Divisible Subgroups of Brauer Groups

and Trace Forms of Central Simple Algebras

Grégory Berhuy, David B. Leep

Received: October 31, 2001

Communicated by Alexander S. Merkurjev

Abstract.

Let F be a field of characteristic different from 2 and assume that F satisfies the strong approximation theorem on orderings (F is a SAP field) and that I^{3}(F) is torsion-free. We prove that the 2-primary component of the torsion subgroup of the Brauer group of F is a divisible group and we prove a structure theorem on the 2-primary component of the Brauer group of F. This result generalizes wellknown results for algebraic number fields. We apply these results to characterize the trace form of a central simple algebra over such a field in terms of its determinant and signatures.

2000 Mathematics Subject Classification: 16K50, 11E81, 11E04 Keywords and Phrases: Central Simple Algebras, Trace Forms, Brauer Groups

1 Introduction and Preliminaries

Let A be a central simple algebra over a field F of characteristic different from 2. The quadratic form q : A → F given by x ↦ Trd_{A}(x^{2}) ∈ F is called the trace form of A, and is denoted by T_{A}. This trace form has been studied by many authors (cf.[Le], [LM], [Ti] and [Se], Annexe §5 for example). In particular, its classical invariants are well-known (loc.cit.). In this article, we prove some divisibility results for the Brauer group of fields F under the assumption that F satisfies the strong approximation theorem on orderings (F is a SAP field) and I^{3}(F) is torsion-free. Then we apply these results to characterize the trace form of a central simple algebra over such a field in terms of its determinant and signatures.

First we review the necessary background for this article. For a field F, Br(F) denotes the Brauer group of F. If p is a prime number, _{p} Br(F) denotes the p-primary component of Br(F). If n ≥ 1, Br_{n}(F) denotes the kernel of multiplication by n in the Brauer group. If A is a central simple algebra over F, the exponent of A, denoted by exp A, is the order of [A] in Br(F) and the index of A, denoted by ind A, is the degree of the division algebra which corresponds to A. We know that exp A divides ind A. If a, b ∈ F^{×}, we denote by (a, b)_{F} the corresponding quaternion algebra, or simply (a, b) if no confusion is possible. We also use the same symbol to denote its class in the Brauer group. We refer to [D], [J], or [Sc] for more information on central simple algebras over general fields. In the following, all quadratic forms are nonsingular. If q is a quadratic form over F, dim q is the dimension of q and det q ∈ F^{×}/F^{×2} is the determinant of q. We denote by ℍ the hyperbolic plane. If q ≃ ⟨a_{1}, ⋯, a_{n}⟩, the Hasse-Witt invariant of q is given by w_{2}(q) =

∑_{i<j}(a_{i}, a_{j}) ∈ Br_{2}(F).

If a_{1}, ⋯, a_{n} ∈ F^{×}, the quadratic form ⟨⟨a_{1}, ⋯, a_{n}⟩⟩ := ⟨1, −a_{1}⟩⊗⋯⊗⟨1, −a_{n}⟩ is called an n-fold Pfister form. If F is a formally real field, the space of orderings of F is denoted by Ω_{F}. We let sign_{v}(q) ∈ ℤ denote the signature of q relative to an ordering v ∈ Ω_{F}. Thus sign_{v}(q) is the difference between the number of positive elements and the number of negative elements in any diagonalization of q. If n ≥ 1, I^{n}(F) is the n^{th} power of the fundamental ideal of the Witt ring W (F) of F. We denote by I^{n}(F)_{t} the kernel of the map I^{n}(F) → ∏_{v∈Ω_{F}} I^{n}(F_{v}). We

