ESSENTIAL DIMENSION, SPINOR GROUPS, AND QUADRATIC FORMS

PATRICK BROSNAN^{†}, ZINOVY REICHSTEIN^{†}, AND ANGELO VISTOLI^{‡}

Abstract. We prove that the essential dimension of the spinor group Spin_{n} grows exponentially with n and use this result to show that quadratic forms with trivial discriminant and Hasse-Witt invariant are more complex, in high dimensions, than previously expected.

1. Introduction

Let K be a field of characteristic different from 2 containing a square root of −1, W(K) be the Witt ring of K and I(K) be the ideal of classes of even-dimensional forms in W(K); cf. [Lam73]. By abuse of notation, we will write q ∈ I^{a}(K) if the Witt class on the non-degenerate quadratic form q defined over K lies in I^{a}(K). It is well known that every q ∈ I^{a}(K) can be expressed as a sum of the Witt classes of a-fold Pfister forms defined over K; see, e.g., [Lam73, Proposition II.1.2]. If dim(q) = n, it is natural to ask how many Pfister forms are needed. When a = 1 or 2, it is easy to see that n Pfister forms always suffice; see Proposition 4.1. In this paper we will prove the following result, which shows that the situation is quite different when a = 3.

Theorem 1.1. Let k be a field of characteristic different from 2 and n ≥ 2 be an even integer. Then there is a field extension K/k and an n-dimensional quadratic form q ∈ I^{3}(K) with the following property: for any finite field extension L/K of odd degree q_{L} is not Witt equivalent to the sum of fewer than

\frac{2^{(n+4)/4} − n − 2}{7}

3-fold Pfister forms over L.

Our proof of Theorem 1.1 is based on new results on the essential dimension of the spinor groups Spin_{n} proven in §3 which are of independent interest. In particular, Theorem 3.3 gives new lower bounds on the essential dimension of Spin_{n} and, in many cases, computes the exact value.

2000 Mathematics Subject Classification. Primary 11E04, 11E72, 15A66.

Key words and phrases. Essential dimension, spinor group, quadratic form, Witt group, Pfister form.

†Supported in part by an NSERC discovery grants and by PIMS Collaborative Research Group in Algebraic geometry, cohomology and representation theory.

‡Supported in part by the PRIN Project “Geometria sulle varietà algebriche”, financed by MIUR.

Acknowledgements. We would like to thank the Banff International Research Station in Banff, Alberta (BIRS) for providing the inspiring meeting place where this work was started. We are grateful to A. Merkurjev and B. Totaro for bringing the problem of computing the Pfister numbers Pf_{k}(a, n) to our attention and for contributing Proposition 4.2. We also thank N. Fakhruddin for helpful correspondence.

2. Essential dimension

Let k be a field. We will write Fields_{k} for the category of field extensions K/k. Let F: Fields_{k} → Sets be a covariant functor.

Let L/k be a field extension. We will say that a ∈ F (L) descends to an intermediate field k ⊆ K ⊆ L if a is in the image of the induced map F (K) → F (L).

The essential dimension ed(a) of a ∈ F (L) is the minimum of the transcendence degrees tr deg_{k} K taken over all fields k ⊆ K ⊆ L such that a descends to K.

The essential dimension ed(a; p) of a at a prime integer p is the minimum of ed(a_{L′}) taken over all finite field extensions L′/L such that the degree [L′ : L] is prime to p.

The essential dimension ed F of the functor F (respectively, the essential dimension ed(F; p) of F at a prime p) is the supremum of ed(a) (respectively, of ed(a; p)) taken over all a ∈ F (L) with L in Fields_{k}.

Of particular interest to us will be the Galois cohomology functors, F_{G} given by K ⇝ H^{1}(K, G), where G is an algebraic group over k. Here, as usual, H^{1}(K, G) denotes the set of isomorphism classes of G-torsors over Spec(K), in the fppf topology. The essential dimension of this functor is a numerical invariant of G, which, roughly speaking, measures the complexity of G-torsors over fields. We write ed G for ed F_{G} and ed(G; p) for ed(F_{G}; p). Essential dimension was originally introduced in this context; see [BR97, Rei00, RY00]. The above definition of essential dimension for a general functor F is due to A. Merkurjev; see [BF03].

Recall that an action of an algebraic group G on an algebraic variety kvariety X is called “generically free” if X has a dense open subset U such that Stab_{G}(x) = \{1\} for every x ∈ U(\overline{k}).

