Dimensions of Anisotropic

Indefinite Quadratic Forms II

To Andrei Suslin on the occasion of his 60th birthday

Detlev W. Hoffmann

Received: September 29, 2009 Revised: April 15, 2010

Abstract. The u-invariant and the Hasse number ũ of a field F of characteristic not 2 are classical and important field invariants pertaining to quadratic forms. They measure the suprema of dimensions of anisotropic forms over F that satisfy certain additional properties. We prove new relations between these invariants and a new characterization of fields with finite Hasse number (resp. finite u-invariant for nonreal fields), the first one of its kind that uses intrinsic properties of quadratic forms and which, conjecturally, allows an ‘algebrogeometric’ characterization of fields with finite Hasse number.

2010 Mathematics Subject Classification: primary: 11E04; secondary: 11E10, 11E81, 14C25 Keywords and Phrases: quadratic form, Pfister form, Pfister neighbor, real field, ordering, strong approximation property, effective diagonalization, u-invariant, Hasse number, Pythagoras number, Rost correspondence, Rost projector

1. Introduction

Throughout this paper, fields are assumed to be of characteristic different from 2 and quadratic forms over a field are always assumed to be finite-dimensional and nondegenerate. The u-invariant of a field F is one of the most important field invariants pertaining to quadratic forms. The definition as introduced by Elman and Lam [EL1] is as follows:

u(F) := sup{dim φ | φ is an anisotropic torsion form over F } ,

where ‘torsion’ means torsion when considered as an element in the Witt ring W F. Note that over a formally real field (or real field for short) torsion forms are exactly the forms of total signature zero, whereas over a nonreal field, all forms are torsion. If F is a real field, for a form φ over F to be isotropic, it is clearly necessary for φ to be indefinite at each ordering of F, i.e., for φ to be totally indefinite or t.i. for short. This leads to another field invariant, the Hasse number ũ defined as

ũ(F) := sup{dim φ | φ is an anisotropic t.i. form} .

One puts ũ(F) = 0 if there are no anisotropic t.i. forms over F. Clearly, u(F) ≤ ũ(F), with equality in the case of nonreal fields since being totally indefinite is then an empty condition. In the present paper, we focus on finiteness criteria for u and ũ and on upper bounds on ũ in terms of u for fields with finite ũ. To formulate these results, we need to introduce further properties. We refer to [L3] for all undefined terminology and basic facts about quadratic forms. Recall that a quadratic form of type ⟨1, −a_{1}⟩⊗…⊗⟨1, −a_{n}⟩ (a_{i} ∈ F^{∗}) is called an n-fold Pfister form, and we write ⟨⟨a_{1}, …, a_{n}⟩⟩ for short. P_{n}F (resp. GP_{n}F) denotes the set of all isometry classes of n-fold Pfister forms (resp. of forms similar to n-fold Pfister forms). A form φ is a Pfister neighbor if there exists a Pfister form π and a ∈ F^{∗} such that φ ⊂ aπ and dim φ > \frac{1}{2} dim π. Pfister forms are either hyperbolic or anisotropic, and if φ is a Pfister neighbor of a Pfister form π then φ is anisotropic iff π is anisotropic. Recall that the n-fold Pfister forms generate additively I^{n}F, the n-th power of the fundamental ideal IF of classes of even-dimensional forms in the Witt ring W F. The Arason-Pfister Hauptsatz [AP], APH for short, states that if φ ∈ I^{n}F, then dim φ < 2^{n} implies that φ is hyperbolic, and dim φ = 2^{n} implies φ ∈ GP_{n}F. Let F be a real field and let X_{F} denote its space of orderings. X_{F} is a compact totally disconnected Hausdorff space with a subbasis of the topology given by the clopen sets H(a) = {P ∈ X_{F} | a >_{P} 0}, a ∈ F^{∗}. φ is called positive (resp. negative) definite at P ∈ X_{F} if sgn_{P} (φ) = dim φ (resp. sgn_{P} (φ) = − dim φ), and indefinite at P if it is not definite at P. A totally positive definite (t.p.d.) form is a form that is positive definite at each P ∈ X_{F}. If φ is a form over F, we denote by D_{F} (φ) those elements in F^{∗} represented by φ, by D_{F} (n) (n ∈ ℕ) those elements in F^{∗} that can be written as a sum of n squares, and we write D_{F} (∞) = ⋃_{n∈ℕ} D_{F} (n) for the nonzero sums of squares in F. If F is nonreal then F^{∗} = D_{F} (∞), and if F is real then D_{F} (∞) is the set of all totally positive elements in F. The Pythagoras number p(F) of a field F is the smallest n such that D_{F} (n) = D_{F} (∞) if such an n exists, otherwise p(F) = ∞. If F is real, then x ∈ D_{F} (φ) clearly implies that x >_{P} 0 (resp. x <_{P} 0) if φ is positive (resp. negative) definite at P. If the converse also holds, i.e. if

D_{F} (φ) = {x ∈ F^{∗} | x >_{P} 0 (resp. x <_{P} 0) if φ is

positive (resp. negative) definite at P }

then φ is called signature-universal (sgn-universal for short). Over a real field, a form is universal (in the usual sense) if and only if it is t.i. and sgn-universal.

One readily sees that if ũ(F) < ∞ then any form φ with dim φ ≥ ũ(F) is sgn-universal. The following properties of fields will be used repeatedly.

Definition 1.1. (i) F is said to satisfy the strong approximation property SAP if given any disjoint closed subsets U, V of X_{F} there exists a ∈ F^{∗} such that U ⊂ H(a) and V ⊂ H(−a).

(ii) A form φ over a real field F is said to have effective diagonalization ED if it has a diagonalization ⟨a_{1}, …, a_{n}⟩ such that H(a_{i}) ⊂ H(a_{i+1}) for 1 ≤ i ≤ n − 1. F is said to be ED if each form over F has ED.

(iii) F is said to have property S_{1} if for every binary torsion form β over F

one has D_{F} (β) ∩ D_{F} (∞) ≠ ∅. (iv) F is said to have property P N(n) for some n ∈ ℕ if each form of

dimension 2^{n} + 1 over F is a Pfister neighbor.

