FINITENESS OF R-EQUIVALENCE GROUPS OF SOME ADJOINT CLASSICAL GROUPS OF TYPE ^{2}D_{3}

GRÉGORY BERHUY

Abstract. Let F be a field of charateristic different from 2. We construct families of adjoint groups G of type ^{2}D_{3} defined over F (but not over k) such that G(F)/R is finite for various fields F which are finitely generated over their prime subfield. We also construct families of examples of such groups G for which G(F)/R ≃ ℤ/2 ℤ when F = k(t), and k is (almost) arbitrary. This gives the first examples of adjoint groups G which are not quasi-split nor defined over a global field, such that G(F)/R is a non-trivial finite group.

Introduction

For an algebraic group G defined over a field F, let G(F)/R be the group of Requivalence classes introduced by Manin in [10]. The algebraic group G is called R-trivial if G(L)/R = 1 for every field extension L/F. It was established by Colliot-Thélène and Sansuc in [4] (see also [11, Proposition 1]) that the group G is R-trivial if the variety of G is stably rational. Moreover, in [4], the following question was raised:

Question: Let F be a field which is finitely generated over its prime subfield, and let G be a connected linear algebraic group defined over F. Assume that F is perfect or G is reductive. Is G(F)/R finite ?

The question was answered positively by Colliot-Thélène and Sansuc if G is quasisplit (cf. Proposition 14, loc.cit) and by Gille for any reductive group G defined over a global field in [5]. Lemma II.1.1 c) of [5] immediately implies that this question has a positive answer if F is a rational extension of a global field k and G is defined over k. Various examples of classical adjoint groups which are not R-trivial were constructed in [1] or [6],[11]. Throughout this paper, we will assume that F is a field of characteristic different from 2 and we will focus on absolutely simple adjoint groups of type ^{2}D_{3}. If F/k is a finitely generated field extension, we construct an infinite family of adjoint groups G of type ^{2}D_{3} defined over F such that G(F)/R is finite as soon as H_{nr}^{3}(F/k, μ_{2}) is finite. If F = k(t), where k is an arbitrary field, we will also give a family of examples of such groups for which G(F)/R ≃ ℤ /2 ℤ. This gives the first examples of adjoint groups G such that G(F)/R which are not quasi-split nor defined over a global field, such that G(F)/R is a non-trivial finite group.

Date: July 11, 2006.

1. Unramified cohomology

Let X be a smooth proper irreducible variety defined over k. We denote by X^{(1)} the set of points of codimension 1 in X. The ring O_{X,x} is then a discrete valuation ring. We will denote by υ_{x} the corresponding discrete valuation and by π_{x} a local parameter. We have a residue map

∂_{x} : H^{n}(k(X), μ_{2}) → H^{n−1}(κ(x), μ_{2}),

where κ(x) denotes the residue field O_{X,x}/(π_{x}). If u ∈ O_{X,x}, we will denote by ū its image in κ(x).

The residue of a cohomology class α ∈ H^{n}(k(X), μ_{2}) can be computed as follows: denote by k(X)_{x} the completion of k(X) with respect to the valuation on O_{X,x}. Then π_{x} is also a local parameter for the unique discrete valuation on k(X)_{x} extending υ_{x}, and we have an injection H^{n}(κ(x), μ_{2}) ↪ H^{n}(k(X)_{x}, μ_{2}). Then we have a decomposition

Res_{k(X)_{x}/k(X)}(α) = α_{0} + (π_{x}) ∪ α_{1},

for some uniquely determined α_{i} ∈ H^{n−i}(κ(x), μ_{2}). We then have the equality ∂_{x}(α) = α_{1}. In particular, for every a_{1}, ⋯, a_{n−1}, b_{1}, ⋯, b_{n−1} ∈ O_{X,x}^{×}, we have

∂_{x}((π_{x}) ∪ (a_{1}) ⋯ ∪ (a_{n−1})) = (\overline{a}_{1}) ⋯ ∪ (\overline{a}_{n−1})
∂_{x}((b_{1}) ⋯ ∪ (b_{n})) = 0

We say that α ∈ H^{n}(k(X), μ_{2}) is unramified at x if ∂_{x}(α) = 0. In this case, the class α_{0} is called the specialization of α at x, and is denoted by s_{x}(α). It does not depend on the choice of π_{x}. If ∂_{x}(α) ≠ 0, we say that α is ramified at x, and that x is a pole of α. It is well-known that the set of poles of α is finite. The unramified cohomology group H_{nr}^{n}(k(X), μ_{2}) is the subgroup of H^{n}(k(X), μ_{2}) consisting of classes which are unramified at every x ∈ X^{(1)}. It is a birational invariant of X. In particular, if X is a rational variety, then the restriction map induces an isomorphism H^{n}(k, μ_{2}) ≃ H_{nr}^{n}(k(X), μ_{2}). Therefore if F/k is a finitely generated extension, we can define the group of unramified elements H_{nr}^{n}(F/k, μ_{2}) by

H_{nr}^{n}(F/k, μ_{2}) = H_{nr}^{n}(k(X), μ_{2}),

where X is any irreducible smooth proper model of F/k. We refer to [2] for more details.

