UNIPOTENT SUBGROUPS OF REDUCTIVE GROUPS

IN CHARACTERISTIC p > 0

PHILIPPE GILLE

Abstract. — We prove two conjectures of Tits on the unipotent elements of semisimple algebraic groups defined over a field with positive characteristic.

Summary

1. — Introduction 1.1. — Good finite subgroups 1.2. — The conjectures of Tits 1.3. — Complements on finite subgroups

2. — Finite unipotent subgroups and Galois cohomology 2.1. — A Galois cohomology class 2.2. — How to see if a finite unipotent subgroup is k–good 2.3. — The case [k : k^{p}] ≤ p 2.4. — Bad unipotents and torsors on the affine line

3. — Conjugacy classes of k–anisotropic automorphisms of split groups

4. — Anisotropic automorphisms are special 4.1. — Special automorphisms 4.2. — Triality 4.3. — Lifting in characteristic 0 4.4. — End of the proof of Theorem 3

1. — Introduction 1.1. — Good finite subgroups. Let k be a field of characteristic p ≥ 0, and then k_{s} be a separable closure of k. Let G/k be a reductive connected algebraic group. For the notions on algebraic groups used here, we refer to the paper by Borel and Tits [BT1] and the book of Demazure and Gabriel [DG]. If G is semisimple and absolutely almost simple, we denote by S(G) the finite set of torsion primes of G as defined by Serre in [Se2]. In the case of G simply connected, the the Dynkin index will be denoted by d_{G} (cf. [LS], § 2). Note taht the prime factors of d_{G} lie in S(G). We recall Tits’ definition of good unipotent elements.

DEFINITION 1 [T2]. — Let u be a unipotent element of G(k). The element u is k–good if u lies in the unipotent radical of a k–parabolic subgroup of G, otherwise u is k–bad.

In characteristic 0, all unipotent elements are k–good, so the definition is relevant only if char(k) = p > 0 ; in that case, a unipotent element u has finite order q = p^{r} and can be viewed as a morphism u^{♯} : ℤ/qℤ → G sending 1 to u. This viewpoint leads to the following definition of goodness for morphisms M → G where M/k is a finite étale group.

DEFINITION 1’. — Let M/k be a finite étale group and ϕ : M/k → G be a morphism. The morphism ϕ is k–good if ϕ(M(k_{s})) lies in the radical of a k–parabolic subgroup of G, otherwise ϕ is k–bad.

2000 Mathematics Subject Classification : 20G15, 12G05.

P. GILLE

If ϕ is injective, we also say that M is a k–good subgroup of G. We say that ϕ : M → G is unipotent if ϕ(M(k_{s})) consists of unipotent elements. Let us prove that the two definitions are compatible.

LEMMA 1. — Assume char(k) = p > 0. Let u be a unipotent element of order q and u^{♯} : ℤ/qℤ → G be the associated morphism. Then u is k–good if and only if u^{♯} is k–good.

Proof : If u is k–good, it is obvious that the morphism u^{♯} is k–good. Conversely, assume that the morphism u^{♯} : ℤ/qℤ → G is k–good. Then u = u^{♯}(1) lies in the radical R(P) of a k–parabolic subgroup P of G, which is an extension of a k–torus S by the unipotent radical R_{u}P of P, i.e. we have an exact sequence 1 → R_{u}(P) → R(P) → S → 1. As S(k) has no element of order p, u lies in R_{u}(P)(k), and u is k–good.

DEFINITION 2 [T2]. — Let α be a k–automorphism of G of finite order. The morphism α is k–isotropic if α normalizes some proper k–parabolic subgroup of G, otherwise ϕ is k–anisotropic.

As above, we can extend this definition to isotropic morphisms M → Aut(G) which will yield the previous one in the case M = ℤ/nℤ.

DEFINITION 2’. — Let ϕ ∈ Hom_{k−gr}(M, Aut(G)). The morphism ϕ is k–isotropic if ϕ(M(k_{s})) normalizes some proper k–parabolic subgroup of G, otherwise ϕ is k–anisotropic.

We say that ϕ ∈ Hom_{k−gr}(M, G) is k–isotropic if the composite Ad∘ϕ : M → G is k–isotropic. Obviously, k–anisotropic implies k–bad.

1.2. — The conjectures of Tits. Our first result is the proof of the following conjecture (see Theorem 2 of § 2).

CONJECTURE 1 [T3]. — Assume p > 0, [k : k^{p}] ≤ p and that G/k is semisimple and simply connected. Then every unipotent subgroup of G(k) (i.e a subgroup consisting of unipotent elements) is k–embeddable into the unipotent radical of a k-parabolic subgroup of G.

The case p = 2 is due to Tits [T2, § 4.5]. Further known cases are when p is not a torsion prime of G, and types A_{n} and C_{n} (loc. cit., § 3.5 and 4.4). In conjecture 1, the condition [k : k^{p}] ≤ p is necessary as the following theorem of Tits shows.

THEOREM 1 [T4, prop. S1] and [T5, th. 7]. — Assume that G is split simply connected and almost simple. If [k : k^{p}] ≥ p^{2} and p divides d_{G}, then the group G(k) contains a k–bad unipotent element of order p.

Our proof of conjecture 1 is based on the reduction to a problem in Galois cohomology for the group G_{k((t))}, which is done in Proposition 3 (§ 2.2), and on known cases of Serre’s conjecture II in Galois cohomology [Gi1]. Our second result is the proof of the following conjecture (see Theorem 3 of § 4), which is related to the shape of conjugacy classes of k–anisotropic automorphisms.

CONJECTURE 2 [T2]. — Assume p > 0 and that G/k is split and almost simple. Let α be a k–anisotropic automorphism of order p. Then α normalizes a maximal k–split torus of G.

Let us recall first the known cases handled by Tits.

1) p = 2, all types,

2) p = 3, G of type D_{4} or E_{6}, using a triality argument due to Harder [H],

3) type A_{n}. It turns out that conjecture 2 and a large part of the present paper make sense in a broader

setting, including the characteristic zero case. To this end, we formulate the following conjecture, which extends the preceeding one.

UNIPOTENT SUBGROUPS

CONJECTURE 2’. — Assume G/k is split almost simple. Let l be a prime of S(G). We assume that l = char(k) or either that k contains a primitive l–root of unity. Let α be a k–anisotropic automorphism of order l. Then α normalizes a maximal k–split torus of G.

Such anisotropic automorphisms are called special automorphisms. In section 4, we prove conjecture 2’ except for the E_{8} case. However, the case of E_{8} and l = p = 5 is proven in [Gi3]. Tits’ original conjecture is thus true.