will say that I^{n}(F) is torsion-free if I^{n}(F)_{t} = 0. A field F satisfies property A_{n} if every torsion n-fold Pfister form defined over F is hyperbolic over F. See [EL2], section 4, for more details on property A_{n}. The absolute stability index of F, denoted st_{a}(F) is the smallest nonnegative integer n such that I^{n+1}(F) = 2I^{n}(F) (or ∞, if no such integer exists). See [EP], p. 1248 for more details. The reduced stability index of F, denoted st_{r}(F) is the smallest nonnegative integer n such that I^{n+1}(F) ≡ 2I^{n}(F) mod W (F)_{t}. See [La2], Chapter 13, for more details. A field F satisfies the strong approximation property (SAP) if for every clopen set X of Ω_{F} there exists a ∈ F^{×} such that a >_{v} 0 if v ∈ X and a <_{v} 0 otherwise. See [La2] for various equivalent definitions and basic properties of SAP fields. If q is a quadratic form defined over F, then q̂ ∈ C(Ω_{F} , ℤ) is the continuous function q̂: Ω_{F} ⟶ ℤ defined by q̂(v) = sign_{v}(q) for every v ∈ Ω_{F}. If M is a discrete torsion Galois-module of exponent m, prime to the characteristic of F , H^{n}(F, M) denotes the n-th cohomology group H^{n}(Gal(F^{sep}/F), M). The group H^{n}(F, M)_{t} denotes the kernel of the map H^{n}(F, M) → ∏_{v∈Ω_{F}} H^{n}(F_{v}, M). If L/F is any field extension, Res_{L/F} denotes

the restriction map. We then have Res_{L/F} (w_{2}(q)) = w_{2}(q_{L}) for any quadratic form q over F. If L/F is a finite Galois extension, Cor_{L/F} denotes the corestriction map. In this paper, we deal only with the case when n is even, because we know that

T_{A} ≃ n < 1 >⊥ \frac{n(n − 1)}{2}ℍ when n is odd (cf.[Se], Annexe §5 for example). An abelian group G is divisible if for all n ≥ 1, we have G = nG. If J is any set, G^{(J)} is the group of families of elements of G indexed by J, with finite supports. In the following, F always denotes a field of characteristic different from 2, and K = F (\sqrt{−1}). We now recall some results about the classical invariants of trace forms of central simple algebras:

Theorem 1.1. Let A be a central simple algebra over F of degree n. Then we have:

1. dim T_{A} = n^{2}

2. det T_{A} = (−1)^{\frac{n(n−1)}{2}}

3. We have sign_{v} T_{A} = ±n for each v ∈ Ω_{F}, and sign_{v} T_{A} = n if and only if Res_{F_{v}/F} ([A]) = 0, where F_{v} is the real closure of (F, v).

4. If n = 2m ≥ 2, then w_{2}(T_{A}) = \frac{m(m − 1)}{2}(−1, −1) + m[A]

The three first statements can be found in [Le], and the last one is proved in [LM] or [Ti] for example.

2 Divisibility results in the Brauer group

Proposition 2.1. Let θ : I^{3}(F) ⟶ ∏_{v∈Ω_{F}} I^{3}(F_{v})/I^{4}(F_{v}). If st_{r}(F) ≤ 4, then

ker(θ) = I^{3}(F)_{t} + I^{4}(F).

Proof. It is clear that ker(θ) ⊇ I^{3}(F)_{t} + I^{4}(F). Now let q ∈ I^{3}(F) and assume q ∈ ker(θ). Then q_{v} ∈ I^{4}(F_{v}) and this implies 16|sign_{v}(q) for each v ∈ Ω_{F}. Thus q̂ ∈ C(Ω_{F} , 16ℤ). Since st_{r}(F) ≤ 4, Theorem 13.1 of [La2] applied to the preorder T = ∑ F^{2} implies there exists q_{0} ∈ I^{4}(F) such that q̂ = q̂_{0}. Then q − q_{0} ∈ I^{3}(F) ∩ W (F)_{t} = I^{3}(F)_{t} and hence q ∈ I^{3}(F)_{t} + I^{4}(F). □

Corollary 2.2. Let θ̄: I^{3}(F)/I^{4}(F) ⟶ ∏_{v∈Ω_{F}} I^{3}(F_{v})/I^{4}(F_{v}). If I^{3}(F)_{t} = 0

and st_{r}(F) ≤ 4, then θ̄ is injective and H^{3}(F, μ_{2})_{t} = 0.