Notice that for any finitely generated field extension F/k, the elements lying in the image of Res_{F/k} : H^{n}(k, μ_{2}) → H^{n}(F, μ_{2}) are unramified. Such elements are called constant. Notice also that if α ∈ H^{n}(F, μ_{2}) is constant, then we have s_{x}(α) = Res_{κ(x)/k}(α) for all x ∈ X^{(1)}.

2. R-equivalence groups of adjoint groups of type ^{2}D_{3}

FINITENESS OF R-EQUIVALENCE GROUPS 3

2.1. A result of Merkurjev. In this section, we recall Merkurjev’s computation of the group of R-equivalence classes of some absolutely simple adjoint classical groups of type ^{2}D_{3} (cf. [11]). Let (A, σ) be a F -central simple algebra of degree 6 with an orthogonal involution, so we can write A = M_{3}(Q), where Q is a quaternion F -algebra, and let PGO^{+}(A, σ) be the connected component of PGO(A, σ), the group-scheme of projective similitudes of (A, σ).

Assume that A is not split, disc(σ) ∈ F^{×}/F^{×2} is not trivial, and that the Clifford algebra C(A, σ) has index 2. If L = F (\sqrt{disc(σ)}) then A_{L} (or equivalently Q_{L}) is split. Hence we can write Q ≃ (disc(σ), α), for some α ∈ F^{×}. Let 1, i, j, ij be the corresponding standard basis for Q, and let γ be the canonical (symplectic) involution on Q. The involution σ is adjoint to a skew-hermitian form (V, h) over (Q, γ), where V is a right Q-vector space of dimension 3.

The skew-hermitian form h represents xi for some x ∈ F^{×}, so we can write h = h′ ⊥ ⟨xi⟩ for some skew-hermitian form (V′, h′) over (Q, γ) of trivial discriminant, where V is a right Q-vector space of dimension 2.

Set (A′, σ′) := (End_{Q}(V′), σ_{h′}). Then C(A′, σ′) = Q_{1} × Q_{2}, for some quaternion F -algebras Q_{1} and Q_{2} satisfying Q_{1} ⊗ Q_{2} = Q in Br(F). Moreover, (Q_{1})_{L} ≃ (Q_{2})_{L} and C(A, σ) = (Q_{i})_{L} in Br(L) (so (Q_{i})_{L} is not split for i = 1, 2).

Proposition 1. Under the previous notation, we have the following group isomorphism:

PGO^{+}(A, σ)(F)/R ≃ N_{L/F} (L^{×}) ∩ Nrd(Q_{1}^{×}) · Nrd(Q_{2}^{×})/N_{L/F} (L^{×}) ∩ Nrd(Q_{i}^{×})

For a proof of all these facts, see [11, Section 3]. Notice that in [11], Merkurjev described more generally the group G(F)/R, when G is an absolutely simple adjoint classical group defined over F.

2.2. Finiteness of some R-equivalence groups.

2.2.1. Some useful lemmas. We will assume that (A, σ) is as in the previous section. We start to investigate the finiteness of PGO^{+}(A, σ)(F)/R. Keeping the notation above, we will identify this group to

N_{L/F} (L^{×}) ∩ Nrd(Q_{1}^{×}) · Nrd(Q_{2}^{×})/N_{L/F} (L^{×}) ∩ Nrd(Q_{i}^{×})

If λ ∈ N_{L/F} (L^{×}) ∩ Nrd(Q_{1}^{×}) · Nrd(Q_{2}^{×}), we will denote by [λ] its class modulo N_{L/F} (L^{×}) ∩ Nrd(Q_{i}^{×}). We start with an easy lemma:

Lemma 2. Let F be any field of characteristic different from 2. Then the map

φ : PGO^{+}(A, σ)(F)/R → H^{3}(F, μ_{2}), [λ] ↦ (λ) ∪ [Q_{1}]

is a well-defined injective group homomorphism.

