WEAKLY COMMENSURABLE S-ARITHMETIC SUBGROUPS IN ALMOST SIMPLE ALGEBRAIC GROUPS OF TYPES B AND C

SKIP GARIBALDI AND ANDREI S. RAPINCHUK

To Kevin McCrimmon on the occasion of his retirement

Abstract. Let G_{1} and G_{2} be absolutely almost simple algebraic groups of types B_{ℓ} and C_{ℓ} respectively defined over a number field K. We determine when G_{1} and G_{2} have the same isomorphism or isogeny classes of maximal Ktori. This leads to the necessary and sufficient conditions for two Zariski-dense S-arithmetic subgroups of G_{1} and G_{2} to be weakly commensurable.

Contents

1. Introduction and the statement of main results 1 2. Steinberg’s theorem for algebras with involution 4 3. Maximal tori in real groups of types B and C 9 4. Local-global principles for embedding étale algebras with involution 13 5. Function field analogue of Theorem 1.4 15 6. Proof of Theorem 1.4 15 7. Weakly commensurable subgroups and proof of Theorem 1.2 19 8. Proofs of Proposition 1.3 and Theorem 1.5 21 9. Alternative proofs via Galois cohomology 22 References 26

1. Introduction and the statement of main results

This paper has two interrelated goals: first, to complete the investigation of weak commensurability of S-arithmetic subgroups of almost simple algebraic groups begun in [PR09], and second, to contribute to the classical problem of characterizing almost simple algebraic groups having the same isomorphism or the same isogeny classes of maximal tori over the field of definition.

Let G_{1} and G_{2} be two semi-simple algebraic groups over a field F of characteristic zero, and let Γ_{i} ⊂ G_{i}(F) be a (finitely generated) Zariski-dense subgroup for i = 1, 2. We recall in §7 below the notion of weak commensurability of Γ_{1} and Γ_{2} introduced in [PR09]. (This notion was inspired by some problems dealing with isospectral and length-commensurable locally symmetric spaces, and we state some geometric consequences of our main results in (7.1) and (7.2).) We further recall that the mere existence of Zariski-dense weakly commensurable subgroups implies that G_{1} and G_{2} either have the same Killing-Cartan type, or one of them is of type B_{ℓ} and the other is of type C_{ℓ}. Moreover, cumulatively the results of [PR09], [PR10] and [Gar12] give, by and large, a complete picture of weak commensurability for S-arithmetic subgroups of almost simple algebraic groups having the same type.

2010 Mathematics Subject Classification. 20G15 (11E57 14L35 20G30).

On the other hand, weak commensurability of S-arithmetic subgroups in the case where G_{1} is of type B_{ℓ} and G_{2} is of type C_{ℓ} has not been investigated so far—it was only pointed out in [PR09] that S-arithmetic subgroups corresponding to the split forms of such groups are indeed weakly commensurable (see also Remark 2.6 below). Our first theorem provides a complete characterization of the situations where Sarithmetic subgroups in the groups of types B and C are weakly commensurable. In its formulation we will employ the description, introduced in [PR09, §1], of Sarithmetic subgroups of G(F), where G is an absolutely almost simple algebraic group over a field F of characteristic zero, in terms of triples (𝒢, K, S) consisting of a number field K ⊂ F, a finite subset S of places of K, and an F/K-form 𝒢 of the adjoint group \overline{G} — we briefly recall this description in §6.

The following definition will enable us to streamline the statements of our results.

Definition 1.1. Let 𝒢_{1} and 𝒢_{2} be absolutely almost simple algebraic groups of types B_{ℓ} and C_{ℓ} with ℓ ⩾ 2, respectively, over a number field K. We say that 𝒢_{1} and 𝒢_{2} are twins (over K) if for each place v of K, both groups are simultaneously either split or anisotropic over the completion K_{v}.

Theorem 1.2. Let G_{1} and G_{2} be absolutely almost simple algebraic groups over a field F of characteristic zero having Killing-Cartan types B_{ℓ} and C_{ℓ} (ℓ ⩾ 3) respectively, and let Γ_{i} be a Zariski-dense (𝒢_{i}, K, S)-arithmetic subgroup of G_{i}(F) for i = 1, 2. Then Γ_{1} and Γ_{2} are weakly commensurable if and only if the groups 𝒢_{1} and 𝒢_{2} are twins.

If Zariski-dense (𝒢_{1}, K_{1}, S_{1})- and (𝒢_{2}, K_{2}, S_{2})-subgroups are weakly commensurable then necessarily K_{1} = K_{2} and S_{1} = S_{2} by [PR09, Th. 3], so Theorem 1.2 in fact treats the most general situation. Furthermore, for ℓ = 2 we have B_{2} = C_{2}, so G_{1} and G_{2} have the same type; then Γ_{1} and Γ_{2} are weakly commensurable if and only if 𝒢_{1} ≃ 𝒢_{2} over K by [PR09, Th. 4]. This shows that the assumption ℓ ⩾ 3 in Theorem 1.2 is essential—the excluded case of ℓ = 2 is treated in Theorem 1.5 below.

Turning to the second problem, of characterizing almost simple algebraic groups having the same (isomorphic classes of) maximal tori, we would like to point out that, as we will see shortly, one gets more satisfactory results if instead of talking about isomorphic groups one talks about isogenous ones. We recall that algebraic K-groups H_{1} and H_{2} are called isogenous if there exists a K-group H with central K-isogenies π_{i}: H → H_{i}, i = 1, 2. For semi-simple K-groups G_{1} and G_{2}, this amounts to the fact that the universal covers G̃_{1} and G̃_{2} are K-isomorphic, and for K-tori T_{1} and T_{2} this simply means that there exists a K-isogeny T_{1} → T_{2}. Furthermore, we say that two semi-simple K-groups G_{1} and G_{2} have the same isogeny classes of maximal K-tori if every maximal K-torus T_{1} of G_{1} is K-isogenous to some maximal K-torus T_{2} of G_{2}, and vice versa. Unsurprisingly, K-isogenous groups have the same isogeny classes of maximal tori. Using the results from [PR09] and [Gar12], we prove the following partial converse for almost simple groups over number fields.

Proposition 1.3. Let G_{1} and G_{2} be absolutely almost simple algebraic groups over a number field K. Assume that G_{1} and G_{2} have the same isogeny classes of maximal K-tori. Then at least one of the following holds:

(1) G_{1} and G_{2} are K-isogenous;

(2) G_{1} and G_{2} are of the same Killing-Cartan type, which is one of the following: A_{ℓ} (ℓ > 1), D_{2ℓ+1} (ℓ > 1), or E_{6};

(3) one of the groups is of type B_{ℓ} and the other of type C_{ℓ} for some ℓ ⩾ 3.

We will prove the proposition in §8. As Theorem 1.5 below shows, it is possible for two isogenous, but not isomorphic, groups to have the same isomorphism classes of maximal K-tori, so the conclusion in (1) cannot be strengthened even if we assume that G_{1} and G_{2} have the same maximal tori. On the other hand, for each of the types listed in (2) one can construct non-isomorphic simply connected, hence non-isogenous, groups of this type having the same tori [PR09, §9], so these types are genuine exceptions. In this paper, we will sharpen case (3). Specifically, we prove the following in §6.

Theorem 1.4. Let G_{1} and G_{2} be absolutely almost simple algebraic groups over a number field K of types B_{ℓ} and C_{ℓ} respectively for some ℓ ⩾ 3.

(1) The groups G_{1} and G_{2} have the same \underline{isogeny} classes of maximal K-tori if and only if they are twins.

(2) The groups G_{1} and G_{2} have the same \underline{isomorphism} classes of maximal Ktori if and only if they are twins, G_{1} is adjoint, and G_{2} is simply connected.

We note that one can give examples of groups G_{1} and G_{2} of types B_{ℓ} and C_{ℓ} respectively over the field ℝ of real numbers, that are neither split nor anisotropic but nevertheless have the same isomorphism classes of maximal ℝ-tori (see Example 3.6). This shows Theorem 1.4, unlike many statements about algebraic groups over number fields, is not a global version of the corresponding theorem over local fields. What is crucial for the proof of Theorem 1.4 (and also Theorem 1.2) is that if the real groups G_{1} and G_{2} are neither split nor anisotropic with G_{1} adjoint and G_{2} simply connected then they cannot have the same maximal ℝ-tori (see Corollary 3.4).

The special case B_{2} = C_{2}. Theorem 1.4 completely settles the question of when the groups of types B_{ℓ} and C_{ℓ} have isogenous tori for ℓ ⩾ 3. The case where ℓ = 2 is special because the root systems B_{2} and C_{2} are the same.

Let G_{1} and G_{2} be groups of type B_{2} = C_{2}. They have the same isogeny classes of maximal tori if and only if they are isogenous by Lemma 8.1 below or [PR09, Th. 7.5(2)]. In particular, when G_{1} and G_{2} are both adjoint or both simply connected, they have the same isogeny classes of maximal tori if and only if G_{1} ≃ G_{2} if and only if they have the same maximal tori. It remains only to give a condition for G_{1} and G_{2} to have the same maximal tori when one is adjoint and the other is simply connected, which we now do.

Theorem 1.5. Let q_{1}, q_{2} be 5-dimensional quadratic forms over a number field K. The groups G_{1} = SO(q_{1}) and G_{2} = Spin(q_{2}) have the same isomorphism classes of maximal K-tori if and only if

(1) q_{1} is similar to q_{2}; and

(2) q_{1} and q_{2} are either both split or both anisotropic at every completion of K.

Notation. For a number field K, we let V^{K} denote the set of all places, and let V_{∞}^{K} (resp., V_{f}^{K}) denote the subset of archimedean (resp., nonarchimedean) places. Given a reductive algebraic group G defined over a field K, for any field extension L/K we let rk_{L} G denote the L-rank of G, i.e., the dimension of a maximal L-split torus.

We write r⟨a⟩ for the symmetric bilinear form (x, y) ↦ a ∑_{i=1}^{r} x_{i}y_{i} on K^{r}, and adopt similar notation for quadratic forms and hermitian forms.

In §6, we systematically use the following: for G_{1} and G_{2} absolutely almost simple groups of types B_{ℓ} and C_{ℓ} respectively, we put G_{1}^{♮} for the adjoint group of G_{1} (“SO”), and G_{2}^{♮} for the simply connected cover of G_{2} (“Sp”).

2. Steinberg’s theorem for algebras with involution

Our proofs of Theorems 1.2 and 1.4 rely on the well-known fact that groups of classical types can be realized as special unitary groups associated with simple algebras with involutions, so their maximal tori correspond to certain commutative étale subalgebras invariant under the involution. This description enables us to apply the local-global principles for the existence of an embedding of an étale algebra with an involutory automorphism into a simple algebra with an involution [PR10]. To ensure the existence of local embeddings, we will use an analogue for algebras with involution of the theorem, due to Steinberg [Ste65], asserting that if G_{0} is a quasi-split simply connected almost simple algebraic group over a field K and G is an inner form of G_{0} over K then any maximal K-torus T of G admits a K-defined embedding into G_{0}. The required analogue roughly states that if (A, τ) is an algebra with involution such that the corresponding group is quasi-split then any commutative étale algebra with involution (E, σ) that can potentially embed in (A, τ) does embed. It can be deduced from the original Steinberg’s theorem along the lines of [Gil04, Prop. 3.2(b)], but in fact one can give a simple direct argument. To our knowledge, this has not been recorded in the literature. Further, the argument for type B_{n} (in Proposition 2.5) extends with minor modifications to other types. So, despite the fact that we will only use this statement for algebras corresponding to groups of type B_{n} and C_{n}, we will give the argument for all classical types. We begin by briefly recalling the types of algebras with involution arising in this context, indicating in each case the étale subalgebras that give maximal tori.

Description of tori in terms of étale algebras. Let A be a central simple algebra of dimension n^{2} over a field L of characteristic ≠ 2, and let τ be an involution of A. Set K = L^{τ}. We recall that τ is said to be of the first (resp., second) kind if the restriction τ|_{L} is trivial (resp., nontrivial). Furthermore, if τ is an involution of the first kind, then it is either symplectic (i.e., dim_{K} A^{τ} = n(n−1)/2) or orthogonal (i.e., dim_{K} A^{τ} = n(n + 1)/2).

We also recall the well-known correspondence between involutions on A = M_{n}(L) and nondegenerate hermitian or skew-hermitian forms on L^{n}, cf. [KMRT98]: given such a form f, there exists a unique involution τ_{f} such that

f(ax, y) = f(x, τ_{f}(a)y)

for all x, y ∈ L^{n} and all a ∈ A; then the pair (M_{n}(L), τ_{f}) will be denoted by A_{f}. Moreover, f is symmetric (resp., skew-symmetric) if and only if τ_{f} is orthogonal (resp., symplectic). Conversely, for any involution τ there exists a form f on L^{n} of appropriate type such that τ = τ_{f}, and any two such forms are proportional. (Note that for involutions of the second kind one can pick the corresponding form to be either hermitian or skew-hermitian as desired.)