Proof. The hypothesis and Proposition 2.1 imply ker(θ) = I^{4}(F). Therefore ker(θ̄) = (0) and θ̄ is injective. Since I^{3}(F)/I^{4}(F) ≃ H^{3}(F, μ_{2}), and I^{3}(F_{v})/I^{4}(F_{v}) ≃ H^{3}(F_{v}, μ_{2}) by [MS1] and [MS2], it follows H^{3}(F, μ_{2})_{t} = 0. □

Proposition 2.3. Assume that I^{3}(F)_{t} = 0 and st_{r}(F) ≤ 4. Let α ∈ H^{2}(F, μ_{2^{r}})_{t} (r ≥ 1). Then there exists β ∈ H^{2}(F, μ_{2^{r+1}}) such that α = 2β.

Proof. The exact sequence

1 → μ_{2} → μ_{2^{r+1}} → μ_{2^{r}} → 1

(where the last map is squaring) induces the following commutative diagram with exact rows

H^{2}(F, μ_{2^{r+1}}) ⟶ H^{2}(F, μ_{2^{r}}) ⟶ H^{3}(F, μ_{2})
↓  ↓  ↓
∏_{v∈Ω_{F}} H^{2}(F_{v}, μ_{2^{r+1}}) ⟶ ∏_{v∈Ω_{F}} H^{2}(F_{v}, μ_{2^{r}}) ⟶ ∏_{v∈Ω_{F}} H^{3}(F_{v}, μ_{2}).

Since the third vertical map is injective by Corollary 2.2, a diagram chase gives the conclusion. □ In Theorem 2.7 below, we need a hypothesis that is slightly stronger than the one occurring in Proposition 2.3. The following result gives a characterization of this hypothesis.

Proposition 2.4. Let K = F (\sqrt{−1}). The following statements are equivalent.

1. F satisfies property A_{3} and st_{a}(F) ≤ 2.

2. I^{3}(F)_{t} = 0 and st_{r}(F) ≤ 2.

3. st_{a}(K) ≤ 2.

4. I^{3}(K) = 0.

5. H^{3}(K, μ_{2}) = 0.

Proof. (4) ⟺ (5): I^{3}(K) = 0 if and only if I^{3}(K)/I^{4}(K) = 0 by the Arason-Pfister Hauptsatz, and I^{3}(K)/I^{4}(K) ≃ H^{3}(K, μ_{2}) by [MS1] and [MS2].

(3) ⟺ (4): st_{a}(K) ≤ 2 means I^{3}(K) = 2I^{2}(K) and this holds if and only if I^{3}(K) = 0, since ⟨1, 1⟩ = 0 implies 2I^{2}(K) = 0.

(1) ⟺ (3): This is [EP], Theorem 3.3.

(1) ⟹ (2): Property A_{3} implies I^{3}(F)_{t} = 0, by [EL1], Theorem 3 and Corollary 3. It is clear that st_{a}(F) ≤ 2 implies st_{r}(F) ≤ 2, by [La2], Theorem 13.1(3).

(2) ⟹ (1): Clearly I^{3}(F)_{t} = 0 implies F satisfies property A_{3}. Let q be a 3-fold Pfister form defined over F. Then there exists q′ ∈ I^{2}(F) such that q − 2q′ ∈ I^{3}(F)_{t} = 0. Thus q = 2q′ with q′ ∈ I^{2}(F) and it follows I^{3}(F) = 2I^{2}(F). □

Proposition 2.5. If st_{r}(F) ≤ 2, then for every β ∈ H^{2}(F, μ_{2^{r+1}}), there exists β′ ∈ H^{2}(F, μ_{2^{r+1}})_{t} such that 2β′ = 2β.