Proof. Since (Nrd_{Q_{1}}(Q_{1}^{×})) ∪ [Q_{1}] = 0, this map is a well-defined group homomorphism . If λ ∈ N_{L/F} (L^{×}) ∩ Nrd_{Q_{1}}(Q_{1}^{×}) · Nrd_{Q_{2}}(Q_{2}^{×}) satisfies (λ) ∪ [Q_{1}] = 0, then λ ∈ Nrd_{Q_{1}}(Q_{1}^{×}) by a well-known theorem of Merkurjev [12], so [λ] = 1. □

Remark 3. In view of this lemma, we just have to investigate the finiteness of the image of φ.

We now assume until the end that X is a smooth irreducible proper model of F defined over k.

Lemma 4. Assume that Q_{1} and Q_{2} have no common pole, and let x ∈ X^{(1)}. Then

∂_{x}((λ) ∪ [Q_{1}]) = \{ \begin{matrix}0 & if x is not a pole of [Q_{1}] or [Q_{2}] \\ 0 or s_{x}[Q_{2}] & if x is a pole of [Q_{1}] \\ 0 or s_{x}[Q_{1}] & if x is a pole of [Q_{2}] \end{matrix}

Proof. Notice that since λ = N_{L/F} (z) for some z ∈ L^{×} and that (Q_{1})_{L} ≃ (Q_{2})_{L}, we get

(λ) ∪ [Q_{1}] = Cor_{L/F} ((z) ∪ [Q_{1}]_{L}) = Cor_{L/F} ((z) ∪ [Q_{2}]_{L}) = (λ) ∪ [Q_{2}].

Let x ∈ X^{(1)}, and assume first that [Q_{1}] and [Q_{2}] are both unramified at x. If (λ) is unramified at x, then (λ)∪[Q_{1}] is also unramified at x, that is ∂_{x}((λ)∪[Q_{1}]) = 0. If (λ) is ramified at x, then write λ = λ_{1}λ_{2}, λ_{i} ∈ Nrd_{Q_{i}}(Q_{i}^{×}). Then (λ_{1}) or (λ_{2}) is ramified at x, since ∂_{x}((λ)) = ∂_{x}((λ_{1})) + ∂_{x}((λ_{2})) and ∂_{x}((λ_{i})) ∈ ℤ/2ℤ. If (λ_{2}) is unramified at x, then ∂_{x}((λ)∪[Q_{1}]) = ∂_{x}((λ_{2})∪[Q_{1}]) = 0. Now assume that (λ_{2}) is ramified at x, so (λ_{1}) is unramified at x. Since [Q_{2}] is unramified at x as well, then ∂_{x}((λ) ∪ [Q_{1}]) = ∂_{x}((λ) ∪ [Q_{2}]) = ∂_{x}((λ_{1}) ∪ [Q_{2}]) = 0. Hence ∂_{x}((λ) ∪ [Q_{1}]) = 0 if x is not a pole of [Q_{1}] or [Q_{2}].

Now assume that x is a pole of [Q_{1}], so [Q_{2}] is unramified at x by assumption. If (λ) is unramified at x then ∂_{x}((λ) ∪ [Q_{1}]) = ∂_{x}((λ) ∪ [Q_{2}]) = 0. If (λ) is ramified at x, then ∂_{x}((λ) ∪ [Q_{1}]) = ∂_{x}((λ) ∪ [Q_{2}]) = s_{x}([Q_{2}]). If x is a pole of [Q_{2}], then similar computations show that ∂_{x}((λ) ∪ [Q_{1}]) = 0 or s_{x}([Q_{1}]).

□

2.2.2. The case where H_{nr}^{3}(F/k, μ_{2}) is finite.

Proposition 5. Assume that [Q_{1}] and [Q_{2}] have no common pole. If H_{nr}^{3}(F/k, μ_{2}) is finite, then PGO^{+}(A, σ)(F)/R is finite.

Proof. By assumption, the kernel of the map

(∂_{x})_{x∈X^{(1)}} : Im(φ) → ∏_{x∈X^{(1)}} H^{2}(κ(x), μ_{2})

is finite. By the previous lemma, its image is finite as well, so we are done by Remark 3.

□

Examples 6. The group H_{nr}^{3}(F/k, μ_{2}) is finite in the following cases (and therefore the previous proposition may be applied):

1) H^{3}(k, μ_{2}) is finite and X is a smooth conic over k

2) k is a finite field and X is a smooth proper variety of dimension 2 over k

3) k is either a local field (i.e. a finite extension of ℚ_{p}), ℝ or ℂ and X is a proper smooth geometrically irreducible curve over k

4) k is a number field and X is a smooth proper geometrically irreducible curve over k.