The proof of conjecture 2’ requires several steps. First we work in characteristic zero and prove by a case-by-case analysis conjecture 2’ (except for the E_{8} case) by using methods inspired by the proof of Hasse principle (§ 4.2). The tame case l ≠ p is then not hard and we can concentrate on the wild case l = p = char(k). We consider a complete discrete valuation ring A with residue field k and fraction field F_{A} of characteristic zero. We remark that special elements lift in characteristic zero, i.e. if α ∈ Aut(G)(k) is a special automorphism of order p, then there exists α̃ ∈ Aut(G)(A) of order p specializing to α. We show that anisotropic automorphisms lift in characteristic 0 (§ 4.3). The proof concludes by a specialization argument.

1.3. — Complements on finite subgroups. In this section, we remark on the extension of some known results to our present setting. The argument as used in the proof of proposition 3.2 of [T2] yields in fact the following.

PROPOSITION 1. — Let ϕ ∈ Hom_{k−gr}(M, G), and let P/k be a k–parabolic subgroup normalized by ϕ. Let L be a Levi subgroup of P defined over k, so that P = R_{u}(P) ⋊ L and let ϕ′ be the composition of M → P → L.

a) The following are equivalent : i) ϕ is k–good in G ii) ϕ′ is k–good in G iii) ϕ′ is k–good in L.

b) If P is minimal among all k–parabolic subgroups normalized by ϕ, then ϕ′ is k– anisotropic in L.

Similarly, we can extend lemma 3.5 of [T2].

LEMMA 2. — Let H/k be a reductive subgroup of G/k. Let ϕ : M → Aut(G, H) ⊂ Aut(G) be a k–anisotropic morphism. Then the morphism ϕ : M → Aut(G, H) → Aut(H) is k–anisotropic.

We also recall :

PROPOSITION 2 ([BT2], prop. 3.6). — Let U ⊂ G(k) be a unipotent subgroup such that every element of U is k_{s}–embeddable in a Borel subgroup. Then U is k–embeddable in the unipotent radical of a k–parabolic subgroup of G.

In particular and in contrast to the tame case, a unipotent element u is k–good iff it is k_{s}–good.

LEMMA 3. — Let g ∈ G(k) be an element of finite order invertible in k. Then g lies in a maximal k–torus of G. In particular g is k_{s}–good.

Proof : Let g ∈ G(k) be an element of finite order invertible in k. Then g is semisimple and g lies in some maximal torus T/k. As T ×_{k} k_{s} is a split maximal torus, g is k_{s}–good.

More generally, if char(k) = 0, Borel-Mostow’s theorem asserts that an automorphism of G of finite order normalizes a maximal torus of G [BM] (see also [P]).

2. — Finite unipotent subgroups and Galois cohomology

2.1. — A Galois cohomology class. Let us set K = k((t)) and denote by K_{mod} a maximal tamely ramified extension of K. Let I = I_{w} ⋊ μ_{m} be a k–étale group, semi–direct product of μ_{m} (m ∈ k^{×}) by a constant p–group I_{w}. Throughout this paper, we assume that

there exists a totally ramified field extension L/K such that Spec(L) → Spec(K) is an I– torsor.

This hypothesis is satisfied in the two extreme cases I = μ_{m} (obviously) and I = I_{w} since the maximal pro–p quotient of Gal(K_{s}/K) is a free pro–p group [Se1, § II.2.2, Cor. 1]. We denote by L̃ = L.K(μ_{m}), which is a Galois extension of K with Galois group Ĩ. So, there is a diagram of field extensions

L̃
⧸  ⧹
L  K(μ_{m}),
⧹  ⧸
L ∩ K(μ_{m})
|
K

and the residue field of L̃ is k(μ_{m}). We denote by f_{θ} ∈ Z^{1}(Ĩ, I(L̃)) the 1–cocycle defined by the I–torsor Spec(L) → Spec(K) and by θ ∈ H^{1}(K, I) its class. For simplicity, the reader can think of I = ℤ/pℤ; this case, which is of main interest for us, was treated in [Gi2].

We consider the set Hom_{k−gr}(I, G) and pick ϕ ∈ Hom_{k−gr}(I, G). The key point, due to Serre, is to associate to ϕ : I → G the 1–cocycle

f_{ϕ} = ϕ_{∗}(f_{θ}) in Z^{1}(Ĩ, G(L̃))

and the cohomology class

γ(ϕ) = ϕ_{∗}(θ) = [f_{ϕ}] in H^{1}(K, G).

Remark 1 : We shall use functoriality for a surjective base change λ : I′ → I and an extension L′/L/K of monodromy group I′ : if there exists θ′ such that λ_{∗}(θ′) = θ, then ϕ_{∗}(θ)_{L′} = λ^{∗}(ϕ∘λ) in H^{1}(L′, G).

2.2. — How to see if a finite unipotent subgroup is k–good

PROPOSITION 3. — a) If ϕ is k–good, then γ(ϕ) = 1 in H^{1}(K, G).

b) Assume that ϕ is unipotent. Then the following assertions are equivalent : i) ϕ is k–good, ii) γ(ϕ) = 1 in H^{1}(K, G), iii) γ(ϕ)_{K_{mod}} = 1 ∈ H^{1}(K_{mod}, G).

Proof : a) : Let P/k be a parabolic subgroup of G/k such that the radical R(P) contains Im(ϕ). The group R(P) is split and we know that H^{1}(K, R(P)) = 1. Thus γ(ϕ) = 1.

b) : As we consider unipotent morphisms, we can assume that I = I_{w}, i.e. I is a finite p–group. There is only the implication iii) ⟹ i) to prove. By hypothesis, there exists a tamely ramified extension K′/K of valuation ring O′/O and residue extension k′/k such that γ(ϕ)_{K′} = 1 in H^{1}(K′, G). One can assume K′ = k′((t′)), (t′)^{e} = t and (e, p) = 1, and that G_{k′} is split. Let B/k′ be a Borel subgroup of G/k′. We consider the extension L′ = L.K′/K′, which is Galois of group I = Gal(L′/K′) and valuation ring O_{L′}. Let us denote by X/k′ = G/B the variety of Borel subgroups of G and by π : G/k′ → X/k′ the canonical map. This variety is projective, the morphism π is smooth and one knows that the map on k′–points G(k′) → X(k′) is onto. Using Hensel’s lemma and the valuative criterion of properness, one obtains easily that the map π : G(O_{L′}) → X(O_{L′}) is onto and the map G(O_{L′})/B(O_{L′}) ⟶^{∼} G(k′)/B(k′) is bijective. We have the following commutative diagram of pointed sets (cf. [Se1], § 1.5.4, prop. 36)

[G(O_{L′})/B(O_{L′})]^{I} ⟶ H^{1}(I, B(O_{L′})) ⟶ H^{1}(I, G(O_{L′}))
↓ ≀  ↓  ↓
[G(L′)/B(L′)]^{I} ⟶ H^{1}(I, B(L′)) ⟶ H^{1}(I, G(L′)).