Type ^{2}A_{ℓ}. Let (A, τ) be a central simple L-algebra of dimension n^{2} with an involution τ of the second kind. Then G = SU(A, τ) is an absolutely almost simple simply connected K-group of type^{2}A_{ℓ} with ℓ = n−1, and conversely any such group corresponds to an algebra with involution (A, τ) of this kind. Any τ-invariant étale commutative subalgebra E ⊂ A gives a maximal K-torus

T = R_{E/K}(GL_{1}) ∩ G = SU(E, τ|_{E})

of G, and all maximal K-tori are obtained this way (see, for example, [PR10, Prop. 2.3]). The group G is quasi-split if and only if A = M_{n}(L) and τ = τ_{h} where h is a nondegenerate hermitian form on L^{n} of Witt index [n/2].

Type B_{ℓ} (ℓ ⩾ 2). Let A = M_{n}(K) with n = 2ℓ + 1, and let τ be an orthogonal involution of A. Then τ = τ_{f} for some nondegenerate symmetric bilinear form f on K^{n}, and

G = SU(A, τ) = SO(f)

is an adjoint group of type B_{ℓ}, and every such group is obtained this way. Furthermore, maximal K-tori T of G bijectively correspond to maximal commutative étale τ-invariant subalgebras E of A (of dimension n) such that dim_{K} E^{τ} = ℓ + 1 under the correspondence given by

T = R_{E/K}(GL_{1}) ∩ G = SU(E, τ|_{E}).

Furthermore, any such algebra admits a decomposition

(2.1)  (E, τ) = (E′, τ′) × (K, id_{K})

where E′ ⊂ E is a τ-invariant subalgebra of dimension 2ℓ. Finally, the group G is quasi-split (in fact, split) if and only if f has Witt index ℓ.

Type C_{ℓ} (ℓ ⩾ 2). Let A be a central simple K-algebra of dimension n^{2} with n = 2ℓ, and let τ be a symplectic involution of A. Then G = SU(A, τ) is an absolutely almost simple simply connected group of type C_{ℓ}, and all such groups are obtained this way. Maximal K-tori of G correspond to maximal commutative étale τ-invariant subalgebras E ⊂ A (of dimension n) such that dim_{K} E^{τ} = ℓ in the fashion described above. The group G is quasi-split (in fact, split) if and only if A = M_{n}(K). Then τ = τ_{f} where f is a nondegenerate skew-symmetric form on K^{n}; there is only one equivalence class of such forms, so in this case G ≃ Sp_{n}.

Type ^{1,2}D_{ℓ} (ℓ ⩾ 4). Let A be a central simple K-algebra K-algebra of dimension n^{2} where n = 2ℓ, and let τ be an orthogonal involution of A. Then G = SU(A, τ) is an almost absolutely simple K-group of type ^{1,2}D_{ℓ} which is neither simply connected nor adjoint, and any K-group of this type is K-isogenous to such a group. Maximal K-tori of G correspond to maximal commutative étale τ-invariant subalgebras E ⊂ A (of dimension n^{2}) such that dim_{K} E^{τ} = ℓ. The group G is quasi-split if and only if A = M_{n}(K) and τ = τ_{f} where f is a symmetric bilinear form on K^{n} of Witt index ℓ − 1 or ℓ.

Summary. Thus, if A is a central simple L-algebra of dimension n^{2} (and L = K for all types except ^{2}A_{ℓ}) then maximal K-tori of the algebraic K-group G = SU(A, τ) correspond in the manner described above to maximal abelian étale τ-invariant subalgebras E ⊂ A with dim_{L} E = n such that for σ = τ|_{E} we have

(2.2)  dim_{K} E^{σ} = \begin{cases}n & if σ|_{L} ≠ id_{L} \\ [\frac{n+1}{2}] & if σ|_{L} = id_{L}. \end{cases}

(Note that the condition is automatically satisfied if σ|_{L} ≠ id_{L}.)

Now, let (E, σ) be an n-dimensional commutative étale L-algebra with an involution satisfying (2.2). Then the question of whether the K-torus T = SU(E, σ)^{∘} can be embedded into G = SU(A, τ) where A is a central simple L-algebra of dimension n^{2} with an involution τ such that σ|_{L} = τ|_{L} translates into the question if there is an embedding (E, σ) ↪ (A, τ) of L-algebras with involution, which we will now investigate in the cases of interest to us. We note that if G is quasi-split then in all cases A = M_{n}(L). In this case, the universal way to construct an embedding (E, σ) ↪ (M_{n}(L), τ) is described in the following well-known statement.

Proposition 2.1. Let (E, σ) be an n-dimensional commutative étale L-algebra with an involution σ.

(i) For any b ∈ E^{×}, the map ϕ_{b}: E × E → K given by

ϕ_{b}(x, y) = tr_{E/L}(x · b · σ(y))

is a nondegenerate sesqui-linear form, which is hermitian or skew-hermitian if and only if b is such.

(ii) Let b ∈ E^{×} be hermitian or skew-hermitian, and let τ_{ϕ_{b}} be the involution on A := End_{L}(E) ≃ M_{n}(L) corresponding to ϕ_{b}; then the regular representation of E gives an embedding

(E, σ) ↪ (A, τ_{ϕ_{b}}) = A_{ϕ_{b}}

of algebras with involution. (iii) Let τ be an involution on A = M_{n}(L), and let f be a hermitian or skew-

hermitian form on L^{n} such that τ_{f} = τ. Then the following conditions are equivalent:

(a) There exists b ∈ E^{×} of the same type as f such that ϕ_{b} is equivalent to f.

(b) There exists a form h on E ≃ L^{n} which is equivalent to f and satisfies

(2.3)  h(ax, y) = h(x, σ(a)y) for all a, x, y ∈ E.

(c) There exists an embedding (E, σ) ↪ (A, τ) as L-algebras with involutions.

Sketch of proof. The nondegeneracy of ϕ_{b} in (i) follows from the fact that the Lbilinear form on E given by (x, y) ↦ tr_{E/L}(xy) is nondegenerate as E/L is étale; other assertions in (i) and (ii) are immediate consequences of the definitions. The implications (a) ⇒ (b) ⇒ (c) in (iii) are obvious, and the equivalence (a) ⇔ (c) (which we will not need) is established in [PR10, Prop. 7.1]. □

We also note that in fact any nondegenerate hermitian/skew-hermitian form h on E satisfying (2.3) is of the form ϕ_{b} for some b ∈ E^{×} of the respective type. Indeed, since the form ϕ_{1} is nondegenerate, we can write h in the form h(x, y) = tr_{E/L}(x · g(σ(y))) for some K-linear automorphism g of E. Then (2.3) implies that g is E-linear, and therefore is of the form g(x) = bx for some b ∈ E^{×}, which will necessarily be of appropriate type.

Example 2.2 (involutions of the first kind). According to Proposition 2.2 in [PR10], if L = K and (E, σ) is a K-algebra with involution of dimension n = 2ℓ satisfying (2.2) then

(E, σ) ≃ (F [δ]/(δ^{2} − d), θ)

where F = E^{σ}, d ∈ F^{×}, and θ(δ) = −δ.

For invertible b ∈ E^{σ} and x_{i}, y_{i} ∈ F, we have

ϕ_{b}(x_{1} + y_{1}δ, x_{2} + y_{2}δ) = tr_{E/K}(bx_{1}x_{2} − bdy_{1}y_{2}) = tr_{F/K}(2b(x_{1}x_{2} − dy_{1}y_{2})),

so ϕ_{b} is the transfer from F to K of the symmetric bilinear form ⟨2b, −2bd⟩. Clearly, if E is F × F, then ϕ_{b} is hyperbolic.

The example gives the entries in the ϕ_{b} column of Table 1.

Proposition 2.3 (type C). Let (E, σ) be an étale K-algebra of dimension n = 2ℓ with involution satisfying (2.2). Then for every symplectic involution τ on M_{n}(K), there is a K-embedding (E, σ) ↪ (M_{n}(K), τ).

Proof. It follows from the structure of (E, σ) in the example that there exists a skewsymmetric invertible b ∈ E (one can take, for example, the element corresponding to δ); then by Proposition 2.1(i), the form ϕ_{b} is nondegenerate and skew-symmetric. On the other hand, since τ is symplectic, we have τ = τ_{f} for some nondegenerate skew-symmetric form f on K^{n}. As any two such forms are equivalent, our assertion follows from Proposition 2.1(iii). □

To handle the algebras corresponding to types B and D, we need the following.

Lemma 2.4. Let (E, σ) be a commutative étale K-algebra with involution of dimension n = 2ℓ satisfying (2.2). Then there exists a nondegenerate symmetric bilinear form h on E that satisfies (2.3) and has Witt index ⩾ ℓ − 1.

Proof. If K is finite then one can take, for example, h = ϕ_{1}, so we can assume in the rest of the argument that K is infinite. It follows from the description of E that, for \overline{K} an algebraic closure of K,

(E ⊗_{K} \overline{K}, σ ⊗ id_{K}) ≃ (M, μ)

where M = ∏_{i=1}^{ℓ}(\overline{K} × \overline{K}) and μ acts on each copy of \overline{K} × \overline{K} by switching components. Viewing M as an affine n-space, consider the K-defined subvariety M_{−} := {x ∈ M | μ(x) = −x}. Clearly, M_{−} is a K-defined vector space, so the K-points E_{−} := M_{−} ∩ E are Zariski-dense in M_{−}. On the other hand, let U ⊂ M be the Zariski-open subvariety of elements with pairwise distinct components; then any x ∈ U generates M as a \overline{K}-algebra. Furthermore, it is easy to see that U ∩ M_{−} ≠ ∅, so U ∩ E_{−} ≠ ∅.

Fix e ∈ U ∩ E_{−}; then 1, e, …, e^{n−1} form a K-basis of E. For x ∈ E we define c_{i}(x) for i = 0, …, n − 1 so that x = ∑_{i=0}^{n−1} c_{i}(x)e^{i}. Set

h(x, y) := c_{n−2}(xσ(y)).

Clearly, h is symmetric bilinear and satisfies (2.3). Let us show that h is nondegenerate. If x = ∑_{i=0}^{n−1} c_{i}(x)e^{i} is in the radical of h, then so is σ(x), and therefore also x_{+} := ∑_{i=0}^{ℓ−1} c_{2i}(x)e^{2i} and x_{−} := ∑_{i=0}^{ℓ−1} c_{2i+1}(x)e^{2i+1}. From h(x_{+}, 1) = 0, h(x_{+}, e^{2}) = 0, etc., we successively obtain that c_{n−2}(x) = 0, c_{n−4}(x) = 0, etc., i.e., x_{+} = 0. Furthermore, we have 0 = h(x_{−}, e^{−1}) = −c_{n−1}(x). Then from h(x_{−}, e) = 0, h(x_{−}, e^{3}) = 0, etc., we successively obtain c_{n−3}(x) = 0, c_{n−5}(x) = 0, etc. Thus, x_{−} = 0, hence x = 0, as required. It remains to observe that the subspace spanned by 1, e, …, e^{ℓ−2} is totally isotropic with respect to h. □

Remark. In an earlier version of this paper, we constructed h in Lemma 2.4 in the form h = ϕ_{b} using some matrix computations. The current proof, which minimizes computations, was inspired by [BG11, §5].

Proposition 2.5 (type B). Let (E, σ) be an étale K-algebra of dimension n = 2ℓ+1 with involution satisfying (2.2). If τ is an orthogonal involution on A = M_{n}(K) such that τ = τ_{f} where f is a nondegenerate symmetric bilinear form on K^{n} of Witt index ℓ then there exists an embedding (E, σ) ↪ (A, τ) of K-algebras with involution.