Lemma 2.1. If an algebraic group G defined over k has a generically free linear k-representation V then ed(G) ≤ dim(V) − dim(G).

Proof. See [Rei00, Theorem 3.4] or [BF03, Lemma 4.11]. ♠

Lemma 2.2. If G is an algebraic group and H is a closed subgroup of codimension e then

(a) ed(G) ≥ ed(H) − e, and

(b) ed(G; p) ≥ ed(H; p) − e for any prime integer p.

Proof. Part (a) is [BF03, Theorem 6.19]. Both (a) and (b) follow directly from [Bro07, Principle 2.10]. ♠

If G is a finite abstract group, we will write ed_{k} G (respectively, ed_{k}(G; p)) for the essential dimension (respectively, for the essential dimension at p) of the constant group scheme G_{k} over the field k. Let C(G) denote the center of G.

Theorem 2.3. Let G be a finite p-group whose commutator [G, G] is central and cyclic. Then ed_{k}(G; p) = ed_{k} G = \sqrt{|G/C(G)|} + rank C(G) − 1 for any base field k of characteristic ≠ p containing a primitive root of unity of degree equal to the exponent of G.

Note that with the above hypotheses, |G/C(G)| is a complete square. Theorem 2.3 was originally proved in [BRV07] as a consequence of our study of essential dimension of gerbes banded by μ_{p^{n}}. Karpenko and Merkurjev [KM07] have subsequently refined our arguments to show that the essential dimension of any finite p-group over any field k containing a primitive p^{th} root of unity is the minimal dimension of a faithful linear k-representation of G. Using [KM07, Remark 4.7] Theorem 2.3 is easily seen to be a special case of their formula. For this reason we omit the proof here.

3. Essential dimension of Spin groups

As usual, we will write ⟨a_{1}, …, a_{n}⟩ for the quadratic form q of rank n given by q(x_{1}, …, x_{n}) = ∑_{i=1}^{n} a_{i}x_{i}^{2}. Let

(3.1)  h = ⟨1, −1⟩

denote the 2-dimensional hyperbolic quadratic form over k. For each n ≥ 0 we define the n-dimensional split form q_{n}^{split} defined over k as follows:

q_{n}^{split} = \begin{cases}h^{⊕n/2}, & if n is even, \\ h^{⊕(n−1/2)} ⊕ ⟨1⟩, & if n is odd. \end{cases}

Let Spin_{n} =^{def} Spin(q_{n}^{split}) be the split form of the spin group. We will also denote the split forms of the orthogonal and special orthogonal groups by O_{n} =^{def} O(q_{n}^{split}) and SO_{n} =^{def} SO(q_{n}^{split}) respectively.

M. Rost [Ros99] computed the following values of ed(Spin_{n}) for n ≤ 14:

ed Spin_{3} = 0 ed Spin_{4} = 0 ed Spin_{5} = 0 ed Spin_{6} = 0
ed Spin_{7} = 4 ed Spin_{8} = 5 ed Spin_{9} = 5 ed Spin_{10} = 4
ed Spin_{11} = 5 ed Spin_{12} = 6 ed Spin_{13} = 6 ed Spin_{14} = 7,

for a detailed exposition of these results; see [Gar08]. V. Chernousov and J.–P. Serre [CS06] recently proved the following lower bounds:

(3.2) ed(Spin_{n}; 2) ≥ \begin{cases}⌊n/2⌋ + 1 & if n ≥ 7 and n ≡ 1, 0 or −1 (mod 8) \\ ⌊n/2⌋ & for all other n ≥ 11. \end{cases}

(The first line is due to B. Youssin and the second author in the case that char k = 0 [RY00].)

The main result of this section, Theorem 3.3 below, shows, in particular, that ed(Spin_{n}) and ed(Spin_{n}; 2) grow exponentially with n.

Theorem 3.3. (a) Let k be a field of characteristic ≠ 2 and n ≥ 15 be an integer.

ed(Spin_{n}; 2) ≥ \begin{cases}2^{(n−1)/2} − \frac{n(n−1)}{2}, if n is odd, \\ 2^{(n−2)/2} − \frac{n(n−1)}{2}, if n ≡ 2 (mod 4), \\ 2^{(n−2)/2} − \frac{n(n−1)}{2} + 1, if n ≡ 0 (mod 4). \end{cases}

(b) Moreover, if char(k) = 0 then

ed(Spin_{n}) = ed(Spin_{n}; 2) = 2^{(n−1)/2} − \frac{n(n−1)}{2}, if n is odd,

ed(Spin_{n}) = ed(Spin_{n}; 2) = 2^{(n−2)/2} − \frac{n(n−1)}{2}, if n ≡ 2 (mod 4), and

ed(Spin_{n}; 2) ≤ ed(Spin_{n}) ≤ 2^{(n−2)/2} − \frac{n(n−1)}{2} + n, if n ≡ 0 (mod 4).