Taking the fixed points of the twisted I–sets by the cocycle f_{ϕ}, one gets

[_{f_{ϕ}}(G(O_{L′})/B(O_{L′}))]^{I} ⟶^{∼} [_{f_{ϕ}}(G(L′)/B(L′))]^{I},

so the class [f_{ϕ}] ∈ H^{1}(I, G(O_{L′})) has a reduction in H^{1}(I, B(O_{L′})). Thus there exists h ∈ G(O_{L′}) such that hϕ(σ)(^{σ}h^{−1}) ∈ B(O_{L′}) for any σ ∈ I. Let \overline{h} be the image of h in G(k′). Then \overline{h}ϕ(σ)(\overline{h})^{−1} ∈ B(k′) for any σ ∈ I (we used here that L/K is totally ramified) and ϕ is k′– good. By Proposition 2, because k′/k is separable, ϕ is k–good.

The wild inertia group Gal(K_{s}/K_{mod}) is a pro–p group. If p is a good prime for G, i.e. if p is not a torsion prime of G, we know that H^{1}(K_{mod}, G) = 1 [Se2, th 4”]. The last proposition together with Proposition 2 gives then another proof of Tits’ result [T2, cor. 2.6].

COROLLARY 1. — Assume p = char(k) ∉ S(G). Then any unipotent subgroup U of G(k) is k–embeddable in the unipotent radical of a k–parabolic subgroup of G.

2.3. — The case [k : k^{p}] ≤ p

THEOREM 2. — Assume char(k) = p > 0, [k : k^{p}] ≤ p and that G is semisimple simply connected. Then every unipotent subgroup of G(k) is k–embeddable in the unipotent radical of a k-parabolic subgroup of G.

We recall Kato’s definition [K] of the p–dimension dim_{p}(k) of k by means of cohomology groups H_{p}^{i}(k). Let Ω_{k} be the k-vector space of the 1–differential forms of the ℤ-algebra k. For any nonnegative integer i, we set Ω_{k}^{i} = ⋀^{i} Ω_{k} and the exterior differential d maps Ω_{k}^{i} to Ω_{k}^{i−1}. There exists an unique additive p–linear application γ : Ω_{k}^{i} → Ω_{k}^{i}/dΩ_{k}^{i−1} such that γ(xω) = x^{p}w for any logarithmic differential form ω = dy_{1}/y_{1} ∧ ⋯ ∧ dy_{i}/y_{i}. The operator ω is inverse of the Cartier operator. We set

H_{p}^{i+1}(k) = Coker(γ − 1 : Ω_{k}^{i} → Ω_{k}^{i}/dΩ_{k}^{i−1}).

If [k : k^{p}] = ∞, one sets dim_{p}(k) = ∞. If [k : k^{p}] = p^{r}, one sets :

- dim_{p}(k) = r if H_{p}^{r+1}(k′) = 0 for any finite extension k′/k,

- dim_{p}(k) = r + 1 otherwise.

Proof of Theorem 2 : Assume [k : k^{p}] ≤ p. By Proposition 2, one can assume that k = k_{s}, that G is split and almost simple and that U is generated by a unipotent element of order q. One has _{p}Br(k′) = H_{p}^{2}(k′) = 0 for any finite extension k′/k because k is separably closed, so dim_{p}(k) ≤ 1. Then dim_{p}(K) ≤ 2 by [K, corollary to th. 3] where K = k((t)) as in § 2. By definition, one has H_{p}^{3}(K′) = 0 for any finite field extension K′/K. As the wild inertia group Gal(K_{s}/K_{mod}) is a pro–p group, the main result of [Gi1] gives then

H^{1}(K_{mod}, G) = 1.

Then Proposition 3.b shows that u is k–good.

2.4. — Bad unipotents and torsors on the affine line. Raghunathan has conjectured that if G is semisimple simply connected, then G–torsors on the affine line 𝔸_{k}^{1} are constant, i.e. come from G–torsors on Spec(k) [R, p. 189]. It turns out that bad unipotent elements yield counterexamples to this conjecture.

PROPOSITION 4. — Assume that G/k admits a bad unipotent element u of order p. Let P : 𝔸_{k}^{1} → 𝔸_{k}^{1} be the Artin–Schreier covering, which is Galois of group 𝔽_{p} and defined by t = P(x) = x^{p} − x. Let us define the 1–cocycle h = (h_{σ})_{σ∈𝔽_{p}} for this covering by

h_{σ} = u^{σ} ∈ G(k[x]) (σ ∈ 𝔽_{p})

Then the cocycle h defines a G–torsor on 𝔸_{k}^{1} which is not isomorphic to a constant torsor.

Proof : One can assume k = k_{s}. So H^{1}(k, G) = 1 and we have to show that the torsor defined by h is not trivial. We lift our cocycle h in G(k((\frac{1}{x}))) and we observe that k(x)/k(t) is wildly ramified at ∞. As u is a bad unipotent element, Proposition 3.b shows that [h]_{k(( \frac{1}{t}))} is not trivial in H^{1}(k((\frac{1}{t})), G), and a fortiori [h] is not trivial in H^{1}(𝔸_{k}^{1}, G).

Remark 2 : Theorem 1 (§ I.2) yields non constant G–torsors on the affine line for G split semisimple simply connected provided p divides d_{G} and [k : k^{p}] ≥ p^{2}.

3. — Conjugacy classes of k–anisotropic automorphisms of split groups From now on, we assume that G/k is split and semisimple. So G is the extension of ℤ to k of

the corresponding Chevalley group scheme G_{/ Spec(ℤ)}. For any ring A, we denote by simplicity G ×_{ℤ} A = G_{J} ×_{Spec(ℤ)} Spec(A).

The goal of this section is to give a powerful cohomological criterion analogous to Proposition 3 for distinguishing conjugacy classes of k–anisotropic automorphisms. We denote by T/ℤ a maximal split torus of G/ℤ and by B/ℤ a Borel subgroup containing T. Let J ⊂ Aut(G, B, T) be a subgroup and let us consider the semi-direct product G_{J}/ℤ = G/ℤ ⋊ J. We have a natural map G_{J}/ℤ → Aut(G)/ℤ with kernel the center of G/ℤ and we note that if G/ℤ is adjoint and J = Aut(G, B, T), one has G_{J}/ℤ = Aut(G)/ℤ. Let us introduce the following notation :

- W = N_{G}(T)/T and W_{J} = N_{G_{J}} (T)/T = W ⋊ J (the Weyl groups),

- T̂ = Hom_{ℤ−gr}(T, 𝔾_{m}), T̂^{0} = Hom_{ℤ−gr}(𝔾_{m}, T).