Proof. Since the characteristic of F is not 2, we have H^{2}(F, μ_{2^{r+1}}) ≃ Br_{2^{r+1}}(F). Let A be a central simple algebra over F such that β = [A], and set X = {v ∈ Ω_{F} , sign_{v} T_{A} = n}, where n = deg A. Then X^{c} = {v ∈ Ω_{F} , sign_{v} T_{A} = −n} by Theorem 1.1. Since the total signature map is continuous with respect to the topology on Ω_{F}, the set X is clopen. Since st_{r}(F) ≤ 2

and X is clopen, there exists q ∈ I^{2}(F) such that sign_{v}(q) = \begin{cases}4, & if v ∉ X \\ 0, & if v ∈ X. \end{cases} In

the Witt ring W F we have q = ∑_{i=1}^{n}⟨⟨a_{i}, b_{i}⟩⟩, with a_{i}, b_{i} ∈ F^{×}. Let B be a central simple algebra over F such that [B] = ∑_{i=1}^{n}(a_{i}, b_{i})_{F}. Let γ ∈ H^{2}(F, μ_{2^{r+1}}) be such that γ = [B] under the isomorphism H^{2}(F, μ_{2^{r+1}}) ≃ Br_{2^{r+1}}(F). Now set β′ = β + γ. We clearly have 2β′ = 2β. Moreover, if v ∈ X, then Res_{F_{v}/F} (β) = 0 by Theorem 1.1 and Res_{F_{v}/F} (γ) = 0 by the choice of B. Similar arguments show that Res_{F_{v}/F} (β′) = 0 for all v ∉ X. It follows that β′ ∈ H^{2}(F, μ_{2^{r+1}})_{t}. □ Remark 2.6. In Proposition 2.5, a stronger conclusion is possible if we also assume that F is a SAP field. This is equivalent to assuming st_{r}(F) ≤ 1. (See [La2].) In this case there exists an element a ∈ F^{×} such that a >_{v} 0 if v ∈ X and a <_{v} 0 if v ∉ X. Let γ ∈ H^{2}(F, μ_{2^{r+1}}) be such that γ = (−1, a)_{F} under the isomorphism H^{2}(F, μ_{2^{r+1}}) ≃ Br_{2^{r+1}}(F). Now set β′ = β + γ. We clearly have 2β′ = 2β. We finish as before. This observation will be used in the proof of Theorem 2.8.

Theorem 2.7. Assume I^{3}(F)_{t} = 0 and st_{r}(F) ≤ 2. Then _{2} Br(F)_{t} is a divisible group.

Proof. It suffices to check that for all [B] ∈ _{2} Br(F)_{t} and all primes p, there exists [A] ∈ _{2} Br(F)_{t} such that p[A] = [B]. Let [B] ∈ _{2} Br(F)_{t}. Then, there exists r ≥ 1 such that 2^{r}[B] = 0. Assume first that p is odd. Then gcd(p, 2^{r}) = 1, so there exist n, m ∈ ℤ such that np + m2^{r} = 1. Then [B] = (np + m2^{r})[B] = p(n[B]). If p = 2, apply Proposition 2.3 and Proposition 2.5. □ We now give a structure theorem on the 2-primary component of the Brauer group. We denote by ∑ F^{2} the multiplicative subgroup of F^{×} of nonzero sums of squares. We use the notation of [K].

Theorem 2.8. Assume that I^{3}(F)_{t} = 0 and F is SAP. Let T (resp. Λ) be an index set of a ℤ/2ℤ-basis of Br_{2}(F)_{t} (resp. of F^{×}/ ∑ F^{×2}). Then we have the following group isomorphism

2 Br(F) ≃ ℤ(2^{∞})^{(T)} × (ℤ/2ℤ)^{(Λ)}.

Proof. Theorem 2.7 implies that _{2} Br(F)_{t} is a divisible group. Since every element of _{2} Br(F)_{t} has 2-power order, the structure theorems on divisible groups (see [K] for example) imply that this group is isomorphic to ℤ(2^{∞})^{(T)}, where T is an index set of a basis of the 2-torsion part of _{2} Br(F)_{t}, namely Br_{2}(F)_{t}.

