Hasse principle for Classical groups over function fields of curves over number fields

R. Parimala, R. Preeti

Abstract

In ([CT]), Colliot-Thélène conjectures the following: Let F be a function field in one variable over a number field, with field of constants k and G be a semisimple simply connected linear algebraic group defined over F. Then the map H^{1}(F, G) → ∏_{v∈Ω_{k}} H^{1}(F_{v}, G) has trivial kernel, Ω_{k} denoting the set of places of k.

The conjecture is true if G is of type ^{1}A^{∗}, i.e., isomorphic to SL_{1}(A) for a central simple algebra A over F of square free index, as pointed out by Colliot-Thélène, being an immediate consequence of the theorems of Merkurjev-Suslin ([S1]) and Kato ([K]). Gille ([G]) proves the conjecture if G is defined over k and F = k(t), the rational function field in one variable over k. We prove that the conjecture is true for groups G defined over k of the types ^{2}A^{∗}, B_{n}, C_{n}, D_{n} (D_{4} nontrialitarian), G_{2} or F_{4}; a group is said to be of type ^{2}A^{∗}, if it is isomorphic to SU(B, τ) for a central simple algebra B of square free index over a quadratic extension k′ of k with a unitary k′|k involution τ.

1 Introduction

Let k be a number field and G a semisimple, simply connected linear algebraic group defined over k. Then the Hasse principle holds for principal homogeneous spaces for G over k, i.e., the natural map H^{1}(k, G) → ∏_{v∈V_{k}} H^{1}(k_{v}, G) is injective, V_{k} denoting the set of real places of k and for v ∈ V_{k}, k_{v} is the completion of k with respect to v, (cf. [PR]).

Let X be a smooth, geometrically integral curve over a number field. Let k(X) be the function field of X, with field of constants k. Let Ω_{k} denote the set of places of k and for v ∈ Ω_{k}, let k_{v}(X) denote the function field of the curve X_{k_{v}}. Let G be a linear algebraic group defined over k(X). Let � ^{1}(k(X), G) be the kernel of the map of pointed sets

H^{1}(k(X), G) → ∏_{v∈Ω_{k}} H^{1}(k_{v}(X), G).

The following conjecture was made by Colliot-Thélène ([CT]) in the 2-dimensional context.

Conjecture: If G is a semisimple, simply connected linear algebraic group defined over k(X), then � ^{1}(k(X), G) is trivial.

In the case when G is defined over k and X is ℙ^{1}, Gille [G] has shown that � ^{1}(k(X), G) is trivial. The fact that � ^{1}(k(X), G) is trivial, if G is of type 1A_{n}, isomorphic to SL_{1}(A) where A is a central simple algebra with square free index, follows immediately from the theorems of Merkurjev-Suslin (cf. 2.1) and Kato (cf. 2.3) and is known to experts for a long time. In this article we study � ^{1}(k(X), G), for G defined over the number field k. We show that this set is trivial if G is of type B_{n}, C_{n} and D_{n} (D_{4} non-trialitarian). We also prove that if G is of type ^{2}A^{∗}, i.e., isomorphic to SU(B, τ) where B is a central simple algebra over a quadratic extension k′ of k of square free index with a k′|k involution τ, then � ^{1}(k(X), G) is trivial. We show from the structure theorems of Cayley algebras and exceptional Jordan algebras due to Springer, that if G is of type G_{2} or F_{4}, then � ^{1}(k(X), G) is again trivial. The main ingredients in the proofs of the theorems stated above are higher dimensional class field theory results due to Kato (cf. [K]) and Jannsen (cf. [J]), results of Arason, Elman and Jacob concerning Witt groups of function fields in one and two variables over number fields (cf. [AEJ2], [AEJ3]), results of Merkurjev-Suslin on reduced norm criterion in terms of cohomology (cf. [S1], §24), theorems of Merkurjev on norm principle for algebraic groups (cf. [M2]) and results of Suresh on the structure of mod 2 Galois cohomology in degree 3 (cf. [Su]). The original conjecture is open for G defined over k(X); it is open even when G is defined over k.

2 Some known results

We record in this section several results which we shall use in this paper. The first theorem is a result of Merkurjev and Suslin. It gives a criterion for an element in a central division algebra over a field E, to be a reduced norm, in terms of the Galois cohomology group H^{3}(E, ℚ/ℤ(2)).

Theorem 2.1 (Suslin, [S1], §24, Theorem.24.4). Let E be a field of characteristic p ≥ 0. Let D be a central division algebra of square free index n over E, with n coprime to p. Then λ ∈ E^{∗} is a reduced norm from D if and only if (λ) ∪ (D) = 0 in H^{3}(E, μ_{n}^{⊗2}).

The next theorem is a norm principle due to Merkurjev for Spin groups. Let A be a central simple algebra of degree 2n ≥ 4 over a field E of characteristic different from 2 and σ be an orthogonal involution on A. Let h be a hermitian form over (A, σ). We have an exact sequence of algebraic groups (cf. §4 and §5 for details),

1 → μ_{2} → Spin(h) → SU(h) → 1

which induces the cohomology exact sequence,

SU(h)(E) →^{δ} E^{∗}/E^{∗2} → H^{1}(E, Spin(h)) → H^{1}(E, SU(h))

The map δ is the spinor norm map and we abbreviate Sn(h_{E}) = image of δ in E^{∗}/E^{∗2}. The norm principle of Merkurjev states:

Theorem 2.2 (Merkurjev, [M2], 6.2) With notation as above, the image of the spinor norm homomorphism Sn(h_{E}) is equal to the subgroup of E^{∗}/E^{∗2} generated by the images of the norm groups N_{L|E}(L^{∗}) over all finite extensions L|E such that the algebra A_{L} is split and the hermitian form h_{L} is isotropic.

We next state a theorem due to Kato. Let X be a proper smooth geometrically integral curve defined over a number field k. Let F be the function field of X and F_{v} the function field of X_{k_{v}}.

Theorem 2.3 (Kato, [K]) With notation as above and for any positive integer n, the canonical map

H^{3}(F, ℤ/n(2)) → ∏_{v∈Ω_{k}} H^{3}(F_{v}, ℤ/n(2))

is injective.

The following theorem due to Jannsen is an analogue of Kato’s theorem for surfaces.

Theorem 2.4 (Jannsen, ([J]) Let E be a function field in two variables over a number field k, then the restriction map

H^{4}(E, ℚ/ℤ(3)) → ⨁_{v∈Ω_{k}} H^{4}(E.k_{v}, ℚ/ℤ(3))

is injective.

Theorem 2.4 is true if we replace ℚ/ℤ(3) by ℤ/2ℤ. This follows from the above result of Jannsen and due to the surjectivity of the map K_{3}^{M}(E) → H^{3}(E, ℤ/2ℤ), where K_{3}^{M}(E) is the Milnor K group, which is a consequence of theorems of Merkurjev-Suslin ([MS]) and Rost.

For a field E we denote the mod 2 Galois cohomology ring H^{∗}(E, ℤ/2ℤ) by H^{∗}(E). Let GW (E) = ⨁_{n=0}^{∞} I^{n}(E)/I^{n+1}(E) be the graded Witt ring of E. We identify H^{1}(E) with E^{∗}/E^{∗2} and for a ∈ E^{∗}, we denote by (a) the corresponding element in H^{1}(E). Arason (cf. [A], Satz 4.8) has shown that the assignment < 1, −a_{1} > ⊗ ⋯ ⊗ < 1, −a_{n} >↦ (a_{1}) ∪ ⋯ ∪ (a_{n}), for a_{1}, ⋯, a_{n} ∈ E^{∗} is a well defined map e_{E}^{n} from the set of n-fold Pfister forms to H^{n}(E). The group I^{n}(E) is generated by n-fold Pfister forms. The Milnor conjecture says that for every positive integer n, the maps e_{E}^{n} on the set of n-fold Pfister forms extend to homomorphisms from I^{n}(E) ↦ H^{n}(E), which are again denoted by e_{E}^{n} and the induced maps ē_{E}^{n} : I^{n}(E)/I^{n+1}(E) → H^{n}(E) are isomorphisms. Arason, Elman and Jacob have proved Milnor conjecture for function fields in two variables over a number field, (cf. [AEJ1], proposition 5.9 and [AEJ3], theorem 1.5). The deep theorems of Merkurjev-Suslin and Rost (cf. [MS]) and Jacob-Rost (cf. [JR]) are used in the proof. In particular, they prove the following:

Theorem 2.5 Let E be a field of transcendence degree at most 2 over a number field. Then the map ē_{E}^{∗} induces an isomorphism of the graded Witt ring GW (E) with the mod 2 Galois cohomology ring H^{∗}(E).

Finally, we shall state a theorem of Suresh which will be used in this paper.

Theorem 2.6 With the same notations as in (2.3), for any element ξ in H^{3}(F) and a ternary form < a, b, c > over F, there exists f ∈ F^{∗} such that

1. f is a value of < a, b, c >

2. For every finite non-dyadic place v of k, ξ_{F_{v}(\sqrt{f})} = 0.

3. For every dyadic place v of k, such that −abc is a square in F_{v}, ξ_{F_{v}(\sqrt{f})} = 0.

For a proof, see [Su].

3 The cases of inner type A_{n} and C_{n}

Let D be a central division algebra of index n over a field E with n coprime to the characteristic of E. We have an invariant (cf. [Se2]), for elements of H^{1}(E, SL_{n,D}) with values in H^{3}(E, μ_{n}^{⊗2}), defined as follows. The set H^{1}(E, SL_{n,D}) is in bijection with E^{∗}/Nrd(D^{∗}). Given λ ∈ E^{∗}, the invariant associated with its class (λ) ∈ E^{∗}/Nrd(D^{∗}) in H^{3}(E, μ_{n}^{⊗2}) is the element (λ) ∪ (D).

Throughout this section, k denotes a number field and F the function field of a smooth geometrically integral curve X over k. Let Ω_{k} denote the set of places of k and for v ∈ Ω_{k}, let F_{v} = k_{v}(X) be the function field of the curve X_{k_{v}}. Let D be a central division algebra of square free index n over F. Then the map H^{1}(F, SL_{n,D}) → H^{3}(F, μ_{n}^{⊗2}) defined by this invariant is injective (cf. 2.1). By a theorem of Kato, the map H^{3}(F, μ_{n}^{⊗2}) → ∏_{v∈Ω_{k}} H^{3}(F_{v}, μ_{n}^{⊗2}) is injective (cf. 2.3). Hence the map H^{1}(F, SL_{n,D}) → ∏_{v∈Ω_{k}} H^{1}(F_{v}, SL_{n,D}) is injective. Thus, we have,

Proposition 3.1 Let k be a number field and X a smooth geometrically integral curve over k. Let F = k(X) be the function field of X. Let G = SL_{n}(D), with D a central division algebra over F with square free index. Then, � ^{1}(F, G) is trivial.

For non zero elements a, b in a field E, with char E ≠ 2, we denote by (a, b)_{E}, the quaternion algebra over E, generated by the elements i, j, with i^{2} = a, j^{2} = b and ij = −ji.

We now consider linear algebraic groups of type C_{n}. Let D be a quaternion division algebra over F and τ_{0} the standard involution on D. Let h be a hermitian form over (D, τ_{0}) and G = Sp(h), the symplectic group of h. Then G is a simply connected group of type C_{n}. The set H^{1}(F, Sp(h)) is in bijection with the set of isomorphism classes of hermitian forms over (D, τ_{0}) of the same rank as h. Given a hermitian form h′ over (D, τ_{0}), there is an associated quadratic form q_{h′} over F defined by q_{h′}(y) = h′(y, y), for y in the underlying space which supports h′. In fact, if h′ is represented by the diagonal matrix < λ_{1}, ⋯, λ_{r} >, q_{h′} is represented by the matrix < λ_{1}, ⋯, λ_{r} > ⊗n_{D}, where n_{D} denotes the norm form on the quaternion algebra D. By a theorem of Jacobson (cf. [S], pg. 352), two hermitian forms h and h′ are isomorphic over (D, τ_{0}) if and only if q_{h} and q_{h′} are isomorphic as quadratic forms.

Let h_{1} and h_{2} be hermitian forms of the same rank as h, representing elements ξ_{1} and ξ_{2} in H^{1}(F, Sp(h)). Then q_{h_{1}} ⊥ (−q_{h_{2}}) is an element of I^{3}(F). If (ξ_{1})_{v} = (ξ_{2})_{v} in H^{1}(F_{v}, Sp(h)), for every v ∈ Ω_{k}, then h_{1} ⊥ (−h_{2}) is hyperbolic over F_{v}, for all v ∈ Ω_{k}. This implies that the class of q_{h_{1}} ⊥ (−q_{h_{2}}) is equal to zero in I^{3}(F_{v}), for all v ∈ Ω_{k}. By ([AEJ2], theorem 4), q_{h_{1}} ⊥ (−q_{h_{2}}) is hyperbolic over F; i.e., h_{1} ≅ h_{2} and ξ_{1} = ξ_{2} in H^{1}(F, Sp(h)). Thus the map H^{1}(F, Sp(h)) → ∏_{v∈Ω_{k}} H^{1}(F_{v}, Sp(h)) is injective. In particular, we have

Proposition 3.2 Let k be a number field and F be the function field of a smooth geometrically integral curve X over k. Let G be a simply connected group of type C_{n} defined over k. Then � ^{1}(F, G) is trivial.

Proof. We just need to remark that the only division algebras with involutions of first kind over number fields are quaternion algebras (cf. [S], 10.2.3). □

4 The case of quadratic and hermitian forms

We continue with the same notation as in §2. The aim of this section is to prove the following two theorems.

Theorem 4.1 Let q be a quadratic form of rank greater than or equal to 3, over a number field k. Then � ^{1}(F, Spin(q)) is trivial.

Let K = k(\sqrt{d}) be a quadratic field extension of k. Let F K = F (\sqrt{d}) and let τ denote the non-trivial automorphism of F K over F.

Theorem 4.2 Let h be a hermitian form over (F K, τ), of rank at least 2. Then � ^{1}(F, SU(h)) is trivial.

We begin with the following

Proposition 4.3 Let q be a quadratic form of rank greater than or equal to 3, over a number field k. The map

\frac{F^{∗}/F^{∗2}}{Sn(q_{F})} → ∏_{v∈Ω_{k}} \frac{F_{v}^{∗}/F_{v}^{∗2}}{Sn(q_{F_{v}})}

is injective.