Proof. Pick a decomposition (2.1), and then use Lemma 2.4 to find a form h′ on E′ with the properties described therein. We can write h′ = h′_{1} ⊥ h′_{2} where h′_{1} is a direct sum of ℓ − 1 hyperbolic planes and h′_{2} is a binary form. Choose a 1dimensional form h′′ so that h′_{2} ⊥ h′′ is isotropic, and consider h = h′ ⊥ h′′ on E = E′ × K. Then h is a nondegenerate symmetric bilinear form on E satisfying (2.3) and having Witt index ℓ. So, h is equivalent to f, hence (E, σ) embeds in (A, τ) by Proposition 2.1(iii). □

Remark 2.6. Let now G_{1} be the K-split adjoint group SO_{2ℓ+1} of type B_{ℓ} and G_{2} be the K-split simply connected group Sp_{2ℓ} of type C_{ℓ} where ℓ ⩾ 2. It was observed in [PR09], Example 6.7, that G_{1} and G_{2} have the same isomorphism classes of maximal K-tori over any field K of characteristic ≠ 2. This was derived from the fact that G_{1} and G_{2} have isomorphic Weyl groups using the results of [Gil04] or [Rag04]. Now, we are in a position to give a much simpler explanation of this phenomenon. Indeed, G_{1} = SU(A_{1}, τ_{1}) where A_{1} = M_{2ℓ+1}(K) and τ_{1} is an orthogonal involution on A_{1} corresponding to a nondegenerate symmetric bilinear form on K^{2ℓ+1} of Witt index ℓ, and G_{2} = SU(A_{2}, τ_{2}) where A_{2} = M_{2ℓ}(K) and τ_{2} is a symplectic involution on A_{2} corresponding to a nondegenerate skew-symmetric form on K^{2ℓ}. Any maximal K-torus T_{2} of G_{2} is of the form SU(E_{2}, σ_{2}) where E_{2} is a 2ℓ-dimensional commutative τ_{2}-invariant subalgebra of A_{2}, σ_{2} = τ_{2}|_{E_{2}}, with (E_{2}, σ_{2}) satisfying (2.2). Set (E_{1}, σ_{1}) = (E_{2}, σ_{2}) × (K, id_{K}). According to Proposition 2.5, there exists an embedding (E_{1}, σ_{1}) ↪ (A_{1}, τ_{1}), which gives rise to a K-isomorphism between T_{2} and the maximal K-torus T_{1} = SU(E_{1}, σ_{1}) of G_{1}. This, combined with the symmetric argument based on Proposition 2.3, yields the required fact. Then, repeating the argument given in loc. cit., we conclude that if K is a number field then for any finite subset S ⊂ V^{K} containing V_{∞}^{K}, the S-arithmetic subgroups of G_{1} and G_{2} are weakly commensurable.

Turning now to type D_{ℓ}, we first observe that if (E, σ) is a K-algebra with involution of dimension n = 2ℓ satisfying (2.2) then the determinant — viewed as an element of K^{×}/K^{×2} — of the symmetric bilinear form ϕ_{b} for invertible b ∈ E^{σ} does not depend on b (cf. [BCKM03, Cor. 4.2]) and will be denoted d(E, σ). Now, if τ is an involution on A = M_{n}(K) that corresponds to a symmetric bilinear form f on K^{n} having determinant d(f) then it follows from Proposition 2.1(iii) that an embedding (E, σ) ↪ (A, τ) can exist only if d(E, σ) = d(f) in K^{×}/K^{×2}.

Proposition 2.7. Let (E, σ) be an étale K-algebra of dimension n = 2ℓ with involution satisfying (2.2). If τ is an orthogonal involution on A = M_{n}(K) such that τ = τ_{f} where f is a nondegenerate symmetric bilinear form on K^{n} of Witt index ⩾ ℓ−1 such that d(E, σ) = d(f) (in K^{×}/K^{×2}) then there exists an embedding (E, σ) ↪ (A, τ) of K-algebras with involution.

Proof. Let h be the symmetric bilinear form on E constructed in Lemma 2.4. As we observed after Proposition 2.1, h is actually of the form h = ϕ_{b} for some invertible b ∈ E^{σ}, so d(h) = d(E, σ). We can write h = h_{1} ⊥ h_{2} where h_{1} is a direct sum of ℓ − 1 hyperbolic planes and h_{2} is a binary form. Similarly, f = f_{1} ⊥ f_{2} where f_{1} is a direct sum of ℓ − 1 hyperbolic planes and f_{2} is binary. Then d(E, σ) = d(f) implies that d(h_{2}) = d(f_{2}), so h_{2} and f_{2} are similar. Thus, a suitable multiple of h is equivalent to f, and our claim follows from Proposition 2.1(iii). □

Finally, we will treat algebras corresponding to the groups of type ^{2}A_{ℓ}. Here L will be a quadratic extension of K and all involutions will restrict to the nontrivial automorphism of L/K.

Proposition 2.8 (type A). Let (E, σ) be an étale n-dimensional L-algebra with involution. If τ is a unitary involution on A = M_{n}(L) such that τ = τ_{f} where f is a hermitian form on L^{n} having Witt index m := [n/2], then there exists an embedding (E, σ) ↪ (A, τ) of L-algebras with involution.

Proof. It is enough to construct a nondegenerate hermitian form on E that satisfies (2.3) and has Witt index m. If K is finite, one can take, for example, h = ϕ_{1}, so we can assume that K is infinite. Set F = E^{σ} so that E = F ⊗_{K} L. Since K is infinite, arguing as in the proof of Lemma 2.4, one can find e ∈ F so that F = K[e]. Then any x ∈ E admits a unique presentation of the form x = ∑_{i=0}^{n−1} e^{i} ⊗ c_{i}(x) with c_{i}(x) ∈ L. Define

h(x, y) := c_{n−1}(xσ(y)).

It is easy to see h is a hermitian form satisfying (2.3); let us show that it is nondegenerate. If x is in the radical of h then from h(x, 1) = 0, h(x, e) = 0, etc., we successively obtain that c_{n−1}(x) = 0, c_{n−2}(x) = 0, etc. Thus, x = 0, proving the nondegeneracy of h. Since 2(m−1) < n−1, the subspace spanned by 1, e, …, e^{m−1} is totally isotropic, hence the Witt index of h is m, as required. □

3. Maximal tori in real groups of types B and C

This section is devoted to determining the isomorphism classes of maximal tori in certain linear algebraic groups, primarily of types B and C, over the real numbers. Recall that every torus T over ℝ is ℝ-isomorphic to the product

(3.1)  (GL_{1})^{α} × (R_{ℂ/ℝ}^{(1)}(GL_{1}))^{β} × (R_{ℂ/ℝ}(GL_{1}))^{γ}

for uniquely determined nonnegative integers α, β, γ [Vos98, p. 64], and then the group T (ℝ) is topologically isomorphic to (ℝ^{×})^{α} × (S^{1})^{β} × (ℂ^{×})^{γ}, where S^{1} is the group of complex numbers of modulus 1. The fact that T is isomorphic to a maximal ℝ-torus of a given reductive ℝ-group G typically imposes serious restrictions on the numbers α, β and γ. To illustrate this, we first consider the following easy example.

Example 3.1. Every maximal ℝ-torus in G = GL_{n,ℍ}, where ℍ is the algebra of Hamiltonian quaternions, is isomorphic to (R_{ℂ/ℝ}(GL_{1}))^{n}. Indeed, every maximal ℝ-torus in G is of the form R_{E/ℝ}(GL_{1}) where E is a maximal commutative 2ndimensional étale subalgebra of A = M_{n}(ℍ). Any commutative 2n-dimensional étale ℝ-algebra E is isomorphic to ℝ^{α} × ℂ^{γ} with α + 2γ = 2n. But in order for E to have an ℝ-embedding in A, we must have α = 0 and then γ = n (cf. [PR10, 2.6]), so our claim follows.

We now recall the standard notations for some classical real algebraic groups. We let SO(r, n − r) denote the special orthogonal group of the n-dimensional quadratic form q = r⟨1⟩ ⊥ (n − r)⟨−1⟩. Similarly, we let Sp(r, n − r) denote the special unitary group of the n-dimensional hermitian form h = r⟨1⟩ ⊥ (n − r)⟨−1⟩ over ℍ with the standard involution. Every adjoint ℝ-group of type B_{ℓ} is isomorphic to some SO(r, n − r) for n = 2ℓ + 1 and some 0 ⩽ r ⩽ n, and every nonsplit simply connected ℝ-group of type C_{ℓ} is isomorphic to Sp(r, ℓ − r) some 0 ⩽ r ⩽ ℓ.

Lemma 3.2 (Adjoint B_{ℓ} over ℝ). The maximal ℝ-tori in G = SO(r, n − r), where n = 2ℓ+1, are of the form (3.1) with α+β +2γ = ℓ and α+2γ ⩽ s := min(r, n−r).

Proof. Let τ be the involution on A = M_{n}(K) that corresponds to the symmetric bilinear form f associated with the quadratic form q = r⟨1⟩ ⊥ (n − r)⟨−1⟩ so that G = SU(A, τ). Let T be a maximal ℝ-torus of G written in the form (3.1). Since the rank of G is ℓ, we immediately obtain

dim T = α + β + 2γ = ℓ.

Furthermore, we have T = SU(E, σ) where E ⊂ A is a τ-invariant maximal commutative étale subalgebra, σ = τ|_{E}, and (2.2) holds. There are exactly 4 isomorphism classes of indecomposable étale ℝ-algebras with involution, which are listed in Table 1. Using this information, we can write

(E, σ) = ℝ^{δ_{1}} × (ℝ × ℝ)^{δ_{2}} × ℂ^{δ_{3}} × (ℂ × ℂ)^{δ_{4}}

where the involutions on factors are as in the table. Comparing this with the structure of T, we obtain δ_{2} = α, δ_{3} = β, and δ_{4} = γ. According to Proposition 2.1(iii), there exists b ∈ E^{σ} such that ϕ_{b} is equivalent to f. But the Witt index of f is s (which equals the ℝ-rank of G), and the Witt index of ϕ_{b} is ⩾ δ_{2} + 2δ_{4}. Thus, α + 2γ ⩽ s. (We note that rk_{ℝ} T = α + γ, immediately yielding the restriction α + γ ⩽ s. So, the restriction we have actually obtained is stronger than one can a priori expect.)

Conversely, suppose α, β, γ satisfy the two constraints, and assume that r > n−r (otherwise we can replace the quadratic form q defining G with −q); in particular, r > ℓ. Consider the étale ℝ-algebra

(E, σ) = ℝ × (ℝ × ℝ)^{α} × ℂ^{β} × (ℂ × ℂ)^{γ} =: (E_{1}, σ_{1}) × ⋯ × (E_{4}, σ_{4})

of dimension 1 + 2α + 2β + 4γ = 2ℓ + 1 = n where the involutions on the factors ℝ, ℝ × ℝ, … are as described in Table 1. (Clearly, E satisfies (2.2).) Let us show that there exists b = (b_{1}, …, b_{4}) ∈ E^{σ} such that ϕ_{b} is equivalent to f. Set b_{2} = ((1, 1), …, (1, 1)) and b_{4} = ((1, 1), …, (1, 1)). Then the quadratic form associated with the bilinear form (ϕ_{2,4})_{(b_{2},b_{4})} on E_{2} × E_{4} is equivalent to (α + 2γ)(⟨1⟩ ⊥ ⟨−1⟩). Since t := (n − r) − (α + 2γ) ⩾ 0, we can choose b_{1} = ±1 and b_{3} = (±1, …, ±1) so that the quadratic form associated with (ϕ_{1,3})_{(b_{1},b_{3})} is equivalent to (2β + 1 − t)⟨1⟩ ⊥ t⟨−1⟩. Then b = (b_{1}, …, b_{4}) is as required. By Proposition

E  σ  ϕ_{b} for b ∈ E^{σ}  SU(E, σ)
ℝ  Id  ⟨b⟩  {1}
ℝ × ℝ  switch  ⟨1, −1⟩  GL_{1}
ℂ conjugation  ⟨b, b⟩  R_{ℂ/ℝ}^{(1)}(GL_{1})
ℂ × ℂ  switch  ⟨1, −1⟩ ⊕ ⟨1, −1⟩ R_{ℂ/ℝ}(GL_{1})

Table 1. Isomorphism classes of indecomposable étale ℝ-algebras with involution and their associated symmetric bilinear forms and unitary groups.

2.1(iii), there exists an embedding (E, σ) ↪ (A, τ), and therefore an ℝ-defined embedding SU(E, σ) ↪ SU(A, τ) = G. Finally, it follows from our construction and Table 1 that T = SU(E, σ) is a torus having the required structure. □

Lemma 3.3 (Simply connected C_{ℓ} over ℝ). The maximal ℝ-tori in the group G = Sp(r, ℓ − r) are of the form (3.1) with α = 0, β + 2γ = ℓ and γ ⩽ s := min(r, ℓ − r).

Proof. Let τ be the involution on A = M_{ℓ}(ℍ) that gives rise to the hermitian form f = r⟨1⟩ ⊥ (ℓ − r)⟨−1⟩, so that G = SU(A, τ). Every maximal ℝ-torus T of G is of the form T = SU(E, σ) for some (2ℓ)-dimensional étale τ-invariant subalgebra E of A, where σ = τ|_{E} and condition (2.2) holds. As in Example 3.1, E ≃ ℂ^{ℓ} as ℝ-algebras, and therefore

(E, σ) = ℂ^{δ_{1}} × (ℂ × ℂ)^{δ_{2}}

where the involutions on ℂ and ℂ × ℂ are as in Table 1. Then in (3.1) for T = SU(E, σ) we have α = 0, β = δ_{1} and γ = δ_{2}. By dimension count, we get β+2γ = ℓ. Furthermore,

γ = rk_{ℝ} T ⩽ rk_{ℝ} G = s.