Let [A] ∈ _{2} Br(F). Remark 2.6 shows that there exists a ∈ F^{×} such that [A′] := [A] + (−1, a) is a torsion element. Choose elements a_{λ} ∈ F^{×} such that

(a_{λ} ∑ F^{×2})_{λ∈Λ} is a ℤ/2ℤ-basis of F^{×}/ ∑ F^{×2}. Then a = b ∏_{λ∈Λ}a_{λ}^{r_{λ}}, where b ∈

∑ F^{2} and r_{λ} = 0 or 1. Since b is a sum of squares, (−1, b) is a torsion element, so we have a decomposition [A] = [B] + ∑ r_{λ}(−1, a_{λ}), where [B] = [A′] + (−1, b) is a torsion element. Now we show that [B] and the r_{λ}’s are uniquely determined. Assume that [B] + ∑ r_{λ}(−1, a_{λ}) = 0. Then (−1, ∏ a_{λ}^{r_{λ}}) = −[B] is a torsion element. This implies that ∏ a_{λ}^{r_{λ}} is positive at all orderings of F, so ∏ a_{λ}^{r_{λ}} is a sum of squares. By choice of the a_{λ}’s, this implies that r_{λ} = 0 for all λ ∈ Λ and hence that [B] = 0. □

3 Trace forms of central simple algebras

In this section, we give realization theorems for trace forms of central simple algebras.

Theorem 3.1. Let n = 2m ≥ 2 be an even integer. Assume that F is SAP and I^{2}(F) is torsion-free. Then a quadratic form q is isomorphic to the trace form of a central simple algebra of degree n if and only if the following conditions are satisfied :

1. dim q = n^{2}

2. det q = (−1)^{\frac{n(n−1)}{2}}

3. sign_{v} q = ±n, for all v ∈ Ω_{F}.

Proof. The necessity follows from Theorem 1.1. Conversely, let q be a quadratic form satisfying the conditions above. Since I^{2}(F) is torsion-free, it is well-known that quadratic forms are classified by dimension, determinant and signatures (see [EL1]). Let X = {v ∈ Ω_{F} , sign_{v} q = n}. This is a clopen set, so the SAP property of F implies there exists a ∈ F^{×} such that a >_{v} 0 if v ∈ X and a <_{v} 0 otherwise. Set A = M_{m}((−1, a)). Then Res_{F_{v}/F} ([A]) = 0 if and only if sign_{v} q = n, so T_{A} and q have the same signatures. Since they also have equal dimension and determinant, they are isomorphic. □ The following proposition gives a characterization of fields that satisfy the hypotheses of Theorem 3.1. Note the similarity to Proposition 2.4.

Proposition 3.2. Let K = F (\sqrt{−1}). The following statements are equivalent.

1. F satisfies property A_{2} and F is a SAP field (st_{a}(F) ≤ 1).

2. I^{2}(F)_{t} = 0 and F is a SAP field (st_{r}(F) ≤ 1).

3. st_{a}(K) ≤ 1.

4. I^{2}(K) = 0.

5. u(K) ≤ 2.

6. ũ(F) ≤ 2.

7. I^{2}(F)_{t} = 0 and F is linked.

Proof. The proof of the equivalence of (1)-(4) is very similar to the proof of the equivalence of the corresponding statements in Proposition 2.4. The equivalence of (4) and (5) is well-known. The equivalence of (6) and (7) appears in [E]. The equivalence of (2) and (6) appears in [ELP]. □ We now give a characterization of fields F such that I^{2}(F) is torsion-free in terms of Brauer groups.

Proposition 3.3. I^{2}(F) is torsion-free if and only if Br(F) has no element of order 4.

