ANOTHER PROOF OF TOTARO’S THEOREM ON SPLITTING FIELDS OF E_{8}-TORSORS

VLADIMIR CHERNOUSOV*

Abstract. We give a short proof of Totaro’s theorem that every E_{8}-torsor over a field k becomes trivial over a finite separable extension of k of degree dividing d(E_{8}) = 2^{6}3^{2}5.

1. Introduction

In the paper we give a short proof of the following theorem due to B. Totaro [7].

Theorem 1.1. Let k be an arbitrary field. Then every E_{8}-torsor defined over k becomes trivial over a finite separable extension of k of degree dividing d(E_{8}) = 2^{6}3^{2}5.

Note that in a second paper on E_{8}-torsors [8], Totaro showed that the bound 2^{6}3^{2}5 is exact, i.e. there is an E_{8}-torsor that can not be split by an extension whose degree is a proper divisor of 2^{6}3^{2}5.

The original proof of Theorem 1.1 is based on an analysis of the subgroup structure of the Weyl group of type E_{8}, Brauer’s theory of blocks, Aschbacher’s theorem on the maximal subgroups of the classical groups over finite fields, and the classification of solvable primitive linear groups. Moreover, some of the computations in [7] were made with the aid of a computer. The aim of the present paper is to simplify the proof. Eventually following the main Totaro’s idea on considering Galois orbits in the corresponding root system Σ(E_{8}) we give a short straightforward proof of Theorem 1.1.

2. Generic case and possible bad cases

Let G_{0} be a split group of type E_{8} over k. Let ξ ∈ Z^{1}(k, G_{0}), and let G = ^{ξ}G_{0} be the corresponding twisted group. Consider a maximal kdefined torus T ⊂ G. Let E/k be a minimal finite extension splitting T.

2000 Mathematics Subject Classification. Primary 11E72; Secondary 14M17, 20G15.

* Supported by the Canada Research Chairs Program, and by NSERC’s Grant G121210944.

The extension K/k is necessarily Galois, and its Galois group Γ acts in a natural way on the root system Σ = Σ(G, T) of G with respect to T. This gives rise to a canonical embedding Γ ↪ W where W = W (E_{8}) is the corresponding Weyl group. If we choose a base of Σ, then the action of Γ on Σ induces an action of Γ on the set R = Σ/(±1). This set has 120 elements, and we always choose positive roots as representatives of the elements of R.

The case of “generic” E_{8}-torsors is easy.

Lemma 2.1. Assume that Γ has an orbit on R of size dividing 120 = 2^{3}·3·5. Then there is a finite separable extension L/k of degree dividing d(E_{8}) such that G splits over L.

Proof. Let α ∈ R be such that |Γ(α)| divides 120. Let Stab_{Γ}(α) be the stabilizer of α in Γ, and consider the subfield L_{1} ⊂ E corresponding to Stab_{Γ}(α). Taking an extension L_{2}/L_{1} of degree 2 if necessary, we may assume that Σ has a root α stable with respect to an (absolute) Galois group of L_{2}. The centralizer Σ′ of α in Σ is the subsystem of type E_{7} which is stable with respect to the Galois group of L_{2}. If H ⊂ G is the subgroup in G of type E_{7} corresponding to Σ′, then H is L_{2}-defined and, by a result of Tits [6], splits over a separable extension L_{3}/L_{2} of degree dividing 2^{2}3. Clearly L_{3} also splits G, and [L_{3} : k] = [L_{3} : L_{2}][L_{2} : L_{1}][L_{1} : k] divides (2^{2}3)2(120) = 2^{6}3^{2}5, as required. □

If Σ contains a proper subroot system stable with respect to Γ, then using known results on groups of classical types and Tits results [6] on splitting fields of groups of types G_{2}, F_{4}, E_{6}, E_{7}, it is easy to conclude that G splits over a finite separable extension of k of degree dividing d(E_{8}). Thus, we may henceforth assume without loss of generality that Σ does not contain root subsystems stable with respect to Γ. In this case, possible “bad” orbit decompositions are given by the following:

Lemma 2.2. ( [7], Lemma 4.1) If Γ has no orbits on R of size dividing 120, then the orbit sizes of Γ are either

(a) 64+ ( multiples of 7 summing to 56);

(b) 50+ (multiples of 7 summing to 70);

(c) 45+ (multiples of 25 summing to 75);

(d) 36+ (multiples of 7 summing to 84) or

(e) (multiples of 16 summing to 48) + (multiples of 9 summing to 72).