Conversely, suppose that T has parameters α, β and γ satisfying our constraints. Consider

(E, σ) = ℂ^{β} × (ℂ × ℂ)^{γ}

with the involutions as above, and assume (as we may) that ℓ − r ⩽ r. Note that

(z, w) ↦ (\begin{matrix}z & 0 \\ 0 & w̄ \end{matrix})

defines an embedding of algebras with involutions ℂ × ℂ ↪ (M_{2}(ℍ), θ) where θ(x) = J^{−1}x̄^{t}J with J = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ), where x̄ is obtained by applying quaternionic conjugation to all entries. Consider the involution θ̂ on A given by θ̂(x) = Ĵ^{−1}x̄^{t}Ĵ where

Ĵ = diag(\underbrace{1, …, 1}_{r−γ} , \underbrace{−1, …, −1}_{β−(r−γ)} , \underbrace{J, …, J}_{γ}).

Then it follows from our construction that there exists an embedding (E, σ) ↪ (A, θ). Noting that (A, τ) ≃ (A, θ), we obtain an embedding (E, σ) ↪ (A, τ). So, there exists an ℝ-embedding SU(E, σ) ↪ SU(A, τ) = G, and it remains to observe that T = SU(E, σ) is a torus having the required structure. □

Alternatively, the results of Lemmas 3.2 and 3.3 can be deduced from the more general classification of maximal ℝ-tori in simple real algebraic groups obtained in [DT94]. For the reader’s convenience we have included the direct proofs above, written in the same language as the rest of the paper.

Corollary 3.4. Let G_{1} be an adjoint real group of type B_{ℓ}, and let G_{2} be a simply connected real group of type C_{ℓ}. The groups G_{1} and G_{2} have the same isomorphism classes of maximal ℝ-tori if and only if G_{1} and G_{2} are either both split or both anisotropic.

Proof. Since every ℝ-anisotropic torus T is of the form (R_{ℂ/ℝ}^{(1)}(GL_{1}))^{dim T}, there is nothing to prove if both groups are anisotropic. If both groups are split, our claim follows from Remark 2.6. Clearly, G_{1} and G_{2} cannot have the same maximal tori if one of the groups is anisotropic and the other is isotropic. So, it remains to consider the case where both groups are isotropic but not split. Then G_{1} contains the torus with α = 1, β = ℓ − 1, and γ = 0 by Lemma 3.2, but G_{2} does not by Lemma 3.3. □

Remark 3.5. Our argument shows that if G_{1} is isotropic and G_{2} is not split then G_{1} has a maximal ℝ-torus that is not isomorphic to any ℝ-torus of G_{2}. Moreover, by Lemma 3.2, a maximal ℝ-torus T_{1} of G_{1} that contains a maximal ℝ-split torus has parameters α = s, β = ℓ − s and γ = 0, hence does not allow an ℝ-embedding into G_{2}. In particular, if G_{1} = SO(n − 1, 1) and G_{2} is not split then every isotropic maximal ℝ-torus of G_{1} is not isomorphic to a subtorus of G_{2}.

Example 3.6 (Absolute rank 3). As an empirical illustration of the landscape over ℝ, we divide the 14 real groups of types B_{3} and C_{3} into equivalence classes under the relation “have isomorphic collections of maximal tori”. For forms of SO_{7} or Sp_{6}, the maximal tori are described by Lemmas 3.2 and 3.3. Also, the four anisotropic (compact) forms obviously make up one equivalence class. For the other groups one can use a computer program such as the Atlas software [AdC09] to find the maximal tori. In summary, the groups SO(1, 6), SO(2, 5), and Spin(2, 5) are each their own equivalence class, and we find the following non-singleton equivalence classes:

{4 anisotropic forms}, {Sp_{6}, SO(4, 3)}, {PSp_{6}, Spin(4, 3)},
and {Sp(1, 2), PSp(1, 2), Spin(1, 6)}.

In particular, Spin(1, 6) and PSp(1, 2) have the same isomorphism classes of maximal tori and yet are neither both split nor both anisotropic. This situation is dual to the one considered and eliminated in Corollary 3.4 (adjoint B_{ℓ} and simply connected C_{ℓ}).

For completeness, we mention the (much easier) analogue of Corollary 3.4 for non-archimedean local fields.

Lemma 3.7. Let G_{1} and G_{2} be absolutely almost simple groups of type B_{ℓ} and C_{ℓ} respectively, with ℓ ⩾ 3, over K a nonarchimedean local field of characteristic ≠ 2. The following are equivalent:

(1) The groups G_{1} and G_{2} have the same isogeny classes of maximal K-tori.

(2) rk_{K} G_{1} = rk_{K} G_{2}.

(3) G_{1} and G_{2} are split.

Proof. (1) obviously implies (2). Suppose (2) and that G_{2} is not split. Then

[ℓ/2] = rk_{K} G_{2} = rk_{K} G_{1} ⩾ ℓ − 1,

but this is impossible because ℓ ⩾ 3, hence (3).

To prove (3) ⇒ (1), we may assume that G_{1} is split adjoint and G_{2} is split simply connected. Combining Propositions 2.3 and 2.5 with (2.1) gives that G_{1} and G_{2} have the same isogeny classes of maximal tori. □

4. Local-global principles for embedding étale algebras with involution

The last ingredient we need to develop before proving Theorem 1.4 in §6 is a result guaranteeing in our situation the validity of the local-global principle for the existence of an embedding of an étale algebra with involution into a simple algebra with involution. This issue was analyzed in [PR10]: although the localglobal principle may fail (cf. Example 7.5 in loc. cit.), it can be shown to hold under rather general conditions. For our purposes we need the following case.

Let (E, σ) be an étale algebra with involution over a number field K of dimension n = 2m and satisfying (2.2). Then E = F [x]/(x^{2} − d) where F = E^{σ} is an mdimensional étale K-algebra and d ∈ F^{×}, with the involution defined by x ↦ −x as in Example 2.2. We write F = ∏_{j=1}^{r} F_{j} where F_{j} is a field extension of K, and suppose that in terms of this decomposition d = (d_{1}, …, d_{r}). Let τ be an orthogonal involution on A = M_{n}(K).

Proposition 4.1. [PR10, Theorem 7.3] Assume that for every v ∈ V^{K} there exists a K_{v}-embedding

ι_{v}: (E ⊗_{K} K_{v}, σ ⊗ id_{K_{v}}) ↪ (A ⊗_{K} K_{v}, τ ⊗ id_{K_{v}}).

If the following condition holds

(⋄)

for every finite subset V ⊂ V^{K}, there exists v_{0} ∈ V^{K} \ V such that
for j = 1, …, r, if d_{j} ∉ F_{j}^{×}^{2}, then d_{j} ∉ (F_{j} ⊗_{K} K_{v_{0}})^{×2};

then there exists an embedding ι: (E, σ) ↪ (A, τ). Furthermore, (⋄) automatically holds if F is a field.

We will now derive from the proposition the following statement, in which n can be odd or even.

Lemma 4.2. Let K be a number field, let (E, σ) be an n-dimensional étale algebra with involution satisfying (2.2), and let τ be an orthogonal involution on A = M_{n}(K). Assume that for every v ∈ V^{K} there is an embedding

ι_{v}: (E ⊗_{K} K_{v}, σ ⊗ id_{K_{v}}) ↪ (A ⊗_{K} K_{v}, τ ⊗ id_{K_{v}}).

Then in each of the following situations (1) n ⩽ 5, or

(2) there is a real v ∈ V^{K} such that (E ⊗_{K} K_{v}, σ ⊗ id_{K_{v}}) is isomorphic to (ℂ,¯ )^{m} or (ℂ,¯ )^{m} × (ℝ, id_{ℝ}) depending on whether n = 2m or n = 2m + 1,

there exists an embedding ι: (E, σ) ↪ (A, τ).

Proof. First, we will reduce the argument to the case of even n, i.e. when E satisfies one of the following conditions:

(1′) n = 2 or 4, or (2′) there is a real v ∈ V^{K} such that (E ⊗_{K} K_{v}, σ ⊗ id_{K_{v}}) is isomorphic to

(ℂ,¯ )^{m}.

Indeed, let n = 2m + 1 and suppose E satisfies condition (1) or (2) of the lemma. Then by [PR10, Prop. 7.2], (E, σ) = (E′, σ′) × (K, id_{K}) and there exists an orthogonal involution τ′ on A′ = M_{n−1}(K) such that for every v ∈ V^{K} there is an embedding

ι′_{v}: (E′ ⊗_{K} K_{v}, σ′ ⊗ id_{K_{v}}) ↪ (A′ ⊗_{K} K_{v}, τ′ ⊗ id_{K_{v}}),

and the existence of an embedding ι′: (E′, σ′) ↪ (A′, τ′) is equivalent to the existence of an embedding ι: (E, σ) ↪ (A, τ). Clearly, E′ satisfies the respective condition (1′) or (2′). So, if we assume that the lemma has already been established for E′, then the existence of ι follows.

Now, suppose that dim_{K} E = 2m and E satisfies (2.2). Write E = F [x]/(x^{2} − d) where F = E^{σ} = ∏_{j=1}^{r} F_{j}, d = (d_{1}, …, d_{r}) with d_{j} ∈ F_{j}^{×}. Assume that there exist K-embeddings φ_{j}: F_{j} ↪ K̄ such that if

M = φ_{1}(F_{1}) ⋯ φ_{r}(F_{r}) and N = M(\sqrt{φ_{1}(d_{1})}, …, \sqrt{φ_{r}(d_{r})})

then there is λ ∈ Gal(N/M) with the property

(4.1)  λ (\sqrt{φ_{j}(d_{j})}) = −\sqrt{φ_{j}(d_{j})} whenever d_{j} ∉ F_{j}^{×} for j = 1, …, r.

Let P be the normal closure of N over K, and let μ ∈ Gal(P/K) be such that μ|_{N} = λ. By Chebotarev’s Density Theorem [CF10, Ch. 7, 2.4], for any finite V ⊂ V^{K}, there exists a nonarchimedean v_{0} ∈ V^{K} \ V that is unramified in P and for which the Frobenius automorphism Fr(w_{0}|v_{0}) is μ for a suitable extension w_{0}|v_{0}. Then it follows from (4.1) that d_{j} ∉ (F_{j}_{w_{0}})^{×2} for any j such that d_{j} ∉ F_{j}^{×2}, and therefore condition (⋄) holds.

Let now (E, σ) be an étale algebra with involution satisfying (1′) or (2′) for which embeddings ι_{v} exist for all v ∈ V^{K}. In order to derive the existence of ι from Proposition 4.1, we need to check (⋄), for which it is enough to find an automorphism λ as in the previous paragraph. Suppose that (1′) hods. Then F = E^{σ} has dimension 1 or 2 respectively. Since we don’t need to consider the case where F is a field (cf. Proposition 4.1), the only remaining case is where F = K × K. Clearly, K(\sqrt{d_{1}}, \sqrt{d_{2}}) always has an automorphism λ such that λ(\sqrt{d_{j}}) = −\sqrt{d_{j}} if d_{j} ∉ K^{×2}, as required. Finally, suppose that (2′) holds. Then F ⊗_{K} K_{v} ≃ ℝ^{m}, and d = (δ_{1}, …, δ_{m}) in ℝ^{m} with δ_{i} < 0 for all i. Then for any embeddings φ_{j}: F_{j} ↪ ℂ we have φ_{j}(F_{j}) ⊂ ℝ and the restriction λ of complex conjugation satisfies λ(\sqrt{d_{j}}) = −\sqrt{d_{j}} for all j, concluding the argument. □

Remark. Example 7.5 in [PR10] shows that there exists (E, σ) with E of dimension 6 for which the local-global principle for embeddings fails, so in terms of dimension the condition (1) in Lemma 4.2 is sharp.

For convenience of further reference, we will also quote the local-global principle for embeddings in the case of symplectic involutions.

Lemma 4.3. [PR10, Th. 5.1] Let A be a central simple K-algebra of dimension n^{2} with a symplectic involution τ (then, of course, n is necessarily even), and let (E, σ) be an n-dimensional étale K-algbra with involution satisfying (2.2). If for every v ∈ V^{K} there exists an embedding

ι_{v}: (E ⊗_{K} K_{v}, σ ⊗ id_{K_{v}}) ↪ (A ⊗_{K} K_{v}, τ ⊗ id_{K_{v}}),

then there exists an embedding (E, σ) ↪ (A, τ).

5. Function field analogue of Theorem 1.4

We recall the following immediate consequence of the rationality of the variety of maximal tori (see [Har68] or [PR94, Cor. 7.3]) which will be used repeatedly: Let G be a reductive algebraic group over a number field K; then given any v ∈ V^{K} and any maximal K_{v}-torus T^{(v)} of G there exists a maximal K-torus T of G that is conjugate to T^{(v)} by an element of G(K_{v}). In particular, for any v ∈ V^{K} there exists a maximal K-torus T of G such that rk_{K_{v}} T = rk_{K_{v}} G. It follows that if G_{1} and G_{2} are reductive K-groups having the same isogeny classes of maximal K-tori then

(5.1)  rk_{K_{v}} G_{1} = rk_{K_{v}} G_{2} for all v ∈ V^{K}.