Proof. Assume that [A] ∈ Br(F) has order 4, so [A] ∈ H^{2}(F, μ_{4}). Then 2[A] ∈ H^{2}(F, μ_{2}) has order 2. Moreover, it is well-known that the image of [A] ∈ H^{2}(F, μ_{4}) ↦ 2[A] ∈ H^{2}(F, μ_{2}) is the kernel of

[B] ∈ H^{2}(F, μ_{2}) ↦ (−1) ∪ [B] ∈ H^{3}(F, μ_{2}) (see for example [LLT], Proposition A2 and Remark A3). So (−1) ∪ 2[A] = 0, that is 2[A] = Cor_{K/F} ([B]) for some [B] ∈ H^{2}(K, μ_{2}). Since H^{2}(K, μ_{2}) is generated by elements of the form (a, b), a ∈ F^{×}, b ∈ K^{×}, the transfer formula shows that 2[A] = ∑(a_{i}, N_{K/F} (b_{i})) for some a_{i} ∈ F^{×} and b_{i} ∈ K^{×}. Since 2[A] has order 2, it is not split, so there exists i such that (a_{i}, N_{K/F} (b_{i})) is not split. Then the norm form of this quaternion algebra is not hyperbolic, and it is a torsion 2-fold Pfister form, since N_{K/F} (b_{i}) is the sum of 2 squares. Conversely, assume that I^{2}(F) is not torsion-free. Then property A_{2} fails (see [EL2], section 4). Theorem 4.3(3) in [EL2] (with x = 1) implies that there exists a binary form ⟨1, −a⟩ and an element b = u^{2} + v^{2}, with u, v ∈ F, such that ⟨1, −a⟩ does not represent b. This means ⟨⟨a, b⟩⟩ is an anisotropic 2-fold Pfister form and b is not a square. Let L := F (\sqrt{b + v\sqrt{b}}). Then L/F is a cyclic quartic extension which contains F (\sqrt{b}). Denote by σ a generator of Gal(L/F) and let A be the cyclic algebra (a, L/F, σ) (see [Sc] for the definition and basic properties of cyclic algebras). It is not difficult to show that 2[A] = (a, b) (for example use [J], Corollary 2.13.20). By construction, the norm form of this quaternion algebra is not hyperbolic, so 2[A] is not split, and [A] has order 4. □ We now apply the results of section 2 to prove the following theorem:

Theorem 3.4. Let n = 2m ≥ 2 be an even integer. Write n = 2^{r+1}s, r ≥ 0, s ≥ 1 odd. Assume that F satisfies the following conditions:

(a) I^{3}(F) is torsion-free

(b) For every [A] ∈ Br(F) such that 2^{r+1}[A] = 0, there exists A′, deg A′ = 2^{r+1} such that [A′] = [A] (c) If r ≥ 1, assume st_{r}(F) ≤ 2.

Then a quadratic form q is isomorphic to the trace form of a central simple algebra of degree n if and only if the following conditions are satisfied :

1. dim q = n^{2}

2. det q = (−1)^{\frac{n(n−1)}{2}}

3. sign_{v} q = ±n, for all v ∈ Ω_{F}.

Before we begin the proof of this theorem, we need the following calculation.

Lemma 3.5. Let n = 2m, m ≥ 1, and assume F is a real closed field.

Let q_{+} = n⟨1⟩ ⊥ \frac{n(n − 1)}{2}ℍ and let q_{−} = n⟨−1⟩ ⊥ \frac{n(n − 1)}{2}ℍ. Then

w_{2}(q_{+}) = \frac{m(m − 1)}{2}(−1, −1)_{F} and w_{2}(q_{−}) = (\frac{m(m − 1)}{2} + m) (−1, −1)_{F}.

In particular, if m is odd, then w_{2}(q_{+}) ≠ w_{2}(q_{−}).

Proof. Let A = M_{n}(F) and let B = M_{m}((−1, −1)). Then deg A = deg B = n and hence Theorem 1.1 implies sign(T_{A}) = n and sign(T_{B}) = −n. This implies T_{A} ≃ q_{+} and T_{B} ≃ q_{−}. In addition, Theorem 1.1 implies

w_{2}(q_{+}) = w_{2}(T_{A}) = \frac{m(m − 1)}{2}(−1, −1) + m[A] = \frac{m(m − 1)}{2}(−1, −1)

and

w_{2}(q_{−}) = w_{2}(T_{B}) = \frac{m(m − 1)}{2}(−1, −1) + m(−1, −1)

= (\frac{m(m − 1)}{2} + m) (−1, −1).

