Documenta Math. 489 Divisible Subgroups of Brauer Groups and Trace Forms of Central Simple Algebras

Let F be a fleld of characteristic difierent from 2 and assume that F satisfles the strong approximation theorem on orderings (F is a SAP fleld) 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 flelds. We apply these results to characterize the trace form of a central simple algebra over such a fleld in terms of its determinant and signatures.


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 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∈ΩF 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 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 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 core-striction map.
In this paper, we deal only with the case when n is even, because we know that [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 ( √ −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: 3. We have sign v T A = ±n for each v ∈ Ω F , and The three first statements can be found in [Le], and the last one is proved in [LM] or [Ti] for example.
The following statements are equivalent.

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 Res Fv/F (β) = 0 by Theorem 1.1 and Res Fv/F (γ) = 0 by the choice of B.

Similar arguments show that Res
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 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 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 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 is a torsion element.This implies that a r λ λ is positive at all orderings of F , so a r λ 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. 2

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 : 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 Fv/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.2 The following proposition gives a characterization of fields that satisfy the hypotheses of Theorem 3.1.Note the similarity to Proposition 2.4.
The following statements are equivalent.
1. F satisfies property A 2 and F is a SAP field (st a (F ) ≤ 1).
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]. 2 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.
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 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. 2 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: 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 : Before we begin the proof of this theorem, we need the following calculation.
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 and The last statement of this Lemma is clear since (−1, −1) F = 0 if F is real closed.2 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 We have sign v (T A ) = n if and only if Res Fv/F ([Q]) = 0, by Theorem 1.1, which is equivalent to w 2 (q Fv ) = m(m − 1) 2 (−1, −1) Fv .This occurs if and only if Documenta Mathematica 6 (2001) 489-500 q Fv 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) + m(m − 1) 2 (−1, −1) F .Since m is even and sign v (q) = ±n, it follows from Lemma 3.5 that 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 [ 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 ), 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 . Thus q and T A are isomorphic, since they have the same dimension, determinant, Hasse-Witt invariant, and signature.2 Corollary 3.6.Assume F satisfies the following conditions.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 : 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 3.6.2 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) det q is a nonzero square over all completions of F , so it is a nonzero square in F , and hence det q = (−1) 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 Fv is isomorphic to the trace form of the split algebra or that of M m ((−1, −1) Fv ).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 . 2 The fact that q Fp T Ap 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 pi such that inv pi [A i ] = 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) for all p, so by [Sc], 6.6.10,there exists a quadratic form q over F such that q Fp 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.