The remark made in the previous paragraph remains valid for global function fields, which can be used to give the following analogue of Theorem 1.4: Suppose G_{1} and G_{2} are absolutely almost simple algebraic groups of types B_{ℓ} and C_{ℓ} (ℓ ⩾ 3) over a global field K of characteristic > 2. The groups G_{1} and G_{2} have the same isogeny classes of maximal K-tori if and only if they are split. Indeed, if the two groups have the same isogeny classes of maximal K-tori, then both groups are K_{v}- split for every v (by (5.1) and Lemma 3.7), hence both groups are K-split (by the Hasse Principle). The converse holds by Remark 2.6.

6. Proof of Theorem 1.4

Throughout this section G_{1} and G_{2} will denote absolutely almost simple algebraic groups of types B_{ℓ} and C_{ℓ} for some ℓ ⩾ 3 defined over a number field K. In 1.1 we defined what it means for G_{1} and G_{2} to be twins. We now observe that since G_{1} and G_{2} cannot be K_{v}-anisotropic for v ∈ V_{f}^{K}, they are twins if and only if both of the following conditions hold:

(6.1) rk_{K_{v}} G_{1} = rk_{K_{v}} G_{2} = ℓ for all v ∈ V_{f}^{K}

(6.2) rk_{K_{v}} G_{1} = rk_{K_{v}} G_{2} = 0 or ℓ for all v ∈ V_{∞}^{K}.

We also note that if G_{1} and G_{2} are twins over K then they remain twins over any finite extension L/K. If K has r real places, then (by the Hasse Principle) there are exactly 4 · 2^{r} pairs of K-groups G_{1}, G_{2} that are twins, equivalently, 2^{r} pairs if one only counts the groups G_{1} and G_{2} up to isogeny.

Now, let G_{1} and G_{2} be as above, with G_{1} adjoint and G_{2} simply connected. Then G_{i} = SU(A_{i}, τ_{i}) for i = 1, 2 where A_{1} = M_{n_{1}}(K), n_{1} = 2ℓ + 1 and the involution τ_{1} is orthogonal, and A_{2} is a central simple K-algebra of dimension n_{2}^{2} with n_{2} = 2ℓ and the involution τ_{2} is symplectic. Any maximal K-torus T_{i} of G_{i} is of the form SU(E_{i}, σ_{i}) where E_{i} ⊂ A_{i} is an n_{i}-dimensional étale τ_{i}-invariant K-subalgebra and σ_{i} = τ_{i}|_{E_{i}} so that (2.2) holds. For i = 1, we can always write (E_{1}, σ_{1}) = (E′_{1}, σ′_{1}) × (K, id_{K}). For i = 2, we set (E_{2}^{+}, σ_{2}^{+}) = (E_{2}, σ_{2}) × (K, id_{K}).

Proposition 6.1. Let (A_{1}, τ_{1}) and (A_{2}, τ_{2}) be algebras with involution as above, and assume that G_{1} = SU(A_{1}, τ_{1}) and G_{2} = SU(A_{2}, τ_{2}) are twins. If (E_{1}, σ_{1}) is isomorphic to an n_{1}-dimensional étale subalgebra of (A_{1}, τ_{1}) satisfying (2.2) then (E′_{1}, σ′_{1}) is isomorphic to a subalgebra of (A_{2}, τ_{2}). Conversely, if (E_{2}, σ_{2}) is isomorphic to an n_{2}-dimensional étale subalgebra of (A_{2}, τ_{2}) satisfying (2.2) then (E_{2}^{+}, σ_{2}^{+}) is isomorphic to a subalgebra of (A_{1}, τ_{1}). Thus, the correspondences

(E_{1}, σ_{1}) ↦ (E′_{1}, σ′_{1}) and (E_{2}, σ_{2}) ↦ (E_{2}^{+}, σ_{2}^{+})

implement mutually inverse bijections between the sets of isomorphism classes of n_{1}- and n_{2}-dimensional étale subalgebras of (A_{1}, τ_{1}) and (A_{2}, τ_{2}) that are invariant under the respective involutions and satisfy (2.2).

Proof. If we have rk_{K_{v}} G_{1} = rk_{K_{v}} G_{2} = ℓ for all v ∈ V_{∞}^{K} then the groups G_{1} and G_{2} are K-split by (6.1) and the Hasse Principle. Then τ_{1} corresponds to a nondegenerate symmetric bilinear form of Witt index ℓ, and A_{2} = M_{n_{2}}(K) with τ_{2} corresponding to a nondegenerate skew-symmetric form. In this case, our claim immediately follows from Propositions 2.3 and 2.5, as in Remark 2.6. So, we may assume that there is a real v_{0} ∈ V_{∞}^{K} such that rk_{K_{v_{0}}} G_{1} = rk_{K_{v_{0}}} G_{2} = 0. Observe that given any real v ∈ V_{∞}^{K} satisfying rk_{K_{v}} G_{1} = rk_{K_{v}} G_{2} = 0, the data in Table 1 shows that that for any n_{1}-dimensional τ_{1}-invariant étale subalgebra E_{1} ⊂ A_{1} satisfying (2.2) and σ_{1} = τ_{1}|_{E_{1}} we have

(6.3)  (E_{1} ⊗_{K} K_{v}, σ_{1} ⊗ id_{K_{v}}) ≃ (ℂ,¯ )^{ℓ} × (ℝ, id_{ℝ}),

and for any n_{2}-dimensional τ_{2}-invariant étale subalgebra E_{2} ⊂ A_{2} satisfying (2.2) and σ_{2} = τ_{2}|_{E_{2}} we have

(6.4)  (E_{2} ⊗_{K} K_{v}, σ_{2} ⊗ id_{K_{v}}) ≃ (ℂ,¯ )^{ℓ}.

Let (E_{1}, σ_{1}) be as in the statement of the proposition. We first show that for any v ∈ V^{K} there is an embedding

ι_{v}: (E′_{1} ⊗_{K} K_{v}, σ′_{1} ⊗ id_{K_{v}}) ↪ (A_{2} ⊗_{K} K_{v}, τ_{2} ⊗ id_{K_{v}}).

If rk_{K_{v}} G_{1} = rk_{K_{v}} G_{2} = ℓ, this follows from Proposition 2.3. Otherwise, v is real, and rk_{K_{v}} G_{1} = rk_{K_{v}} G_{2} = 0, so we see from (6.3) that

(E′_{1} ⊗_{K} K_{v}, σ′_{1} ⊗ id_{K_{v}}) ≃ (ℂ,¯ )^{ℓ}.

Then the existence of ι_{v} follows from the argument given in the proof of Lemma 3.3. Now, applying Lemma 4.3 we obtain the existence of an embedding ι: (E′_{1}, σ′_{1}) ↪ (A_{2}, τ_{2}), as required.

Conversely, let (E_{2}, σ_{2}) be as in the proposition. Then arguing as above (using Proposition 2.5 and the proof of Lemma 3.2) we obtain the existence of local embeddings

ι_{v}: (E_{2}^{+} ⊗_{K} K_{v}, σ_{2}^{+} ⊗ id_{K_{v}}) ↪ (A_{1} ⊗_{K} K_{v}, τ_{1} ⊗ id_{K_{v}})

for all v ∈ V^{K}. It follows from (6.4) that

(E_{2}^{+} ⊗_{K} K_{v_{0}}, σ_{2}^{+} ⊗ id_{K_{v_{0}}} ) ≃ (ℂ,¯ )^{ℓ} × (ℝ, id_{ℝ}).

This enables us to use Lemma 4.2 which yields the existence of an embedding (E_{2}^{+}, σ_{2}^{+}) ↪ (A_{1}, τ_{1}), completing the argument. □

The following consequence of the proposition proves the “if” component in both parts, (1) and (2), of Theorem 1.4.

Corollary 6.2. Let G_{1} and G_{2} be absolutely almost simple algebraic groups of types B_{ℓ} and C_{ℓ} respectively that are twins. Then

(i) G_{1} and G_{2} have the same \underline{isogeny} classes of maximal K-tori.

(ii) If G_{1} is adjoint and G_{2} is simply connected then G_{1} and G_{2} have the same \underline{isomorphism} classes of maximal K-tori.

Proof. (ii) easily follows from the proposition, and (i) is an immediate consequence of (ii). □

Remark 6.3. The assumption ℓ ⩾ 3 was never used in Proposition 6.1 and Corollary 6.2. So, these statements remain valid also for ℓ = 2, which will be helpful in § 8.

We now turn to the proof of the “only if” direction in both parts of Theorem 1.4 where the assumption ℓ ⩾ 3 becomes essential and will be kept throughout the rest of the section. This direction requires a bit more work and involves the notion of generic tori. To recall the relevant definitions, we let G denote a semi-simple algebraic K-group, and fix a maximal K-torus T of G. Furthermore, we let Φ(G, T) denote the corresponding root system, and let K_{T} denote the minimal splitting field of T over K. The natural action of Gal(K_{T} /K) on the group of characters X(T) gives rise to an injective group homomorphism

θ_{T} : Gal(K_{T} /K) ⟶ Aut(Φ(G, T)).

Then T is called generic (over K) if θ_{T} (Gal(K_{T} /K)) contains the Weyl group W (G, T). As the following statement shows, generic tori with prescribed local properties always exist.

Proposition 6.4. [PR09, Corollary 3.2] Let G be an absolutely almost simple algebraic K-group, and let V ⊂ V^{K} be a finite subset. Suppose that for each v ∈ V we are given a maximal K_{v}-torus T^{(v)} of G. Then there exists a maximal K-torus T of G which is generic over K and which is conjugate to T^{(v)} by an element of G(K_{v}) for all v ∈ V.

We now return to the situation where G_{1} and G_{2} are absolutely almost simple K-groups of types B_{ℓ} and C_{ℓ} (ℓ ⩾ 3) respectively. We let G_{1}^{♮} denote the adjoint group of G_{1}, and G_{2}^{♮} the simply connected cover of G_{2}. Furthermore, given a maximal K-torus T_{i} of G_{i}, we let T_{i}^{♮} denote the image of T_{i} in G_{i}^{♮} if i = 1 and the preimage of T_{i} in G_{i}^{♮} if i = 2.

Proposition 6.5. Let T_{i} be a generic maximal K-torus of G_{i} where i = 1, 2. If there exists a K-isogeny π: T_{i} → T_{3−i} onto a maximal K-torus of G_{3−i} then there exist a K-isomorphism T_{i}^{♮} ≃ T_{3−i}^{♮}.

The proof below is an adaptation of Lemma 4.3 and Remark 4.4 in [PR09].

Proof. We have K_{T_{1}} = K_{T_{2}} =: L, and let G = Gal(L/K). Then θ_{T_{j}} is an isomorphism of G on W_{j} = W (G_{j}, T_{j}) for j = 1, 2. The isogeny π induces a G-equivariant homomorphism of character groups π^{∗}: X(T_{3−i}) → X(T_{i}). Let X_{j}^{♮} = X(T_{j}^{♮}); we need to prove that there is a G-equivariant isomorphism ψ: X_{3−i}^{♮} → X_{i}^{♮}. (We recall that X_{1}^{♮} is the subgroup of X(T_{1}) generated by all the roots in Φ_{1} = Φ(G_{1}, T_{1}), and X_{2}^{♮} is generated by the weights of the root system Φ_{2} = Φ(G_{2}, T_{2}).)

To avoid cumbersome notations, we will assume that i = 1. (This does not restrict generality as along with π there is always a K-isogeny π′: T_{3−i} → T_{i}.) Consider

ϕ = π^{∗} ⊗ id_{ℝ}: V_{2} = X(T_{2}) ⊗_{ℤ} ℝ ⟶ X(T_{1}) ⊗_{ℤ} ℝ = V_{2}

and μ: W_{2} → W_{1} defined by μ = θ_{T_{1}} ∘ θ_{T_{2}}^{−1}. Then the fact that π^{∗} is G-equivariant implies that

(6.5)  ϕ(w · v) = μ(w) · ϕ(v) for all v ∈ V_{2}, w ∈ W_{2}.

On the other hand, it follows from the explicit description of the root systems as in [Bou02] that there exists a linear isomorphism ϕ_{0} : V_{2} → V_{1} and a group isomorphism μ_{0} : W_{2} → W_{1} such that

(6.6)  ϕ_{0}(w · v) = μ_{0}(w) · ϕ_{0}(v) for all v ∈ V_{2}, w ∈ W_{2},

ϕ_{0} takes the short roots of Φ_{2} to the long roots of Φ_{1}, and (1/2)ϕ_{0} takes the long roots of Φ_{2} to the short roots of Φ_{1}, consequently ϕ_{0}(X_{2}^{♮}) = X_{1}^{♮}. (Note that we identify W_{j} with the Weyl group of the root system Φ_{j}.)

We claim that there exists a nonzero λ ∈ ℝ and z ∈ W_{1} such that

ϕ(v) = λ · z · ϕ_{0}(v) and μ(w) = z · μ_{0}(w) · z^{−1} for all v ∈ V_{2}, w ∈ W_{2}.