Note that if F is a nonreal field, i.e., F has no orderings, then F^{∗} = D_{F} (∞) and all forms over F are torsion, so F is SAP, ED and S_{1}. The paper is structured as follows. In §2 we give a new proof of the fact that ED is equivalent to SAP plus S_{1}, a result originally due to Prestel-Ware [PW]. In §3 we prove that for a field, having finite Hasse number is equivalent to having finite u-invariant plus having property ED. This result is originally due to Elman-Prestel [EP], but we give a proof that also allows us to derive various estimates for ũ in terms of u that are better than any previously known such estimates. In §4, we prove that having finite Hasse number is equivalent to having property P N(n) for some n ≥ 2, in which case we give estimates on ũ in terms of n. Since property P N(2) is equivalent to F being linked (see Lemma 4.3), we will thus also recover as corollary a famous result on the uinvariant and the Hasse number of linked fields due to Elman-Lam [EL2], [E] (Corollary 4.12). We also explain how our results, conjecturally, provide an ‘algebro-geometric’ criterion for the finiteness of ũ (resp. u in case of nonreal fields).

Acknowledgment. I am grateful to the referee for various suggestions that helped to streamline the paper considerably. The revised version of this paper has been completed during a stay at Emory University. I thank Skip Garibaldi and Emory University for their hospitality during that stay.

2. ED equals SAP plus S_{1}

The following theorem is due to Prestel-Ware [PW]. We give a new proof based mainly on the study of binary forms.

Theorem 2.1. F has ED if and only if F has SAP and S_{1}.

To prove this, we use alternative descriptions of the properties involved.

Lemma 2.2. Let F be a real field.

(i) F is SAP if and only if for all a, b ∈ F^{∗} there exists c ∈ F^{∗} such that H(c) = H(a) ∩ H(b) (or, equivalently, there exists d ∈ F^{∗} such that H(d) = H(a) ∪ H(b)).

(ii) F is ED if and only if for all a, b ∈ F^{∗}, there exist c, d ∈ F^{∗} such that ⟨a, b⟩ ≅ ⟨c, d⟩ and H(c) = H(a) ∩ H(b) (or, equivalently, H(d) = H(a) ∪ H(b)).

(iii) F has property S_{1} if and only if, for all a ∈ F^{∗}, s ∈ D_{F} (∞), and x ∈ D_{F} (⟨1, as⟩), there exists t ∈ D_{F} (∞) such that tx ∈ D_{F} (⟨1, a⟩).

Proof. (i) This is well known, see, e.g., [L1, Prop. 17.2]. (ii) The ‘only if’ is nothing else but ED for binary forms. As for the converse, we use induction on the dimension n of forms. Forms of dimension ≤ 2 have ED by assumption. So let φ be a form of dimension n ≥ 3. Then we can write φ = ⟨a_{1}, …, a_{n}⟩ and we may assume that ⟨a_{2}, …, a_{n}⟩ is already an ED. Write ⟨a_{1}, a_{2}⟩ ≅ ⟨b_{1}, b_{2}⟩ with H(b_{1}) = H(a_{1})∩H(a_{2}) (so ⟨b_{1}, b_{2}⟩ is an ED of ⟨a_{1}, a_{2}⟩). Then φ ≅ ⟨b_{1}, b_{2}, a_{3}, …, a_{n}⟩. Now let ⟨c_{2}, …, c_{n}⟩ be an ED of ⟨b_{2}, a_{3}, …, a_{n}⟩. Then one readily checks that ⟨b_{1}, c_{2}, …, c_{n}⟩ is an ED of φ. (iii) ‘if’: Let ⟨u, v⟩ ≅ u⟨1, uv⟩ be torsion. Then uv = −s with s ∈ D_{F} (∞). Put a = −s. Then ⟨1, −1⟩ ≅ ⟨1, as⟩ which is hyperbolic and hence represents u. But then, by assumption, there exists t ∈ D_{F} (∞) such that tu is represented by ⟨1, a⟩ ≅ ⟨1, −s⟩ and hence t is represented by u⟨1, −s⟩ ≅ ⟨u, v⟩. ‘only if’: x ∈ D_{F} (⟨1, sa⟩) implies that there exists y ∈ F^{∗} such that ⟨1, sa⟩ ≅ ⟨x, y⟩. Now the torsion form xa⟨s, −1⟩ represents some u ∈ D_{F} (∞) by S_{1}. Hence ⟨sa, −a⟩ ≅ ⟨xu, −xus⟩ and hence

⟨1, sa, −a⟩ ≅ ⟨1, xu, −xus⟩ ≅ ⟨−a, x, y⟩

Thus, ⟨1, a⟩ = ⟨x, xus, −xu, y⟩ in W F, so x⟨1, us, −u, xy⟩ is isotropic and there exists v ∈ D_{F} (⟨1, us⟩) ∩ D_{F} (⟨u, −xy⟩). Note that us ∈ D_{F} (∞), so v ∈ D_{F} (∞). Hence, ⟨1, us⟩ ≅ ⟨v, vus⟩ and ⟨−u, xy⟩ ≅ ⟨−v, vuxy⟩, and we get ⟨1, a⟩ ≅ x⟨vus, vuxy⟩ ≅ ⟨xvus, vuy⟩, thus xt ∈ D_{F} (⟨1, a⟩) with t := vus ∈ D_{F} (∞). □

Proof of Theorem 2.1. ‘only if’: Clearly, ED implies SAP. Now let ⟨a, b⟩ be any binary torsion form. Then sgn_{P} (⟨a, b⟩) = 0, so H(a) ∩ H(b) = ∅, and by ED, there exists c ∈ −D_{F} (∞) and d ∈ D_{F} (∞) such that ⟨a, b⟩ ≅ ⟨c, d⟩, in particular, d is a totally positive element represented by ⟨a, b⟩ and we have established S_{1}. ‘if’: Let F be SAP and S_{1}. We will verify the alternative description of ED from Lemma 2.2(ii). Let ⟨a, b⟩ be any binary form. By SAP, there exists d′ ∈ F^{∗} such that H(a) ∪ H(b) = H(d′). Then ⟨a, b, −d′⟩ is t.i., thus the form φ ≅ ⟨a, b, −d′, −d′ab⟩ ≅ −d′⟨⟨ad′, bd′⟩⟩ has total signature zero and is therefore torsion. Hence, there exists some n ∈ F such that for σ_{n} ≅ ⟨⟨−1⟩⟩^{⊗n} ≅ ⟨1, 1⟩^{⊗n}, we have that σ_{n} ⊗ ⟨a, b, −d′, −d′ab⟩ ∈ GP_{n+2}F is hyperbolic. But then its Pfister neighbor σ_{n} ⊗ ⟨a, b⟩ ⊥ ⟨−d′⟩ is isotropic. It follows that there exist u, v ∈ D_{F} (σ_{n}) ⊂ D_{F} (∞) such that d′ ∈ D_{F} (⟨ua, vb⟩), and hence ad′u ∈ D_{F} (⟨1, abuv⟩). Now uv ∈ D_{F} (∞), and by Lemma 2.2(iii), there exists w ∈ D_{F} (∞) such that ad′uw ∈ D_{F} (⟨1, ab⟩), i.e. d := d′uw ∈ D_{F} (⟨a, b⟩).