Note that while the proof of part (a) below goes through for any n ≥ 3, our lower bounds become negative (and thus vacuous) for n ≤ 14.

Proof. (a) Since replacing k by a larger field k′ can only decrease the value of ed(Spin_{n}; 2), we may assume without loss of generality that \sqrt{−1} ∈ k. The n-dimensional split quadratic form q_{n}^{split} is then k-isomorphic to

(3.4)  q(x_{1}, …, x_{n}) = −(x_{1}^{2} + ⋯ + x_{n}^{2}).

over k and hence, we can write Spin_{n} as Spin(q), O_{n} as O_{n}(q) and SO_{n} as SO_{n}(q).

Let Γ_{n} ⊆ SO_{n} be the subgroup consisting of diagonal matrices. This subgroup is isomorphic to μ_{2}^{n−1}. Let G_{n} be the inverse image of Γ_{n} in Spin_{n}; this is a constant group scheme over k. By Lemma 2.2(b)

ed(Spin_{n}; 2) ≥ ed(G_{n}; 2) − \frac{n(n − 1)}{2} .

Thus in order to prove the lower bounds of part (a), it suffices to show that

(3.5)  ed(G_{n}; 2) = ed(G_{n}) = \begin{cases}2^{(n−1)/2}, if n is odd, \\ 2^{(n−2)/2}, if n ≡ 2 (mod 4), \\ 2^{(n−2)/2} + 1, if n is divisible by 4. \end{cases}

The structure of the finite 2-group G_{n} is well understood; see, e.g., [Woo89]. Recall that the Clifford algebra A_{n} of the quadratic form q, as in (3.4) is the algebra given by generators e_{1}, …, e_{n}, and relations e_{i}^{2} = −1, e_{i}e_{j} +e_{j}e_{i} = 0 for all i ≠ j. For any I = {i_{1}, …, i_{r}} ⊆ {1, …, n} with i_{1} < i_{2} < ⋯ < i_{r} set e_{I} =^{def} e_{i_{1}} … e_{i_{r}}. Here e_{∅} = 1. The group G_{n} consists of the elements of A_{n} of the form ±e_{I}, where the cardinality r = |I| of I is even. The element −1 is central, and the commutator [e_{I}, e_{J}] is given by [e_{I}, e_{J}] = (−1)^{|I∩J|} . It is clear from this description that G_{n} is a 2-group of order 2^{n}, the commutator subgroup [G_{n}, G_{n}] = {±1} is cyclic, and the center C(G) is as follows:

C(G_{n}) = \begin{cases}\{±1\} ≃ ℤ/2ℤ, if n is odd, \\ \{±1, ±e_{\{1,...,n\}}\} ≃ ℤ/4ℤ, if n ≡ 2 (mod 4), \\ \{±1, ±e_{\{1,...,n\}}\} ≃ ℤ/2ℤ × ℤ/2ℤ, if n is divisible by 4. \end{cases}

Formula (3.5) now follows from Theorem 2.3.

(b) Clearly ed(Spin_{n}; 2) ≤ ed(Spin_{n}). Hence, we only need to show that for n ≥ 15

(3.6)  ed(Spin_{n}) ≤ \begin{cases}2^{(n−1)/2} − \frac{n(n−1)}{2}, if n is odd, \\ 2^{(n−2)/2} − \frac{n(n−1)}{2}, if n ≡ 2 (mod 4), \\ 2^{(n−2)/2} − \frac{n(n−1)}{2} + n, if n ≡ 0 (mod 4). \end{cases}

In view of Lemma 2.1 it suffices to show that Spin_{n} has a generically free linear representation V of dimension

dim(V) = \begin{cases}2^{(n−1)/2}, if n is odd, \\ 2^{(n−2)/2}, if n ≡ 2 (mod 4), \\ 2^{(n−2)/2} + n if n ≡ 0 (mod 4). \end{cases}

In the case where n is not divisible by 4 such a representation is given by the following lemma.

Lemma 3.7. (cf. [PV94, Theorem 7.11]) If n ≥ 15 then, over a field of characteristic 0, the following representations of Spin_{n} of characteristic 0 are generically free:

(i) the spin representation, of dimension 2^{(n−1)/2}, if n is odd,

(ii) either of the two half-spin representation, of dimension 2^{(n−2)/2}, if n ≡ 2 (mod 4).