The last statement of this Lemma is clear since (−1, −1)_{F} ≠ 0 if F is real closed. □ Proof of Theorem 3.4 Notice that property (a) implies that quadratic forms are classified by dimension, determinant, Hasse-Witt invariant and signatures (see [EL1]). The necessity follows from Theorem 1.1. Now suppose q satisfies (1)-(3). Assume first that r = 0, so m is odd. By hypothesis, there exists a quaternion

algebra Q such that [Q] = w_{2}(q) + \frac{m(m − 1)}{2}(−1, −1)_{F}. Let A = M_{m}(Q). Then

w_{2}(T_{A}) = \frac{m(m − 1)}{2}(−1, −1)_{F} + m[Q] = \frac{m(m − 1)}{2}(−1, −1)_{F} + [Q] = w_{2}(q).

We have sign_{v}(T_{A}) = n if and only if Res_{F_{v}/F} ([Q]) = 0, by Theorem 1.1, which

is equivalent to w_{2}(q_{F_{v}}) = \frac{m(m − 1)}{2}(−1, −1)_{F_{v}}. This occurs if and only if q_{F_{v}} ≃ q_{+}, by Lemma 3.5, since m is odd and sign_{v}(q) = ±n. Thus q and T_{A} have the same signatures. Since q and T_{A} also have the same dimension, determinant and Hasse-Witt invariant, it follows that they are isomorphic. Assume now that r ≥ 1. Let B be a central simple algebra over F such that

[B] = w_{2}(q) + \frac{m(m − 1)}{2}(−1, −1)_{F}. Since m is even and sign_{v}(q) = ±n, it follows from Lemma 3.5 that

Res_{F_{v}/F} ([B]) = Res_{F_{v}/F} (w_{2}(q) + \frac{m(m − 1)}{2}(−1, −1)_{F_{v}}) = 0

for all v ∈ Ω_{F}. By Theorem 2.7, there exists [A_{1}] ∈ _{2} Br(F)_{t} such that 2^{r}[A_{1}] = [B]. Let X = {v ∈ Ω_{F} , sign_{v} q = n}. Since X is clopen and st_{r}(F) ≤ 2, we can use the ideas in the proof of Proposition 2.5 to find a central simple algebra D over F such that 2[D] = 0 and such that [A_{2}] = [A_{1}] + [D] satisfies Res_{F_{v}/F} [A_{2}] = 0 if and only if sign_{v}(q) = n. Then 2^{r}[A_{2}] = [B] since r ≥ 1. Since 2[B] = 0, we have 2^{r+1}[A_{2}] = 0, and so by assumption there exists a central simple algebra A_{3}, deg A_{3} = 2^{r+1}, such that [A_{3}] = [A_{2}]. Now set A = M_{s}(A_{3}), and note that A has degree n. Since A and A_{2} are Brauer equivalent, q and T_{A} have equal signatures by construction of A_{2}. Since

m[A] = 2^{r}s[A_{2}] = s[B] = [B] = \frac{m(m − 1)}{2}(−1, −1)_{F} + w_{2}(q),

it follows that w_{2}(T_{A}) = \frac{m(m − 1)}{2}(−1, −1)_{F} + m[A] = w_{2}(q). Thus q and T_{A} are isomorphic, since they have the same dimension, determinant, Hasse-Witt invariant, and signature. □

Corollary 3.6. Assume F satisfies the following conditions.

(a) I^{3}(F) is torsion-free

(b′) For every r ≥ 0 and for every [A] ∈ Br(F) such that 2^{r+1}[A] = 0, there exists A′, deg A′ = 2^{r+1} such that [A′] = [A].

Then a quadratic form q is isomorphic to the trace form of a central simple algebra of degree n if and only if the following conditions are satisfied :

1. dim q = n^{2}

2. det q = (−1)^{\frac{n(n−1)}{2}}

3. sign_{v} q = ±n, for all v ∈ Ω_{F}.