Proof. case.1. rank(q) = 3 : For any λ ∈ F^{∗}, since Sn(λq) = Sn(q), after scaling we may assume that q =< 1, a, b >, for some a, b ∈ k^{∗}. Let D = (−a, −b)_{F}. Then Sn(q_{F}) = Nrd(D^{∗}) modulo squares. If α ∈ F^{∗} is a local spinor norm then α is a reduced norm from D locally and by (3.1), α is a reduced norm from D and hence a spinor norm from q_{F}.

case.2. rank(q) = 4 : Suppose disc(q) = 1. After scaling we assume that q =< 1, a, b, ab >. Then Sn(q_{F}) =Nrd((−a, −b)_{F}^{∗}) modulo squares and the proof follows as in case 1.

Suppose disc(q) = d. By scaling we may assume that q =< 1, a, b, abd >. We have Sn(q_{F}) = Nrd((−a, −b)_{F (\sqrt{d})}) ∩ F^{∗} modulo squares (cf. [KMRT], 15.11). Let α ∈ F^{∗} be such that α ∈ Sn(q_{F_{v}}), for every v ∈ Ω_{k}. Then α is a reduced norm from (−a, −b)_{(F (\sqrt{d}))_{w}}, for all w ∈ Ω_{k(\sqrt{d})}. By (3.1), α ∈ Nrd(−a, −b)_{F (\sqrt{d})} ∩ F^{∗} = Sn(q_{F}) modulo squares.

case.3. rank(q) = 5 : Let d = disc(q). Then the form q ⊥< −d > is a six dimensional form over the number field k, which is indefinite and hence is isotropic (cf. [S], 6.6.6). Thus, q represents d and after scaling, we may assume that q ≅< d, 1, a, b, ab >. Hence q is a Pfister neighbour for the Pfister form q_{1} =< 1, a > ⊗ < 1, b > ⊗ < 1, d >. By the norm principle (cf. 2.2), spinor norms for q_{F} are products of norms from finite extensions of F where q_{F} is isotropic. As q_{F} is isotropic if and only (q_{1})_{F} is hyperbolic, spinor norms for q_{F} are products of norms from finite extensions of F where (q_{1})_{F} is hyperbolic. Let α ∈ F^{∗} be a spinor norm locally for all v ∈ Ω_{k}, for q_{F}. Then for every v ∈ Ω_{k}, α is a similarity factor for (q_{1})_{F_{v}} (cf. [L], Ch. 7, 4.5). Hence the form < 1, −α > q_{1} in I^{4}(F) is zero in I^{4}(F_{v}), for every v ∈ Ω_{k}. As I^{4}(F) → ∏_{v∈Ω_{k}} I^{4}(F_{v}) is injective (cf. [AEJ2], theorem 4), we have < 1, −α > q_{1} is zero in W (F), i.e., α is a similarity factor for q_{1} over F. Hence α is represented by q_{1} over F. As q_{1} is a Pfister form, α is a spinor norm of q_{1} over F. By the norm principle (cf. 2.2), Sn(q_{1}_{F}) = Sn(q_{F}) and hence α is a spinor norm of q over F.

case.4. rank(q) ≥ 6 : We complete the proof by induction on rank(q). Let q = q_{1} ⊥ q_{2}, with rank(q_{1}) = 5. Let disc(q_{1}) = d. After scaling q, we assume that q_{1} ≅< d, 1, a, b, ab >, as in case.3. Let α ∈ F^{∗} be a spinor norm locally for q_{F}. Let l(Y) = F (\sqrt{−α}), with l denoting the field of constants in F (\sqrt{−α}) and Y a curve over l.

Let q′ =< d, 1, a, b >⊥ q_{2}. Since rank(q′) ≥ 5, q′ is isotropic over l_{w} and hence over l_{w}(Y), for every finite place w of l. Let w be a real place, where q′ is definite. Since q′ represents 1, the elements a, b and hence ab are all positive at l_{w} and hence over k_{v}, where v is the restriction of w to k. Since α is a spinor norm of q over F_{v}, α is a sum of squares in F_{v} and hence in l_{w}(Y). Since −α is a square in l_{w}(Y), it follows that −1 is a sum of squares in l_{w}(Y), i.e., l_{w}(Y) has no ordering. This implies that cd(l_{w}(Y)) ≤ 1, (cf. [Se1]). Thus q′ is isotropic over l_{w}(Y). In particular, for each w ∈ Ω_{l}, every element of l_{w}(Y)^{∗} is a spinor norm for (q′)_{l_{w}(Y)}. By induction hypothesis, Sn(q′) = l(Y)^{∗}/l(Y)^{∗2}. By the norm principle (cf. 2.2), α being a norm from l(Y), is a spinor norm for q′ and hence for q. □

Remark 4.4 In the case of quadratic forms of rank 3 or 4, the proposition 4.3 holds more generally for forms over the function field F, i.e., if q is a quadratic form over F of rank 3 or 4, then the map

\frac{F^{∗}/F^{∗2}}{Sn(q_{F})} → ∏_{v∈Ω_{k}} \frac{F_{v}^{∗}/F_{v}^{∗2}}{Sn(q_{F_{v}})}

is injective. The proof given in the proposition works as well in these cases.

Proof of theorem 4.1. We have an exact sequence of algebraic groups:

\xymatrix{1 \ar[r] & μ_{2} \ar[r] & Spin(q) \ar[r]^{η} & SO(q) \ar[r] & 1}

which gives rise to an exact sequence of pointed sets:

\xymatrix{SO(q)(F) \ar[r]^{δ^{0}} & F^{∗}/F^{∗2} \ar[r] & H^{1}(F, Spin(q)) \ar[r]^{η} & H^{1}(F, SO(q)) \ar[r]^{δ^{1}} & H^{2}(F, μ_{2}).}

The map δ^{0} is induced by the spinor norm. The set H^{1}(F, SO(q)) classifies isomorphism classes of quadratic forms, with the same rank and discriminant as q. For a class [q′] ∈ H^{1}(F, SO(q)), δ^{1}([q′]) = c(q′ ⊥ (−q)), where c is the Clifford invariant of (q′ ⊥ (−q)). Thus the image H^{1}(F, Spin(q)) → H^{1}(F, SO(q)), consists of classes of quadratic forms q′ with the same rank, discriminant and Clifford invariant as q; in particular, q′ ⊥ (−q) ∈ I^{3}(F). We have a commutative diagram with exact rows:

[figure: 1; \frac{F^{∗}/F^{∗2}}{Sn(q_{F})}; δ^{0}; H^{1}(F, Spin(q)); η; H^{1}(F, SO(q)); 1; _{∏v∈Ω_{k} \frac{F_{v}^{∗}/F_{v}^{∗2}}{Sn(q_{Fv})}}; δ^{0}; ∏_{v∈Ω_{k}} H^{1}(F_{v}, Spin(q_{F_{v}})); η; ∏_{v∈Ω_{k}} H^{1}(F_{v}, SO(q_{F_{v}}))]

Let ξ ∈ H^{1}(F, Spin(q)) be such that ξ_{v} = 1, for all v ∈ Ω_{k}. The element η(ξ) corresponds to the class of a quadratic form q′ over F with q′ ⊥ (−q) ∈ I^{3}(F). By the commutativity of the above diagram, (q′ ⊥ (−q))_{F_{v}} is zero in I^{3}(F_{v}), for all v ∈ Ω_{k}. By ([AEJ2], theorem 4), we have an injection I^{3}(F) → ∏_{v∈Ω_{k}} I^{3}(F_{v}). Thus q′ ⊥ (−q) is equal to zero in I^{3}(F). By Witt’s cancellation theorem, q′ ≅ q and ξ lies in the kernel of η. Hence there exists α ∈ F^{∗}, such that δ^{0}([α]) = ξ. From the commutativity of the above diagram, it follows that α is locally a spinor norm, for all v ∈ Ω_{k}. The theorem now follows from the proposition 4.3. □

Recall that if E is a field of characteristic different from 2 and L is a quadratic extension of E, with σ denoting the non trivial automorphism of L over E, W (L|E, σ) denotes the Witt group of σ-hermitian forms. We have a homomorphism of groups W (L|E, σ) → W (E), given by associating to any h ∈ W (L|E, σ), the quadratic form q_{h} defined as q_{h}(x, x) = h(x, x), for any x in the space supporting h. This gives rise to the following exact sequence:

1 → W (L|E, σ) → W (E) → W (L)

where the map W (E) → W (L) is given by scalar extension from E to L. In fact if L = E(\sqrt{d}), for some d ∈ E^{∗}, then the image of W (L|E, σ) in W (E) is the subgroup W (E). < 1, −d >, (cf. [S], 10.1.3).

Proof of theorem 4.2. We have the following exact sequence of algebraic groups

1 → SU(h) → U(h) → R_{F K|F}^{1} (G_{m}) → 1

where for any extension L of F,

R_{F K|F}^{1} (G_{m})(L) = (LK)^{∗1} = {x ∈ (LK)^{∗} | N_{LK|L} (x) = 1 }.

As Nrd : U(h)(F) → (F K)^{∗1} is surjective, the above sequence gives rise to the following exact sequence of pointed sets,

1 → H^{1}(F, SU(h)) →^{η} H^{1}(F, U(h)).

The set H^{1}(F, U(h)) classifies isomorphism classes of hermitian forms, with the same rank as h. An element of H^{1}(F, SU(h)) maps under η to the class of a hermitian form with the same rank and discriminant as h. We have the following commutative diagram,

\xymatrix{1 \ar[r] & H^{1}(F, SU(h)) \ar[d] \ar[r]^{η} & H^{1}(F, U(h)) \ar[d] \\ 1 \ar[r] & ∏_{v∈Ω_{k}} H^{1}(F_{v}, SU(h)) \ar[r]^{η} & ∏_{v∈Ω_{k}} H^{1}(F_{v}, U(h))}

Let ξ ∈ H^{1}(F, SU(h)) be locally trivial in H^{1}(F_{v}, SU(h)), for every v ∈ Ω_{k}. The element η(ξ) corresponds to the class of a hermitian form h′ over (F K, τ) with rank and discriminant of h′ same as those of h. Moreover, (h ⊥ (−h′))_{F_{v}} is the hyperbolic form locally, for every v ∈ Ω_{k}. The hermitian forms h and h′ correspond to quadratic forms q_{h} and q_{h′} over F respectively such that the rank, discriminant and Clifford invariants of q_{h′} are the same as those of q_{h}. Hence the form q_{h} ⊥ (−q_{h′}) ∈ I^{3}(F). Further, the form q_{h} ⊥ (−q_{h′}) is locally zero in I^{3}(F_{v}), for every v ∈ Ω_{k}. By ([AEJ2], theorem 4), q_{h} ⊥ (−q_{h′}) is zero in I^{3}(F). Hence h ≅ h′ over (F K, τ) and η(ξ) is trivial. Since kernel(η) is trivial, ξ is trivial. □

5 A classification theorem for hermitian forms over division algebras with an orthogonal involution

Let E be a field of characteristic different from 2 and L a quadratic field extension of E with σ denoting the nontrivial automorphism of L over E. Let

U_{2n}(L, σ) denote the unitary group of the hyperbolic form ( \begin{matrix}0 & I_{n} \\ −I_{n} & 0 \end{matrix} ) over

(L, σ). If h is a hermitian form over (L, σ) of rank 2n, it defines an element ξ_{h} ∈ H^{1}(E, U_{2n}(L, σ)). The set H^{1}(E, SU_{2n}(L, σ)) injects into H^{1}(E, U_{2n}(L, σ)), the image consisting of hermitian forms over (L, σ) of rank 2n and trivial discriminant. Hence if h has trivial discriminant, ξ_{h} defines an element in H^{1}(E, SU_{2n}(L, σ)). The Rost invariant of ξ_{h} is the Arason invariant of the quadratic form q_{h} associated to h (see §4 and [BP2], §3); i.e., the Rost invariant of an even rank hermitian form over (L, σ), with trivial discriminant is the same as the Arason invariant of the associated quadratic form in I^{3}(E).

We next recall (cf. [BP2], §3) the Rost invariant associated to a hermitian form over a central division algebra D over any field E, with an orthogonal involution τ. Let h be a hermitian form over (D, τ). We denote by R_{h} the Rost invariant on H^{1}(E, Spin(h)) which takes values in H^{3}(E, ℚ/ℤ(2)). Its values on the subset \frac{E^{∗}/E^{∗2}}{Sn(h_{E})} ⊂ H^{1}(E, Spin(h)) are given by [λ] ↦ (λ)∪(D), (cf. [KMRT], §31.B, pp. 437). If h is a hermitian form of rank 2n, trivial discriminant and trivial Clifford invariant, the class of h defines an element in H^{1}(E, U_{2n}(D, τ)),

where U_{2n}(D, τ) is the unitary group of the hyperbolic form ( \begin{matrix}0 & I_{n} \\ −I_{n} & 0 \end{matrix} ),

which admits a lift ξ ∈ H^{1}(E, Spin_{2n}(D, τ)) under the composite map :

H^{1}(E, Spin_{2n}(D, τ)) → H^{1}(E, SU_{2n}(D, τ)) → H^{1}(E, U_{2n}(D, τ))

The Rost invariant of h, denoted as R(h) is defined to be R(h) = [R(ξ)] ∈ H^{3}(E, ℚ/ℤ(2))/H^{1}(E, μ_{2}) ∪ (D), (cf. [BP2], §3). If D = E this invariant coincides with the Arason invariant. We recall the following lemma, (cf. [BP2], 3.6).

Lemma 5.1 Let (D, τ) be a central division algebra with an orthogonal involution over a field E. Let h be a hermitian form over (D, τ). Let ξ ∈ H^{1}(E, Spin(h)) and h′ the hermitian form over (D, τ), associated to the image of ξ in H^{1}(E, U(h)). Then [R_{h}(ξ)] = R(h′ ⊥ (−h)) in H^{3}(E, ℚ/ℤ(2))/H^{1}(E, μ_{2}) ∪ (D).

Let k be a number field. We denote by V_{k}, the set of real places of k.

Lemma 5.2 Let k be a number field and M a function field in two variables over k. Then the map H^{n}(M) → ∏_{v∈V_{k}} H^{n}(M.k_{v}) is injective, for n ≥ 5.

Proof. Let n ≥ 5. Let ξ ∈ H^{n}(M) be trivial in H^{n}(M.k_{v}), for every v ∈ V_{k}. As every real closure of M contains a real closure of k, by ([AEJ1], 2.2), ξ is a (−1)-torsion element in H^{n}(M). We have the following exact sequence,

\xymatrix{H^{n}(M(\sqrt{−1})) \ar[r]^{cores} & H^{n}(M) \ar[d]^{(−1) ∪} \\ H^{n+1}(M(\sqrt{−1})) & H^{n+1}(M) \ar[l]^{res}}

As k is a number field, vcd(k) ≤ 2 and hence vcd(M) ≤ 4 and H^{r}(M(\sqrt{−1})) = 0, for r ≥ 5. In view of the above exact sequence, as n ≥ 5, we have (−1) ∪ : H^{n}(M) → H^{n+1}(M) is an isomorphism. As ξ is (−1)-torsion in H^{n}(M), ξ is zero in H^{n}(M). □

We record the following lemma, which is a consequence of a theorem of Jannsen (cf. 2.4) and a theorem of Arason-Elman-Jacob (cf. [AEJ1], 2.2).