PROPOSITION 5. — Let ϕ ∈ Hom_{gr}(I, G_{J}(k)).

a) The following assertions are equivalent i) ϕ is k–anisotropic,

ii) [_{f_{ϕ}}G_{J}(O_{L})]_{I} = (_{f_{ϕ}}G_{J})(K),

iii) the group _{f_{ϕ}}G_{J}/K is K–anisotropic.

b) Assume ϕ is k–anisotropic. Let ϕ′ : I → G_{J} be another group morphism. The following assertions are equivalent :

i) ϕ′ is conjugate to ϕ under G_{J}(k), ii) γ(ϕ) = γ(ϕ′) in H^{1}(K, G_{J}).

c) Let k′/k be a Galois extension such that ϕ/k′ : I → G_{J} is a k′–anisotropic morphism. With the notation Z_{G}(ϕ) = Z_{G}(Im(ϕ)), one has an injection

H^{1}(k′/k, Z_{G_{J}} (ϕ)) ↪ H^{1}(K.k′/K,_{f_{ϕ}} G_{J}).

Proof : Let us begin by the second assertion.

b) The implication i) ⟹ ii) is obvious. Conversely, assume that γ(ϕ) = γ(ϕ′) in H^{1}(L̃/K, G). Then there exists g ∈ G(L) such that

g^{−1}ϕ(σ)^{σ}g = ϕ′(σ) (σ ∈ Ĩ).

Let B be the Bruhat–Tits building of the group G_{L} [BrT,§ I.7.4] which is a metric space. This building is equipped with an (isometric) left action of the group G_{J}(L̃) and an action of the Galois group Ĩ = Gal(L̃/K) denoted by x ↦ ^{σ}x for any σ ∈ Ĩ. We denote by c the center of the building, i.e. the vertex of B fixed by G(O_{L}). One defines a twisted action of Ĩ on B by

σ.x = ϕ(σ).^{σ}x (σ ∈ Ĩ).

This action is isometric and simplicial (i.e. maps a facet on a facet). Clearly, because Im(ϕ) ⊂ G_{J}(k), the center c is fixed under the twisted action of Ĩ. Due to

σ.(g.c) = ϕ(σ)(^{σ}g).c = gϕ′(σ).c = g.c,

we see that g.c is also fixed by Ĩ for the twisted action. Assume that g.c ≠ c. As the segment [c, g.c] is fixed pointwise by Ĩ, there exists a facet of B containing strictly c which is stabilized by the twisted action of Ĩ. Hence there exists (by the isomorphim between the link of c in B and the spherical building of G_{J}/k(μ_{m})) a facet of the spherical building of G_{J}/k(μ_{m}) which is stabilized by Ĩ. In other words, there exists a proper k–parabolic subgroup P of G such that

[_{f_{ϕ}}(G_{J}(k(μ_{m}))/N_{G_{J}} (P)(k(μ_{m})))]_{I} ≠ ∅, i.e

{[φ] ∈ G_{J}(k(μ_{m}))/N_{G_{J}} (P)(k(μ_{m})) | φ^{−1} Im(ϕ)φ ∈ N_{G_{J}} (P)(k(μ_{m}))}^{Gal(k(μ_{m})/k)} ≠ ∅,

which means that ϕ normalizes the k–parabolic subgroup φ(P). This contradicts the anisotropy of ϕ. We conclude that g.c = c, i.e. g ∈ G_{J}(O_{L}). Reducing the identity g^{−1}ϕ(σ)(^{σ}g) = ϕ′(σ) to G_{J}(k(μ_{m})), one gets \overline{g}^{−1}ϕ\overline{g} = ϕ′ with \overline{g} ∈ G(k).

a) i) ⟹ ii) : Assume ϕ is k–anisotropic. The same argument as before with ϕ′ = ϕ shows that

(_{f_{ϕ}}G_{J})(K) = {g ∈ G_{J}(L̃) | ϕ(s)(^{σ}g)ϕ(σ)^{−1} = g ∀σ ∈ Ĩ} ⊂ G_{J}(O_{L}),

so (_{f_{ϕ}}G_{J})(K) = [_{f_{ϕ}}G_{J}(O_{L})]_{I}.

ii) ⟹ iii) : As (_{f_{ϕ}}G)(K) ⊂ G_{J}(O_{L}) is bounded, the group _{f_{ϕ}}G/K does not contain any non trivial K–split torus, hence it is K–anisotropic. iii) ⟹ i) : We prove not i) ⟹ not iii). Assume ϕ is k–isotropic, i.e. there exists a k– parabolic subgroup P such that Im(ϕ) ⊂ N_{G_{J}} (P)(k). Then the class γ(ϕ) in H^{1}(K, G_{J}) comes from H^{1}(K, N_{G_{J}} (P)), and the twisted group _{f_{ϕ}}G/K is isotropic.

c) We set K′ = K ⊗_{k} k′, O′ = L ⊗_{k} k′, etc.. The first assertion shows that (_{f_{ϕ}}G_{J})(K′) =

[_{f_{ϕ}}G_{J}(O_{L′})]_{I}. But the map of Gal(k′/k) groups Z_{G_{J}} (ϕ)(k′) → [_{f_{ϕ}}G_{J}(O_{L′})]_{I} is split, so the

map H^{1}(k′/k, Z_{G_{J}} (ϕ)) → H^{1}(K.k′/K,_{f_{ϕ}}G_{J}) is injective.

First, we give the following corollary of the proposition and of Lemma 3 (§ 1.3).

COROLLARY 2. — Assume that the adjoint group G_{ad} of G is simple. Assume m is an integer such that k contains a primitive m-th root of unity and such that there exists a k–anisotropic automorphism of order m. Then m ∈ S(G).

Proof : One can assume that G = G_{ad}. Let α be a k–anisotropic automorphism of order m and assume that m ∉ S(G). The exact sequence 1 → G → Aut(G) → Out(G) → 1 and the fact that Out(G) is a finite group such that the prime divisors of ♯ Out(G) are bad primes shows that α is an inner automorphism. By Lemma 3, there exists a maximal torus S which contains α. The choice of a primitive m–root of unity ζ induces an isomorphim ℤ/mℤ ⟶^{∼} μ_{m}; taking L = k( ^{m} \sqrt{t})/K and θ = (t) ∈ H^{1}(K, μ_{m}) as in section 2, we see that γ(α) ∈ Im(H^{1}(L/K, S) → H^{1}(L/K, G)). But H^{1}(L/K, S) is an S(G)-primary torsion group, so H^{1}(L/K, S) = 1 and γ(α) = 1. By Proposition 5.a, α is then k–isotropic. Contradiction.