Indeed, it was shown in [PR09, Lemma 4.3] (using that ℓ ⩾ 3) that a suitable multiple ϕ′ = λ^{−1} · ϕ takes the short roots of Φ_{2} to the long roots of Φ_{2}, and (1/2)ϕ_{0} takes the long roots of Φ_{2} to the short roots of Φ_{1}. Then z := ϕ′ ∘ ϕ_{0}^{−1} is an automorphism of Φ_{1}, hence can be identified with an element of W_{1}. This gives the formula for ϕ, and then the formula for μ follows from (6.5) and (6.6).

Put ψ := λ^{−1} · ϕ. Then

ψ(X_{2}^{♮}) = z(ϕ_{0}(X_{2}^{♮})) = X_{1}^{♮},

and ψ is G-equivariant, as required. □

Corollary 6.6. Let T_{i} be a generic maximal K-torus of G_{i}. If there exists v ∈ V such that T_{i}^{♮} does not allow a K_{v}-defined embedding into G_{3−i}^{♮} then T_{i} is not Kisogenous to any maximal K-torus T_{3−i} of G_{3−i}. Thus, if G_{1} and G_{2} have the same isogeny classes of maximal K-tori then G_{1}^{♮} and G_{2}^{♮} have the same isomorphism classes of maximal K_{v}-tori for all v ∈ V.

Proof. The first assertion immediately follows from the proposition. To derive the second assertion from the first, we observe that given v ∈ V and a maximal K_{v}- torus T_{i} of G_{i}^{♮} that does not allow a K_{v}-embedding into G_{3−i}^{♮}, we can find a maximal K-torus T_{i} of G_{i} such that T_{i}^{♮} is conjugate to T_{i} by an element G_{i}^{♮}(K_{v}). □

Proof of Theorem 1.4, “only if”. Assume G_{1} and G_{2} have the same isogeny classes of maximal K-tori. Then by Corollary 6.6, G_{1}^{♮} and G_{2}^{♮} have the same isomorphism classes of maximal K_{v}-tori for all v. It follows that G_{1} and G_{2} are twins (by Corollary 3.4 for v real and Lemma 3.7 for v finite), completing the proof of part (1) of Theorem 1.4.

Now suppose that G_{1} and G_{2} have the same isomorphism classes of maximal K-tori, in particular, there is a K-isomorphism π: T_{1} → T_{2} between two generic K-tori. Then as in the proof of Proposition 6.5, π^{∗} induces ϕ: V_{2} → V_{1} which necessarily satisfies ϕ(X(T_{2})) = X(T_{1}) and ϕ(X(T_{2}^{♮})) = X(T_{1}^{♮}). Since X(T_{1}^{♮}) ⊆ X(T_{1}) and X(T_{2}^{♮}) ⊇ X(T_{2}), this is possible only if both inclusions are in fact equalities, i.e., G_{1} = G_{1}^{♮} and G_{2} = G_{2}^{♮}. This completes the proof of part (2) of Theorem 1.4. □

7. Weakly commensurable subgroups and proof of Theorem 1.2

We begin by recalling the notion of weak commensurability of Zariski-dense subgroups introduced in [PR09]. Let G_{1} and G_{2} be semi-simple algebraic groups over a field F of characteristic zero, and let Γ_{i} ⊂ G_{i}(F) be a Zariski-dense subgroup for i = 1, 2. Semi-simple elements γ_{i} ∈ Γ_{i} are weakly commensurable if there exist maximal F -tori T_{i} of G_{i} such that γ_{i} ∈ T_{i}(F) and for some characters χ_{i} ∈ X(T_{i}) we have

χ_{1}(γ_{1}) = χ_{2}(γ_{2}) ≠ 1.

Furthermore, the subgroups Γ_{1} and Γ_{2} are weakly commensurable if every semisimple element γ_{1} ∈ Γ_{1} of infinite order is weakly commensurable to some γ_{2} ∈ Γ_{2} of infinite order, and vice versa.

The focus in [PR09] was on analyzing when two Zariski-dense S-arithmetic subgroups in absolutely almost simple algebraic groups are weakly commensurable. This analysis was based on a description of such S-arithmetic groups in terms of triples, which we will now briefly recall. Let G be a (connected) absolutely almost simple algebraic group defined over a field F of characteristic zero, \overline{G} be its adjoint group, and π: G → \overline{G} be the natural isogeny. Suppose we are given the following data:

• a number field K with a fixed embedding K ↪ F;

• a finite set S of valuations of K containing all archimedean valuations; and • an F/K-form 𝒢 of \overline{G} (i.e., a K-defined algebraic group such that there exists an F -defined isomorphism of algebraic groups _{F} 𝒢 ≃ \overline{G}, where _{F} 𝒢 is the group obtained from 𝒢 by the extension of scalars F/K).

(Note that it is assumed in addition that S does not contain any nonarchimedean valuations v such that 𝒢 is K_{v}-anisotropic.) We then have an embedding ι: 𝒢(K) ↪ \overline{G}(F) and a natural S-arithmetic subgroup 𝒢(O_{K}(S)), where O_{K}(S) is the ring of Sintegers in K, defined in terms of a fixed K-embedding 𝒢 ↪ GL_{n}, i.e. 𝒢(O_{K}(S)) = 𝒢(K)∩GL_{n}(O_{K}(S)). A subgroup Γ of G(F) such that π(Γ) is commensurable with ι(𝒢(O_{K}(S))) is called (𝒢, K, S)-arithmetic. (It should be pointed out that we do not fix an F -defined isomorphism _{F} 𝒢 ≃ \overline{G} in this definition, and by varying it we obtain a class of subgroups invariant under F -defined automorphisms of 𝒢 in the obvious sense.)

It was shown in [PR09] that if G_{i} is absolutely almost simple and Γ_{i} is Zariskidense and (𝒢_{i}, K_{i}, S_{i})-arithmetic for i = 1, 2 then the weak commensurability of Γ_{1} and Γ_{2} implies that K_{1} = K_{2} =: K and S_{1} = S_{2} =: S, and additionally either G_{1} and G_{2} are of the same type or one of them is of type B_{ℓ} and the other is of type C_{ℓ} for some ℓ ⩾ 3. That paper also contains many precise conditions for two Sarithmetic subgroups to be weakly commensurable in the case where G_{1} and G_{2} are of the same type. The goal of this section is to prove Theorem 1.2 which provides such conditions when one of the groups is of type B_{ℓ} and the other of type C_{ℓ} (ℓ ⩾ 3). In conjunction with the previous results, this completes the investigation of weak commensurability of S-arithmetic subgroups in absolutely almost simple groups over number fields.

Proof of Theorem 1.2. Let G_{1} and G_{2} be absolutely almost simple algebraic groups of types B_{ℓ} and C_{ℓ} (ℓ ⩾ 3) respectively defined over a number field K, and let Γ_{i} be a Zariski-dense (𝒢_{i}, K, S)-arithmetic subgroup of G_{i}.

Suppose that 𝒢_{1} and 𝒢_{2} are twins. Then by Theorem 1.4, they have the same isogeny classes of maximal K-tori. This automatically implies that Γ_{1} and Γ_{2} are weakly commensurable. To see this, we basically need to repeat the argument given in [PR09, Example 6.5], which we also give here for the reader’s convenience. First, we may assume without any loss of generality that G_{1} and G_{2} are adjoint (cf. Lemma 2.4 in [PR09]), hence Γ_{i} ⊂ 𝒢_{i}(K). Let γ_{1} ∈ Γ_{1} be a semi-simple element of infinite order, and let T_{1} be a maximal K-torus of 𝒢_{1} that contains γ_{1}. Then there exists a K-isogeny φ: T_{1} → T_{2} onto a maximal K-torus T_{2} of 𝒢_{2}. The subgroup φ(T_{1}(K) ∩ Γ_{1}) is an S-arithmetic subgroup of T_{2}(K), so there exists n > 0 such that γ_{2} := φ(γ_{1})^{n} ∈ Γ_{2}. Let χ_{1} ∈ φ^{∗}(X(T_{2})) be a character such that χ_{1}(γ_{1}) is not a root of unity, and let χ_{2} ∈ X(T_{2}) be such that φ^{∗}(χ_{2}) = χ_{1}. Then

(nχ_{1})(γ_{1}) = χ_{1}(γ_{1})^{n} = χ_{2}(γ_{2}) ≠ 1,

which implies that Γ_{1} and Γ_{2} are weakly commensurable.

Conversely, suppose that Γ_{1} and Γ_{2} are weakly commensurable. According to [PR09, Theorem 6.2], this in particular implies that

rk_{K_{v}} 𝒢_{1} = rk_{K_{v}} 𝒢_{2} for all v ∈ V^{K}.

As we have seen in Lemma 3.7, for v ∈ V_{f}^{K} and the groups under consideration, the equality of ranks implies that both groups are actually K_{v}-split, verifying condition (6.1) in § 6. Assume that condition (6.2) fails for a real v_{0} ∈ V_{∞}^{K}. Then by Corollary 3.4, there is an i ∈ {1, 2} and a maximal K_{v_{0}}-torus T_{i} of 𝒢_{i}^{♮} that does not allow a K_{v_{0}}-embedding into 𝒢_{3−i}^{♮}; obviously T_{i} is K_{v_{0}}-isotropic. Let T_{i}^{(v_{0})} be a maximal K_{v_{0}}-torus of 𝒢_{i} such that (T_{i}^{♮})^{(v_{0})} = T_{i}. Furthermore, for v ∈ S \ {v_{0}} we let T_{i}^{(v)} denote a maximal K_{v}-torus of 𝒢_{i} such that rk_{K_{v}} T_{i}^{(v)} = rk_{K_{v}} 𝒢_{i}. Using Proposition 6.4, we can find a maximal K-torus T_{i} of 𝒢_{i} that is generic and that is conjugate to T^{(v)} by an element of 𝒢_{i}(K_{v}) for all v ∈ S ∪ {v_{0}}. Then clearly

rk_{S} T_{i} := ∑_{v∈S} rk_{K_{v}} T_{i} > 0

as rk_{S} 𝒢_{i} > 0. By Dirichlet’s Theorem [PR94, Theorem 5.12], the group of Sintegral points T_{i}(O_{K}(S)) has the following structure: H × ℤ^{d} where d = rk_{S} T_{i} − rk_{K} T_{i}. Since T_{i} is obviously K-anisotropic, we conclude that there exists γ_{i} ∈ T_{i}(K) ∩ Γ_{i} of infinite order (as in the previous paragraph, we are assuming that G_{1} and G_{2} are adjoint, hence Γ_{j} ⊂ 𝒢_{j}(K) for j = 1, 2). Then γ_{i} is weakly commensurable to some semi-simple γ_{3−i} ∈ Γ_{3−i} of infinite order. Let T_{3−i} be a maximal K-torus of 𝒢_{3−i} containing γ_{3−i}. By the Isogeny Theorem [PR09, Theorem 4.2], the tori T_{i} and T_{3−i} are K-isogenous. Using Proposition 6.5, we conclude that T_{i}^{♮} and T_{3−i}^{♮} are K-isomorphic. This implies that over K_{v_{0}}, the torus T_{i} ≃ T_{i}^{♮} has an embedding into 𝒢_{3−i}. A contradiction, proving (6.2), and completing the proof of Theorem 1.2. □

As we already mentioned, the notion of weak commensurability was introduced in order to tackle some differential-geometric problems dealing with length-commen-surable and isospectral locally symmetric spaces, and we would like to conclude this section with a sample of geometric consequences of the results of the current paper established in [PR11]. For a Riemannian manifold M, we let L(M) denote the weak length spectrum of M, i.e., the collection of lengths of all closed geodesics in M. Two Riemannian manifolds M_{1} and M_{2} are called length-commensurable if ℚ · L(M_{1}) = ℚ · L(M_{2}).

(7.1)

Let M_{1} be an arithmetic quotient of the real hyperbolic space ℍ^{p} (p ⩾ 5), and M_{2} be an arithmetic quotient of the quaternionic hyperbolic space ℍ_{H}^{q} (q ⩾ 2). Then M_{1} and M_{2} are not length-commensurable.

Theorem 1.2 is used to handle the case p = 2n and q = n − 1 for n ⩾ 3; for other values of p and q, the claim follows from [PR09, Th. 8.15].

Now, let 𝔛_{1} be the symmetric space of the real Lie group G_{1} = SO(n + 1, n), and let 𝔛_{2} be the symmetric space of the real Lie group G_{2} = Sp_{2n} where n ⩾ 3.

(7.2)

Let M_{i} be the quotient of 𝔛_{i} by a (𝒢_{i}, K)-arithmetic subgroup of G_{i} for i = 1, 2. If 𝒢_{1} and 𝒢_{2} are twins then

ℚ · L(M_{2}) = λ · ℚ · L(M_{1}) where λ = \sqrt{\frac{2n + 2}{2n − 1}}.

(We refer to [PR09], § 1, for the notion of arithmeticity and the explanation of other terms used here.) We finally note that even though one can make 𝔛_{1} and 𝔛_{2} length-commensurable by scaling the metric on one of them, this will never make them isospectral [Yeu11].