Case 1) readily follows from Proposition 3 and Proposition A.1 of [7]. Case 2) follows from Theorem 0.8 of [8]. Now let us consider Case 3): if k is a local field, it follows from Corollary 2.9. of [8]. If k = ℝ, it follows from a result of Colliot-Thélène and Parimala (see [3]). Finally, if k = ℂ, then k(X) has cohomological dimension at most 1 and therefore H^{3}(k(X), μ_{2}) = 0. In case 4), it readily follows from Theorem 0.8 of [8] that we have an injective homomorphism

H_{nr}^{3}(k(X), μ_{2}) ↪ ∏_{υ∈P (k)} H_{nr}^{3}(k_{υ}(X), μ_{2}),

where P (k) denotes the set of all places of k. By Corollary 2.9 of [8], H_{nr}^{3}(k_{υ}(X), μ_{2}) is zero if X has good reduction with respect to υ. Since X has good reduction with respect to all but finitely many places, it follows from Case 3) that H_{nr}^{3}(k(X), μ_{2}) = H_{nr}^{3}(F/k, μ_{2}) is finite.

The reader may find more finiteness results for H_{nr}^{3}(F/k, μ_{2}) in [2].

2.2.3. The case where H_{nr}^{3}(F/k, μ_{2}) ≃ H^{3}(k, μ_{2}). We give here another family of examples. Keeping notation of the previous sections, we will assume that Q_{1} and Q_{2} have no common poles. We then set

S_{1} = {x ∈ X|x is a pole of Q_{2} such that s_{x}([Q_{1}]) ≠ 0}
S_{2} = {x ∈ X|x is a pole of Q_{1} such that s_{x}([Q_{2}]) ≠ 0}

Proposition 7. Assume that [Q_{1}] and [Q_{2}] have no common pole, and let n_{i} be the number of elements of S_{i}. Assume that H_{nr}^{3}(F/k, μ_{2}) ≃ H^{3}(k, μ_{2}) (e.g. F/k is rational) and that there exists x_{0} ∈ X^{(1)} satisfying the following conditions:

1) One of the class [Q_{i}] is unramified at x_{0} and the corresponding specialization is zero

2) The restriction map Res_{κ(x_{0})/k} : H^{3}(k, μ_{2}) → H^{3}(κ(x_{0}), μ_{2}) is injective.

Then PGO^{+}(A, σ)(F)/R is finite, and its cardinality is at most 2^{n_{1}+n_{2}}.

Proof. Without any loss of generality, we may assume for example that [Q_{1}] is unramified at x_{0} ∈ X^{(1)} and that s_{x_{0}}([Q_{1}]) = 0. Assume that (λ) ∪ [Q_{1}] ∈ Im(φ) lies in the kernel of the map

(∂_{x})_{x∈X^{(1)}} : Im(φ) → ∏_{x∈X^{(1)}} H^{2}(κ(x), μ_{2})

By assumption (λ) ∪ [Q_{1}] is constant, so we have

s_{x}((λ) ∪ [Q_{1}]) = Res_{κ(x)/k}((λ) ∪ [Q_{1}]) for all x ∈ X^{(1)}

Since ∂_{x_{0}}([Q_{1}]) = s_{x_{0}}([Q_{1}]) = 0, we have Res_{k(X)_{x_{0}} /k(X)}((λ) ∪ [Q_{1}]) = 0, and therefore s_{x_{0}}((λ) ∪ [Q_{1}]) = Res_{κ(x_{0})/k}((λ) ∪ [Q_{1}]) = 0. Since the restriction map Res_{κ(x_{0})/k} : H^{3}(k, μ_{2}) → H^{3}(κ(x_{0}), μ_{2}) is injective, we get (λ)∪[Q_{1}] = 0. Therefore [λ] = 1 ∈ PGO^{+}(A, σ)(F)/R by Lemma 2. It follows that we have an injection

PGO^{+}(A, σ)(F)/R ↪ ∏_{x∈X^{(1)}} H^{2}(κ(x), μ_{2})

The use of Lemma 4 leads to the conclusion. □

Let us now consider the case where F = k(t), where t is an indeterminate over k, so one may take X = 𝔸_{k}^{1}. A point x of 𝔸_{k}^{1} of codimension 1 then corresponds to a unique monic irreducible polynomial π k[t] and κ(x) ≃ k[t]/(π). In this case, we will say that a cohomology class is (un)ramified at π, and ∂_{x} and s_{x} will be respectively denoted by ∂_{π} and s_{π}. If π has odd degree, a classical restrictioncorestriction argument show that the restriction map H^{3}(k, μ_{2}) → H^{3}(k[t]/(π), μ_{2}) is injective. Hence, from the previous proposition, we obtain:

Corollary 8. Let F = k(t) and assume that [Q_{1}] and [Q_{2}] have no common pole. Let n_{i} be the number of elements of S_{i}. Assume that there exists a monic irreducible polynomial π ∈ k[t] of odd degree such that one of the class [Q_{i}] is unramified at π and the corresponding specialization is zero. Then PGO^{+}(A, σ)(F)/R is finite, and its cardinality is at most 2^{n_{1}+n_{2}}.