Lemma 5.3 Let k be a number field and M a function field in two variables over k. Then the map I^{4}(M) → ∏_{v∈V_{k}} I^{4}(M.k_{v}) is injective.

Proof. Let q ∈ I^{4}(M) with q_{M.k_{v}} = 0 locally for all v ∈ Ω_{k}. Since e_{M}^{n} is well defined (cf. [AEJ1], 1.2), we have the following commutative diagram for each n:

\xymatrix{I^{n}(M) \ar[r] \ar[d]_{e_{M}^{n}} & ∏_{v∈Ω_{k}} I^{n}(M.k_{v}) \ar[d]_{e_{M}^{n}} \\ H^{n}(M) \ar[r] & ∏_{v∈Ω_{k}} H^{n}(M.k_{v})}

In view of this commutative diagram, the remark following (2.4) and since ē_{M}^{4} is an isomorphism (2.5), it follows that q ∈ I^{5}(M). Since q is locally zero, using the above commutative diagram for n = 5, we see that e_{M}^{5}(q) is locally trivial in H^{5}(M.k_{v}), for every v ∈ Ω_{k}. By the preceding lemma (5.2), we have e_{M}^{5}(q) is zero in H^{5}(M). Hence q ∈ I^{6}(M). Repeating this argument, we get that q ∈ ⋂_{n≥5} I^{n}(M) and hence is zero, by Arason-Pfister’s theorem (cf. [S], 4.5.6). □ Theorem 5.4 Let k be a number field and let F = k(X) be the function field of a smooth, geometrically integral curve X over k. Let D be a quaternion division algebra over F, with an orthogonal involution σ. Let h_{1} and h_{2} be two hermitian forms over (D, σ) with the same rank and discriminant. Suppose further that c(h_{1} ⊥ (−h_{2})) = 0 and R(h_{1} ⊥ (−h_{2})) = 0. Suppose h_{1} and h_{2} are equivalent over F_{v} for all v ∈ Ω_{k}, then h_{1} ≅ h_{2}.

Proof. Let L be a quadratic extension of F contained in D such that σ restricted to L is identity. Let μ ∈ D^{∗} be such that σ(μ) = −μ and Int(μ) restricted to L is the non-trivial automorphism τ_{0} of L over F (cf. [BP2], §3.2). The involution τ = Int(μ) ∘ σ on D, being symplectic is the canonical involution on D. Let L = l(Y), where l is the field of constants in L. For v ∈ Ω_{k}, let F_{v} = k_{v}(X) be the function field of the curve X_{k_{v}} and L_{v} = L ⊗_{F} F_{v}. We have the following commutative diagram with exact rows, (cf. [BP2], 3.2).

\xymatrix{W (D, τ) \ar[r]^{π_{1}} \ar[d] & W (L|F, τ_{0}) \ar[r]^{ρ̃} \ar[d] & W (D, σ) \ar[r]^{π̃_{2}} \ar[d] & W (L) \ar[d] \\ ∏_{v∈Ω_{k}} W (D_{F_{v}}, τ) \ar[r]^{π_{1}} & ∏_{v∈Ω_{l}} W (L_{v}|F_{v}, τ_{0}) \ar[r]^{ρ̃} & ∏_{v∈Ω_{v}} W (D_{F_{v}}, σ) \ar[r]^{π̃_{2}} & ∏_{v∈Ω_{l}} W (L_{v})}

Let h = h_{1} ⊥ (−h_{2}). Then h has even rank, trivial discriminant, trivial Clifford invariant and trivial Rost invariant. Further h is zero in W (D_{F_{v}}, σ), for every v ∈ Ω_{k}. The element π̃_{2}(h) ∈ W (L) has even rank, trivial discriminant and trivial Clifford invariant and hence belongs to I^{3}(L). Further, π̃_{2}(h) is zero in W (L_{w}), for every w ∈ Ω_{l}. By ([AEJ2], theorem 4), π̃_{2}(h) is zero in I^{3}(L). Thus there exists h_{0} ∈ W (L|F, τ_{0}) such that ρ̃(h_{0}) = h. The rank of h_{0} is even. We show that the lift h_{0} ∈ W (L|F, τ_{0}) may be modified so as to have trivial discriminant. Let α = disc(h_{0}) ∈ F^{∗}/N_{L|F} (L^{∗}). We have c(ρ̃(h_{0})) = (L)∪(α) ∈ Br(F)/(D), (cf. [BP1], 3.2.3). Since c(ρ̃(h_{0})) = c(h) = 0, we have (L) ∪ (α) = 0 or (L) ∪ (α) = (D) ∈ Br(F). If (L) ∪ (α) = 0, then disc(h_{0}) = 1. Suppose (L) ∪ (α) = (D). Let L = F (\sqrt{a}) so that D = (a, α)_{F}. The image of the form < 1 >∈ W (D, τ) under the map π_{1} in W (L|F, τ_{0}), is simply (< 1, −α >), which has discriminant α in F^{∗}/N_{L|F} (L^{∗}). Modifying h_{0} by π_{1}(< 1 >), we may assume that disc(h_{0}) = 1.

We now show that the lift h_{0} of h may be modified to have trivial Rost invariant. Let rank(h_{0}) = 2n. Let SU(μ^{−1}\sqrt{a}H_{2n}) be the special unitary group with respect to the hermitian form μ^{−1}\sqrt{a}H_{2n} over (D, σ). The inclusion SU_{2n}(L|F, τ_{0}) → SU(μ^{−1}\sqrt{a}H_{2n}) gives rise to an injection SU_{2n}(L|F, τ_{0}) → SU_{2n}(D, σ) (by a choice of an isomorphism μ^{−1}\sqrt{a}H_{2n} ≅ H_{2n} (cf. [BP2], pg. 671). This lifts to a homomorphism ρ_{0} : SU_{2n}(L|F, τ_{0}) → Spin_{2n}(D, σ). We have the following commuting diagram:

\xymatrix{SU_{2n}(L|F, τ_{0}) \ar[rr]^{ρ_{0}} \ar[dr]_{˜ρ̃} &  & Spin_{2n}(D, σ) \ar[dl] \\  & U_{2n}(D, σ)}

which yields a corresponding diagram:

\xymatrix{H^{1}(F, SU_{2n}(L|F, τ_{0})) \ar[rr]^{ρ_{0}} \ar[dr]_{˜ρ̃} &  & H^{1}(F, Spin_{2n}(D, σ)) \ar[dl] \\  & H^{1}(F, U_{2n}(D, σ))}

The map ρ̃ at the level of Witt groups is induced by the map ˜ ρ̃, (for varying n). Indeed for the hermitian form h_{0}, R(h_{0}) = R_{Spin_{2n}(D,τ)}(ρ_{0}(h_{0})), (cf. [BP2], 3.20). Since R(˜ρ̃(h_{0})) = R(h) = 0, there exists λ ∈ F^{∗}, such that R(h_{0}) = (λ) ∪ (D). The element, π_{1}(< 1, −λ >) has the associated quadratic form < 1, −λ > ⊗n_{D}, n_{D} denoting the norm form of D over F and has Rost invariant (λ) ∪ (D). Modifying h_{0} by π_{1}(< 1, −λ >), we may assume that R(h_{0}) = 0. Thus, the quadratic form associated to h_{0}, q_{h_{0}}, defines an element in I^{4}(F).

The image of π_{1} consists of hermitian forms f whose associated quadratic forms q_{f}, are multiples of n_{D}. Since h = ρ̃(h_{0}) is locally trivial over F_{v}, for every v ∈ Ω_{k}, h_{0}_{F_{v}} is in the image of π_{1} and hence q_{h_{0}} is a multiple of n_{D} over F_{v}, for every v ∈ Ω_{k}.

Let C be the conic defined by aX_{1}^{2} + bX_{2}^{2} − 1 over F. Then F (C) is a 2 dimensional field over k and n_{D} is zero over F (C) (cf. [S], 5.2, (iv)). Hence the class of q_{h_{0}} in I^{4}(F_{v}(C)) is zero, for all v ∈ Ω_{k}. The map I^{4}(F (C)) → ∏_{v∈Ω_{k}} I^{4}(F_{v}(C)) being injective (cf. 5.3), q_{h_{0}} is zero in I^{4}(F (C)) and hence is a multiple of of n_{D} (cf. [S], 5.4, (iv)). It follows that h_{0} is in the image of π_{1} and hence ρ(h_{0}) = h = 0 in W (D, σ). □

6 Hasse principle for groups of type D_{n} (D_{4} nontrialitarian)

Let (D, σ) be a central simple algebra over a field E with an orthogonal involution. Let L|E be an extension which splits D and let ϕ : (D, σ) ⊗_{E} L ≅ (M_{n}(L), τ_{q_{0}}) be a splitting with σ ⊗ 1 transported to the adjoint involution on M_{n}(L) corresponding to a quadratic form q_{0} over L. The form q_{0} is determined upto a scalar. Let h be a hermitian form over (D, σ) ⊗_{E} L. Then by Morita theory with respect to ϕ, h is equivalent to a quadratic form q over L. The similarity class of q is uniquely determined by h and is independent of the choice of ϕ and q_{0}. The form h is isotropic if and only if q is isotropic. In particular, Sn(h_{L}) = Sn(q_{L}).

Lemma 6.1 Let (D, σ) be a quaternion algebra with an orthogonal involution over a local field k. Let h be a hermitian form of rank 3 over (D, σ) and σ_{h} the involution on M_{3}(D), adjoint with respect to h. Suppose disc(σ_{h}) ∉ k^{∗2}. Then h is isotropic.

Proof. Let τ be the canonical symplectic involution on D. Let σ = Int u∘τ, for some u ∈ D^{∗}, such that τ(u) = −u. The hermitian form h corresponds under scaling by u, to a skew hermitian form h_{1} with respect to τ (cf. [BP1], §1.3). The involution τ_{h_{1}} on M_{3}(D) adjoint with respect to h_{1}, corresponds with σ_{h}. Then det(h_{1}) = disc(τ_{h_{1}}) = disc(σ_{h}) (cf. [KMRT], 7.2). By the hypothesis on h, disc(σ_{h}) ∉ k^{∗2}. Hence det(h_{1}) is not in k^{∗2} and by ([S], 10.3.6), h_{1} and hence h is isotropic. □

Theorem 6.2 Let (D, σ) be a quaternion division algebra over a number field k with an orthogonal involution σ and let h be a hermitian form over (D, σ) of rank at least 2. Let F = k(X) be the function field of a smooth geometrically integral curve X over k. For each v ∈ Ω_{k}, let F_{v} be the function field of the curve X_{k_{v}}. Then the map

\frac{F^{∗}/F^{∗2}}{Sn(h_{F})} → ∏_{v∈Ω_{k}} \frac{F_{v}^{∗}/F_{v}^{∗2}}{Sn(h_{F_{v}})}

is injective.

Proof. Suppose rank(h) = 2. Let δ = disc(h) ∈ k^{∗}/k^{∗2}. The Clifford algebra C = C(M_{2}(D), τ_{h})), is a quaternion algebra over k(\sqrt{δ}) and Sn(h_{F}) = Nrd(C_{F (\sqrt{δ})}) ∩ F^{∗} modulo squares, (cf. [KMRT], 15.11). Let λ ∈ F^{∗} be a local spinor norm for h_{F}. Then λ is a reduced norm from C ⊗_{F} F_{v}, for every place v of k and by (3.1), C being a quaternion algebra, λ is a reduced norm from C_{F (\sqrt{δ})} and belongs to Nrd(C_{F (\sqrt{δ})}) ∩ F^{∗} = Sn(h_{F}) modulo squares.

Let rank(h) = n ≥ 3. Let λ ∈ F^{∗} be a local spinor norm for h_{F}. Then λ is a reduced norm from D_{F}, (cf. 2.2 and 3.1). Let L be a quadratic extension of F such that D_{L} is split and λ = N_{L|F} (μ), for some μ ∈ L^{∗}. The element λ is also a norm from F (\sqrt{−λ}). By ([W], Lemma 2.13), there exists θ ∈ L(\sqrt{−λ}) such that N_{L(\sqrt{−λ})|F} (θ) = ν^{2}λ, for some ν ∈ F^{∗}. By (2.2), it suffices to show that every element of L(\sqrt{−λ})^{∗} modulo squares is contained in Sn(h_{L(\sqrt{−λ})}). We note that for every ordering v of k where D_{k_{v}} is split and h_{k_{v}} is definite, λ ∈ F_{v}^{∗} being a spinor norm of h_{F_{v}} is a sum of squares so that L(\sqrt{−λ}) . k_{v} has no orderings. In particular, if l is the field of constants of L(\sqrt{−λ}) and L(\sqrt{−λ}) = l(Y), Y a curve over l, for any ordering w of l extending v, l_{w}(Y) has no ordering. We rename l = k and Y = X and assume that D ⊗_{k} k(X) is split and for every ordering v of k where D_{k_{v}} is split and h_{k_{v}} is definite, k_{v}(X) has no orderings; in particular, cd(k_{v}(X)) ≤ 1. We then show that every λ ∈ k(X)^{∗} is a spinor norm for h_{k(X)}. This is done by induction on rank(h).