In particular, there exists c ∈ F^{∗} such that ⟨a, b⟩ ≅ ⟨c, d⟩. Since uw ∈ D_{F} (∞), we have H(d) = H(d′) = H(a) ∪ H(b) as required. □

3. Relations between the Hasse number and the u-invariant

In this section, we will only consider real fields since for nonreal fields u = ũ, and most of the statements below are trivially true. It is quite possible for a real field F that u(F) is finite but ũ(F) is infinite. Elman-Prestel [EP, Th. 2.5] gave the following necessary and sufficient criterion for the finiteness of ũ(F):

Theorem 3.1. ũ(F) < ∞ if and only if u(F) < ∞ and F has ED.

The main purpose of this section is to give a new and elementary proof of this statement that in the case of ED-fields will allow us at the same time to derive upper bounds for ũ in terms of u that considerably improve previous upper bounds obtained by Elman-Prestel [EP, Prop. 2.7] and Hornix [Hor1, Th. 3.9]. The following remark is well known and will be useful.

Remark 3.2. For any field F, if p(F) > 2^{n} then ũ(F) ≥ u(F) ≥ 2^{n+1}. In particular, p(F) ≤ u(F) ≤ ũ(F).

Proposition 3.3. Suppose that F has ED and that there exists an ndimensional t.p.d. sgn-universal form ρ. Then

ũ(F) ≤ \frac{n}{2}(u(F) + 2) .

Proof. We may clearly assume that u(F) (and hence p(F)) is finite. The form p(F) × ⟨1⟩ is t.p.d. and sgn-universal, so we may assume that n ≤ p(F). If n = 1 then F is obviously pythagorean and u(F) = 0. Since F has ED, any t.i. form φ over F contains a binary torsion form β as a subform. But then β is isotropic as u(F) = 0, hence φ is isotropic. It follows that ũ(F) = 0 and the above inequality is clearly satisfied. So we may assume that 2 ≤ n ≤ p(F) = p and we have ũ(F) ≥ u(F) ≥ p ≥ n by Remark 3.2. It suffices to consider the case ũ(F) > u(F). Let φ_{0} be any anisotropic t.i. form with dim φ_{0} > u(F), and write dim φ_{0} = m = rn + k + 1 with r ≥ 1 and 0 ≤ k ≤ n − 1. Since F is ED and thus SAP, we may assume after scaling that 0 ≤ sgn_{P} φ_{0} ≤ dim φ_{0} − 2 = rn + k − 1 for all orderings P on F. Let φ_{1} = a_{0}(φ_{0} ⊥ −ρ)_{an}, where a_{0} is chosen such that 0 ≤ sgn_{P} φ_{1} for all orderings P. If i_{W} denotes the Witt index, we have i_{W} (φ_{0} ⊥ −ρ) ≤ n − 1, for otherwise one could write φ_{0} ≅ ρ ⊥ τ for some form τ. Since φ_{0} is t.i. and since F has ED, this implies that there exists x ∈ D_{F} (∞) such that −x is represented by τ. But then the form φ_{0} contains the subform ρ ⊥ ⟨−x⟩ which is isotropic as ρ is t.p.d. and sgn-universal, clearly a contradiction. This implies that

dim φ_{1} ≥ dim φ_{0} + n − 2(n − 1) = (r − 1)n + (k + 1) + 2 .

Note also that sgn_{P} (φ_{0} ⊥ −ρ) = sgn_{P} φ_{0} − n for each ordering P. Hence, one obtains

sgn_{P} φ_{1} ≤ max{(r − 1)n + k − 1, n}

for each ordering P. Note that if r ≥ 2, then φ_{1} is again t.i. as 0 ≤ sgn_{P} φ_{1} < dim φ_{1} for all orderings P. Applying this procedure altogether r − 1 times, we get a form φ_{r−1} which is anisotropic, t.i., and such that

dim φ_{r−1} ≥ n + (k + 1) + 2(r − 1) ,
0 ≤ sgn_{P} φ_{r−1} ≤ max{n + k − 1, n} for all orderings P.

We therefore have

dim φ_{r−1} − sgn_{P} φ_{r−1} ≥ min{2r, k + 2r − 1} .

Since dim φ_{r−1}−sgn_{P} φ_{r−1} is even, this yields dim φ_{r−1}−sgn_{P} φ_{r−1} ≥ 2r for all orderings P. By ED, the anisotropic form φ_{r−1} contains a torsion subform φ_{t} of dimension ≥ 2r. Hence u(F) ≥ 2r and thus u(F)+2 ≥ 2(r+1). On the other hand, by assumption m = rn+k+1 ≤ n(r+1). These two inequalities together imply m ≤ \frac{n}{2}(u(F) + 2). It follows readily that ũ(F) ≤ \frac{n}{2}(u(F) + 2). □

Proof of Theorem 3.1. The ‘only if’ part is easy and left to the reader. As for the ‘if’ part, we have ∞ > u(F) ≥ p(F) by Lemma 3.2, and if we put ρ = p(F) × ⟨1⟩, then Proposition 3.3 immediately yields ũ(F) ≤ \frac{p(F)}{2}(u(F) + 2) < ∞. □

For a real field F, let m̃(F) be the smallest integer n ≥ 1 such that there exists an n-dimensional t.p.d. sgn-universal form, and m̃(F) = ∞ if there are no t.p.d. sgn-universal forms (cf. [GV] where an analogous invariant m(F) for anisotropic universal forms was introduced). If p(F) < ∞, we have that p(F) × ⟨1⟩ is sgn-universal. Hence m̃(F) ≤ p(F). With this new invariant, Proposition 3.3 immediately implies

Corollary 3.4. Suppose that ũ(F) < ∞. Then

ũ(F) ≤ \frac{m̃(F)}{2}(u(F) + 2) .