Using this corollary, it is easy to construct an infinite family of non quasi-split adjoint groups G of type^{2}D_{3} defined over k(t) (but not over k) such that G(k(t))/R is finite for an (almost) arbitrary field k.

Example 9. Let k be a field of characteristic different from 2 and let F = k(t). Let a, α ∈ k^{×} and let π ∈ k[t] be a monic irreducible polynomial satisfying the following conditions:

1) (−1) ∪ (a) ∪ (α) = 0

2) The quaternion k-algebra (a, α) is not split over κ(π) (In particular (a, α) is not split over k, and therefore is not split over F, and α ∉ k^{×2}).

3) There exists b ∈ k such that π(b) is a non-zero norm in k(\sqrt{α}).

Let Q_{1} = (a, α) ⊗_{k} F, Q_{2} = (π, α), Q = (aπ, α) and L = F (\sqrt{aπ}). Let 1, i, j, ij be the standard basis of Q and γ its canonical involution. Notice that Q is a division algebra, since ∂_{π}([Q]) = Res_{κ(π)/k}(α) ≠ 0 (otherwise (a, α) would be split over κ(π)).

Let σ be the involution on A = M_{3}(Q) adjoint to the skew-hermitian form ⟨j, −aj, i⟩ over (Q, γ). The skew-hermitian form h′ := ⟨j, −aj⟩ has trivial discriminant and the corresponding adjoint involution σ′ on A′ := M_{2}(Q) can be written

σ′ ≃ σ_{⟨1,−a⟩} ⊗ ρ,

where ρ is the orthogonal involution on Q defined by

ρ(1) = 1, ρ(i) = i and ρ(j) = −j

It is then easy to check that C(A′, σ′) = Q_{1} × Q_{2}, using the formulas describing Clifford algebras of tensor products of involutions (see[9], p.150 for example or [13]), and the fact that disc(ρ) = α ∈ F^{×}/F^{×2}.

Claim: PGO^{+}(A, σ)(F)/R ≃ ℤ/2ℤ .

Indeed, [Q_{1}] has no pole and [Q_{2}] has exactly one pole. Notice also that π is not a scalar multiple of t − b, since π(b) ≠ 0 by assumption. Hence [Q_{2}] is unramified at t − b. Moreover we have s_{t−b}([Q_{2}]) = (π(b)) ∪ (α) = 0 by assumption. By Corollary 8, we then get that | PGO^{+}(A, σ)(F)/R| ≤ 2. Now it is enough to find a non trivial-class in PGO^{+}(A, σ)(F)/R. First of all, we clearly have −aπ ∈ N_{L/F} (L^{×}). Moreover, since (−1) ∪ (a) ∪ (α) = 0, we have −1 ∈ Nrd_{Q_{1}}(Q_{1}^{×}), so a = (−1) · (−a) ∈ Nrd(Q_{1}^{×}). Since −π ∈ Nrd_{Q_{2}}(Q_{2}^{×}), we get −aπ = a · (−π) ∈ N_{L/F} (L^{×}) ∩ Nrd(Q_{1}^{×}) · Nrd(Q_{2}^{×}). It remains to show that the R-equivalence class of −aπ is not trivial. For, it suffices to prove that φ([−aπ]) ≠ 0; this is the case since ∂_{π}((−aπ) ∪ [Q_{1}])) = (a, α)_{κ(π)} ≠ 0.

Remark 10. The group PGO^{+}(A, σ) obtained is not quasi-split since Q is a division algebra. Moreover, it is not defined over k. Otherwise [Q] would be unramified at π, which is not the case as we have seen above. To obtain concrete examples, one may take for k any field such that −1 ∈ k^{×2} such there exists a non split quaternion algebra (a, α) over k, and for π any arbitrary monic irreducible polynomial of odd degree satisfying π(0) = 1.

Acknowledgements: We would like to thank P. Gille and J.-L. Colliot-Thélène for stimulating conversations, and the anonymous referee for his constructive comments and suggestions.

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University of Southampton, School of Mathematics, Highfield, SO17 1BJ Southampton, United Kingdom

E-mail address: berhuy@soton.ac.uk