For convenience of the reader we give a sketch of the proof due to Totaro. It is based on the following result.

ANOTHER PROOF OF TOTARO’S THEOREM ON E_{8}-TORSORS 3

Lemma 2.3. (i) A 7-Sylow subgroup of W has only one fixed point in R. (ii) A 5-Sylow subgroup of W has 4 orbits of size 25 and 4 orbits of size 5 in R.

Proof. This is easy to check by direct inspection. □

Proof of Lemma 2.2. Let us first assume that 7 divides |Γ|. Then, by Lemma 2.3, all orbits of Γ in R have sizes divisible by 7 except for one whose size is ≡ 1 modulo 7. The size of this exceptional orbit is either 36, 50 or 64, since by our assumption there is no orbit of size dividing 120. Thus, assuming that |Γ| is a multiple of 7 we have cases (a), (b), (d).

Assume next that |Γ| is not divisible by 7, but divisible by 25. Since the sum of sizes of all orbits of Γ in R is 120, and sizes of orbits do not divide 120 we find, by Lemma 2.3, that all orbits of Γ have size divisible by 25 except for one whose size is 45. Hence we have case (c).

Finally, assume that the order of Γ is divisible by neither 7 nor 25. Recall that |W | = 2^{14}3^{5}5^{2}7. Since there is no orbit of Γ whose size divide 120, all of them have sizes a multiple of 16 or 9. The only way it can happen is case (e). Lemma 2.2 is proved. □

By [7], Lemma 6.1, cases (b), (c) are impossible. By [7], Lemma 4.2, in case (a) the complementary subset to the orbit of size 64 forms a subsystem of type D_{8}. The remaining cases (d) and (e), which caused most of the complications in [7], will be dealt with in a simple fashion in the following two sections.

For later use, we need the following fact related to the Rost invariant for E_{7}. For the definition and properties of the Rost invariant R_{G} of an algebraic group G we refer to [4].

Proposition 2.4. Let H_{0} be a split simple simply connected algebraic group of type E_{7} defined over an arbitrary field K, and let

R_{H_{0}} : H^{1}(K, H_{0}) → H^{3}(K, Q/Z(2))

be the Rost invariant of H_{0}. Let ξ ∈ H^{1}(K, H_{0}) be such that the 3component of R_{H_{0}}(ξ) is trivial. Then there is a separable extension L/K of degree dividing 4 such that ξ is trivial over L.

Proof. By [6], there is a quasi-split subgroup H′ ⊂ H_{0} of type E_{6} such that ξ is in the image of H^{1}(K, H′) → H^{1}(K, H_{0}). Taking a proper quadratic extension E/K if necessary, we may assume that H′ is split over E. One knows that for a split group H′_{E} of type E_{6} the 2component of R_{H′}(ξ_{E}), where ξ_{E} is the image of ξ under the restriction map H^{1}(K, H_{0}) → H^{1}(E, H_{0}), is a symbol. Taking again a separable quadratic extension L/E killing this symbol we may assume that the 2component of R_{H′}(ξ_{L}) is trivial over L. Then ξ_{L} ∈ Ker R_{H′}. It remains to observe that Ker R_{H′} = 1, by [3] (see also [2]). □

3. An orbit of size 36

Let R_{1} ⊂ R be an orbit of Γ of size 36, and let R_{2} = R \ R_{1}. Take a positive root α ∈ R_{1} and consider Γ_{1} = Stab_{Γ}(α). Note that in the definition of Γ_{1}, α is viewed as an element of R, but not of Σ. Let E′_{1} ⊂ E be the subfield corresponding to Γ_{1}. Taking a proper quadratic extension E_{1}/E′_{1} if necessary, we may assume that α viewed as a root in Σ is stable with respect to an (absolute) Galois group of E_{1}. Since |R_{1}| = 36, the index [E_{1} : k] is either 2^{2}3^{2} or 2^{3}3^{2}.

Lemma 3.1. If the 3-component of R_{G_{0}}([ξ]) is trivial over E_{1}, then there is a separable extension E_{2}/k of degree dividing 2^{5}3^{2} which kills ξ.