Next, we give another bound which will lead to further improvements.

Proposition 3.5. Suppose that u(F) < ∞ and that F has ED (or, equivalently, that ũ(F) < ∞). Let ρ = ⟨1⟩ ⊥ ρ′ be a t.p.d. m-fold Pfister form, m ≥ 1, such that its pure part ρ′ is sgn-universal. Then

ũ(F) ≤ 2^{m−2}(u(F) + 6) .

If m = 2 then ũ(F) ≤ u(F) + 4.

Proof. If m = 1, then dim ρ′ = 1 and the assumptions imply that F is pythagorean, hence ũ = u = 0 and there is nothing to show. So we may assume m ≥ 2. Furthermore, if d is an integer such that 2^{d} ≤ p(F) = p ≤ 2^{d+1}−1, then we may assume that m ≤ d+1. For we have that (2^{d+1}−1)×⟨1⟩ is the pure part of ⟨⟨−1, …, −1⟩⟩ ∈ P_{d+1}F and it is totally positive definite and sgn-universal. We proceed similarly as before, but this time we put ũ = ũ(F) = r2^{m} + k + 1 with r ≥ 0 and 0 ≤ k ≤ 2^{m} − 1. If r = 0 then we have ũ ≤ 2^{m}. If 2^{d} + 1 ≤ p ≤ 2^{d+1} − 1 then u ≥ 2^{d+1} ≥ 2^{m} by Remark 3.2, and thus necessarily u = ũ and there is nothing to show.

Suppose that p = 2^{d} so that in particular u ≥ 2^{d}. Our previous bound yields ũ ≤ 2^{d−1}(u + 2). If m = d + 1, then 2^{d−1}(u + 2) < 2^{m−2}(u + 6) and there is nothing to show. If m ≤ d, then we have ũ = k + 1 ≤ 2^{m} ≤ 2^{d} ≤ u and thus ũ = u, again there is nothing to show. So we may assume that r ≥ 1. Let φ_{0} be an anisotropic t.i. form of dimension ũ. As before, we may this time assume that dim φ_{0} − 2 = r2^{m} + k − 1 ≥ sgn_{P} φ_{0} ≥ 0 for all orderings P. We claim that i_{W} (φ_{0} ⊥ −ρ) ≤ 2^{m} − 2. Indeed, otherwise φ_{0} would contain a subform ρ̃ of dimension 2^{m} − 1 with ρ̃ ⊂ ρ. Now it is well known that all codimension 1 subforms of a Pfister form are similar to its pure part. Hence, φ_{0} would contain a subform similar to ρ′, and since φ_{0} is t.i. and by ED, φ_{0} would contain a subform similar to ρ′ ⊥ ⟨−x⟩ for some x ∈ D_{F} (∞). By assumption, ρ′ ⊥ ⟨−x⟩ is isotropic, a contradiction. Thus, we obtain as in the proof of the previous lemma an anisotropic t.i. form φ_{1} such that

dim φ_{1} ≥ (r − 1)2^{m} + k + 1 + 4 ,
0 ≤ sgn_{P} φ_{1} ≤ max{(r − 1)2^{m} + k − 1, 2^{m}} ,

and reiterating this construction r − 1 times, we get an anisotropic t.i. form φ_{r−1} such that

dim φ_{r−1} ≥ 2^{m} + k + 1 + 4(r − 1) ,
0 ≤ sgn_{P} φ_{r−1} ≤ max{2^{m} + k − 1, 2^{m}} for all orderings P.

This yields dim φ_{r−1} −sgn_{P} φ_{r−1} ≥ 4r−2 for all orderings P, and thus, by ED, the existence of an anisotropic torsion subform φ_{t} of φ_{r−1} with dim φ_{t} ≥ 4r−2. In particular, u + 6 ≥ 4(r + 1). On the other hand, ũ ≤ 2^{m}(r + 1) and thus ũ ≤ 2^{m−2}(u + 6). Now if m = 2, we have dim φ_{r−1} ≥ 4r + k + 1 = dim φ_{0} and 0 ≤ sgn_{P} φ_{r−1} ≤ max{4 + k − 1, 4}. In particular, since all the forms φ_{i} are anisotropic and t.i., it follows readily from the construction and the fact that ũ = 4r + k + 1 that dim φ_{0} = dim φ_{1} = … = φ_{r−1} = ũ. Note also that 0 ≤ k ≤ 3, so that by repeating our construction one more time, we obtain an anisotropic t.i. form φ_{r} such that dim φ_{r} = ũ and sgn_{P} φ_{r} ≤ 4 for all orderings P. Thus, φ_{r} contains a torsion subform of dimension ≥ ũ − 4 and therefore ũ ≤ u + 4. □

Proposition 3.6. Suppose that I_{t}^{3}F = 0, and that u(F) < ∞ and F has ED (or, equivalently, that ũ(F) < ∞). If there exists a t.p.d. sgn-universal binary form ρ over F, then u(F) = ũ(F).

Proof. By [ELP, Th. H], I_{t}^{3}F = 0 implies that ũ = ũ(F) is even. By Proposition 3.3, ũ ≤ u + 2. So let us assume that ũ ≠ u, i.e. ũ = u + 2. The proof of Proposition 3.3 then shows that there exists an anisotropic t.i. form φ (which is nothing but the form φ_{r−1} in the proof) with dim φ = ũ and which contains a torsion subform φ_{t}, dim φ_{t} = dim φ − 2 = u. After scaling, we may assume that φ_{t} ⊥ ⟨1⟩ ⊂ φ. Let d = d_{±}φ_{t}. Then φ_{t} ⊥ ⟨1, −d⟩ ∈ I^{2}F, and since sgn_{P} φ_{t} = 0 and sgn_{P} φ_{t} ⊥ ⟨1, −d⟩ ∈ 4ℤ, it follows that φ_{t} ⊥ ⟨1, −d⟩ ∈ I_{t}^{2}F. As dim φ_{t} ⊥ ⟨1, −d⟩ = u + 2, this form must be isotropic. Thus, φ_{t} ⊥ ⟨1⟩ ≅ ψ ⊥ ⟨d⟩. Comparing discriminants and signatures, it follows that ψ ∈ I_{t}^{2}F. So ⟨1, −x⟩ ⊗ ψ ∈ I_{t}^{3}F = 0 for all x ∈ F^{∗}, thus ψ ≅ xψ which implies that ψ is universal, hence the subform ψ ⊥ ⟨d⟩ of φ is isotropic, a contradiction. □

The following is an immediate consequence.