Proof. This follows immediately from the Theorem 3.4 and the following observation. Condition (b′) with r = 0 implies that F is a linked field. That is, a sum of quaternion algebras defined over F is similar to another quaternion algebra defined over F. A theorem of Elman ([E]) states that a field F is linked and has I^{3}(F)_{t} = 0 if and only if ũ(F) ≤ 4. It is known that if ũ(F) < ∞, then F is a SAP field (see [ELP]). Thus condition (c) in Theorem 3.4 holds automatically in the situation of Corollary 3.6. □

Remark 3.7. Condition (b) is realized for example when exp A = ind A for every central simple algebra. In particular, it is the case when every central simple algebra is cyclic. For example, condition (b) holds for local fields, global fields or quotient fields of excellent two-dimensional local domains with algebraically closed residue fields of characteristic zero, e.g. finite extensions of ℂ((X, Y)) (see [CTOP], Theorem 2.1 for the last example and [CF] for the others). Such fields also satisfy condition (a). This is well-known for local fields and global fields (see [CF]). If F is a field of the last type, then I^{3}(F) = 0 (see [CTOP], Corollary 3.3). We finish this paper giving a local-global principle for trace forms over global fields.

Corollary 3.8. Let F be a global field of characteristic different from 2, and let n = 2m ≥ 2 be an even integer. Then a quadratic form q over F is isomorphic to the trace form of a central simple algebra of degree n defined over F if and only if q is isomorphic to the trace form of a central simple algebra of degree n defined over all completions of F.

Proof. Assume that q is a trace form over all completions of F. Then dim q = n^{2}. By assumption, (−1)^{\frac{n(n−1)}{2}} det q is a nonzero square over all completions of F, so it is a nonzero square in F, and hence det q = (−1)^{\frac{n(n−1)}{2}} ∈ F^{×}/F^{×2}. Since q is a trace form over all real completions of F, we have sign_{v} q = ±n for all real places v of F, according to whether q_{F_{v}} is isomorphic to the trace form of the split algebra or that of M_{m}((−1, −1)_{F_{v}}). Now apply Theorem 3.4. The other implication is clear, since (T_{A})_{L} ≃ T_{A⊗L} for every central simple algebra over F, and every field extension L/F. □ The fact that q_{F_{𝔭}} ≃ T_{A_{𝔭}} for all places 𝔭 implies that q ≃ T_{A} does not mean that A ⊗ F_{𝔭} ≃ A_{𝔭} for all places. We sketch below the construction of a counterexample. Example 3.9. We refer to [CF] for the definition of inv_{𝔭} and the theorems concerning central simple algebras over global fields. Assume n ≡ 0 [8]. Let 𝔭_{1}, 𝔭_{2} be two places of F. For i = 1, 2, let A_{i} be a

central simple of degree n over F_{𝔭_{i}} such that inv_{𝔭_{i}}[A_{i}] = \frac{1}{n}, and let A_{𝔭} be M_{n}(F_{𝔭}) for the other places over F. Now let q_{𝔭} be the trace form of A_{𝔭}. We have w_{2}(q_{𝔭}) ≠ 0 if and only if 𝔭 = 𝔭_{1}, 𝔭_{2}. Moreover det q_{𝔭} = (−1)^{\frac{n(n−1)}{2}} for all 𝔭, so by [Sc], 6.6.10, there exists a quadratic form q over F such that q_{F_{𝔭}} ≃ q_{𝔭}. So q is locally a trace form, then q is the trace form of some central simple algebra A over F, but we can never have A ⊗ F_{𝔭} ≃ A_{𝔭} for all 𝔭. Otherwise, we will have ∑ inv_{𝔭}([A]) = 0 ∈ ℚ/ℤ, which is not the case by choice of the A_{𝔭}’s.

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Grégory Berhuy Ecole Polytechnique Fédérale de Lausanne, D.M.A. CH-1015 Lausanne Switzerland gregory.berhuy@epfl.ch David B. Leep Department of Mathematics University of Kentucky Lexington, KY 40506-0027, USA leep@ms.uky.edu