Proof. Let Σ′ be the root subsystem of Σ consisting of roots orthogonal to α. Consider the subgroup H of G corresponding to Σ′. It has type E_{7} and is defined over E_{1} since so is α. Since H contains a semisimple anisotropic E_{1}-kernel of G, by a result due to R. Steinberg (cf. [2], Theorem 3.2), there is a cocycle ξ_{1} ∈ Z^{1}(E_{1}, H_{0}), where H_{0} ⊂ G_{0} is a canonical E_{1}-split subgroup of type E_{7}, such that ξ is equivalent to ξ_{1} over E_{1}. Note that R_{G_{0}}(ξ) = R_{H_{0}}(ξ_{1}). Then, by Proposition 2.4, there is a separable extension E_{2}/E_{1} of degree dividing 4 which kills ξ_{1}, and hence ξ. Its degree over k divides 4(2^{3}3^{2}), as required. □

By Lemma 3.1, we may henceforth assume without loss of generality that the 3-component of R_{G_{0}}([ξ]) is nontrivial over E_{1}.

Lemma 3.2. Let β ∈ R_{2}. Then |Γ_{1}(β)| is multiple of 21.

Proof. Since Γ_{1} contains a 7-Sylow subgroup of W, the size of Γ_{1}(β) is divisible by 7 by Lemma 2.3 (i). Assume that |Γ_{1}(β)| is not divisible by 3. Take the extension E_{2}/E_{1} of degree prime to 3 corresponding to the stabilizer Γ_{2} = Stab_{Γ_{1}}(β). By a counting argument, there are at least two roots in R_{2} different from β whose Γ_{2}-orbits have sizes not divisible by 3. Repeating the above construction 2 times, we can find a finite extension E/E_{1} of degree prime to 3 with the property that an (absolute) Galois group of E stabilizers α and at least 3 roots in R_{2}. Then it follows from Tits’ classification [5] that the E-rank of G is at most 5. Again, by Tits’ classification, all simple groups which could appear in a semisimple E-anisotropic kernel of G have trivial 3components of the Rost invariant, implying therefore that R_{G_{0}}(ξ_{E}) has also trivial 3-component. On the other hand, since [E : E_{1}] is prime to 3, the 3-component of R_{G_{0}}(ξ_{E}) is still nontrivial – a contradiction. □

Recall that we assumed that Σ has no subroot systems stable with respect to Γ; in particular we may assume that R_{1} is not a subroot system. It follows that there is δ ∈ R_{1} such that either α + δ or α − δ is a root, call it β = α ± δ, belonging to R_{2}. Since the size of Γ_{1}(β) is divisible by 21, so is |Γ_{1}(δ)|. Since R_{1} consists of 36 elements, the size of Γ_{1}(δ), hence that of Γ_{1}(β), is exactly 21.

Let R′_{1} = Γ_{1}(δ), R′′_{1} = R_{1} \ R′_{1}, R′_{2} = Γ_{1}(β), R′′_{2} = R_{2} \ R′_{2}. Recall that we denote the subsystem of Σ of type E_{7} consisting of all roots in Σ orthogonal to α by Σ′. Lemma 3.3. ±R′′_{2} coincides with Σ′. Proof. Since (α, β) = ±1 and (α, δ) = ±1, the intersection of Σ′/ ± 1 with R′_{1} and R′_{2} is empty, hence

(Σ′/ ± 1) = ((Σ′/ ± 1) ∩ R′′_{1}) ∪ ((Σ′/ ± 1) ∩ R′′_{2}.)

The order of (Σ′/ ± 1) ∩ R′′_{2} being Γ_{1}-stable is divisible by 21. Since R′′_{1} has order 16 and |Σ′/ ± 1| = 63, we have (Σ′/ ± 1) ∩ R′′_{1} = ∅. □

As a direct consequence of the above lemma we have Corollary 3.4. (i) (α, γ) = ±1, if γ ∈ R_{1} and γ ≠ α. (ii) α ± γ_{1} ∈ R′′_{1}, if γ_{1} ∈ R′′_{1}. (iii) (γ_{1}, γ_{2}) = ±1, if γ_{1}, γ_{2} ∈ R_{1}, γ_{1} ≠ γ_{2}. Proof. Properties (i) and (ii) are clear since (Σ′/ ± 1) ⊂ R_{2}. Property (iii) follows from (i), since α was an arbitrary root in R_{1}. □ Lemma 3.5. ±R′′_{1} is a subroot system of Σ. Proof. Let γ ∈ R′′_{1}. We have to show that γ ± γ′ ∈ R′′_{1} for all γ′ ∈ R′′_{2} different from γ. Arguing as above, we see that there exists a subset R′_{1,γ} of R_{1}, with 21 elements, comprised of roots whose sum with γ is in R_{2}. By Corollary 3.4, the remaining 14 roots in R_{1} \ R′_{1,γ} have sum with γ in R_{1} \ R′_{1,γ}. We will be finished if we show that R′_{1,γ} = R′_{1}.