Corollary 3.7. Suppose that p(F) = 2 and ũ(F) < ∞. If I_{t}^{3}F = 0 then u(F) = ũ(F). In particular, if u(F) ≤ 6 or ũ(F) ≤ 8, then ũ(F) = u(F).

Remark 3.8. Let F be a real field with ũ(F) < ∞. Suppose that d is an integer with 2^{d} + 1 ≤ p ≤ 2^{d+1} − 1. The Pfister form ⟨⟨−1, …, −1⟩⟩ ∈ P_{d+1}F is t.p.d. and its pure part is sgn-universal, so we can use Proposition 3.5 for m = d + 1. For p = 2^{d} + 1, d ≥ 1, we get 2^{d−1}(u + 6) − \frac{p}{2}(u + 2) = 2^{d+1} − \frac{1}{2}u − 1. In this case, Proposition 3.3 gives a better bound when u ≤ 2^{d+2} −4 (note that we will have u ≥ 2^{d+1}), the bounds are the same for u = 2^{d+2} − 2, and for u ≥ 2^{d+2} Proposition 3.5 gives a sharper bound. Summarizing our best bounds in the various cases, we obtain

(i) p(F) = 1 if and only if ũ(F) = u(F) = 0.

(ii) If p(F) = 2 then ũ(F) ≤ u(F) + 2. If in addition I_{t}^{3}F = 0 then

ũ(F) = u(F) = 2n for some integer n ≥ 1. (iii) If p(F) = 3 then ũ(F) ≤ u(F) + 4. (iv) If p(F) = 2^{m} then ũ(F) ≤ 2^{m−1}(u(F) + 2).

(v) If p(F) = 2^{m} +1 then ũ(F) ≤ (2^{m−1} + \frac{1}{2})(u(F)+2) if u(F) ≤ 2^{m+2} −2, and ũ(F) ≤ 2^{m−1}(u(F) + 6) if u(F) ≥ 2^{m+2} − 2.

(vi) If 2^{m} + 2 ≤ p(F) ≤ 2^{m+1} − 1, then ũ(F) ≤ 2^{m−1}(u(F) + 6).

Remark 3.9. It is difficult to say at this point how good our bounds really are. In fact, we know extremely little about fields with u(F) < ũ(F) < ∞. The only values which could be realized so far are fields where u(F) = 2n and ũ(F) = 2n + 2 for any n ≥ 2 (see [L2], [Hor2], [H3]), and fields with u(F) = 8 and ũ(F) = 12, see [H2, Cor. 6.4].

For the balance of this section, we finish with stating results about all possible pairs of values for (p(F), u(F)) for real fields, in particular real fields satisfying SAP but not S_{1} or vice versa (such fields will always have ũ = ∞). The construction of such fields with prescribed values (p, u) uses Merkurjev’s method of iterated function fields and is rather technical. We omit the proof and refer the interested reader to [H4].

Theorem 3.10. Let N′ be the set of pairs of integers (p, u) such that either p = 1 and u = 0 or u = 2n ≥ 2^{m} ≥ p ≥ 2 for some integers m and n. Let N = N′ ∪ {(p, ∞); p ≥ 2 or p = ∞}.

(i) If F is a real field, then (p(F), u(F)) ∈ N.

(ii) Let E be a real field and let (p, u) ∈ N. Then there exists a real field extension F/E such that F is non-SAP, F has property S_{1} and (p(F), u(F)) = (p, u). In particular, ũ(F) = ∞.

(iii) If F is a real SAP field with ũ(F) = ∞, then u(F) ≥ 4 and (p(F), u(F)) ∈ N.

(iv) Let E be a real field and let (p, u) ∈ N with u ≥ 4. Then there exists a real field extension F/E such that F is SAP, F does not have property S_{1} and (p(F), u(F)) = (p, u). In particular, ũ(F) = ∞.

4. Linkage of fields and the Pfister neighbor property

The purpose of this section is to derive a criterion for the finiteness of the Hasse number. Real fields with finite Hasse number are relatively scarce but interesting nonetheless. But our results are just as valid for nonreal fields, we thus get also a criterion for the finiteness of u for nonreal fields. Recall that the field F is said to have the Pfister neighbor propery P N(n), n ≥ 0, if every form of dimension 2^{n} + 1 over F is a Pfister neighbor. This property is a somewhat stronger version of the notion of n-linkage whose definition we now recall:

Definition 4.1. Let n ≥ 1 be an integer. A field F is called n-linked if to any n-fold Pfister forms π_{1} and π_{2} over F there exist a_{1}, a_{2} ∈ F^{∗} and an (n − 1)- fold Pfister form σ such that π_{i} ≅ ⟨⟨a_{i}⟩⟩ ⊗ σ, i = 1, 2. F is called linked if F is 2-linked.

Remark 4.2. (i) Trivially, every field is 1-linked and satisfies P N(0) and P N(1). (ii) Let n ≥ 2. Every isotropic form of dimension 2^{n}+1 is a Pfister neighbor. In fact, if dim φ = 2^{n} + 1 and φ is isotropic, then φ ≅ ℍ ⊥ ψ with dim ψ = 2^{n} − 1. Then φ ⊥ −ψ ≅ π ∈ P_{n+1}F, where π denotes the hyperbolic (n+1)-fold Pfister form. So in particular, if F is nonreal and u(F) ≤ 2^{n}, then F has property P N(n)

Lemma 4.3. Let n ≥ 2.

(i) If F is n-linked then F is m-linked for all m ≥ n and I_{t}^{n+2}F = 0. (ii) F is n-linked iff to each form φ ∈ I^{n}F there exists a form π ∈ P_{n}F such that φ ≡ π mod I^{n+1}F iff to each anisotropic φ ∈ I^{n}F there exist τ ∈ P_{n−1}F and an even-dimensional form σ such that φ ≅ τ ⊗ σ.