Questions : Assume p > 0. We lack an analogue of Proposition 2 (§ 3.1) for k–anisotropic automorphisms of order p. More precisely, we ask the following :

1) Let α be a k–anisotropic automorphism of G of order p. Is α_{k_{s}} k_{s}–anisotropic ?

2) Let α, α′ be k–anisotropic automorphisms of G of order p such that α and α′ are conjugate under Aut(G)(k_{s}). Are α and α′ conjugate under Aut(G)(k) ?

4. — Anisotropic automorphisms are special In this section, we prove conjecture 2’ ; as mentionned earlier, this proves Tits original

conjecture is proven as the case E_{8}, p = 5 has already been done in [Gi3].

THEOREM 3. — Conjecture 2’ is true for all types excepting possibly case E_{8}, l = 5 ≠ p.

The steps have been described in section 1.2 and we take into account cases done by Tits in the wild case. Their proof works also in the tame case, which is much simpler.

4.1. — Special automorphisms

DEFINITION 3. — A k–automorphism f of a torus S/k is anisotropic if the k–group S^{f} of fixed points of S by f is finite.

DEFINITION 3’. — a) An element w of W_{J} is special if the automorphism w : T → T defined by t ↦ w.t is anisotropic.

b) An element of G_{J}(k) of order n is special (relative to the split torus T) if it lies in N_{G_{J}} (T)(k) and its image in W_{J} is a special element of order n.

Some special elements of W_{J} are listed in [Sp] for any simple type.

LEMMA 4. — Let w be a special element of W_{J} of order n.

a) The homomorphism N(w) = 1 + w + ... + w^{n−1} : T → T is trivial.

b) Let g be a special element relative to T mapping to w. Then for any τ ∈ T (k), τg is a special element of G_{J}(k) of order n and is N_{G}(T)(\overline{k})–conjugate to g.

c) Let g be an element in N_{G}(T)(k) such that g is k–anisotropic in G. Then g acts anisotropically on T. Moreover, if char(k) = p > 0 and g has order p^{r}, then g is special (relative to T).

d) Assume that n is odd. Let A be a complete discrete valuation ring with residue field k and fraction field F_{A}. Let g be a special element of G_{J}(k) (relative to T) mapping to w. Then there exists g̃ ∈ N_{G_{J} ×ℤA}(T)(A) of order n mapping to g in G_{J}(k) such that g̃ is a special element of G_{J}(F_{A}) (relative to T) and mapping to w in W_{J}.

Proof : a) As (1 − w).N(w) = 0 and 1 − w is an invertible automorphism of T̂ ⊗_{ℤ} ℚ, then N(w) = 0 as an automorphism of T̂ ⊗_{ℤ} ℚ, hence N(w) : T → T is trivial.

b) Let us denote by w the image of g in W_{J}. We have to show that τg has order n; it is given by the computation

(τg)^{n} = τ × gτg^{−1} × g^{2}τg^{−2} ⋯ × g^{n−1}τg^{(−n+1)} = τ × (w.τ) × ⋯ × (w^{n−1}.τ) = N(w).τ = 1.

So the order of τg divides n and as w has order n, the order of τg is exactly n. We consider the equation

τg = xgx^{−1} (x ∈ T (\overline{k})),

or equivalently

τ = xgx^{−1}g^{−1} = x × w.x^{−1} (x ∈ T (\overline{k})).

As the group morphism T → T given by t ↦ t × w.t^{−1} has finite kernel, it is surjective, so there exists x ∈ T (\overline{k}) such that τg = xgx^{−1}.

c) Let g be an element of N_{G_{J}} (T)(k) such that g is k–anisotropic. Assume that g acts isotropically on T, then T^{⟨g⟩} = Ker(1 − g : T → T) has dimension greater than 1, so there exists a k–split subtorus S of T such that g ∈ Z_{G}(S)(k). As Z_{G}(S) is a k–Levi subgroup of a proper k–parabolic subgroup, g is k–isotropic. So g acts anisotropically on T. Assume moreover that g has order p^{r}; we have to show that the image of g in W_{J}, say w_{0}, has order p^{r}. If not, g^{p^{r−1}} is an element of order p of T (k), and as μ_{p}(k) = 1, this leads to a contradiction.

d) As G_{J} is a Chevalley group defined over ℤ, the sequence 1 → T → N_{G_{J}} (T_{J}) → W_{J} → 1 can be defined over ℤ, and as H_{ét}^{1}(ℤ, T) ⟶^{∼} (Pic(ℤ))^{rank(G)} = 1, one has the exact sequence

1 → T (ℤ) → N_{G_{J}} (T_{J})(ℤ) → W_{J} → 1

and an isomorphism T (ℤ) ≈ (ℤ/2ℤ)^{rank(G)} (see [T1]). So the class of the preceeding extension is killed by 2. Let g ∈ N_{G_{J}} (T_{J})(k) be of odd order n. Then there exists g_{0} ∈ N_{G_{J}} (T_{J})(ℤ) of order n with same image in W_{J} as g. There thus exists τ ∈ T (k) such that α = τ\overline{α}_{0}, where \overline{α}_{0} is the reduction in k of α_{0} via the map N_{G_{J}} (T_{J})(ℤ) → N_{G_{J}} (T_{J})(A) → N_{G_{J}} (T_{J})(k). As T ×_{ℤ} A is a smooth group scheme, the map T (A) → T (k) is surjective by Hensel’s lemma. We pick τ̃ ∈ T (A) mapping to τ. The element g̃:= τ̃α_{0} of N_{G_{J}} (T_{J})(A) lifts g = τ\overline{α}_{0}, maps to w in W_{J} and by assertion b), g̃ is a special element of G_{J}(F_{A}) relative to T.

4.2. — Triality

In this section, we handle the case l = 3 assuming then that k contains a primitive third root of unity. First, let us remark that we can assume that G is adjoint, so the group Aut(G) = G ⋊ Aut(G, B, T) is of the type we studied before. Let α ∈ Aut(G)(k) be a k– anisotropic automorphism of order 3. We shall use the following triality argument.

LEMMA 5. — Assume that G has type D_{4} (resp. F_{4}, E_{6}, E_{7}, E_{8}). Then α normalizes a k–Levi subgroup of a k–parabolic subgroup of G of type A_{2} (resp. A_{2}, A_{2} × A_{2}, E_{6}, E_{7}).

Proof : Let P be a k–parabolic subgroup of G such that P_{red} has type A_{3} (resp. C_{3}, D_{5}, E_{6}, E_{7}). Then codim_{G}(P) = 6 (resp. 15, 16, 27, 57). Let us define the k–group

C = P ∩ α(P) ∩ α^{2}(P).