Suppose rank(h) = 3. Let S_{1} be the set of real places of k such that D_{k_{v}} is split and h_{k_{v}} is indefinite. Let S_{2} be the set of dyadic places of k such that D_{k_{v}} is split and disc(σ_{h}) ∉ k_{v}^{∗2}. Let S_{3} be the set of dyadic places of k such that D_{k_{v}} is not split and disc(σ_{h}) ∉ k_{v}^{∗2}. For v ∈ S_{1} ∪ S_{2}, h_{k_{v}} corresponds under Morita equivalence to a quadratic form of rank 6 over k_{v}, which is isotropic. We choose a rank 1 subform < X_{3v} > of h_{k_{v}}, such that under Morita equivalence, < X_{3v} > corresponds to the quadratic form < 1, −1 > over k_{v}. For v ∈ S_{1} ∪ S_{2}, let < X_{1v}, X_{2v} > denote the orthogonal complement of < X_{3v} > in h_{k_{v}}. For v ∈ S_{3}, since D_{k_{v}} is not split and disc(σ_{h}) ∉ k_{v}^{∗2}, h_{k_{v}} is isotropic in view of 6.1. We choose a rank 1 subform < X_{1v} > of h_{k_{v}} such that < X_{1v} >^{⊥}≅< X_{2v}, X_{3v} > is hyperbolic. Using weak approximation, one can find a rank 1 subform < X_{1} > of h over k, such that for each v ∈ S_{1} ∪ S_{2} ∪ S_{3}, < X_{1} >_{k_{v}} ≅< X_{1v} >. One can choose a subform < X_{2} > in < X_{1} >^{⊥} such that < X_{2} >_{k_{v}} ≅< X_{2v} >, for each v ∈ S_{1} ∪ S_{2} ∪ S_{3}. Let < X_{1}, X_{2} >^{⊥}≅< X_{3} >. Clearly, < X_{3} >_{k_{v}} ≅< X_{3v} >, for v ∈ S_{1}∪S_{2}∪S_{3}. Thus h ≅< X_{1}, X_{2}, X_{3} >. Since D is split over F, we choose an isomorphism ϕ : (D_{F} , σ) → (M_{2}(F), τ_{q_{0}}), q_{0} being a rank 2 quadratic form over F. The isomorphism ϕ yields a Morita correspondence between hermitian forms over D_{F} and quadratic forms over F. Let < X_{1} >_{F} correspond to < a′, b′ > over F, < X_{2} >_{F} correspond to < c′, d′ > over F and < X_{3} >_{F} correspond to < e′, f′ > over F. Thus h_{F} corresponds to the rank 6 quadratic form q =< a′, b′, c′, d′, e′, f′ >. Since the spinor norm group is insensitive to scaling, we replace q by the form (a′b′c′) . q =< b′c′, c′a′, a′b′, d′a′b′c′, e′a′b′c′, f′a′b′c′ >. Renaming, we set q =< −a, −b, ab, c, d, −cdδ >, δ = disc(q) = disc(σ_{h}) ∈ k^{∗}/k^{∗2}. We note that the form < d, −cdδ >= a′b′c′ < e′, f′ >. We choose g ∈ F^{∗} such that g is a value of the quadratic form < a δ, b δ, −ab δ > and such that for ξ = (λ) ∪ (c δ) ∪ (d δ) ∈ H^{3}(F), ξ_{F_{v}(\sqrt{g})} = 0, for every finite nondyadic v ∈ Ω_{k} and for every dyadic v ∈ Ω_{k} where δ ∈ k_{v}^{∗2}, (cf. 2.6). Set α = g δ ∈ F^{∗}. Then α is a value of the quadratic form < a, b, −ab > over F. The form < −a, −b, ab > being isotropic over F (\sqrt{α}), we have, q ≅ γ < 1, −α >⊥< −α >⊥< c, d, −cdδ >, for some γ ∈ F^{∗}. Let q_{1} =< −α, c, d, −cdδ >. Then disc(q_{1}) = g ∈ F^{∗}/F^{∗2}. We claim that λ is a spinor norm for q_{1} locally, for every v ∈ Ω_{k}. Over F (\sqrt{g}), q_{1} ≅< −δ, c, d, −cd δ > and the Clifford algebra C(q_{1}) ≅ (c δ, d δ)_{F (\sqrt{g})}. For a finite v ∈ Ω_{k} such that v is nondyadic or v is dyadic and δ ∈ k_{v}^{∗2}, over F_{v}(\sqrt{g}), (λ) ∪ C(q_{1}) = ξ_{F_{v}(\sqrt{g})} = 0. As C(q_{1}) is a quaternion algebra over F_{v}(\sqrt{g}), λ is a reduced norm from C(q_{1}) and hence [λ] ∈ Sn((q_{1})_{F_{v}}), (cf. [KMRT], 15.11). For v ∈ S_{1} ∪ S_{2}, by choice, the form < d, −cdδ >= a′b′c′ < e′, f′ >≅ a′b′c′ < X_{3v} >≅< 1, −1 > over F_{v}. Hence q_{1} being isotropic over F_{v}, λ ∈ Sn((q_{1})_{F_{v}}). For v ∈ S_{3}, over F_{v}, a′b′c′ < ab, c, d, −cdδ > corresponds under Morita equivalence to < X_{2v}, X_{3v} >. The form < X_{2v}, X_{3v} > being hyperbolic, < ab, c, d, −cdδ > is hyperbolic and hence < c, d, −cd δ > is isotropic over F_{v}. In particular, q_{1} is isotropic and λ ∈ Sn((q_{1})_{F_{v}}. For a real v ∈ Ω_{k} such that D_{k_{v}} is split and h_{k_{v}} is equivalent to a definite quadratic form, cd(k_{v}(X)) ≤ 1 and (q_{1})_{k_{v}} being 4 dimensional is isotropic. Hence λ ∈ Sn((q_{1})_{F_{v}}). Let v ∈ Ω_{k} be a real place such that D_{k_{v}} is not split. We claim that (q_{1})_{F_{v}} is isotropic. Since every form of rank greater than 1 over D_{k_{v}} is isotropic, we have < X_{3v} >≅< −X_{3v} >. As < X_{3v} > corresponds to the quadratic form < e′, f′ > over F_{v}, we have 2 < e′, f′ >= 0. Since < d, −cdδ >≅ a′b′c′ < e′, f′ >, we have < d, −cdδ > is torsion in W (F_{v}). To show that (q_{1})_{F_{v}} is isotropic, it is enough to show that q_{1} is isotropic over F_{v}(\sqrt{g}). Over F_{v}(\sqrt{g}), q_{1} ≅< −δ, c, d, −cdδ >≅ d (< 1, −c δ > ⊗ < 1, cd >).

As < 1, −c δ > is torsion, we have < 1, −c δ > ⊗ < 1, cd > is torsion over F_{v}(\sqrt{g}). As vcd(F_{v}(\sqrt{g})) ≤ 1, I^{2}(F_{v}(\sqrt{g})) is torsion free. Hence q_{1} is isotropic over F_{v}(\sqrt{g}) and hence over F_{v}. Thus λ is a spinor norm for q_{1} over F_{v}, for every place v of k and hence by (4.4), λ is a spinor norm for q_{1} and hence for h.

Suppose rank(h) = n ≥ 4. Let S_{1} be the set of real places of k where D_{k_{v}} is split and h_{k_{v}} is isotropic. Let S_{2} be the set of finite places of k where D_{k_{v}} is not split. Let v ∈ S_{2}. The form h_{k_{v}} being n dimensional, n ≥ 4, is isotropic over D_{k_{v}}. Let < α_{v} > be a 1 dimensional subform of h_{k_{v}} such that < α_{v} >^{⊥} is isotropic. Let v ∈ S_{1}. Since h_{k_{v}} is isotropic, choose a 1 dimensional subform, < α_{v} > of h_{k_{v}}, such that < α_{v} >^{⊥} is isotropic. By weak approximation, one may choose a 1 dimensional subform < α > of h such that < α >_{F_{v}} ≅< α_{v} >, for v ∈ S_{1} ∪ S_{2}. Let h_{1} =< α >^{⊥}. We claim that (h_{1})_{F_{v}} is isotropic over F_{v}, for every place v ∈ Ω_{k}. This is by choice for v ∈ S_{1} ∪ S_{2}; in fact, (h_{1})_{k_{v}} itself is isotropic. If v ∉ S_{1} ∪ S_{2}, v real and D_{k_{v}} is split, then h_{k_{v}} is definite, cd(F_{v}) ≤ 1 and (h_{1})_{F_{v}} being equivalent to a quadratic form of rank ≥ 3, is isotropic. If v ∉ S_{1} ∪ S_{2}, v real and D_{k_{v}} is not split, (h_{1})_{F_{v}} being of rank ≥ 2 is isotropic. If v ∉ S_{1} ∪ S_{2}, v finite, D_{k_{v}} being split, (h_{1})_{F_{v}} corresponds to a quadratic form of rank at least 6 and hence is isotropic. Thus (h_{1})_{F_{v}} is isotropic and since D_{F_{v}} is split, Sn((h_{1})_{F_{v}}) = F_{v}^{∗} modulo squares, for every v ∈ Ω_{k}. By induction, Sn((h_{1})_{F}) = F^{∗}/F^{∗2}. This completes the proof of the theorem. □

Corollary 6.3 With the same notation as in (6.2), let B be a central simple algebra of degree 4 over k. If λ ∈ F^{∗} is such that λ^{2} is a reduced norm from B_{F_{v}}, for all v ∈ Ω_{k}, then λ^{2} is a reduced norm from B_{F}.

Proof. With notation as in [KMRT], there is an equivalence of categories 1A_{3} ≅ ^{1}D_{3}, (cf. [KMRT], 15.32). Under this equivalence, let the degree 4 algebra (B × B^{op}) over (k × k), with the switch involution, correspond to the degree 6 algebra A over k with an orthogonal involution σ, i.e., C(A, σ) ≅ (B × B^{op}). We note that (A, σ) ≅ (M_{3}(H), τ_{h}), H a quaternion algebra over k and h a rank 3 skew hermitian form over (H, τ), τ denoting the standard involution of H. Further, Spin(A, σ) = Spin(h). We denote the extension of these algebras with involution to F by (B_{F} × B_{F}^{op}) and (A_{F} , σ) respectively. Then,

Sn(h_{F}) = {ρ ∈ F^{∗} | ρ^{2} ∈ Nrd_{B_{F}} (B_{F}^{∗}) }, modulo squares,

(cf. [KMRT], 15.34). Hence, the element λ as in the statement of the corollary, is locally a spinor norm for (A_{F_{v}}, σ), for every v ∈ Ω_{k}. By the above theorem (6.2), λ is a spinor norm for (A_{F} , σ). By the description for the spinor norms of (A_{F} , σ) given above, λ^{2} is a reduced norm for B_{F}. This completes the proof of the corollary. □

Remark 6.4 One does not know, even in the setting of the corollary, whether local reduced norms are reduced norms from B_{F}.

Theorem 6.5 With the same notation as in (6.2), let G be a semisimple simply connected linear algebraic group defined over k, of type D_{n} (non-trialitarian). Then the map

H^{1}(F, G) → ∏_{v∈Ω_{k}} H^{1}(F_{v}, G)

has trivial kernel.

Proof. We may assume without loss of generality that G is absolutely almost simple. Hence G is isomorphic to Spin(h), where h is a hermitian form over (D, σ), for some central division algebra D with an orthogonal involution σ over k. Since D is 2 torsion, D is either a quaternion division algebra over k or D = k. If D = k, then h is a quadratic form over k with rank(h) ≥ 3 and the theorem is proved in (4.1). So we may assume that D is a division algebra over k. Let rank(h) = n. We have an exact sequence of linear algebraic groups,

1 → μ_{2} → Spin(h) → SU(h) → 1

which in turn gives rise to the following commutative diagram with exact rows:

\xymatrix{SU(h)(F) \ar[d] \ar[r] & F^{∗}/F^{∗2} \ar[d] \ar[r] & H^{1}(F, Spin(h)) \ar[d] \ar[r] & H^{1}(F, SU(h)) \ar[d] \\ ∏_{v∈Ω_{k}} SU(h)(F_{v}) \ar[r] & ∏_{v∈Ω_{k}} F_{v}^{∗}/F_{v}^{∗2} \ar[r] & ∏_{v∈Ω_{k}} H^{1}(F_{v}, Spin(h)) \ar[r] & ∏_{v∈Ω_{k}} H^{1}(F_{v}, SU(h))}

Let ξ ∈ H^{1}(F, Spin(h)) be locally trivial in H^{1}(F_{v}, Spin(h)), for all v ∈ Ω_{k}. Then under the composite map,

H^{1}(F, Spin(h)) → H^{1}(F, SU(h)) → H^{1}(F, U(h))

the image of ξ in H^{1}(F, U(h)), defines a hermitian form h′ which has the same rank and discriminant as h and further c(h′ ⊥ (−h)) = 0. Let Spin_{2n}(D, σ) and U_{2n}(D, σ) denote respectively the spin and unitary groups of the hyper-

bolic form ( \begin{matrix}0 & I_{n} \\ −I_{n} & 0 \end{matrix} ) . Let ξ′ ∈ H^{1}(F, Spin_{2n}(D, σ)) be a lift of h′ ⊥ (−h)

in H^{1}(F, U_{2n}(D, σ)). Then R(ξ′) = R_{h}(ξ), where R_{h} : H^{1}(F, Spin(h)) → H^{3}(F, ℚ/ℤ(2)) is the Rost invariant map (cf. 5.1). Since ξ is locally trivial, R_{h}(ξ) ∈ H^{3}(F, ℚ/ℤ(2)) is locally trivial. Since D is a quaternion algebra, R_{h}(ξ) in fact belongs to H^{3}(F, ℤ/4ℤ) and the map H^{3}(F, ℤ/4ℤ) → ∏_{v∈Ω_{k}} H^{3}(F_{v}, ℤ/4ℤ) is injective (cf. 2.3). Hence R_{h}(ξ) is trivial in H^{3}(F, ℤ/4ℤ). Hence by the classification theorem (cf. 5.4), h ≅ h′ and the image of ξ in H^{1}(F, U(h)) is trivial. Let η be the image of ξ in H^{1}(F, SU(h)). Since the nontrivial element in H^{1}(F, SU(h)) which maps to the trivial element in H^{1}(F, U(h)) is not in the image of H^{1}(F, Spin(h)) (cf. [BP2], 7.11), it follows that η is trivial and hence in view of the exact sequence above, ξ comes from an element ξ̃ ∈ \frac{F^{∗}/F^{∗2}}{Im(Sn(h_{F}))}. By the commutative diagram above, ξ̃ is locally trivial and by (6.2), ξ̃ and hence ξ is trivial. □

7 Rost invariant for special unitary groups

Let E be a field of characteristic different from 2 and L a quadratic field extension of E. Let (D, τ) be a quaternion division algebra over L with a unitary L|E involution. Let D_{0} ⊂ D be a quaternion division algebra over E such that D = D_{0}. L and τ restricted to D_{0} is the canonical symplectic involution on D_{0}. For a hermitian form h over (D, τ), we denote the unitary and the special unitary group with respect to h by U(h) and SU(h) respectively. We have the following exact sequence of algebraic groups,

1 → SU(h) → U(h) → R_{L|E}^{1}(G_{m}) → 1

which gives rise to the following exact sequence in Galois cohomology,

U(h)(E) →^{Nrd} L^{∗1} →^{δ} H^{1}(E, SU(h)) → H^{1}(E, U(h)) (⋆)

The next proposition computes the Rost invariant on the image of δ. The proposition is also a consequence of ([MPT], theorem 1.9) (see Appendix).

Proposition 7.1 With the notation as above, for μ ∈ L^{∗1}, R(δ(μ)) = N_{L|E}(ν)∪ (D_{0}) ∈ H^{3}(E, ℚ/ℤ(2)), where ν ∈ L^{∗} is such that μ = ν^{−1} τ(ν).