(iii) F has property P N(n) if and only if there exists to every form φ over F a form ψ such that dim ψ ≤ 2^{n} if dim φ even (resp. dim ψ ≤ 2^{n} − 1 if dim φ odd) such that φ ≡ ψ mod I^{n+1}F.

(iv) If F has property P N(n) then F is n-linked. In particular, I_{t}^{n+2}F = 0.

Furthermore, F is ED.

(v) F has property P N(2) iff F is linked.

Proof. (i) and (ii) are well known, see [EL2, § 2], [H1]. (iii) ‘only if’: If dim φ ≤ 2^{n}, then put ψ ≅ φ. So suppose dim φ ≥ 2^{n} + 1. Write φ ≅ ψ ⊥ τ with dim ψ = 2^{n} + 1. By P N(n), ψ is a Pfister neighbor and there exists ψ′, dim ψ′ = 2^{n} − 1 such that ψ ⊥ −ψ′ ≅ π ∈ GP_{n+1}F. Then, in W F, we have

φ ≡ φ − π ≡ ψ′ ⊥ τ mod I^{n+1}F .

Now dim ψ′ ⊥ τ = dim φ − 2 and the result follows by an easy induction on the dimension.

‘if’: Let dim φ = 2^{n} + 1. By assumption, there exists a form ψ, dim ψ = 2^{n} − 1 (possibly after adding hyperbolic planes) such that φ ⊥ −ψ ∈ I^{n+1}F. Then dim(φ ⊥ −ψ) = 2^{n+1} and thus φ ⊥ −ψ ∈ GP_{n+1}F by APH, which implies that φ is a Pfister neighbor. (iv) To show that F is n-linked, let φ ∈ I^{n}F. By (iii), there exists ψ such that dim ψ = 2^{n} (possibly after adding hyperbolic planes) and φ ≡ ψ mod I^{n+1}F. But clearly ψ ∈ I^{n}F, and thus ψ ∈ GP_{n}F by APH. Let x ∈ F^{∗} be such that xψ ∈ P_{n}F. We then have ψ ≡ xψ mod I^{n+1}F, and n-linkage together with I_{t}^{n+2}F = 0 follows from (i) and (ii). Now n-linked fields, n ≥ 2, are easily seen to be SAP. So to establish ED, it suffices to establish property S_{1} by Theorem 2.1. Let ⟨a, b⟩ be any torsion form. Let γ ≅ ⟨\underbrace{1, …, 1}_{2^{n}−1}⟩. Then by P N(n), the form γ ⊥ ⟨−a, −b⟩ is a t.i. Pfister

neighbor of a Pfister form π ∈ P_{n+1}F. Since π contains γ which is a Pfister neighbor (and in fact subform) of σ_{n} ≅ ⟨1, 1⟩^{⊗n}, one necessarily has that σ_{n} divides π, so there exists c ∈ F^{∗} such that π ≅ σ_{n} ⊗ ⟨1, c⟩. Now π contains a t.i. Pfister neighbor and is therefore also t.i. and hence torsion. But then ρ ≅ ⟨1, 1⟩ ⊗ σ_{n} ⊗ ⟨1, c⟩ ∈ P_{n+2}F is torsion as well and therefore hyperbolic by (i). Now σ_{n} ⊥ γ ⊥ ⟨−a, −b⟩ is a Pfister neighbor of ρ. Since ρ is hyperbolic, its neighbor σ_{n} ⊥ γ ⊥ ⟨−a, −b⟩ is isotropic. Hence there exists x ∈ D_{F} (⟨a, b⟩) ∩ D_{F} (σ_{n} ⊥ γ). But clearly, D_{F} (σ_{n} ⊥ γ) ⊂ D_{F} (∞) which shows that the binary torsion form ⟨a, b⟩ represents the totally positive element x.

(v) This follows immediately from the fact that a field is linked iff the classes of quaternion algebras form a subgroup in Br(F) together with the characterization of 5-dimensional Pfister neighbors by their Clifford invariant (see [Kn, p. 10]). □

The following observation is essentially due to Fitzgerald [F, Lemma 4.5(ii)].

Lemma 4.4. Suppose that ũ(F) ≤ 2^{n}. Let φ be a form over F of dimension 2^{n} + 1. Then φ is a Pfister neighbor. In particular, F has P N(n).

Proof. By Remark 4.2(ii) the result is clear if φ is isotropic. Thus, we may assume φ anisotropic, so necessarily F must be real. Since ũ(F) < ∞ implies that F is SAP, we may assume that after scaling, sgn_{P} (φ) ≥ 0 for all P ∈ X_{F}, and that there exists c ∈ F^{∗} such that H(c) = {P ∈ X_{F} | sgn_{P} (φ) = dim φ}. In particular, the Pfister form ⟨⟨\underbrace{−1, …, −1}_{n}, −c⟩⟩ ∈ P_{n+1}F is positive definite

at all those P ∈ X_{F} at which φ is positive definite, and it has signature zero at all those P ∈ X_{F} at which φ is indefinite. Let ψ ≅ (π ⊥ −φ)_{an}. It follows that | sgn_{P} (ψ)| ≤ 2^{n} − 1 for all P ∈ X_{F}. But since ũ(F) ≤ 2^{n}, the anisotropic form ψ must therefore have dim ψ ≤ 2^{n}, so in particular,

i_{W} (π ⊥ −φ) = \frac{1}{2}(dim(π ⊥ −φ) − dim ψ) ≥ \frac{1}{2}(2^{n+1} + 1) ,

and therefore i_{W} (π ⊥ −φ) ≥ 2^{n} + 1 = dim φ, which implies that φ ⊂ π. In particular, φ is a Pfister neighbor of π. □

Theorem 4.5. If a field F has property P N(n), n ≥ 2, then either u(F) ≤ ũ(F) ≤ 2^{n}, or 2^{n+1} ≤ u(F) ≤ ũ(F) ≤ 2^{n+1} + 2^{n} − 2.