8. Proofs of Proposition 1.3 and Theorem 1.5

Proof of Proposition 1.3. We can assume that G_{1} and G_{2} are connected absolutely almost simple adjoint K-groups having the same isogeny classes of maximal K-tori. Assume that provisions (2) and (3) of the proposition do not hold; let us show that (1) must hold. First, by [PR09, Theorem 7.5], G_{1} and G_{2} have the same Killing-Cartan type. Furthermore, if L_{i} is the minimal Galois extension of K over which G_{i} becomes an inner form then L_{1} = L_{2}; in other words, G_{1} and G_{2} are inner twists of the same quasi-split K-group. So, the required assertion is a consequence of the following lemma. □

Lemma 8.1. Let G_{1} and G_{2} be connected absolutely almost simple \underline{adjoint} Kgroups of the same Killing-Cartan type which is different from A_{ℓ} (ℓ > 1), D_{2ℓ+1} (ℓ > 1) or E_{6}. Assume that G_{1} and G_{2} are inner twists of the same quasi-split K-group (which holds automatically if G_{1} and G_{2} are not of type D). If G_{1} and G_{2} have the same isogeny classes of maximal K-tori then G_{1} ≃ G_{2}.

Proof. First, suppose that the groups are not of type D. As we have seen in § 5, the fact that G_{1} and G_{2} have the same isogeny classes of maximal K-tori implies that rk_{K_{v}} G_{1} = rk_{K_{v}} G_{2} for all v ∈ V^{K}. For groups of one of the types under consideration, this implies that G_{1} ≃ G_{2} over K_{v} for all v ∈ V^{K} and then our assertion follows from the Hasse principle for Galois cohomology of adjoint groups (see [PR09, § 6] for details of the argument).

Now, suppose the groups are of type D_{2ℓ} for some ℓ ⩾ 2. There exists a maximal K-torus T_{1} of G_{1} that is generic and such that rk_{K_{v}} T_{1} = rk_{K_{v}} G_{1} at every place v where at least one of G_{1} or G_{2} is not quasi-split. (Note that the set of such v’s is finite, cf. [PR94, Theorem 6.7].) By hypothesis, T_{1} is isogenous to a maximal Ktorus T_{2} of G_{2}, which is necessarily also generic. Following Lemma 4.3 and Remark 4.4 in [PR09], one finds a K-isomorphism T_{1} → T_{2} that extends to a K̄-isomorphism G_{1} → G_{2}. Then our assertion follows from Theorem 20 in [Gar12]. □

Proof of Theorem 1.5. The “if” direction is actually contained in Corollary 6.2— see Remark 6.3. For the “only if” direction, we first observe that if G_{1} and G_{2} have the same isomorphism classes of maximal K-tori then by Lemma 8.1 the groups SO(q_{1}) and SO(q_{2}) are isomorphic, hence the forms q_{1} and q_{2} are similar, yielding assertion (1). Thus, we can assume that G_{1} = SO(q) and G_{2} = Spin(q) for a single quadratic form q.

To prove assertion (2), it is enough to show that if v ∈ V^{K} is such that the Witt index of q over K_{v} is 1 then there exists a 2-dimensional K_{v}-torus T_{1} that has a K_{v}-embedding into G_{1} but does not allow a K_{v}-embedding into G_{2}. For this we pick a quadratic extension L/K_{v} and set

T_{1} = GL_{1} ×R_{L/K_{v}}^{(1)}(GL_{1}).

We can write q = q′ ⊥ q′′ where q′ is a hyperbolic plane. Then SO(q′) = GL_{1} and SO(q′′) = PSL_{1,D} where D is a quaternion division algebra over K_{v}. Since L embeds in D, the torus R_{L/K_{v}}^{(1)}(GL_{1}) embeds in SL_{1,D} and then also in PSL_{1,D}. It follows that T_{1} embeds in G_{1} = SO(q). On the other hand, let T_{2} ⊂ G_{2} be a maximal K_{v}-torus that splits over L. We can identify G_{2} with SU(A, τ) where A = M_{2}(D) with D a quaternion division algebra over K and τ is a symplectic involution on A. Let E_{2} be the K_{v}-subalgebra of A generated by T_{2}(K_{v}). Then E_{2}⊗_{K_{v}} L ≃ L^{4}. As in § 3, we conclude that (E_{2}, τ|_{E_{2}}) is isomorphic to (L, σ)×(L, σ) where σ is the nontrivial automorphism of L, or to (L × L, λ) where λ is the switch involution. Then T_{2} = SU(E_{2}, τ|_{E_{2}}) is isomorphic respectively to R_{L/K_{v}}^{(1)}(GL_{1})^{2} or R_{L/K_{v}}(GL_{1}). Neither such torus can be isomorphic to T_{1}. □

9. Alternative proofs via Galois cohomology

Although the main body of the paper demonstrates the effectiveness (and in fact the ubiquity) of the technique of étale algebras in dealing with maximal tori of classical groups, it is worth pointing out that some parts of the argument can also be given in the language of Galois cohomology of algebraic groups. In this section, we will illustrate such an exchange by giving a cohomological proof of the “if” direction of Theorem 1.4(2), i.e., of Corollary 6.2(ii).

Our main tool is Proposition 9.1, for which we need some notation. Let G be a connected semi-simple algebraic group over a number field K. Fix a maximal K-torus T of G, and let N = N_{G}(T) and W = N/T denote respectively its normalizer and the corresponding Weyl group. For any field extension P/K, we let θ_{P} : H^{1}(P, N) → H^{1}(P, W) denote the map induced by the natural K-morphism N → W, and let

𝒞(P) := Ker (H^{1}(P, N) ⟶ H^{1}(P, G));

its elements are in one-to-one correspondence with the G(P)-conjugacy classes of maximal P -tori in G, see for example [PR09, Lemma 9.1] where this correspondence is described explicitly. There is an obvious K-defined map W → Aut T, so for any ξ ∈ H^{1}(K, W) one can consider the corresponding twisted K-torus _{ξ}T.

Proposition 9.1. Assume that there exists a subset V_{0} ⊂ V_{∞}^{K} such that G is K_{v}- anisotropic for all v ∈ V_{0} and is K_{v}-split for all v ∈ V^{K} \ V_{0}. Then the sequence

(9.1)  𝒞(K) ⟶^{θ_{K}} H^{1}(K, W) \underrightarrow{∏ ρ_{v}} ∏_{v∈V_{0}} H^{1}(K_{v}, W)

is exact.

Here ρ_{v} denotes the natural restriction map H^{1}(K, W) → H^{1}(K_{v}, W).

Proof. If V_{0} is empty then it follows from the Hasse principle for adjoint groups [PR94, Theorem 6.22] that G is K-split. In this case it was shown by Gille [Gil04] and Raghunathan [Rag04] (or earlier by Kottwitz [Kot82]) that θ_{K}(𝒞(K)) = H^{1}(K, W), and our claim follows. So, we will assume in the rest of the argument that V_{0} is not empty.

We first prove that ρ_{v}θ_{K} = 0 for all v ∈ V_{0}. Given ξ ∈ 𝒞(K), one can pick g ∈ G(K̄) such that n(σ) := g^{−1}σ(g) belongs to N(K̄) for all σ ∈ Gal(K̄/K), and the cocycle σ ↦ n(σ) represents ξ. Then the maximal torus T′ = gT g^{−1} is defined over K. Now, let v ∈ V_{0}. According to our definitions, G is anisotropic over K_{v} = ℝ, so it follows from the conjugacy of maximal tori in compact Lie groups that T and T′ are conjugate by an element of G(K_{v}). Then the one-to-one correspondence between the elements of 𝒞(K_{v}) and the G(K_{v})-conjugacy classes of maximal K_{v}-tori in G (or a simple direct computation) implies that the image of ξ under the restriction map 𝒞(K) → 𝒞(K_{v}) is trivial, and hence the image of θ_{K}(ξ) under the restriction map H^{1}(K, W) → H^{1}(K_{v}, W) is trivial as well.

Now suppose that G is simply connected; we verify that every ξ ∈ ∩_{v∈V_{0}} ker ρ_{v} is in the image of θ_{K}. Pick v ∈ V_{0}. Since ξ lies in the kernel of H^{1}(K, W) → H^{1}(K_{v}, W), the twisted torus _{ξ}T is K_{v}-isomorphic to T, hence K_{v} anisotropic (as G is K_{v}-anisotropic). Thus,

Ker (H^{2}(K,_{ξ}T) → ∏_{v∈V^{K}} H^{2}(K_{v},_{ξ}T)) = 0

by [PR09, Prop. 6.12]. Invoking now [PR09, Th. 9.2], we see that to prove the inclusion ξ ∈ θ_{K}(𝒞(K)), it is enough to show that ρ_{v}(ξ) ∈ θ_{K_{v}}(𝒞(K_{v})) for all v ∈ V^{K}. If v ∈ V_{0} then by construction ρ_{v}(ξ) is trivial, and there is nothing to prove. Otherwise, the group G is K_{v}-split, so by the result of Gille-Kottwitz-Raghunathan we have θ_{K_{v}}(𝒞(K_{v})) = H^{1}(K_{v}, W), and the inclusion ρ_{v}(ξ) ∈ θ_{K_{v}}(𝒞(K_{v})) is obvious. Since ξ was arbitrary, we have proved that ∩ ker ρ_{v} contains the image of θ_{K}.

In case G is not simply connected, we fix a K-defined universal cover π: G̃ → G of G and use the tilde to denote the objects associated with G̃. Then π yields a K-isomorphism of W̃ and W and we have a commutative diagram

𝒞̃(K) \underrightarrow{θ̃_{K}} H^{1}(K, W̃) \underrightarrow{∏ ρ̃_{v}} ∏_{v∈V_{0}} H^{1}(K_{v}, W̃)
↓  ∥  ∥
𝒞(K) ⟶^{θ_{K}} H^{1}(K, W) \underrightarrow{∏ ρ_{v}} ∏_{v∈V_{0}} H^{1}(K_{v}, W).

The top row is exact by the previous paragraph, hence ∩ ker ρ_{v} contains the image of θ_{K}. □

We now begin to work our way towards the proof of Theorem 1.4(2)/Corollary 6.2(ii). Let G_{1} be adjoint of type B_{ℓ} and let G_{2} be simply connected of type C_{ℓ} for some ℓ ⩾ 2. We will use a subscript i ∈ {1, 2} to denote the objects associated with G_{i}. In particular, we let T_{i} denote a maximal torus of G_{i}, and let N_{i} = N_{G_{i}}(T_{i}) and W_{i} = N_{i}/T_{i} be its normalizer and the Weyl group. Then W_{i} naturally acts on T_{i} by conjugation. We say that the morphisms of algebraic groups φ: T_{1} → T_{2} and ψ: W_{1} → W_{2} are compatible if

φ(w · t) = ψ(w) · φ(t) for all t ∈ T_{1}, w ∈ W_{1}.

Lemma 9.2. One can pick maximal K-tori T_{i} of G_{i} for i = 1, 2 so that there exist compatible K-defined isomorphisms

φ: T_{1} → T_{2} and ψ: W_{1} → W_{2}.

Proof. Imitating the argument given in [PR94, Proposition 6.16], it is easy to see that there exists a quadratic extension L/K that splits both G_{1} and G_{2}. Indeed, let V_{i} be the (finite) set of places v ∈ V^{K} such that G_{i} does not split over K_{v}, and let V = V_{1} ∪V_{2}. Pick a quadratic extension L/K so that the local degree [L_{w} : K_{v}] = 2 for all v ∈ V and w|v. We claim that L is as required. By the Hasse principle, it is enough to show that both G_{1} and G_{2} split over L_{w} for any w ∈ V^{L}. For a given w, we let v ∈ V^{K} be the place that lies below w. If v ∉ V then by our construction G_{1} and G_{2} split already over K_{v}, and there is nothing to prove. If v ∈ V then [L_{w} : K_{v}] = 2, and then the proof of [PR94, Proposition 6.16] that G_{1} and G_{2} split over L_{w}, as required.

Now, let σ ∈ Gal(L/K) be a generator. According to [PR94, Lemma 6.17], for each i ∈ {1, 2}, there exists an L-defined Borel subgroup B_{i} of G_{i} such that T_{i} := B_{i} ∩ B_{i}^{σ} is a maximal K-torus of G_{i} that splits over L. Considering the action of σ on the root system Φ(G_{i}, T_{i}), we see that it takes the system of positive roots corresponding to B_{i} into the system of negative roots. For groups of types B_{ℓ} and C_{ℓ}, this implies that σ acts on the character group X(T_{i}) as multiplication by (−1). It easily follows from the description of the corresponding root systems (cf. [Bou02]) that there exist compatible (in the obvious sense) isomorphisms φ^{∗}: X(T_{2}) → X(T_{1}) (of abelian groups) and ψ: W_{1} → W_{2} (of abstract groups considered as subgroups of GL(X(T_{1})) and GL(X(T_{2}))). Then φ^{∗} gives rise to an isomorphism φ: T_{1} → T_{2} of algebraic groups that is compatible (as defined above) with ψ (which can be considered as a morphism of algebraic groups). It remains to observe that since σ acts on X(T_{1}) and X(T_{2}) as multiplication by (−1), both φ and ψ are K-defined (in fact, σ acts on W_{1} and W_{2} trivially). □

Remark. If both groups G_{1} and G_{2} are K-split then one can, of course, take for T_{1} and T_{2} their maximal K-split tori.