Proof. The element N_{L|E}(ν) ∪ (D_{0}) is well defined with respect to μ, since for any λ ∈ E^{∗}, N_{L|E}(ν) ∪ (D_{0}) = N_{L|E}(λν) ∪ (D_{0}) in H^{3}(E, ℚ/ℤ(2)). Let X_{μ} be the torsor corresponding to δ(μ). Let E(X_{μ}) denote the function field of X_{μ}. Rost has shown (cf. [G1], §2.3, theorem 1) that the kernel K_{μ} of the map

H^{3}(E, ℚ/ℤ(2)) →^{res} H^{3}(E(X_{μ}), ℚ/ℤ(2)),

is a finite cyclic group generated by R(δ(μ)). We claim that R(δ(μ)) has order at most 2. We choose a quadratic extension field M of E such that D_{0}_{M} is split. Set ML = M ⊗_{E} L. Then D_{ML} is split and Nrd : U(h)(M) → (ML)^{∗1} is surjective. Hence res(R(δ(μ))) is trivial in H^{3}(M, ℚ/ℤ(2)) and cores(res(R(δ(μ)))) = 2. R(δ(μ)) = 0.

As the torsor X_{μ} has a rational point over the field E(X_{μ}), δ(μ) is trivial in H^{1}(E(X_{μ}), SU(h)). Hence μ ∈ Nrd(U(h)(E(X_{μ}))) and by (cf. [KMRT], pg. 202), μ = θ^{−1} τ(θ), for some θ ∈ Nrd(D_{E(X_{μ})}). Thus, N_{L|E}(ν) ∪ (D_{0}_{E(X_{μ})}) = N_{L⊗_{E} E(X_{μ}) |E(X_{μ})}(θ)∪(D_{0}_{E(X_{μ})}) in H^{3}(E(X_{μ}), ℚ/ℤ(2)). Since θ ∈ Nrd(D_{E(X_{μ})}), by the norm principle (2.2), N_{L⊗_{E} E(X_{μ}) |E(X_{μ})}(θ) ∈ Nrd(D_{0}_{E(X_{μ})}). Hence N_{L|E}(ν)∪(D_{0}_{E(X_{μ})}) = 0 in H^{3}(E(X_{μ}), ℚ/ℤ(2)) and N_{L|E}(ν)∪(D_{0}) ∈ K_{μ}. Since K_{μ} is generated by R(δ(μ)), N_{L|E}(ν) ∪ (D_{0}) = R(δ(μ)) or N_{L|E}(ν) ∪ (D_{0}) = 0. Suppose N_{L|E}(ν) ∪ (D_{0}) = 0. Then there exists a quadratic extension P of E, such that D_{0} is split over P and N_{L|E}(ν) = N_{P |E}(α), for some α ∈ P^{∗}. Set P L = P ⊗_{E} L. By (cf. [W], lemma 2.13), there exist β ∈ (P L)^{∗} and δ ∈ E^{∗}, such that N_{P L |L}(β) = ν. δ. As D is split over P L, by the norm principle, (2.2), ν. δ ∈ Nrd(D). As μ = (ν. δ)^{−1}τ(ν. δ), by (cf. [KMRT], pg. 202), μ ∈ Nrd(U(h)(E)), δ(μ) is trivial and R(δ(μ)) = 0. Hence if μ ∈ L^{∗1} is not in Nrd(U(h)(E)), then N_{L|E}(ν) ∪ (D_{0}) is not zero and hence coincides with R(δ(μ)). Thus in either case, N_{L|E}(ν) ∪ (D_{0}) = R(δ(ν)). □

Let U_{2n}(D_{0}, τ_{0}) denote the unitary group of the hyperbolic form ( \begin{matrix}0 & I_{n} \\ I_{n} & 0 \end{matrix} )

over (D_{0}, τ_{0}). We denote the unitary group and the special unitary group with

respect to the hyperbolic form ( \begin{matrix}0 & I_{n} \\ I_{n} & 0 \end{matrix} ) by U_{2n}(D, τ) and SU_{2n}(D, τ) re-

spectively. We have a natural inclusion U_{2n}(D_{0}, τ_{0}) ↪ U_{2n}(D, τ). Since τ_{0} is symplectic, the reduced norm of an element in U_{2n}(D_{0}, τ_{0}) has reduced norm 1 and we have the following diagram

\xymatrix{U_{2n}(D_{0}, τ_{0}) \ar[rr]^{ρ_{0}} \ar[dr]_{˜ρ̃} &  & SU_{2n}(D, τ) \ar[dl] \\  & U_{2n}(D, τ)}

which induces the following commutative diagram

\xymatrix{H^{1}(E, U_{2n}(D_{0}, τ_{0})) \ar[rr]^{ρ_{0}} \ar[dr]_{˜ρ̃} &  & H^{1}(E, SU_{2n}(D, τ)) \ar[dl] \\  & H^{1}(E, U_{2n}(D, τ))}

Proposition 7.2 With the notation as above, if [h] ∈ H^{1}(E, U(D_{0}, τ_{0})) then R(h) = R(ρ_{0}(h)).

Proof. By (cf. [KMRT], pg. 436), there exists an integer n_{ρ_{0}} such that n_{ρ_{0}} R(h) = R(ρ_{0}(h)). We show that n_{ρ_{0}} = 1. Let X = R_{L|E}(X_{D}) where X_{D} is the Brauer Severi variety of D over L. Let M = E(X)(X_{1}, ⋯, X_{2n}). Then D_{0}_{M} is not split, since Br(E) → Br(E(X)) is injective, (cf. [MT], corollary 2.12) and D_{0}_{ML} = D_{M} is split. Let L = E(\sqrt{d}). Then D_{0}_{M} = (a, d)_{M}, for some a ∈ M^{∗}. Let i, j ∈ D_{0}_{M} be such that i^{2} = a, j^{2} = d, ij = −ji. We have the splitting ϕ : D_{0}_{M} ⊗_{M} ML ≅ M_{2}(ML), defined by,

ϕ(i ⊗ 1) = ( \begin{matrix}0 & 1 \\ a & 0 \end{matrix} ) , ϕ(j ⊗ 1) = ( \begin{matrix}\sqrt{d} & 0 \\ 0 & −\sqrt{d} \end{matrix} ) .

An explicit computation shows that ϕ ∘ τ_{ML} ∘ ϕ^{−1} = Int(q_{1}) ∘ T, where

T ( \begin{matrix}x & y \\ z & w \end{matrix} ) = ( \begin{matrix}τ(x) & τ(z) \\ τ(y) & τ(w) \end{matrix} )

and q_{1} =< 1, −a >. Under Morita equivalence, through ϕ, every τ-hermitian form over (D_{ML}, τ) corresponds to a ML|M hermitian form. The (D_{ML}, τ) hermitian form h =< X_{1}, ⋯, X_{2n} > corresponds to an ML|M hermitian form represented by < X_{1}, ⋯, X_{2n} > ⊗ < 1, −a >, whose Rost invariant is ((−1)^{n} X_{1}. · · · X_{2n}) ∪ (a) ∪ (d) = P f(h) ∪ (D_{0}_{M}) ≠ 0, where P f(h) is the Pfaffian norm of h (cf. [KMRT], pg. 19). Since R(h) = P f(h) ∪ (D_{0}) (cf. [KMRT], pg. 440), it follows that n_{ρ_{0}} = 1. □

8 Classification theorems for hermitian forms over quaternion division algebras with a unitary involution

Let K = k(\sqrt{d}) be a quadratic field extension of a field k of characteristic different from 2 and (D, τ) be a quaternion algebra over K with a K|k involution τ. Let D_{0} ⊂ D be a quaternion k algebra such that τ restricted to D_{0} is τ_{0}, the canonical involution of D_{0} and D = D_{0} K. We have D = D_{0} ⊕ D_{0}\sqrt{d}. For any hermitian form h over (D, τ), let

h(x, y) = h_{1}(x, y) + h_{2}(x, y)\sqrt{d}, h_{i}(x, y) ∈ D_{0}, for i = 1, 2.

Since τ(h(y, x)) = h(x, y) and τ(\sqrt{d}) = −\sqrt{d}, it follows that τ_{0}(h_{1}(y, x)) = h_{1}(x, y) and τ_{0}(h_{2}(y, x)) = −h_{2}(x, y). Thus h_{1} is a hermitian form over (D_{0}, τ_{0}) and h_{2} is a skew-hermitian form over (D_{0}, τ_{0}). Let p_{1}(h) = h_{1} and p_{2}(h) = h_{2}. Clearly p_{i}(h ⊥ h′) = p_{i}(h) ⊥ p_{i}(h′) for i = 1, 2. Suppose that h is hyperbolic. Let W be a totally isotropic subspace of h, then W is also a totally isotropic subspace for p_{i}(h), for i = 1, 2. Thus we have homomorphisms

p_{1} : W (D, τ) → W (D_{0}, τ_{0})

and

p_{2} : W (D, τ) → W^{−1}(D_{0}, τ_{0}).

Let ρ̃ : W (D_{0}, τ_{0}) → W (D, τ) be the homomorphism defined as follows: Let f be a hermitian form over D_{0} and V_{0} its underlying D_{0} vector space. Let V = V_{0} ⊗_{k} K and write V = V_{0} ⊕ V_{0}\sqrt{d}. Define

ρ̃(f)(x_{1} ⊕y_{1}\sqrt{d}, x_{2} ⊕y_{2}\sqrt{d}) = f(x_{1}, x_{2})+f(x_{1}, y_{2})\sqrt{d}−f(y_{1}, x_{2})\sqrt{d}−f(y_{1}, y_{2})d.

It is easy to check that ρ̃ is a well defined homomorphism. We also have homomorphisms π_{i} : W (K) → W (k), for i = 1, 2, defined as follows. For any quadratic form q over K, write q(x, y) = q_{1}(x, y) + q_{2}(x, y)\sqrt{d}, where, q_{i}(x, y) ∈ k, for i = 1, 2. Then q_{1} and q_{2} are quadratic forms over k and π_{i}(q) = q_{i}, for i = 1, 2. Let π̃_{1} be the composition W (K) →^{π_{1}} W (k) → W (D_{0}, τ_{0}), where the map W (k) → W (D_{0}, τ_{0}) is induced by base change.

Proposition 8.1 (Suresh) The following sequence:

W (K) →^{π̃_{1}} W (D_{0}, τ_{0}) →^{ρ̃} W (D, τ) →^{p_{2}} W^{−1}(D_{0}, τ_{0}) (⋆⋆)

is exact.

Proof. Let f be a hermitian form over D_{0} and V_{0} its underlying D_{0}-vector space. Then the underlying vector space for p_{2}ρ̃(f) is V_{0} ⊗_{k} K = V_{0} ⊕V_{0}\sqrt{d} and p_{2}ρ̃(f)(x_{1} ⊕y_{1}\sqrt{d}, x_{2} ⊕y_{2}\sqrt{d}) = f(x_{1}, y_{2})−f(y_{1}, x_{2}). Thus the space W = \{x⊕ 0 | x ∈ V_{0}} is a totally isotropic subspace for p_{2}ρ̃(f) and W^{⊥} = W. Therefore p_{2}ρ̃(f) = 0. Let h be an anisotropic hermitian form over D such that p_{2}(h) = 0. In particular, there exists a vector x ≠ 0 such that p_{2}(h)(x, x) = h_{2}(x, x) = 0. This implies that h(x, x) = h_{1}(x, x) = α ∈ k. Since h is anisotropic α ≠ 0. Therefore we can write h =< α >⊥ h′. It is easy to see that ρ̃(< α >) =< α > and induction on the rank of h, yields the exactness at W (D, τ). We next show that ρ̃π̃_{1} = 0. For θ = a + b\sqrt{d} ∈ K^{∗}, with a, b ∈ k^{∗}, π̃_{1}(< θ >) ∈ W (D_{0}, τ_{0})

is represented by the matrix ( \begin{matrix}a & bd \\ bd & ad \end{matrix} ), which is equivalent to the diagonal

form < a, adN_{K|k}(θ) >. The form ρ̃π̃_{1}(< θ >) ∈ W (D, τ), is also represented by the form < a, adN_{K|k}(θ) >. Since < 1, dN_{K|k}(θ) > is equivalent to < 1, −1 > over (D, τ), ρ̃π̃_{1}(< θ >) = 0. Thus ρ̃π̃_{1} = 0. Suppose (V_{0}, h) is an anisotropic hermitian form over (D_{0}, τ_{0}) such that ρ̃(h) = 0. Then there exists a vector x_{1} + y_{1}\sqrt{d} ≠ 0 ∈ V_{0} ⊕ V_{0}\sqrt{d} such that ρ̃(h)(x_{1} + y_{1}\sqrt{d}, x_{1} + y_{1}\sqrt{d}) = 0. Then h(x_{1}, x_{1}) = h(y_{1}, y_{1})d and h(x_{1}, y_{1}) = h(y_{1}, x_{1}). Set a = h(y_{1}, y_{1}) and bd =

h(x_{1}, y_{1}). Then π̃_{1}(< a + b\sqrt{d} >) is represented by the matrix ( \begin{matrix}a & bd \\ bd & ad \end{matrix} ),

which is the matrix representing h restricted to the subspace of V_{0} spanned by (x_{1}, y_{1}). The proof of the proposition now follows by induction on the rank of h. □

Let K = k(\sqrt{d}) be a quadratic field extension of a field k of characteristic different from 2 and let D be a central division algebra over K with an involution τ of second kind over K|k. Let SU_{2n}(D, τ) be the special unitary group with

respect to the hyperbolic form H_{2n} = ( \begin{matrix}o & I_{n} \\ I_{n} & 0 \end{matrix} ). Let h be a hermitian

form over (D, τ) of even rank 2n and trivial discriminant. Then there exists ξ ∈ H^{1}(k, SU_{2n}(D, τ)), such that the image of ξ in H^{1}(k, U_{2n}(D, τ)) is the class of h. We say that the Rost invariant R(h) of h is zero, if there exists a ξ ∈ H^{1}(k, SU_{2n}(D, τ)) lifting the class of h and such that R(ξ) = 0, where R(ξ) is the Rost invariant associated to ξ.

Lemma 8.2 Let K be a field such that vcd(K) = n. For any field extension E of K, with [E : K] ≤ 2 assume that the maps \overline{e_{r}} : I^{r}(E)/I^{r+1}(E) → H^{r}(E) are well defined isomorphisms for all r ≥ 0. Then the map I^{n+1}(K) → C(X_{K}, 2^{n+1}ℤ) is surjective, X_{K} denoting the space of orderings of K.