Then codim_{G}(C) ≤ 3 × codim_{G}(P), so dim_{G}(C) ≥ 28 − 10 (resp. 7, 30, 52, 77) (cf. [PR] p. 380). We claim that C is a k–Levi subgroup of some k–parabolic subgroup of G. Let Q be a k– parabolic subgroup such that C ⊂ Q ⊂ P and minimal for this property. Then the k–parabolic subgroup R_{u}(Q).(Q ∩ α(Q)) contains C and is contained in Q, so R_{u}(Q).(Q ∩ α(Q)) = Q and M := Q∩α(Q) is a k–Levi subgroup of G ([BT1], proposition 4.10). But C = M ∩α^{2}(Q) is a k– parabolic subgroup of M; if C ≠ M, then α normalizes the split k–unipotent group R_{u}(C), and by Proposition 3.1 of [BT2], α normalizes a k–proper parabolic subgroup of G, which contradicts the assumption of anisotropy. So C is a k–Levi subgroup of Q/k and the restriction of α is still k–anisotropic (Lemma 2). As dim_{k}(C) ≥ 10, the only possibility is that C has type A_{2}. Other cases are considered on a case-by-case basis using Corollary 2 and we leave the details of the argument to the reader.

Let us recall that Theorem 3 is true for A_{2} and prove it inductively in the cases considered in Lemma 5. Lemma 5 gives a k–Levi subgroup C of a k–parabolic proper subgroup of G normalized by α. So the connected center Z_{G}(C)^{0} is a k–split torus normalized by α and, by induction, the derived group DC contains a k–split maximal torus S normalized by α. It follows that Z_{G}(C)^{0}.S is a maximal k-split torus of C (and G) normalized by α.

4.3. — Lifting in characteristic 0.

LEMMA 6. — Assume p = char(k) > 0. Let A be a complete discrete valuation ring with residue field k and fraction field F_{A}. Let ϕ : I → G_{J} be a group homomorphism which lifts to ϕ̃: I → G_{J} ×_{Spec(ℤ)} Spec(A).

a) If ϕ̃: I → G_{J} ×_{ℤ} F_{A} is a F_{A}–isotropic morphism, then ϕ is a k–isotropic morphism.

b) Assume ϕ is k–anisotropic. If ϕ̃ normalizes some maximal F_{A}–split torus of G_{J,F_{A}}, then ϕ normalizes some maximal k–split torus of G_{J}/k. Moreover, let ϕ′ : I → G_{J} be a group homomorphism which lifts to ϕ̃ : I → G_{J} ×_{Spec(ℤ)} Spec(A) such that ϕ̃ and ϕ̃′ are conjugate under G_{J}(F_{A}). Then ϕ and ϕ′ are conjugate under G_{J}(k) and ϕ′ normalizes some maximal k–split torus of G_{J}/k.

Proof : a) Assume that ϕ̃ is a k–isotropic element of G_{J}(F_{A}). This means that there exists a standard parabolic subgroup P/ℤ of G/ℤ such that I (through ϕ̃) has a fixed point in X(F_{A}), where X = G_{J}/N_{G_{J}} (P) is the projective ℤ–scheme of parabolic subgroups of G_{J} of the same type as P. As X(A) = X(F_{A}), then I has a fixed point in X(k), and ϕ is k–isotropic.

b) Assume that ϕ is k–anisotropic and that ϕ̃ normalizes a F_{A}–maximal split torus, i.e. there exists g ∈ G_{J}(F_{A}) such that

(∗)  g^{−1}ϕ̃(σ)g = f̃(σ) ∈ N_{G_{J}} (T)(F_{A}) (σ ∈ I),

where f̃: I → N_{G_{J}} (T)(F_{A}) is a group homomorphism. We consider the Bruhat–Tits building B of the group G_{J,F_{A}} with center c, which is the vertex stabilized by G_{J}(A). We denote by B^{I} the fixed point of X by I (through ϕ̃), which is convex and contains c. The torus T defines an apartment A = T̂ ⊗_{ℤ} ℝ ⊂ X. From (∗), we get g.A ⊂ B^{I}. Let us denote by π(c) the projection of c on the apartment g.A.

Claim π(c) = g.c = c : We write c = g.a with a ∈ A. For any σ ∈ I, one has

g.a = ϕ(σ)g.a = gf̃(σ).a,

so f̃(σ).a = a. As I acts on A trough f̃, we get a ∈ (T̂^{0} ⊗_{ℤ} ℝ)^{I} = A^{I}. But (T̂^{0} ⊗_{ℤ} ℝ)^{I} = (T̂^{0})^{I} ⊗_{ℤ} ℝ, so if a ≠ c, it means then f̃ is k–isotropic, which gives a contradiction to the anisotropy of ϕ by a). We conclude that π(c) = g.c.

As B^{I} is convex, the segment [c, π(c)] is fixed pointwise by α. If c ≠ π(c), there exists a facet F of B strictly containing c which is stabilized by I. By the isomorphism between the link of c and the spherical building of G_{J}/k, there exists a proper k–parabolic subgroup P of G such that ϕ(P) = P, which again gives a contradiction to the anisotropy of ϕ. We conclude that π(c) = c, i.e. c ∈ g.A. So g.c = c, i.e. g ∈ G_{J}(A) and f̃(σ) ∈ N_{G_{J}} (T)(A). Reducing (∗) in k, it yields \overline{g}^{−1}ϕ(σ)\overline{g} ∈ N_{G_{J}} (T)(k), so ϕ normalizes the maximal k–split torus \overline{g}.T. One can then assume that ϕ and ϕ̃ normalizes T.

Now, let ϕ′ : I → G_{J} be a group homomorphism which lifts to ϕ̃: I → G_{J} ×_{ℤ} A and such that ϕ̃ and ϕ̃′ are conjugate under G_{J}(F_{A}). Let g ∈ G_{J}(F_{A}) be an element such that ϕ̃′ = gϕ̃g^{−1}. Then ϕ̃′ normalizes the F_{A}–torus gT g^{−1}. But ϕ̃ is anisotropic by assertion a), so ϕ̃′ is anisotropic and the proof of assertion b) shows that g ∈ G_{J}(A). We have then ϕ̃′ = gϕ̃g^{−1} with g ∈ G_{J}(A), so ϕ = \overline{g}ϕ(\overline{g})^{−1} with \overline{g} ∈ G_{J}(k), and ϕ′ normalizes the maximal k–split torus \overline{g}T \overline{g}^{−1}.