For the rest of the paper, we fix compatible K-defined isomorphisms

φ^{0} : T_{1}^{0} → T_{2}^{0} and ψ^{0} : W_{1}^{0} → W_{2}^{0}.

(Thus, we henceforth slightly change the notations used in Lemma 9.2.) Given arbitrary maximal K-tori T_{i} of G_{i} for i = 1, 2, we pick elements g_{i} ∈ G(K̄) so that

T_{i} = g_{i}T_{i}^{0}g_{i}^{−1},

and then for any σ ∈ Gal(K̄/K), the element n_{i}(σ) := g_{i}^{−1}σ(g_{i}) belongs to N_{i}^{0}(K̄). Let φ = φ(g_{1}, g_{2}) be the morphism T_{1} → T_{2} defined by

φ(t) = g_{2}φ^{0}(g_{1}^{−1}tg_{1})g_{2}^{−1},

and let ν_{i}^{0} : N_{i}^{0} → W_{i}^{0} denote the canonical morphism.

Lemma 9.3. If

(9.2)  ψ^{0}(ν_{1}^{0}(n_{1}(σ))) = ν_{2}^{0}(n_{2}(σ)) for all σ ∈ Gal(K̄/K)

then φ = φ(g_{1}, g_{2}) is defined over K.

Proof. We need to show that φ commutes with every σ ∈ Gal(K̄/K). Since φ^{0} is defined over K, for any t ∈ T_{1}(K̄), we have

σ(φ(t)) = σ(g_{2})φ^{0}(σ(g_{1})^{−1}σ(t)σ(g_{1}))σ(g_{2})^{−1}
= g_{2}n_{2}(σ)φ^{0}(n_{1}(σ)^{−1}g_{1}^{−1}σ(t)g_{1}n_{1}(σ))n_{2}(σ)^{−1}g_{2}^{−1}
= g_{2} [ (ν_{2}^{0}(n_{2}(σ))) · φ^{0}((ν_{1}^{0}(n_{1}(σ))) · (g_{1}^{−1}σ(t)g_{1})) ] g_{2}^{−1}.

Since φ^{0} is compatible with ψ^{0}, condition (9.2) implies that the latter reduces to

g_{2}φ^{0}(g_{1}^{−1}σ(t)g_{1})g_{2}^{−1} = φ(σ(t)).

It follows that σ(φ(t)) = φ(σ(t)), i.e. φ commutes with σ, as required. □

Pursuant to the above notations, for an extension P/K and i = 1, 2, we set

𝒞_{i}(P) = Ker (H^{1}(P, N_{i}^{0}) → H^{1}(P, G_{i})) ,

and let θ_{iP} : H^{1}(P, N_{i}^{0}) → H^{1}(P, W_{i}^{0}) denote the canonical map (induced by ν_{i}). The isomorphism H^{1}(K, W_{1}^{0}) → H^{1}(K, W_{2}^{0}) induced by ψ^{0} will still be denoted by ψ^{0}.

Lemma 9.4. Assume that

(9.3)  ψ^{0}(𝒞_{1}(K)) = 𝒞_{2}(K).

Then for i = 1 or 2, given any maximal K-torus T_{i} of G_{i} and an element g_{i} ∈ G_{i}(K̄) such that T_{i} = g_{i}T_{i}^{0}g_{i}^{−1}, there exists g_{3−i} ∈ G_{3−i}(K̄) such that the maximal torus

T_{3−i} := g_{3−i}T_{3−i}^{0}g_{3−i}^{−1}

and the isomorphism

φ(g_{1}, g_{2}): T_{1} → T_{2}

are K-defined. Thus, in this case G_{1} and G_{2} have the same isomorphism classes of maximal K-tori.

Proof. To keep our notations simple, we will give an argument for i = 1 (the argument in the case i = 2 is totally symmetric). As above, we set n_{1}(σ) = g_{1}^{−1}σ(g_{1}) ∈ N_{1}^{0}(K̄) for σ ∈ Gal(K̄/K), observing that these elements define a cohomology class n_{1} ∈ 𝒞_{1}(K). Then (9.3) implies that there exists h_{2} ∈ G_{2}(K̄) such that for the cohomology class m_{2} ∈ 𝒞_{2}(K) defined by the elements m_{2}(σ) = h_{2}^{−1}σ(h_{2}) ∈ N_{2}^{0}(K̄), we have

ψ^{0}(θ_{1K}(n_{1})) = θ_{2K}(m_{2}) in H^{1}(K, W_{2}).

Then there exists w_{2} ∈ W_{2}(K̄) such that

(9.4)  ψ^{0}(ν_{1}^{0}(n_{1}(σ))) = w_{2}^{−1}ν_{2}^{0}(m_{2}(σ))σ(w_{2}) for all σ ∈ Gal(K̄/K).

Picking z_{2} ∈ N_{2}^{0}(K̄) so that ν_{2}^{0}(z_{2}) = w_{2}, and setting

g_{2} = h_{2}z_{2} and n_{2}(σ) = g_{2}^{−1}σ(g_{2}) ∈ N_{2}^{0}(K̄) for σ ∈ Gal(K̄/K),

we obtain from (9.4) that (9.2) holds. Then g_{2} is as required. Indeed, the fact that n_{2}(σ) ∈ N_{2}^{0}(K̄) implies that T_{2} = g_{2}T_{2}^{0}g_{2}^{−1} is defined over K, and Lemma 9.3 yields that the morphism φ(g_{1}, g_{2}): T_{1} → T_{2} is also defined over K. □

Proof of Corollary 6.2(ii). Suppose that G_{1} and G_{2} are twins, and let V_{0} be the set of all archimedean places v ∈ V^{K} such that G_{1} and G_{2} are both K_{v}-anisotropic. Then for any v ∈ V^{K} \ V_{0}, both G_{1} and G_{2} are K_{v}-split. Then according to Proposition 9.1 we have

θ_{iK}(𝒞_{i}(K)) = ker (H^{1}(K, W_{i}^{0}) → ∏_{v∈V_{0}} H^{1}(K_{v}, W_{i}^{0}))

for i = 1, 2, and as ψ_{0} : W_{1}^{0} → W_{2}^{0} is an isomorphism, condition (9.3) holds, and the claim follows from Lemma 9.4. □

Remark. It follows from the explicit description of the root systems of types B_{ℓ} and C_{ℓ} that the isomorphism φ in Lemma 9.2 can be chosen so that for t ∈ T_{1}(K̄) there exist λ_{1}, …, λ_{ℓ} ∈ K̄^{×} such that the values of the roots α ∈ Φ(G_{1}, T_{1}) on t are

λ_{i}^{±1}, i = 1, …, ℓ, and λ_{i}^{±1} · λ_{j}^{±1}, i, j = 1, …, ℓ, i ≠ j,

and the values of the roots α ∈ Φ(G_{2}, T_{2}) on ϕ(t) are

λ_{i}^{±2}, i = 1, …, ℓ, and λ_{i}^{±1} · λ_{j}^{±1}, i, j = 1, …, ℓ, i ≠ j.

Then any identification of the form φ(g_{1}, g_{2}) also has this property, which was used in [PR11].

Alternatively, suppose that G_{i} for i = 1, 2 is realized as SU(A_{i}, τ_{i}) as described in the beginning of §6. Let E_{1} be a (τ_{1} ⊗ idK̄)-invariant maximal commutative étale K̄-subalgebra of A_{1} ⊗_{K} K̄ satisfying (2.2), and let σ_{1} = τ_{1}|_{E_{1}}. Then in the notations of §6, the algebra (E′_{1}, σ′_{1}) admits a K̄-embedding embedding into (A_{2} ⊗_{K} K̄, τ_{2} ⊗ idK̄), and we let (E_{2}, σ_{2}) the image of this embedding. It is easy to see that if we let T_{i} denote the maximal torus of G_{i} defined by (E_{i}, σ_{i}) then the isomorphism T_{1} ≃ T_{2} coming from the isomorphism of algebras (E′_{1}, σ′_{1}) ≃ (E_{2}, σ_{2}) is the same as the isomorphism coming from the description of the root systems (cf. the proof of Lemma 9.2); in particular, it is compatible with the natural isomorphism of the Weyl groups. So, the assertion of Lemma 9.2 means that given any K-algebras with involution (A_{1}, τ_{1}) and (A_{2}, τ_{2}) as above, there exists a τ_{1}- invariant maximal commutative étale K-subalgebra E_{1} of A_{1} that satisfies (2.2) and is such that for σ_{1} = τ_{1}|_{E_{1}}, the algebra (E′_{1}, σ′_{1}) admits an embedding into (A_{2}, σ_{2}). Moreover, by Corollary 6.2(ii), if the corresponding groups G_{1} and G_{2} are twins then the correspondence (E_{1}, σ_{1}) ↦ (E′_{1}, σ′_{1}) gives a bijection between the sets of isomorphism classes of maximal commutative étale K-subalgebras of (A_{1}, τ_{1}) and (A_{2}, τ_{2}) that are invariant under the respective involutions and satisfy (2.2). Thus, we recover Proposition 6.1.

Acknowledgements. SG’s research was partially supported by NSA grant H98230-11-1-0178 and the Charles T. Winship Fund. AR’s research was partially supported by NSF grant DMS-0965758 and the Humboldt Foundation. During the preparation of the final version of this paper, he was visiting the Mathematics Department of the University of Michigan as a Gehring Professor; the hospitality and generous support of this institution are gratefully acknowledged.

References

[AdC09] J. Adams and F. du Cloux, Algorithms for representation theory of real reductive groups, J. Inst. Math. Jussieu 8 (2009), no. 2, 209–259.

[BCKM03] R. Brusamarello, P. Chuard-Koulmann, and J. Morales, Orthogonal groups containing a given maximal torus, J. Algebra 266 (2003), 87–101.

[BG11] M. Bhargava and B.H. Gross, Arithmetic invariant theory, preprint, July 2011.

[Bou02] N. Bourbaki, Lie groups and Lie algebras: Chapters 4–6, Springer-Verlag, Berlin, 2002.

[CF10] J.W.S. Cassels and A. Fröhlich (eds.), Algebraic number theory, 2nd ed., London Math. Soc., 2010.

[DT94] D.Ž. Doković and N.Q. Thǎńg, Conjugacy classes of maximal tori in simple real algebraic groups and applications, Canad. J. Math. 46 (1994), no. 4, 699–717.

[Gar12] S. Garibaldi, Outer automorphisms of algebraic groups and determining groups by their maximal tori, Michigan Math. J. 61 (2012), no. 2.

[Gil04] P. Gille, Type des tores maximaux des groupes semi-simples, J. Ramanujan Math. Soc. 19 (2004), no. 3, 213–230.

[Har68] G. Harder, Eine Bemerkung zum schwachen Approximationssatz, Arch. Math. 19 (1968), 465–471.

[KMRT98] M.-A. Knus, A.S. Merkurjev, M. Rost, and J.-P. Tignol, The book of involutions, Colloquium Publications, vol. 44, Amer. Math. Soc., 1998.

[Kot82] R.E. Kottwitz, Rational conjugacy classes in reductive groups, Duke Math. J. 49 (1982), no. 4, 785–806.

[PR94] V.P. Platonov and A. Rapinchuk, Algebraic groups and number theory, Academic Press, Boston, 1994.

[PR09] G. Prasad and A.S. Rapinchuk, Weakly commensurable arithmetic groups and isospectral locally symmetric spaces, Publ. Math. IHES 109 (2009), no. 1, 113–184.

[PR10] , Local-global principles for embedding of fields with involution into simple algebras with involution, Comm. Math. Helv. 85 (2010), 583–645.

[PR11] , On the fields generated by the lengths of closed geodesics in locally symmetric spaces, arxiv:1110.0141v1, October 2011.

[Rag04] M. S. Raghunathan, Tori in quasi-split groups, J. Ramanujan Math. Soc. 19 (2004), no. 4, 281–287.

[Ste65] R. Steinberg, Regular elements of semisimple algebraic groups, Publ. Math. IHES (1965), no. 25, 49–80.

[Vos98] V.E. Voskresenskiĭ, Algebraic groups and their birational invariants, Translations of Mathematical Monographs, vol. 179, AMS, Providence, RI, 1998.

[Yeu11] S.-K. Yeung, Isospectral problem of locally symmetric spaces, IMRN (2011), no. 12, 2810–2824.

Department of Mathematics and Computer Science, MSC W401, 400 Dowman Dr., Emory University, Atlanta, GA 30322, USA

   E-mail address: skip at member.ams.org

Department of Mathematics, University of Virginia, Charlottesville, VA 22904-4137, USA

   E-mail address: asr3x at virginia.edu