Proof. Let ϕ ∈ C(X_{K}, 2^{n+1}ℤ). By ([S], 3.6.1), there exists a quadratic form q ∈ W (K), such that sgn(q) = 2^{m} ϕ, for some m ≥ 0. Multiplying q by < 1, 1 >^{⊗s}, if necessary, we may assume that q ∈ I^{n+1}(K). Suppose m > 0. We have the following commutative diagram:

I^{n+1}(K) →^{sgn} C(X_{K}, 2^{n+1}ℤ)
↓ e_{n+1}  ↓ mod2^{n+2}
H^{n+1}(K) →^{h_{n+1}} C(X_{K}, ℤ/2ℤ)

where h_{n+1} is as defined in (cf. [AEJ1], remark following theorem 2.3). Since m > 0, the signature of q modulo 2^{n+2} is zero. We have an exact sequence in Galois cohomology,

H^{r}(K(\sqrt{−1})) →^{cores} H^{r}(K) →^{∪(−1)} H^{r+1}(K) → H^{r+1}(K(\sqrt{−1})).

Since vcd(K) ≤ n, H^{r}(K(\sqrt{−1})) = 0, for r ≥ n + 1, so that ∪(−1) is an isomorphism. Thus H^{n+1}(K) is (−1)-torsion free. By ([AEJ1], 2.2 and 2.3), h_{n+1} is injective. Since h_{n+1}(e_{n+1}(q)) = 0, e_{n+1}(q) = 0. Since \overline{e_{n+1}} is an isomorphism, q ∈ I^{n+2}(K). Since the map I^{n+1}(K) →^{⊗<1,1>} I^{n+2}(K) is surjective (cf.[AEJ1], pg. 22, remark following 1.16), there exists q_{1} ∈ I^{n+1}(K), such that [< 1, 1 > ⊗q_{1}] = [q]. We have sgn(q_{1}) = 2^{m−1}ϕ. Repeating the process, we arrive at q ∈ I^{n+1}(K) with sgn(q) = ϕ. □

We have the following classification theorem for hermitian forms.

Theorem 8.3 Let K = k(\sqrt{d}) be a quadratic extension of a number field k. Let k(X) be the function field of a smooth geometrically integral curve X over k and K(X) = K ⊗_{k} k(X). Let (D, τ) be a quaternion division algebra over K(X), with a K(X)|k(X) unitary involution τ. Let h_{1} and h_{2} be hermitian forms over (D, τ) which have the same rank, discriminant and such that R(h_{1} ⊥ (−h_{2})) = 0. Suppose further that h_{1} and h_{2} are equivalent over k_{v}(X), for every v ∈ Ω_{k}. Then h_{1} ≅ h_{2}.

Proof. Let h = h_{1} ⊥ (−h_{2}). Let D_{0} = (a, b)_{k(X)} ⊂ D be a quaternion algebra over k(X), such that D = D_{0} .K(X) and τ restricted to D_{0} is τ_{0}, τ_{0} denoting the canonical involution on D_{0}. Let C be the conic, aX_{1}^{2} + bX_{2}^{2} − 1 = 0. The algebra D ⊗_{k(X)} k(X)(C) is split and the hermitian form h over D_{k(X)(C)} corresponds by Morita equivalence to a hermitian form over K(X)(C)|k(X)(C), which in turn corresponds to a quadratic form q(h) over k(X)(C), of even rank, trivial discriminant and trivial Clifford and Rost invariants. Hence [q(h)] ∈ I^{4}(k(X)(C)). Further, [q(h)] is zero in W (k_{v}(X)(C)), for every v ∈ Ω_{k}. By (5.3), I^{4}(k(X)(C)) → ∏_{v∈Ω_{k}} I^{4}(k_{v}(X)(C)) is injective. Hence h is zero in W (D_{k(X)(C)}, τ). We have the following commutative diagram:

W (D, τ)  →^{p_{2}}  W^{−1}(D_{0}, τ_{0})
↓  ↓
W (D_{k(X)(C)}, τ) →^{p_{2}} W^{−1}(D_{0_{k(X)(C)}}, τ_{0})

with the second vertical map injective by (cf. [PSS]), so that p_{2}(h) is zero in W^{−1}(D_{0}, τ_{0}). Hence by 8.1, there exists h′ ∈ W (D_{0}, τ_{0}), such that ρ̃(h′) = h.

We show that h′ can be chosen to have trivial Pfaffian norm (cf. [KMRT], pg. 19). Since R(h) = 0, there exists a lift ξ ∈ H^{1}(k(X), SU_{2n}(D, τ)) of h such that R(ξ) = 0. Since ρ_{0}(h′) is also a lift of h in H^{1}(k(X), SU_{2n}(D, τ)), by (cf. [KMRT], pg. 387, last paragraph), there exists μ ∈ K(X)^{∗1} such that ρ_{0}(h′)ξ̃ = δ(μ), where ξ̃ is a cocycle representing the cohomology class ξ and δ is the connecting map in (⋆) for the groups (SU_{2n}(h))ξ̃ and (U_{2n}(h))ξ̃. By (cf. [G1], §2.3, lemma 7), R(ρ_{0}(h′)ξ̃) = R(ρ_{0}(h′)) + R(ξ). As R(ξ) = 0 we have, R(δ(μ)) = R(ρ_{0}(h′)). By (7.2), R(ρ_{0}(h′) = P f(h′) ∪ (D_{0}). Let μ = ν^{−1}τ(ν), for some ν ∈ K(X)^{∗}. Then by (7.1), R(δ(μ)) = N_{K(X) |k(X)}(ν) ∪ (D_{0}) = P f(h′) ∪ (D_{0}). Hence P f(h′) = N_{K(X) |k(X)}(ν). Nrd(x), for some x ∈ D_{0}. If h′ ≅< λ_{1}, ⋯, λ_{2n} >, then replacing h′ by the equivalent form < λ_{1} xτ(x), ⋯, λ_{2n} >, we assume that P f(h′) = N_{K(X) |k(X)}(ν). Now replacing h′ by the form h′ ⊥< 1, −N_{K(X) |k(X)}(ν) >, we assume that P f(h′) is trivial, noting that ρ̃(< 1, −N_{K(X) |k(X)}(ν) >) = 0 in W (D, τ).

We have, W (D_{0}, τ_{0}) ≅ W (k(X)).n_{D_{0}}, under the map f ↦ q_{f}, where q_{f}(x, x) = f(x, x) and n_{D_{0}} denotes the norm form of D_{0}, (cf. §3). If f ≅< λ_{1}, ⋯, λ_{n} >∈ W (D_{0}, τ_{0}) then q_{f} =< λ_{1}, ⋯, λ_{n} > ⊗n_{D_{0}}. We set Q_{f} =< λ_{1}, ⋯, λ_{n} > as an element of W (k(X)). We note that for f ∈ W (D_{0}, τ_{0}), P f(f) = disc(Q_{f}).

As P f(h′) = 1, we have Q_{h′} ∈ I^{2}(k(X)). We claim that h′ is in the image of π̃_{1}.

Consider the exact sequence (⋆⋆) locally, for a real place v of k such that K_{v} = K ⊗ k_{v} is a proper quadratic extension of k_{v}. Since ρ̃((h′)_{k_{v}(X)}) = 0, there exists f_{v} ∈ W (K_{v}(X)) such that [(h′)_{k_{v}(X)}] = [π̃_{1}(f_{v})]. Hence [q_{h′}] = [(Q_{h′} ⊗ n_{D_{0}})_{k_{v}(X)}] = [π_{1}(f_{v}) ⊗ n_{D_{0}}]. Since cd(K_{v}(X)) ≤ 1, Br(K_{v}(X)) = 0, so that D_{0}_{K_{v}(X)} is split. Hence π_{1}(f_{v}) ⊗ n_{D_{0}} = π_{1}(f_{v} ⊗ n_{D_{0}_{Kv (X)}}) = 0. In particular, (h′)_{k_{v}(X)} = 0. Consider a real place v of k, such that K_{v} = K ⊗k_{v} is isomorphic to K_{w_{1}} × K_{w_{2}}, where w_{1} and w_{2} are two orderings of K, extending the ordering v of k. Then the map I^{2}(K_{v}(X)) →^{π_{1}} I^{2}(k_{v}(X)) is surjective, so that there exists f_{v} ∈ I^{2}(K_{v}(X)), such that π_{1}(f_{v}) = (Q_{h′})_{k_{v}(X)}. Let f_{v} = (f_{w_{1}}, f_{w_{2}}). We define a continuous function ϕ on X_{K(X)}, as follows. The space X_{K(X)} is the union of open and closed sets X_{K_{w}(X)}, w varying over the real orderings of K. For an ordering w of K lying over an ordering v of k, we set ϕ_{w} = sgn_{w}(f_{v} ⊗ (n_{D_{0}})_{K_{v}(X)}). Since f_{v} ∈ I^{2}(K_{v}(X)), ϕ_{w} ∈ C(X_{K_{w}(X)}, 16ℤ), for every w ∈ X_{K_{w}(X)}. By (8.2), there exists a quadratic form q_{2} ∈ I^{4}(K(X)), such that sgn_{w}(q_{2}) = ϕ_{w}. We claim that q_{2} is a multiple of n_{D_{0}}. Consider the following commutative diagram:

I^{4}(K(X))  →^{i_{C}}  I^{4}(K(X)(C))
↓  ↓
∏_{w∈X_{K}} I^{4}(K_{w}(X)) → ∏_{w∈X_{K}} I^{4}(K_{w}(X)(C))

If w is a finite place of K, I^{4}(K_{w}(X)) is zero, so that, (i_{C}(q_{2}))_{w} is zero. Let w be a real place of K. Since sgn_{w}(q_{2}) = sgn_{w}(f_{w} ⊗ n_{D_{0}}), q_{2} is Witt equivalent to f_{w} ⊗ n_{D_{0}}, since the signature is the only invariant for quadratic forms in I^{4}(K_{w}(X)). Hence q_{2} is split over K_{w}(X)(C) and the element i_{C}(q_{2}) ∈ I^{4}(K(X)(C)) is locally zero, for every w ∈ X_{K}. By (5.3), i_{C}(q_{2}) = 0. Hence q_{2} = q_{3} ⊗ n_{D_{0}}, for some q_{3} ∈ W (K(X)). Clearly, q_{3} is even dimensional. Since q_{2} = q_{3} ⊗ n_{D_{0}} ∈ I^{4}(K(X)) and (q_{3} ⊥< 1, −disc(q_{3}) >) ⊗ n_{D_{0}} ∈ I^{4}(K(X)), < 1, −disc(q_{3}) > ⊗n_{D_{0}} ∈ I^{4}(K(X)) and being of rank 8 is zero. Replacing q_{3} by q_{3} ⊥< 1, −(disc(q_{3})) > if necessary, we assume that q_{3} ∈ I^{2}(K(X)). We have,

sgn_{v}(π̃_{1}(q_{3})) =  sgn_{v}(π_{1}(q_{3}) ⊗ n_{D_{0}})
=  sgn_{v}(π_{1}(q_{3} ⊗ n_{D_{0}}))
=  sgn_{v}(π_{1}(q_{2}))
=  sgn_{v}(π_{1}(f_{v} ⊗ n_{D_{0}}))
= sgn_{v}((Q_{h′})_{k_{v}(X)} ⊗ n_{D_{0}}).

Hence the form q_{h′} ⊥ (−q_{π̃_{1}(q_{3})}) ∈ I^{4}(K(X)) is torsion. Since I^{4}(K(X)) is torsion free (cf. [AEJ2], cor.3), q_{h′} ⊥ (−q_{π̃_{1}(q_{3})}) is equivalent to zero. Hence h′ = π̃_{1}(q_{3}) and ρ̃(h′) = h is zero in W (D, τ). □

9 A classification theorem for hermitian forms over division algebras of odd degree with a unitary involution

Let k be a number field and X a smooth geometrically integral curve over k. Let k(X) be the function field of X and for v ∈ Ω_{k}, let k_{v}(X) denote the function field of the curve X_{k_{v}}. Let K be a quadratic field extension of k and K(X) = K ⊗_{k} k(X) and for v ∈ Ω_{k}, let K_{v}(X) = K ⊗_{k} k_{v}(X). Let (D, τ) denote a central division algebra of odd degree over K(X) with a K(X)|k(X) unitary involution τ. We prove the following classification theorem:

Theorem 9.1 Let the notation be as in the previous paragraph. Let h_{1} and h_{2} in W (D, τ) be hermitian forms of the same rank and discriminant and such that h_{1} ≅ h_{2}, locally over k_{v}(X), for every v ∈ Ω_{k}. Then h_{1} ≅ h_{2} over k(X).

Proof. Let h = h_{1} ⊥ (−h_{2}). Then h has even rank, trivial discriminant and is locally zero in W (D_{K_{v}(X)}, τ). Let L be an odd degree field extension of k(X) such that D_{L⊗_{k(X)}K(X)} is split, (cf. [BP1], 3.3.1). Let L = l(Y), where l is the field of constants of L. By Morita equivalence h corresponds to a hermitian form over L ⊗_{k(X)} K(X) |L and hence to a quadratic form q(h) over L. Moreover, q(h) has even rank, trivial discriminant, trivial Clifford invariant and is locally zero in W (l_{w}(Y)), for every w ∈ Ω_{l}. Hence q(h) ∈ I^{3}(l(Y)) and is locally zero in I^{3}(l_{w}(Y)), for every w ∈ Ω_{l}. By ([AEJ2], theorem 4), q(h) is zero in W (l(Y)). As L is an odd degree extension of k(X), by ([BL], theorem 2.1), h is zero in W (D, τ). Hence h_{1} ≅ h_{2}. □

10 Hasse principle for some groups of type ^{2}A_{n}

We begin with a result on the Hasse principle for special unitary groups of hermitian forms over quaternion algebras with unitary involutions.

Theorem 10.1 Let (D, τ) be a quaternion division algebra over a number field K, with a K|k unitary involution τ. Let X be a smooth geometrically integral curve over k. Let k(X) be the function field of X and for each v ∈ Ω_{k}, let k_{v}(X) be the function field of the curve X_{k_{v}}. Let K(X) = K⊗_{k}k(X) and for v ∈ Ω_{k}, let K_{v}(X) = K ⊗_{k} k_{v}(X). Let h be a hermitian form over (D, τ). Let SU(h) denote the special unitary group of h. Then the natural map H^{1}(k(X), SU(h)) → ∏_{v∈Ω_{k}} H^{1}(k_{v}(X), SU(h)) has trivial kernel.