Let δ ∈ R′_{1}. By Corollary 3.4 (iii), either γ + δ or γ − δ is a root. Call it β. Since (α, β) ≡ 0 modulo 2, we have either α = ±β or β ∈ Σ′ = R′′_{2}. The first case is impossible, since the Γ_{1}-orbits of δ and γ consist of 21 and at most 14 elements respectively. Then β ∈ R_{2}, so that δ ∈ R′_{1,γ}. □

To finish the consideration of orbits of size 36, it remains to note that the subroot system R′′_{1} is Γ_{1}-stable, hence it has an automorphism of order 7. However the minimal simple root system having an automorphism of order 7 has type A_{6} and consists of 42 elements.

4. An orbit of size a multiple of 16

We start with an explicit description of a 3-Sylow subgroup of W, denoted below by Ψ, and its action on the root system Σ. Recall that |Ψ| = 3^{5}. Let Π = {α_{1}, …, α_{8}} be a fixed basis of Σ. Here and below we label roots as in [1]. Consider the subroot system of type E_{6} ×A_{2} in Σ generated by Σ_{1} = ⟨ α_{1}, …, α_{6} ⟩ and Σ_{2} = ⟨ α_{8}, −α ⟩ where α is the highest root of Σ^{+}. Comparing the orders of the Weyl groups of type E_{6}, A_{2}, E_{8}, we find that the direct product Ψ = Ψ_{1} × Ψ_{2} of 3-Sylow subgroups Ψ_{1} of W (E_{6}) and Ψ_{2} of W (A_{2}) is a 3-Sylow subgroup of W.

Recall that Ψ_{2} has order 3. As for Ψ_{2}, we choose the subgroup in W (A_{2}) generated by the element e which takes α_{8} into −α and −α into −(α_{8} − α).

The root system Σ_{1} contains a subroot system Σ_{3} of type A_{2}×A_{2}×A_{2} generated by the roots ⟨α_{1}, α_{3}⟩, ⟨α_{5}, α_{6}⟩ and ⟨α_{2}, −β⟩ respectively, where β is the positive root of maximal length in Σ_{1} with respect to the basis α_{1}, …, α_{6}. Let w_{0}, w_{1} ∈ W (E_{6}) be the elements of maximal length with respect to the bases {α_{1}, …, α_{6}} and {α_{1}, α_{3}, α_{4}, α_{2}, −β, α_{5}} respectively. Let d = w_{0}w_{1}. It is easy to see that d has order 3 and takes the roots α_{1}, α_{3}, α_{5}, α_{6}, α_{2}, −β into α_{6}, α_{5}, α_{2}, −β, α_{3}, α_{1} respectively. Therefore d permutes the components of Σ_{3} and their Weyl groups.

Let a be an arbitrary element of order 3 in the Weyl group of the first component of Σ_{3}. Denote b = dad^{−1} and c = dbd^{−1}. Clearly, a, b, c commute and d permutes them. Consider the subgroup Ψ_{1} in W (E_{6}) generated by a, b, c, d. Since Ψ_{1} has order 3^{4}, it is a 3-Sylow subgroup of W (E_{6}).

One easily checks that there are 4 orbits of Ψ on R which are as follows. The Ψ-orbit of α_{7} consists of 81 elements in Σ^{+} \ {Σ_{1}^{+} ∪ Σ_{2}^{+}}. The Ψ-orbit of α_{1} consists of 9 elements and coincides with Σ_{3}^{+}. The Ψ-orbit of α_{8} consists of 3 elements in Σ_{2}^{+} = {α_{8}, α, α − α_{8}}. Lastly, the Ψ-orbit of α_{4} consists of the remaining 27 elements in Σ_{1}^{+} \ Σ_{3}^{+}.

We also need information about the stabilizer Stab_{Ψ}(β) of a root β ∈ R. It is easy to see that for each root β ∈ Ψ(α_{7}) = Σ^{+} \{Σ_{1}^{+} ∪Σ_{2}^{+}} one has Stab_{Ψ}(β) ⊂ ⟨ a ⟩∪⟨ b ⟩∪⟨ c ⟩. Furthermore, for each β ∈ Ψ(α_{4}), Stab_{Ψ_{1}}(β) has order 3 and is generated by an element of the form da^{ϵ_{1}}b^{ϵ_{2}}c^{ϵ_{3}} where ϵ_{i} is 0, 1 or 2.