LEMMA 7. — Let ϕ ∈ Hom_{gr}(I, G_{J}(k)) be a k–anisotropic morphism. Then there exists g ∈ G_{J}(k_{s}) such that the following hold

i) g^{−1} Im(ϕ)g ⊂ N_{G_{J}} (T)(k_{s}),

ii) g^{−1}^{s}g ⊂ N_{G_{J}} (T)(k_{s}) ∀s ∈ Gal(k_{s}/k).

The torus g.T is defined over k and normalized by ϕ. Moreover, in the case I is cyclic, one can replace condition i) by

i′) g^{−1} Im(ϕ)g ∈ N_{G_{J}} (T)(k),

Remark 3 : Lemma 7 yields conjecture 2 of Tits by an uniform argument in the case k is separably closed.

Proof of Lemma 7 : It is well–known that the map

H^{1}(K, N_{G_{J}} (T)) → H^{1}(K, G_{J})

is surjective. In other words, there exists a finite Galois extension M/K such that K ⊂ L ⊂ M ⊂ K_{s}, an element g ∈ G_{J}(M) and a cocycle h ∈ Z^{1}(Gal(M/K), N_{G_{J}} (T)(K_{s})) such that

(∗)  g^{−1}ϕ(s)^{s}g = h_{s}  (s ∈ Gal(M/K)).

Let us denote by K′/K the maximal unramified subextension of M, by O′ its valuation ring and by k′ its residue field Set L′ = L.K. We may assume K′ = k′((t)).

M
|
L′
⧸  ⧹
K′  L
⧹  ⧸
K

Let B be the Bruhat–Tits building of the group G_{M} [BrT1, § I.7.4]. This building is equipped with an action of the group G_{J}(M) and an action of the Galois group Gal(M/K) denoted by x ↦ ^{s}x for any s ∈ Gal(M/K). We denote by c the center of the building, i.e. the vertex of B fixed by G_{J}(O_{M}), where O_{M} is the valuation ring of M. The torus T ×_{k} M defines an apartment A = T̂^{0} ⊗_{ℤ} ℝ ⊂ B, which contains the center c, and which is fixed pointwise by Gal(M/K). One defines a twisted action of Gal(M/K) on B by

s.x = ϕ(s).^{s}x (s ∈ Gal(M/K)).

This action is isometric and simplicial (i.e. maps a facet on a facet). Clearly, since Im(ϕ) ⊂ G_{J}(k′), the center c is fixed under the twisted action of Gal(M/K). We claim that the apartment gA is stabilised by Gal(M/K) for the twisted action. This follows from the following computation ; for any x = g.a ∈ A, one has

s.x = ϕ(s).^{s}(g.a) = g.(h_{s}.a) ∈ gA.

Let us denote by π(c) = g.a the projection of c to the apartment gA.

Claim : π(c) = g.c = c : As c is fixed under the twisted action of Gal(L/K), the projection π(c) is also fixed. Moreover (observe ^{s}a = a), one has π(c) = g.a = s.(g.a) = ϕ(s).^{s}g.^{s}a = gh_{s}.a (s ∈ Gal(L/K)), so

a = h_{s}.a (s ∈ Gal(M/K)).

Now, we consider the maximal torus

nT /K ⊂ _{n}G/K ≈ _{f_{ϕ}}G/K.

The group _{n}G/K ≈_{f_{ϕ}} G/K is anisotropic by Proposition 5, so the torus _{n}T is anisotropic and

0 = _{ n̂}T^{0}(K) ⊗_{ℤ} ℝ = {x ∈ A | h_{s}.x = x}. One deduces that a = c, i.e. π(c) = h.c.

Let us assume that π(c) ≠ c. Then the segment [c, π(c)] is fixed pointwise by the twisted action of Gal(M/K). So there exists a facet of B containing strictly c which is stabilized by the twisted action of Gal(M/K) and there exists (by the isomorphim between the link of c in B and the spherical building of G/k) a facet of the spherical building of G/k which is pointwise fixed by Gal(M/K). In other words, there exists a proper k–parabolic subgroup of G such that

[_{f_{ϕ}}(G_{J}(k′)/N_{G_{J}} (P)(k′))]^{Gal(M/K)} ≠ ∅.

The group Gal(M/K) acts on the previous set through Gal(L′/K) = I × Gal(k′/k). One has

[_{f_{ϕ}}(G_{J}(k′)/N_{G_{J}} (P)(k′))]^{I} = {[d] ∈ G_{J}(k′)/N_{G_{J}} (P)(k′) | d^{−1} Im(ϕ)d ∈ N_{G_{J}} (P)(k′)},

[_{f_{ϕ}}(G_{J}(k′)/N_{G_{J}} (P)(k′))]^{Gal(k′/k)}
= {[d] ∈ G_{J}(k′)/N_{G_{J}} (P)(k′) | d^{−1}(^{s}d) ∈ N_{G_{J}} (P)(k′) ∀s ∈ Gal(k′/k)}.

Let d ∈ N_{G_{J}} (P)(k′) be a coset in the intersection of the two previous sets and let us denote by z ∈ Z^{1}(k′/k, N_{G_{J}} (P)) the cocycle defined by z_{s} = d^{−1}(^{s}d). The element d gives rise to a trivialization β_{d} :_{z} G_{J} ⟶^{∼} G_{J} such that Im(ϕ) ⊂ β_{d}(_{z}N_{G_{J}} (P)), which gives a contradiction to the anisotropy of ϕ. We conclude that π(c) = c. Let us denote by a ↦ \overline{a} the map G_{J}(O_{M}) → G_{J}(k′). The identity (∗) holds in G_{J}(O_{M}) and replacing d by \overline{d} ∈ G_{J}(k′), one can assume that g ∈ G_{J}(k′) ⊂ G_{J}(L′), that L′ = M and h_{s} ∈ G_{J}(k′) ⊂ G_{J}(L′) for every s ∈ Gal(L′/K). So we get i) and ii).

Assume now that I is cyclic with a generator σ. Identity (∗) for σ yields

(∗∗)  g^{−1}ϕ(σ)g = h_{σ} ∈ N_{G_{J}} (T)(k′).

Since Gal(k′/k) acts trivially on the Weyl group W_{J}, the map s ↦ h_{σ}^{−1}(^{s}h_{σ}) defines a Gal(k′/k)–cocycle in T (k′) and by Hilbert’s theorem 90, there exists τ ∈ T (k′) such that h_{σ}^{−1}(^{s}h_{σ}) = τ^{−1}(^{s}τ). Replacing g by gτ in (∗∗), one can assume that h_{σ} ∈ N_{G_{J}} (k) and taking h = h_{σ}, the proposition is proved.