Proof. Let ξ ∈ H^{1}(k(X), SU(h)) be such that ξ is locally trivial in H^{1}(k_{v}(X), SU(h)), for every v ∈ Ω_{k}. Under the map H^{1}(k(X), SU(h)) → H^{1}(k(X), U(h)), let ξ map to the hermitian form h′. Then the hermitian form h′ ⊥ (−h) has even rank, trivial discriminant and is locally trivial. We claim that the Rost invariant, R(h′ ⊥ (−h)) is trivial. We first note that as ξ is locally trivial, R(ξ) is locally trivial in H^{3}(k_{v}(X), ℚ/ℤ(2)) for every v ∈ Ω_{k}. Hence R(ξ) is zero in H^{3}(k(X), ℚ/ℤ(2)), by (2.3). We now consider the map SU(h) → SU(h ⊥ (−h)), given by, f ↦ (f, 1). This gives rise to a map from H^{1}(F, SU(h)) →^{i} H^{1}(F, SU(h ⊥ (−h))), and the image of ξ under this map corresponds to the hermitian form h′ ⊥ −h in H^{1}(k(X), U(h ⊥ −h)). By (cf. [KMRT], pg. 436), there exists an integer n_{i}, such that n_{i}R(ξ) = R(i(ξ)). By going over to a suitable field extension of k, where D is split and the Rost invariant is computed, we see that n_{i} = 1. Hence R(i(ξ)) = 0 and in particular, R(h′ ⊥ (−h)) = 0. Since h′ ⊥ (−h) is a hermitian form of even rank, trivial discriminant, trivial Rost invariant and is locally trivial, by (8.3), we have h′ ≅ h in W (D, τ). We have the following exact sequence of algebraic groups,

1 → SU(h) → U(h) → R_{K(X)|k(X)}^{1}(G_{m}) → 1

The above sequence gives rise to the following cohomology exact sequence,

U(h)(k(X)) →^{Nrd} K^{∗1} → H^{1}(k(X), SU(h)) → H^{1}(k(X), U(h)).

Since ξ maps to the trivial element in H^{1}(k(X), U(h)), there exists ν ∈ K(X)^{∗1} such that under the connecting map K(X)^{∗1} → H^{1}(k(X), SU(h)), the image of ν is ξ. Since ξ is locally trivial, we have ν ∈ Nrd(U(h)(k_{v}(X))) for every v ∈ Ω_{k}. We show that the natural map

K(X)^{∗1}/ Nrd(U(h)(k(X))) → ∏_{v∈Ω_{k}} K_{v}(X)^{∗1}/ Nrd(U(h)(k_{v}(X)))

is an injection. By (cf. [KMRT], pg. 202), we have,

\begin{aligned}Nrd(U(h)(k(X))) & = & \{z τ(z)^{−1} | z ∈ Nrd(D)\} \\ & = & Nrd(U_{2}(D, τ)(k(X))), \end{aligned}

where U_{2}(D, τ) is the unitary group of the hyperbolic form ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ), in di-

mension 2. We have the following commutative diagram,

\xymatrix{1 \ar[r] & K(X)^{∗1}/ Nrd(U(h)(k(X))) \ar[d] \ar[r] & H^{1}(k(X), SU_{2}(D, τ)) \ar[d]^{(⋆ ⋆ ⋆)} \\ 1 \ar[r] & ∏_{v∈Ω_{k}} K_{v}(X)^{∗1}/ Nrd(U(h)(k_{v}(X))) \ar[r] & ∏_{v∈Ω_{k}} H^{1}(k_{v}(X), SU_{2}(D, τ))}

Thus, to complete the proof of the theorem, we show that the natural map

H^{1}(k(X), SU_{2}(D, τ)) → ∏_{v∈Ω_{k}} H^{1}(k_{v}(X), SU_{2}(D, τ))

has trivial kernel.

Let D = D_{0} .K with the restriction of τ to D_{0} being the canonical involution on D_{0}. By (cf. [KMRT], 15.35 and 15.36), we have SU_{2}(D, τ) = Spin(q), where q =< 1, −d >⊥ n_{D_{0}}, where K = k(\sqrt{d}) and n_{D_{0}} denotes the norm form on the quaternion algebra D_{0}. Hence there is a bijection

i : H^{1}(k(X), SU_{2}(D, τ)) →^{≅} H^{1}(k(X), Spin(q))

and by (cf. 4.1), H^{1}(k(X), Spin(q)) → ∏_{v∈Ω_{k}} H^{1}(k_{v}(X), Spin(q)) has trivial kernel and hence H^{1}(k(X), SU_{2}(D, τ)) → ∏_{v∈Ω_{k}} H^{1}(k_{v}(X), SU_{2}(D, τ)) has trivial kernel. In particular, in diagram (⋆ ⋆ ⋆), the left vertical map is injective. This completes the proof of the theorem. □

The following proposition will be used in the proof of (10.4).

Proposition 10.2 Let L be a quadratic field extension of a field E of characteristic not 2. Let (A, σ) be a central division algebra over L of even degree, with a L|E unitary involution. Let h be a hermitian form over (A, σ). Then for any field extension M of E, we have,

N_{M⊗_{E}L |L}(Nrd(U(h)(M))) ⊂ Nrd(U(h)(E)).

Proof. Set ML = M ⊗_{E} L. Let ϕ_{L|E} and ϕ_{ML |M} denote the non trivial automorphisms of L over E and ML over M respectively. By (cf. [KMRT], pg. 202), Nrd(U(h)(M)) = {z ϕ_{ML |M}(z)^{−1} | z ∈ Nrd(D_{ML})}. Let x ∈ N_{ML |L}(Nrd(U(h)(M))). Then x = N_{ML |L}(y ϕ_{ML |M}(y)^{−1}), for some y ∈ Nrd(D_{ML}). We note that N_{ML |L}(ϕ_{ML |M}(y)) = ϕ_{L|E}(N_{ML |L}(y)). As N_{ML |L}(Nrd(D_{ML})) ⊂ Nrd(D), setting t = N_{ML |L}(y), we have t ∈ Nrd(D) and x = t ϕ_{L|E}(t^{−1}), proving the proposition. □

Let (D, τ) be a division algebra with square free index over a number field K, with a K|k unitary involution τ. Let X be a smooth geometrically integral curve over k. Let k(X) be the function field of X and for each v ∈ Ω_{k}, let k_{v}(X) be the function field of the curve X_{k_{v}}. Let K(X) = K ⊗_{k} k(X) and for v ∈ Ω_{k}, let K_{v}(X) = K ⊗_{k} k_{v}(X). In the next part of this section we prove the Hasse principle for groups of the form SU(h), where h is a hermitian form over (D, τ). We begin with the following proposition.

Proposition 10.3 With notation as above, suppose further that (D, τ) has odd degree over K. Let h be a hermitian form over (D, τ). Let K(X)^{∗1} = {x ∈ K(X)^{∗} | N_{K(X)|k(X)} (x) = 1 }. Then the natural map

K(X)^{∗1}/ Nrd(U(h)(k(X))) → ∏_{v∈Ω_{k}} K_{v}(X)^{∗1}/ Nrd(U(h)(k_{v}(X)))

is injective.

Proof. Let λ ∈ K(X)^{∗1} be locally in Nrd(U(h)(k_{v}(X))), for every v ∈ Ω_{k}. As degree D is odd, by a result of Suresh, (cf. [KMRT], pg. 202), Nrd(U(h)(k(X))) = Nrd(D_{k(X)}^{∗}) ∩ K(X)^{∗1}. As D has square free index and λ is locally a reduced norm from D_{k_{v}(X)}, for every v ∈ Ω_{k}, by (3.1), λ is a reduced norm for D_{k(X)}. Hence λ ∈ Nrd(D_{k(X)}^{∗}) ∩ K(X)^{∗1} = Nrd(U(h)(k(X))). □

Theorem 10.4 Let (D, τ) be a division algebra with square free index over a number field K, with a K|k unitary involution τ. Let X be a smooth geometrically integral curve over k. Let k(X) be the function field of X. Let h be a hermitian form over (D, τ). Let SU(h) denote the special unitary group of h. Then the natural map H^{1}(k(X), SU(h)) → ∏_{v∈Ω_{k}} H^{1}(k_{v}(X), SU(h)) has trivial kernel.

Proof. Let ξ ∈ H^{1}(k(X), SU(h)) be such that ξ is locally trivial in H^{1}(k_{v}(X), SU(h)), for every v ∈ Ω_{k}. Under the map H^{1}(k(X), SU(h)) → H^{1}(k(X), U(h)), let ξ map to the hermitian form h′. Then the hermitian form h′ ⊥ (−h) has even rank, trivial discriminant and is locally trivial. As ξ is locally trivial, the Rost invariant of ξ, R(ξ) is locally trivial in H^{3}(k_{v}(X), ℚ/ℤ(2)) for every v ∈ Ω_{k}. Hence R(ξ) is zero in H^{3}(k(X), ℚ/ℤ(2)), by (2.3). Consider the map SU(h) → SU(h ⊥ (−h)), given by, f ↦ (f, 1), which gives rise to a map from H^{1}(F, SU(h)) →^{i} H^{1}(F, SU(h ⊥ (−h))). The image of ξ under this map corresponds to the hermitian form h′ ⊥ −h in H^{1}(k(X), U(h ⊥ −h)). As in the proof of 10.1, one shows that R(i(ξ)) = 0. In particular, R(h′ ⊥ (−h)) = 0. Hence h′ ⊥ (−h) is a hermitian form of even rank, trivial discriminant, trivial Rost invariant and is locally trivial. We claim that h ≅ h′ over k(X).

Suppose the degree of D is odd. Then by the classification theorem (9.1), h ≅ h′.

Suppose the degree of D is even. Let D ≅ H ⊗_{K} D′, where H is a quaternion division algebra over K and D′ is an odd degree division algebra over K. Let L be an odd degree extension of k such that (D⊗_{k}L, τ) ≅ (M_{r}(H ⊗_{k}L), σ_{f}), where σ is a unitary L ⊗_{k} K |L involution on H ⊗_{k} L and σ_{f}, the adjoint involution on M_{r}(H ⊗_{k} L) with respect to the hermitian form f over (H ⊗_{k} L, σ), (cf. [BP1], 3.3.1). Let l(Y) = L ⊗_{k} k(X), where l is the field of constants in l(Y). Over l(Y), by Morita theory, h′ ⊥ (−h) corresponds to a hermitian form h_{1} over (H_{l(Y)}, σ) of even rank, trivial discriminant, trivial Rost invariant and such that h_{1} is locally zero in W (H_{l_{w}(Y)}, σ), for every w ∈ Ω_{l}. By (8.3), h_{1} is zero in W (H_{l(Y)}, σ) and hence h′ ⊥ (−h) is zero in W (D_{l(Y)}, τ). Since [l(Y) : k(X)] = [L : k] is odd, by ([BL], theorem 2.1), h′ ⊥ (−h) is zero in W (D_{k(X)}, τ) and hence h ≅ h′ and ξ maps to the trivial element in H^{1}(k(X), U(h)).

We have the following exact sequence of algebraic groups,

1 → SU(h) → U(h) → R_{K(X)|k(X)}^{1}(G_{m}) → 1

The above sequence gives rise to the following cohomology exact sequence,

U(h)(k(X)) →^{Nrd} K^{∗1} → H^{1}(k(X), SU(h)) → H^{1}(k(X), U(h)).

Since ξ maps to the trivial element in H^{1}(k(X), U(h)), there exists ν ∈ K(X)^{∗1} such that under the natural map K(X)^{∗1} → H^{1}(k(X), SU(h)), the image of ν is ξ. Since ξ is locally trivial, we have ν ∈ Nrd(U(h)(k_{v}(X))) for every v ∈ Ω_{k}. We show that the natural map from

K(X)^{∗1}/ Nrd(U(h)(k(X))) → ∏_{v∈Ω_{k}} K_{v}(X)^{∗1}/ Nrd(U(h)(k_{v}(X)))

is injective. If the degree of D is odd, then this follows from proposition 10.3. Hence we assume that the degree of D is even. Let λ ∈ K(X)^{∗1} be locally in Nrd(U(h)(k_{v}(X))), for every v ∈ Ω_{k}. Let H, D′, L, l(Y) and σ be as in the previous paragraph. As H^{1}(l(Y), SU(h)) → ∏_{w∈Ω_{l}} H^{1}(l_{w}(Y), SU(h)) has trivial kernel, (10.1), λ considered as an element of l(Y)^{∗} is in Nrd(U(h)(l(Y))). By proposition (10.2), we have N_{l(Y)⊗_{k(X)}K(X) |K(X)}(U(h)(l(Y))) ⊂ Nrd(U(h)(k(X))). As the dimension of L over k is odd, λ^{2r+1} ∈ Nrd(U(h)(k(X))), for some positive integer r. We show that λ^{2} ∈ Nrd(U(h)(k(X))). We choose a quadratic field extension N of k such that H_{N⊗_{k}K} is split. Then (D_{N⊗_{k}K}, τ) ≅ (M_{2}(D′_{N⊗_{k}K}), τ′), for some N⊗_{k}K|N unitary involution τ′. The division algebra D′ has odd degree and arguing as in the case of odd degree algebras, we have, λ ∈ Nrd(U(h)(N ⊗_{k} k(X))). Hence λ^{2} ∈ Nrd(U(h)(k(X))). Thus, λ ∈ Nrd(U(h)(k(X))) and the proof of the theorem is complete. □

11 The groups G_{2} and F_{4}

For any field E, characteristic E ≠ 2, if G is a semisimple simply connected absolutely almost simple linear algebraic group defined over E of type G_{2}, G is isomorphic to Aut(C) where C is a Cayley algebra defined over E. The pointed set H^{1}(E, G) classifies isomorphism classes of Cayley algebras over E. Given two Cayley algebras C and C′, they are isomorphic if and only if their norm forms n_{C} and n_{C′} are isomorphic. The norm form of a Cayley algebra is a 3-fold Pfister form over E.

Let k be a number field and X be a smooth geometrically integral curve defined over k. Let F = k(X) be its function field and for every v ∈ Ω_{k} let F_{v} = k_{v}(X) be the function field of X_{k_{v}}. Let G be as above of type G_{2} over the field F. Then G ≅ Aut(C) for some Cayley algebra C over F. Let ξ be an element in H^{1}(F, G) which is trivial in H^{1}(F_{v}, G), for every v ∈ Ω_{k}. The element ξ corresponds to a Cayley algebra C(ξ) over F. By hypothesis, n_{C} ≅ n_{C(ξ)} over F_{v} for every v ∈ Ω_{k}. Since the map I^{3}(F) → ∏_{v∈Ω_{k}} I^{3}(F_{v}) is injective, (cf. [AEJ2], theorem 4), n_{C} ≅ n_{C(ξ)} over F so that C ≅ C(ξ) i.e., ξ is trivial.