Let R_{1} and R_{2} be unions of orbits of Γ whose sizes are divisible by 16 and 9 respectively. Let Γ_{3} ≤ Γ be a 3-Sylow subgroup. Without loss of generality we may assume that Γ_{3} is a subgroup of Ψ.

Lemma 4.1. |Γ_{3}| ≤ 3^{3}.

Proof. If |Γ_{3}| = 3^{5}, then Γ_{3} = Ψ and hence Γ_{3} has the orbit Γ_{3}(α_{7}) = Ψ(α_{7}) of size 81; which is impossible.

Assume that |Γ_{3}| = 3^{4} = 81. Then Γ_{3} is a normal subgroup in Ψ and hence Ψ acts in a natural way on Γ_{3}-orbits. Since Ψ has the orbit Ψ(α_{7}) of size 81, Γ_{3} has at least three orbits of size 27. Since R_{1} and R_{2} contain at most one and two orbits of size 27 respectively, we find that Γ_{3} has exactly 3 orbits of size 27 and their union is necessarily Σ^{+} \ {Σ_{1}^{+} ∪ Σ_{2}^{+}}. It follows that for each β ∈ Σ^{+} \ {Σ_{1}^{+} ∪ Σ_{2}^{+}} we have Stab_{Ψ}(β) ⊂ Γ_{3} and this implies ⟨ a, b, c ⟩ ⊂ Γ_{3}. But then the orbit Γ_{3}(α_{4}) contains at least 27 elements giving thus the fourth orbit of size 27 – a contradiction. □

We are ready to finish the proof. Since |Γ_{3}| ≤ 27, the Γ_{3}-orbits of roots in R_{2} have sizes divisible by 9 or 27. Since |R_{2}| = 72, there is at least one β ∈ R_{2} such that the size of its Γ_{3}-orbit is not divisible by 27. As in § 3, consider Γ′ = Stab_{Γ}(β) and let E_{1} ⊂ E be the subfield corresponding to Γ′. If the 3-component of R_{G_{0}}(ξ) is trivial over E_{1}, then the same argument as in Lemma 3.1 completes the proof. Thus we may assume without loss of generality that |Γ_{3}| = 27, and that for each root β ∈ R_{2}, whose Γ_{3}-orbit has size divisible by 9 but not by 27, the 3-component of R_{G_{0}}(ξ) is nontrivial over the corresponding field E_{1}.

Note that in this possible “bad” case we have that Stab_{Γ_{3}}(β), being a group of order 3, is a 3-Sylow subgroup of Γ′. By arguing as in Lemma 3.2, we may therefore additionally assume that a nontrivial x ∈ Stab_{Γ_{3}}(β) has at most 3 invariant positive roots with respect to the canonical action of Γ_{3} ⊂ W on Σ. In particular, this assumption implies that for each root in R_{2} ∩ (Σ^{+} \ {Σ_{1}^{+} ∪ Σ_{2}^{+}}) its Γ_{3}-orbit has size 27, hence that β with the above property is in Σ_{1}^{+}. We also have e ∉ Γ_{3}, since each root in Σ_{1} is stable with respect to e.

Consider the canonical morphism

f : Ψ → Ψ/⟨ e ⟩ ≃ Ψ_{1} = ⟨ a, b, c, d ⟩.

Since e ∉ Γ_{3}, the image f(Γ_{3}) has order 27, hence it is a normal subgroup in Ψ_{1}. As in Lemma 4.1, we find that Ψ_{1} acts on Γ_{3}-orbits of Γ_{3} on Σ_{1}^{+}. Thus Σ_{1}^{+} \ Σ_{3}^{+}, being a unique Ψ_{1}-orbit of size 27, is a disjoint union of 3 Γ_{3}-orbits of size 9. Then for each root β ∈ Σ_{1}^{+} \ Σ_{3}^{+}, Stab_{Ψ_{1}}(β), being a group of order 3, is contained in Γ_{3}. However it is easy to see that all such stabilizers generate Ψ_{2}, whose order is 3^{4}. This contradicts our assumption that |Γ_{3}| = 27.

Acknowledgement

Work on the present paper started in 2002 at the Centre Bernoulli, E.P.F.L., Lausanne, and continued at Bielefeld University in 2003. The author thanks both centres for their hospitality and their stimulating atmosphere. The author would also like to thank Forschergruppe “Spektrale Analysis, asymptotische Verteilungen und stochastische Dynamik” for its support.

References

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[7] B. Totaro, Splitting fields for E_{8}-torsors, Duke Math. J. 121 (2004), no. 3, 425–455.

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Department of mathematical sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1

E-mail address: chernous@math.ualberta.ca