LEMMA 8. — Assume that p = char(k) is an odd prime ; let A be a complete discrete valuation ring with residue field k and fraction field F_{A}. Let H/k be a k–form of G_{J}, ℌ a A–form of G_{J}×_{ℤ}A with special fiber H/k Let S be a maximal k–torus of H, and let 𝔖 be a maximal A–torus of ℌ with special fiber S. Let h ∈ N_{H}(S)(k) be an element of finite order q = p^{r}. Then h lifts to an element h̃ ∈ N_{ℌ}(𝔖)(A) of order q.

Remark 4 : Such an ℌ exists by Hensel isomorphism H_{ét}^{1}(A, Aut(G_{J})) ⟶^{∼} H^{1}(k, Aut(G_{J})) and such an A–torus 𝔖 exists by [SGA3, exp. XV, § 8].

Proof : Let A_{s}/A be the étale extension associated to k_{s}/k. We remark first that we can assume that h generates H/H^{0}. Let us denote by w the image of h in (N_{H}(S)/S)(k), which has order q.

First case : w acts anisotropically on S ×_{k} k_{s} : First, remark that the lifting assumption in N_{ℌ}(𝔖)(A_{s}) is satisfied by Lemma 4.d (§ 4.1, we use here that p is odd). Let us denote the Weyl group scheme N_{ℌ}(𝔖)/𝔖 by W (𝔖)/A; that is a twisted finite group, so isomorphic to N_{G}(S)/S. We also denote by ⟨w⟩ the cyclic subgroup of W (𝔖) generated by w and by N_{w}/A the preimage of ⟨w⟩ in N_{ℌ}(𝔖). By Hensel’s lemma, one has the following exact commutative diagram of pointed sets

1 ⟶ 𝔖(A) ⟶ N_{w}(A) ⟶ ⟨w⟩ ⟶ H^{1}(A, 𝔖)
↓  ↓  ∥  ↓ ≀
1 ⟶ S(k) ⟶ N_{w}(k) ⟶ ⟨w⟩ ⟶ H^{1}(k, S),

and

1  1
↓  ↓
𝕊 = 𝕊
↓  ↓
1 ⟶ 𝔖(A) ⟶ N_{w}(A) ⟶ ⟨w⟩ ⟶ 1
sp ↓  sp ↓  ∥
1 ⟶ S(k) ⟶ N_{w}(k) ⟶ ⟨w⟩ ⟶ 1,
↓  ↓
1  1

where 𝕊 := Ker(𝔖(A) → S(k)). So, the obstruction to lifting h to an element of order q in N_{ℌ}(𝔖)(A) is the class, say η in H^{2}(⟨h⟩, 𝕊), of the restriction to the cyclic group ⟨h⟩ of the vertical extension. We put μ/A = Ker(1 − w : 𝔖 → 𝔖) which is a finite A–multiplicative group, and we have

H^{2}(⟨h⟩, 𝕊) = (𝕊)^{⟨w⟩}/N(w).𝕊 = Ker(μ(A) → μ(k)),

because N(w) = 0 (Lemma 4.a). The same approach for k = k_{s} gives an obstruction η_{s} in μ(A_{s}) which vanishes. As μ(A) ⊂ μ(A_{s}), one has η = 0, and h lifts to N_{ℌ}(𝔖)(A).

Second case : h acts isotropically on S ×_{k} k_{s} : This means that dim(S^{⟨h⟩}) ≥ 1, i.e. there exists a non–trivial subtorus S_{0} of S such that h ∈ Z_{H}(S_{0})(k). We have Z_{H^{0}}(S_{0}) = Z_{H}(S_{0})^{0}. As h generates H/H^{0}, we have an exact sequence 1 → Z_{H^{0}}(S_{0}) → Z_{H}(S_{0}) → H/H^{0} → 1 which is split by h. The element h acts trivially on the coradical torus (cf. [SGA3], exp. XXII, § 6.2) C =: corad(Z_{H^{0}}(S_{0})) of Z_{H^{0}}(S_{0}), and it defines an extension of the canonical morphism Z_{H^{0}}(S_{0}) → C to a morphism Z_{H}(S_{0})^{0} → C. So one can define the group H′/k by the following diagram

1  1
↑  ↑
C  =  C
↑  ↑
1 ⟶ Z_{H^{0}}(S_{0}) ⟶ Z_{H}(S_{0}) ⟶ H/H^{0} ⟶ 1
↑  ↑  ∥
1 ⟶ DZ_{H^{0}}(S_{0}) ⟶  H′  ⟶ H/H^{0} ⟶ 1.
↑  ↑
1  1

So H′ is a k–form of some G′_{J′} and h ∈ H(k) because C(k) has no k–points of p–power order. Denoting S′ = Ker(S → C) ⊂ H′, h acts anisotropically on the maximal torus S′ ×_{k} k_{s} of H′. Defining ℌ′/A and 𝔖′/A by Hensel’s lift, the first case shows that h lifts in N_{ℌ′}(𝔖′)(A) = N_{ℌ}(𝔖)(A) ∩ ℌ′(A), so h lifts in N_{ℌ}(𝔖)(A).

4.4. — End of the proof of Theorem 3

Recall that char(k) = p > 0 and we consider a k–anisotropic automorphism α of prime order l of G. We can assume that l is odd. Let A be a complete discrete valuation ring with residue field k and with fraction field F_{A} such that F_{A} contains a primitive p–root of unity. First, we show that α lifts to Aut(G)(A), i.e. there exists α̃ ∈ Aut(G)(F_{A}) of order l lifting α. If l = p, this is a consequence of Lemmas 7 and 8. If l ≠ p, then ℤ/lℤ is a finite group of multiplicative type ; as Aut(G) is smooth, the Grothendieck rigidity theorem [SGA3, exp. VI, Cor. 7.3] says that the map

Hom_{A−gr}(ℤ/lℤ, Aut(G)) → Hom_{k−gr}(ℤ/lℤ, Aut(G))

is surjective, so α lifts in characteristic zero. By Lemma 6.a, α̃ is an F_{A}–anisotropic automorphism of G/F_{A}, the characteristic zero case shows that α̃ is special and finally Lemma 6.b shows that α is special.

Acknowledgments : It is a pleasure to thank Tamás Szamuely for his careful reading of a preliminary version of this paper.

I wish to express my hearty thanks to Jean–Pierre Serre and Jacques Tits, who introduced me to this subject.

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[T4] J. TITS. — Résumés de cours 93–94, Annuaire du Collège de France.

[T5] J. TITS. — Résumés de cours 98–99, Annuaire du Collège de France.

Philippe GILLE

Département de Mathématiques, CNRS Bât. 425, Université de Paris–Sud F–91405 ORSAY CEDEX Courrier électronique : gille@math.u-psud.fr