Proof. Let F be a field with property P N(n) for some n ≥ 2. Suppose that ũ(F) > 2^{n}, i.e. there exists an anisotropic t.i. φ with dim φ = m > 2^{n}. By Lemma 4.3(iv), F has ED and so φ can be diagonalized as φ ≅ ⟨a_{1}, …, a_{m}⟩ with −a_{1}, a_{m} ∈ D_{F} (∞). By removing some of the a_{i}, 2 ≤ i ≤ m−1 if necessary, we will retain a t.i. form, so we may assume that φ is t.i. and dim φ = 2^{n} + 1. But then, by P N(n), φ is a Pfister neighbor of some π ∈ P_{n+1}F which in turn is torsion and anisotropic as its Pfister neighbor φ is t.i. and anisotropic. This shows that 2^{n+1} ≤ u(F) ≤ ũ(F). Now suppose that ũ(F) > 2^{n+1} + 2^{n} − 2. By a similar argument as above, we conclude that there exists an anisotropic t.i. form φ with dim φ = 2^{n+1}+2^{n}−1. By Lemma 4.3(iii), there exists an anisotropic form ψ of dimension ≤ 2^{n} − 1 such that φ ≡ ψ mod I^{n+1}F. Let π ≅ (φ ⊥ −ψ)_{an} ∈ I^{n+1}F. Then by dimension count and since φ is anisotropic, we have 2^{n+1} ≤ dim π ≤ 2^{n+2} − 2. Since F is (n + 1)-linked, Lemma 4.3(ii) implies dim π = 2^{n+1}, and thus, by APH, π ∈ GP_{n+1}F. Also, by dimension count, we have φ ≅ π ⊥ ψ. After scaling, we may assume that π ∈ P_{n+1}F, so that sgn_{P} (π) ∈ {0, 2^{n+1}}. Now φ is t.i., and since F has ED by Lemma 4.3(iv), we can write ψ ≅ ⟨a, …⟩ with a <_{P} 0 whenever sgn_{P} (π) = 2^{n+1}. But then π ⊥ ⟨a⟩ is a t.i. subform of φ. On the other hand, π ⊥ ⟨a⟩ is also a Pfister neighbor of π ⊗ ⟨1, a⟩ ∈ P_{n+2}F. Since π ⊥ ⟨a⟩ is t.i., this implies that π ⊗ ⟨1, a⟩ is torsion and therefore hyperbolic since I_{t}^{n+2}F = 0 by Lemma 4.3(ii). But then the Pfister neighbor π ⊥ ⟨a⟩ is isotropic and therefore also φ, a contradiction. □

Remark 4.6. (i) The above proof also shows that if F has P N(n), n ≥ 2, then the case ũ(F) ≤ 2^{n} occurs iff there are no anisotropic torsion (n+1)-fold Pfister forms iff I_{t}^{n+1}F = 0. (ii) If we were only considering nonreal fields then the proofs could be shortened by essentially deleting arguments referring to or making use of ED, signatures, etc..

Corollary 4.7. ũ(F) < ∞ if and only if F has P N(n) for some n ≥ 2. In particular, if F is nonreal then u(F) < ∞ if and only if F has P N(n) for some n ≥ 2

Proof. The ‘if’-part in the first statement follows from Theorem 4.5, the converse from Lemma 4.4. The statement for nonreal fields is then clear because in that case u = ũ. □

Remark 4.8. If F is real, then we still get a sufficient criterion for the finiteness of u(F) even if ũ(F) = ∞. Indeed, for real F, one has that if u(F (\sqrt{−1})) is finite then u(F) is finite, more precisely, one has u(F) < 4u(F (\sqrt{−1})) (see [EKM, Th. 37.4]). Thus, we get the following: If F (\sqrt{−1}) has property P N(n) for some n ≥ 2, then u(F) < 2^{n+3} + 2^{n+2} − 8.

Conjecture 4.9. If a field F has property P N(n), n ≥ 2, then u(F) ≤ ũ(F) ≤ 2^{n}, or u(F) = ũ(F) = 2^{n+1}.

Corollary 4.10. For n ≥ 2, P N(n) implies P N(m) for all m ≥ n + 2. Furthermore, the following are equivalent:

(i) Conjecture 4.9 holds.

(ii) For n ≥ 2, P N(n) implies P N(n + 1).

Proof. If n ≥ 2, then P N(n) implies that ũ(F) ≤ 2^{n+2}, and P N(m) for m ≥ n + 2 follows from Lemma 4.4. Now suppose that F has P N(n) and that Conjecture 4.9 holds. Then P N(n+1) follows from Lemma 4.4. Conversely, suppose that n ≥ 2 and that P N(n) implies P N(n + 1). Then we have u(F) ≤ ũ(F) ≤ 2^{n} or 2^{n+1} ≤ u(F) ≤ ũ(F) ≤ 2^{n+1} + 2^{n} − 2 because of P N(n), and also u(F) ≤ ũ(F) ≤ 2^{n+1} or 2^{n+2} ≤ u(F) ≤ ũ(F) ≤ 2^{n+2} + 2^{n+1} − 2 because of P N(n + 1). Putting the two together, we obtain u(F) ≤ ũ(F) ≤ 2^{n} or u(F) = ũ(F) = 2^{n+1}. □

The only evidence we have as to the veracity of Conjecture 4.9 is the following.

Lemma 4.11. P N(2) implies P N(3). In particular, if F has P N(2), then u(F) ≤ ũ(F) ≤ 4 or u(F) = ũ(F) = 8.

Proof. Suppose F has P N(2) and let φ be any 9-dimensional form over F. Write φ ≅ α ⊥ β with dim α = 5. Since α is a Pfister neighbor, there exists π ∈ GP_{2}F such that π ⊂ α ⊂ φ (see, e.g., [L3, Ch. X, Prop. 4.19]). Write φ ≅ π ⊥ γ. Then dim γ = 5 and γ is also a Pfister neighbor, so there exists ρ ∈ GP_{2}F such that ρ ⊂ γ. Hence, there exist a, b, c, d, e, f, g ∈ F^{∗} such that φ ≅ a⟨⟨b, c⟩⟩ ⊥ d⟨⟨e, f⟩⟩ ⊥ ⟨g⟩. Since P N(2) implies that F is linked by Lemma 4.3(v), we may assume that b = e, and after scaling (which doesn’t change the property of being a Pfister neighbor), we may also assume a = 1, so

φ ≅ ⟨⟨b, c⟩⟩ ⊥ d⟨⟨b, f⟩⟩ ⊥ ⟨g⟩ ⊂ ⟨⟨b⟩⟩ ⊗ (⟨⟨c⟩⟩ ⊥ d⟨⟨f⟩⟩ ⊥ ⟨g⟩) .