For any field E of characteristic not 2 or 3, if G is a semisimple simply connected absolutely almost simple linear algebraic group defined over E, of type F_{4}, G is isomorphic to Aut(J), J being a 27 dimensional central simple Jordan algebra over E. The set H^{1}(E, G) classifies isomorphism classes of exceptional central simple Jordan algebras over E. Given such a Jordan algebra J over E, there are three invariants, f_{3}(J) ∈ H^{3}(E), f_{5}(J) ∈ H^{5}(E) and g_{3}(J) ∈ H^{3}(E, ℤ/3ℤ), (cf. [Se2], §9). The algebra J is reduced if and only if g_{3}(J) = 0. If J is reduced, the two invariants f_{3}(J) and f_{5}(J) completely determine the isomorphism class of J, thanks to the classification theorems of Springer (cf. [Sp], theorem 1).

Let k be an algebraic number field and k(X) as above. Let J be a 27 dimensional exceptional central simple Jordan algebra over k and G = Aut(J). Since H^{1}(k(\sqrt{−1}), F_{4}) = (1), (cf. [Se2], §9.4), J is split over k(\sqrt{−1}). Hence g_{3}(J) = 0 and J is reduced. Let ξ ∈ H^{1}(F, G) be trivial locally at all places of k. Let ξ correspond to an exceptional Jordan algebra J′ over F. Since J′ ≅ J ⊗ F_{v} locally for all v in Ω_{k}, g_{3}(J′) = g_{3}(J ⊗ F_{v}), for all v ∈ Ω_{k}. Since H^{3}(F, ℤ/3ℤ) → ∏_{vΩ_{k}} H^{3}(F_{v}, ℤ/3ℤ) is injective (cf. 2.3), g_{3}(J′) = g_{3}(J ⊗ F) = 0. Hence J′ is reduced. Similarly, as f_{3}(J′) = f_{3}(J ⊗ F_{v}), for every v ∈ Ω_{k}, we have f_{3}(J′) = f_{3}(J ⊗ F). Since f_{5}(J′) = f_{5}(J ⊗ F_{v}), for every v ∈ Ω_{k}, we have f_{5}(J′)−f_{5}(J ⊗F) is in the kernel of the natural map H^{5}(F) → ∏_{w∈X_{F}} H^{5}(F_{w}), X_{F} denoting all the orderings of F and hence is torsion. As vcd(F) = 3, H^{5}(F) is torsion free. Hence f_{5}(J′) = f_{5}(J ⊗ F), so that by Springer’s theorem, J′ ≅ J ⊗ F and ξ is trivial.

12 The Hasse principle

The aim of this section is to prove the Hasse principle stated in the introduction. We say that a semisimple simply connected absolutely simple group over a field E is of type A^{∗} if it is isomorphic to SL_{1}(A) for a central simple algebra A over E of square free index or if it is isomorphic to SU(B, τ) for a central simple algebra B over a quadratic extension L of E of square free index with an L|E involution τ.

Theorem 12.1 Let k be a number field and X a smooth geometrically integral curve defined over k. Let k(X) denote the function field of X and for every v ∈ Ω_{k}, let k_{v}(X) denote the function field of the curve X_{k_{v}}. Let G be a semisimple simply connected linear algebraic group defined over k, which is the product of the Weil restrictions of absolutely simple groups of types A^{∗}, B_{n}, C_{n}, D_{n} (D_{4} non-trialitarian), G_{2}, and F_{4}. Then the map

H^{1}(k(X), G) → ∏_{v∈Ω_{k}} H^{1}(k_{v}(X), G)

has trivial kernel.

Proof. Recall that for a finite field extension L of a field E, if G = R_{L|E}(G′) is the Weil restriction of a linear algebraic group G′ defined over L, then H^{1}(E, G) = H^{1}(L, G′). The theorem is now a consequence of (3.1, 3.2, 4.1, 4.2, 6.5, 10.1, 10.4 and §11). □

Appendix

Rost invariant for the special unitary groups

Let E be a field of characteristic different from 2 and L = E(\sqrt{d}) be a quadratic field extension of E. Let (D, τ) be a central division algebra over L with a unitary L|E involution. For a hermitian form h over (D, τ), we denote the unitary and the special unitary groups with respect to h by U(h) and SU(h) respectively. We have the following exact sequence of algebraic groups,

1 → SU(h) → U(h) → R_{L|E}^{1}(G_{m}) → 1

which gives rise to the following exact sequence in Galois cohomology,

U(h)(E) →^{Nrd} L^{∗1} →^{δ} H^{1}(E, SU(h)) → H^{1}(E, U(h)).

The next theorem computes the Rost invariant on the image of δ.

Theorem With the notation as above, for μ ∈ L^{∗1},

R(δ(μ)) = Cores_{L|E}((ν) ∪ (D)) ∈ H^{3}(E, ℚ/ℤ(2)),

where ν ∈ L^{∗} is such that μ = ν τ(ν)^{−1}. Proof. We first show that Cores_{L|E}((ν) ∪ (D)) is well defined. Indeed, for λ ∈ E^{∗}, we have

Cores_{L|E}((ν λ) ∪ (D)) = Cores_{L|E}((ν) ∪ (D)) + Cores_{L|E}((λ) ∪ (D))
= Cores_{L|E}((ν) ∪ (D)) + (λ) ∪ Cores_{L|E}(D)
= Cores_{L|E}((ν) ∪ (D)),

since Cores_{L|E}(D) = 0. Set ξ = Cores_{L|E}((ν) ∪ (D)). If δ(μ) = 1, i.e., μ ∈ Nrd(U(h)(E))^{∗} then ν can be chosen to be in Nrd(D)^{∗} (cf. [KMRT], pg.

202). Hence (ν) ∪ (D) = 0 and ξ = 0. Further, R(δ(μ)) = 0. Hence, in this case, R(δ(μ)) = ξ = 0. We now assume that δ(μ) ≠ 1. By ([KMRT], pg.438), we have, R(δ(μ))_{L} = (μ) ∪ (D) = (ν) ∪ (D) + (τ(ν)) ∪ (D^{−1}) = ξ_{L}. Hence corestricting to E, we get, 2. R(δ(μ)) = 2. ξ.

case.1. Suppose degree (D) is odd. We choose a field extension M of E of degree n, with n odd, such that D ⊗_{E} (M ⊗_{E} L) is split. Set ML = M ⊗_{E} L. Since D is split over ML, ξ_{M} = 0. Further, U(h)(M) →^{Nrd} (ML)^{∗1} is surjective, so that δ(μ)_{M} = 1. Hence R(δ(μ))_{M} = 0. Since Cores_{M|E} ∘ res coincides with multiplication by n, we have n. ξ = n. R(δ(μ)) = 0. As 2. ξ = 2. R(δ(μ)), we have ξ = R(δ(μ)).

case.2. Suppose degree (D) = 2^{n}, for some positive integer n. Let ν = a+b\sqrt{d}, for some a, b ∈ E. As μ ∉ Nrd(U(h)(E)), we have, b ≠ 0. Consider the rational function field E(t). We extend the base field E to E(t). Set μ_{t} = \frac{t+b\sqrt{d}}{t−b\sqrt{d}} and

ν_{t} = t + b\sqrt{d}. Let X_{μ_{t}} be the torsor corresponding to δ(μ_{t}) ∈ H^{1}(E(t), SU(h)). Let E(t)(X_{μ_{t}}) denote the function field of X_{μ_{t}}. By a result of Rost (cf. [G1], §2.3, theorem 1), the kernel K_{μ_{t}}, of the map

H^{3}(E(t), ℚ/ℤ(2)) →^{res} H^{3}(E(t)(X_{μ_{t}}), ℚ/ℤ(2)),

is a finite cyclic group generated by R(δ(μ_{t})). Since δ(μ_{t}) is trivial over E(t)(X_{μ_{t}}), μ_{t} ∈ Nrd(U(h)(E(t))). Hence there exists λ ∈ E(t)(X_{μ_{t}})^{∗} such that λ. ν_{t} ∈ Nrd(D_{E(t)(X_{μ_{t}})}) (cf. [KMRT], pg. 202). Set ξ_{t} = Cores_{L(t)|E(t)}((ν_{t}) ∪ (D)). Then over E(t)(X_{μ_{t}}), we have,

ξ_{t}_{E(t)(X_{μ_{t}})} = Cores_{L(t)(X_{μ_{t}})|E(t)(X_{μ_{t}})}((λ. ν_{t}) ∪ (D)) = 0.

Therefore ξ_{t} ∈ K_{μ_{t}}. Let s be the order of R(δ(μ_{t})). Then there exists a positive integer r ≤ s such that ξ_{t} = r. R(δ(μ_{t})). Since ξ_{t}_{L(t)} = R(δ(μ_{t}))_{L(t)}, 2. ξ_{t} = 2. R(δ(μ_{t})) and hence (2r−2) R(δ(μ_{t})) = 0. Hence 2r−2 = sl, for some positive integer l and r = \frac{sl}{2} + 1. If l is even, we have ξ_{t} = R(δ(μ_{t})). Suppose l is an odd integer. Then ξ_{t} = (\frac{s}{2} + 1)R(δ(μ_{t})). In this case, we show that s = 2m, where m denotes the exponent of D. Suppose s ≠ 2m. We first note that \frac{s}{2}. R(δ(μ_{t}))_{L(t)} = (ξ_{t} −R(δ(μ_{t})))_{L(t)} = 0. We have, m. R(δ(μ_{t}))_{L(t)} = m. ξ_{L(t)} = m. ((μ_{t}) ∪ (D)) = (μ_{t}) ∪ (D^{m}) = 0. Hence over E(t), 2m. R(δ(μ_{t})) = 0. As s is the order of R(δ(μ_{t})), s divides 2m. As m is a power of 2, \frac{s}{2}. R(δ(μ_{t}))_{L(t)} = 0 and s ≠ 2m, we have \frac{m}{2}. R(δ(μ_{t}))_{L(t)} = 0. Let ∂_{(t−a)} : H^{3}(L(t), ℚ/ℤ(2)) → H^{2}(L, ℚ/ℤ(1)) denote the residue with respect to the prime (t − a) in L(t) (cf. [G1], §1.3). We have, ∂_{(t−a)}((μ_{t}) ∪ (D)) = (D). Since R(δ(μ_{t}))_{L(t)} = (μ_{t}) ∪ (D) and \frac{m}{2}. R(δ(μ_{t}))_{L(t)} = 0, we have D^{\frac{m}{2}} = 0 in Br(L), which is a contradiction. Hence s = 2m. Since m. ξ_{t} = Cores_{L(t)|E(t)}((ν_{t}) ∪ (D^{m})) = 0, we have

(m + 1). ξ_{t} = ξ_{t}
= (\frac{s}{2} + 1). R(δ(μ_{t}))
= (m + 1). R(δ(μ_{t})).

As 2. ξ_{t} = 2. R(δ(μ_{t})) and m + 1 is odd, we have ξ_{t} = R(δ(μ_{t})).

Let O be the ring of integers of the completion L((t−a)) of L(t) with respect to the discrete valuation corresponding to the prime (t − a) on L(t). Let G be a semi simple simply connected O group scheme with the special fibre isomorphic to SU(h) over the residue field L at the prime (t − a). We have the following commutative diagram (cf. [G1], theorem 2)

H^{1}(L((t − a)), G_{L((t−a))}) →^{R_{L((t−a))}} H^{3}(L((t − a)), ℚ/ℤ(2))
↑  ↑
H_{et}^{1}(O, G)
≀ ↓
H^{1}(L, SU(h))  →^{R_{L}}  H^{3}(L, ℚ/ℤ(2))

The torsor δ(μ_{t}) over L((t − a)) comes from a torsor for G over O, since μ_{t} is a unit in O and it specialises to δ(μ) in H^{1}(L, SU(h)). In view of the above commutative diagram, R(δ(μ))_{L((t−a))} = R(δ(μ_{t})) = Cores_{L((t−a))|E((t−a))}((ν_{t}) ∪ (D)). Since characteristic E is coprime to m, ν_{t} = b\sqrt{d} + t = b\sqrt{d} + a + (t − a) = (a + b\sqrt{d}). α^{m}, for some α ∈ L((t − a)). Set M = E((t − a)) and ML = L((t − a)). Hence Cores_{ML|M}((ν_{t}) ∪ (D)) = Cores_{ML|M}(((a + b\sqrt{d}). α^{m}) ∪ (D)) = Cores_{L|E}((a + b\sqrt{d}) ∪ (D))_{ML} + Cores_{ML|M}((α^{m}) ∪ (D)). Since Cores_{ML|M}((α^{m})∪(D)) = Cores_{ML|M}((α)∪(D^{m})) = 0, we have R(δ(μ))_{ML} = Cores_{L|E}((a+b\sqrt{d})∪(D))_{ML}. Since the map H^{3}(L, ℚ/ℤ(2)) → H^{3}(ML, ℚ/ℤ(2)) is injective, (cf. [G1], §1.3), we have R(δ(μ)) = Cores_{L|E}((a + b\sqrt{d}) ∪ (D)).

case.3. Suppose degree (D) = 2^{l}. m, where m is odd. In this case, we choose an extension M of E of odd degree n such that D_{M⊗_{E}L} has degree some power of 2. Set ML = M ⊗_{E} L. By the previous case, R(δ(μ))_{M} = Cores_{ML|M}((ν)∪(D_{ML})) = Cores_{L|E}((ν)∪(D))_{M}. Since Cores_{ML|M} ∘res coincides with multiplication by n, we have n. R(δ(μ)) = n. Cores_{L|E}((ν) ∪ (D)). As 2. R(δ(μ)) = 2. Cores_{L|E}((ν)∪(D)), we have R(δ(μ)) = Cores_{L|E}((ν)∪(D)). □

Remark The above result is also a consequence of a theorem of Merkurjev-Parimala-Tignol, (cf. [MPT], theorem 1.9), in view of the following commutative diagram

\xymatrix{U(h)(E) \ar[r]^{Nrd} \ar[d] & L^{∗1} \ar[r]^{δ} \ar[d] & H^{1}(E, SU(h)) \ar[r] \ar[d] & H^{1}(E, U(h)) \ar[d] \\ P GU(h)(E) \ar[r]^{δ} & H^{1}(E, μ_{n[L]}) \ar[r] & H^{1}(E, SU(h)) \ar[r] & H^{1}(E, U(h))}

where P GU(h) is the projective unitary group with respect to h and μ_{n[L]} =

kernel(R_{L|E}(μ_{n}) →^{N_{L|E}} μ_{n}). The proof of Merkurjev-Parimala-Tignol, uses invariants of quasi-trivial tori.

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School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400 005, India. E-mail addresses:

parimala@math.tifr.res.in
preeti@math.tifr.res.in