{1}, K_{2}, …, K_{n} of k such that X is trivial over K_{i} for all i. Fix a pair (α, λ) in H^{1}(k, H) × H^{1}(k, Z) mapping to X. Then r_{H}(α) equals r_{G}(X), and it is killed by K_{i} for all i. Let K be the extension constructed from H, α, and the K_{i} as in §3. The extension K kills r_{G}(X), hence trivializes X by the proposition. By construction, K is separable over k of degree dividing [K_{i} : k] for all i. This completes the proof of the theorem.

6. Final remarks

In the special case where d = 1 and X is a G-torsor, Totaro’s question overlaps an earlier question by Serre [Se 95, p. 233]. But even this simpler case is wide open for general groups of type F_{4}. For k of characteristic ≠ 2, 3, the following are equivalent:

For every k-group G of type F_{4}, every G-torsor X, and d = 1, the answer to (0.2) is “yes”.

(6.1)

Albert k-algebras are classified by their invariants f_{3}, f_{5}, and

g_{3}. (6.2)

Indeed, the equivalence of (6.1) and (6.2) is an easy consequence of a theorem of Rost from [Ro 02], which says: If J, J′ are Albert k-algebras such that f_{3}, f_{5}, and g_{3} agree for J and J′, then there exist extensions K_{1}, K_{2} such that J, J′ are isomorphic over K_{1} and K_{2} and such that the degrees [K_{1} : k] and [K_{2} : k] are coprime. But proving or disproving (6.2) is Question #1 on Petersson-Racine’s list of open problems in [PR 94]; it is viewed by many as the principal outstanding problem concerning Albert algebras.

Suppose for the moment that (6.1) is true. To extend (6.1) to the case of general d, one would want an analogue of Corollary 1.7 for the prime 3.

If the answer to Totaro’s question is yes for G simply connected of type 1E_{6} with trivial Tits algebras, then one can deduce (in a manner very similar to the above) that two Albert algebras are isotopic if and only if they have the same f_{3} and g_{3} invariants. Whether or not that is true is Question #4 in Petersson-Racine’s list.

Acknowledgements. The first author thanks FernUniversität Hagen for its hospitality while a first draft of this article was written, and Holger Petersson for the particularly simple argument for the prime 3 in §3.

The second author has been supported in part by the European research network “Algebraic K-Theory, Linear Algebraic Groups and Related Structures” HPRN-CT-2002-00287. He thanks the University of Ghent for its hospitality during a stay where some of the work on this article has been carried out.

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

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

Division of Pure Mathematics, School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, UK

   E-mail address: Detlev.Hoffmann@Nottingham.ac.uk