Now δ ≅ ⟨⟨c⟩⟩ ⊥ d⟨⟨f⟩⟩ ⊥ ⟨g⟩ has dimension 5 and is therefore again a Pfister neighbor, so as above there exist h, k, l, m ∈ F^{∗} such that δ ≅ h⟨⟨k, l⟩⟩ ⊥ ⟨m⟩. We thus get that

φ ⊂ ⟨⟨b⟩⟩ ⊗ δ ≅ h⟨⟨b, k, l⟩⟩ ⊥ m⟨⟨b⟩⟩ ⊂ h⟨⟨b, k, l, −hm⟩⟩ ∈ GP_{4}F ,

which shows that φ is a Pfister neighbor. The remaining statement now follows from Corollary 4.10. □

Since linked fields are exactly the fields that have P N(2), one readily recovers the following result due to Elman and Lam [EL2] and Elman [E, Th. 4.7]. We leave it as an exercise to the reader to fill in the details.

Corollary 4.12. Let F be a linked field. Then u(F) = ũ(F) ∈ {0, 1, 2, 4, 8}. In particular, I_{t}^{4}F = 0. Furthermore, let n ∈ {0, 1, 2}. Then ũ(F) ≤ 2^{n} iff I_{t}^{n+1}F = 0.

Note that u(F) = ũ(F) = 0 can only occur when F is real, whereas u(F) = ũ(F) = 1 implies that F is nonreal.

Remark 4.13. It is not difficult to see that the iterated power series field F = ℂ((X_{1}))((X_{2})) … ((X_{n})) is a (nonreal) field with property P N(n) and u(F) = 2^{n+1}. Using Merkurjev’s method of iterated function fields, it is also possible to construct to any n ≥ 2 a real field F with property P N(n) and ũ(F) = 2^{n+1}. For details, see [H4].

Remark 4.14. Merkurjev [M] constructed to each positive integer n a field F with I^{3}F = 0 and u(F) = 2n (resp. a field F with I^{3}F = 0 and u(F) = ∞). Trivially, such a (nonreal) field is 3-linked. So the n-linkage property, n ≥ 3, does not give any indication on how large u might be, whereas the stronger property P N(n) does.

We finish this paper with some remarks on a possible geometric interpretation of the property P N(n) which can be formulated in the language of Chow groups. We refer to [Kar], [EKM, §80]. Let φ be a (nondegenerate) quadratic form of dimension n + 2 ≥ 3, and let X = X_{φ} be the smooth projective n-dimensional quadric {φ = 0} over F. We call X (an)isotropic if φ is (an)isotropic. Let \overline{F} denote the algebraic closure of F and let \overline{X} = X\overline{F}. Let l_{0} be the class of a rational point in CH^{n}(\overline{X}), the Chow group of 0-dimensional cycles, and let 1 ∈ CH^{0}(X) be the class of X. A Rost correspondence on X is an element ρ ∈ CH^{n}(X × X) which, over \overline{F}, is equal to l_{0} × 1 + 1 × l_{0} ∈ CH^{n}(\overline{X} × \overline{X}). A Rost projector is a Rost correspondence that is also an idempotent in the ring of correspondences on X. It is known that if a quadric has a Rost correspondence, then it has in fact also a Rost projector (see [Kar, Rem. 1.4]). The study of Rost correspondences/projectors has proven to be crucial in the motivic theory of quadrics. It is known that if X is isotropic, then l_{0} × 1 + 1 × l_{0} is actually the unique Rost projector on X (see [Kar, Lem. 5.1]). For anisotropic forms, the situation is much more complicated. The following is known:

Theorem 4.15. Let φ be an anisotropic form over F of dimension ≥ 3.

(i) If X_{φ} possesses a Rost projector, then dim φ = 2^{n} + 1 for some n ≥ 1 (see Karpenko [Kar, Prop. 6.2, 6.4]).

(ii) If φ is a Pfister neighbor of dimension 2^{n} + 1 then X_{φ} has a unique Rost projector (considered as element in CH^{r}(X_{φ} × X_{φ}), r = 2^{n} − 1) (see Izhboldin-Vishik [IV, Th. 1.12] for char(F) = 0, Elman-Karpenko-Merkurjev [EKM, Cor. 80.11] in the general case).

In view of part (i), it is natural to ask whether or not the converse of part (ii) also holds. This is still an open problem (see also [Kar, Conj. 1.6]):

Conjecture 4.16. If an anisotropic quadric X_{φ} possesses a Rost correspondence, then φ is a Pfister neighbor of dimension 2^{n} + 1 for some n ≥ 1.

Of course, by Theorem 4.15(ii), to prove the conjecture, one may assume that dim φ = 2^{n} + 1 for some n ≥ 1. Since 3-dimensional forms are always Pfister neighbors, trivially the conjecture holds in that case. The conjecture is also true in the cases n = 2, 3 as shown by Karpenko (see [Kar, Prop. 10.8, Th. 1.7]):

Theorem 4.17. Let φ be an anisotropic form over F of dimension 2^{n} + 1, n = 2, 3. If X_{φ} possesses a Rost correspondence, then φ is a Pfister neighbor.

It is now natural to introduce the property RP (n) for n ≥ 1: RP (n): F has the property RP (n) for n ≥ 1 if every form φ over F of dimen-

sion 2^{n} + 1 has a Rost projector. In view of the above, we immediately get

Proposition 4.18. Let n ≥ 1.

(i) P N(n) implies RP (n).

(ii) If n ≤ 3, then RP (n) implies P N(n).

(iii) If Conjecture 4.16 holds, then RP (n) implies P N(n) for all n ∈ ℕ.

Conjecturally and in view of Theorem 4.5, we therefore get an ‘algebrogeometric’ criterion for the finiteness of the Hasse number:

Corollary 4.19. If Conjecture 4.16 holds, then ũ(F) < ∞ (resp. u(F) < ∞ for nonreal F) if and only if F has property RP (n) for some n ≥ 2.

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Detlev Hoffmann

School of Mathematical Sciences University of Nottingham University Park Nottingham NG7 2RD UK detlev.hoffmann@nottingham.ac.uk 266

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