Proof. For n ≥ 27 this follows directly from [AP71, Theorem 1]. For n between 15 and 25 this is proved in [Po85]. ♠

In the case where n ≥ 16 is divisible by 4, we define V as the sum of the half-spin representation W of Spin_{n} and the natural representation k^{n} of SO_{n}, which we will view as a Spin_{n}-representation via the projection Spin_{n} → SO_{n}. It remains to check that V = W × k^{n} is a generically free representation of Spin_{n}. Indeed, for a ∈ k^{n} in general position, Stab(a) is conjugate to Spin_{n−1} (embedded in Spin_{n} in the standard way). Thus it suffices to show that the restriction of W to Spin_{n−1} is generically free. Since W restricted to Spin_{n−1} is the spin representation of Spin_{n−1} (see, e.g., [Ada96, Proposition 4.4]), and n ≥ 16, this follows from Lemma 3.7(i). This completes the proof of Theorem 3.3. ♠

Remark 3.8. The characteristic 0 assumption in part (b) is used only in the proof of Lemma 3.7. It seems likely that Lemma 3.7 (and thus Theorem 3.3(b)) remain true if char(k) = p > 2 but we have not checked this.

If char(k) ≠ 2 and \sqrt{−1} ∈ k, we have the weaker (but asymptotically equivalent) upper bound ed(Spin_{n}) ≤ ed(G_{n}), where ed(G_{n}) is given by (3.5). This is a consequence of the fact that every Spin_{n}-torsor admits reduction of structure to G_{n}, i.e., the natural map H^{1}(K, G_{n}) → H^{1}(K, Spin_{n}) is surjective for every field K/k; cf. [BF03, Lemma 1.9].

Remark 3.9. A. S. Merkurjev (unpublished) recently strengthened our lower bound on ed(Spin_{n}; 2), in the case where n ≡ 0 (mod 4) as follows:

ed(Spin_{n}; 2) ≥ 2^{(n−2)/2} − \frac{n(n − 1)}{2} + 2^{m} ,

where 2^{m} is the highest power of 2 dividing n. If n ≥ 16 is a power of 2 and char(k) = 0 this, in combination with the upper bound of Theorem 3.3(b), yields

ed(Spin_{n}; 2) = ed(Spin_{n}) = 2^{(n−2)/2} − \frac{n(n − 1)}{2} + n .

In particular, ed(Spin_{16}) = 24. The first value of n for which ed(Spin_{n}) is not known is n = 20, where 326 ≤ ed(Spin_{20}) ≤ 342.

Remark 3.10. The same argument can be applied to the half-spin groups yielding

ed(HSpin_{n}; 2) = ed(HSpin_{n}) = 2^{(n−2)/2} − \frac{n(n − 1)}{2}

for any integer n ≥ 20 divisible by 4 over any field of characteristic 0. Here, as in Theorem 3.3, the lower bound

ed(HSpin_{n}; 2) ≥ 2^{(n−2)/2} − \frac{n(n − 1)}{2}

is valid for over any base field k of characteristic ≠ 2. The assumptions that char(k) = 0 and n ≥ 20 ensure that the half-spin representation of HSpin_{n} is generically free; see [PV94, Theorem 7.11].

Remark 3.11. Theorem 3.3 implies that for large n, Spin_{n} is an example of a split, semisimple, connected linear algebraic group whose essential dimension exceeds its dimension. Previously no examples of this kind were known, even for k = ℂ.

Note that no complex connected semisimple adjoint group G can have this property. Indeed, let 𝔤 be the adjoint representation of G on its Lie algebra. If G is an adjoint group then V = 𝔤 × 𝔤 is generically free; see, e.g., [Rich88, Lemma 3.3(b)]. Thus ed G ≤ dim(G) by Lemma 2.1.

Remark 3.12. Since ed SO_{n} = n − 1 for every n ≥ 3 (cf. [Rei00, Theorem 10.4]), it follows that, for large n, Spin_{n} is also an example of a split, semisimple, connected linear algebraic group G with a central subgroup Z such that ed G > ed G/Z. To the best of our knowledge, this example is new as well.

4. Pfister numbers

Let K be a field of characteristic not equal to 2 and a ≥ 1 be an integer. We will continue to denote the Witt ring of K by W (K) and its fundamental ideal by I(K). If non-singular quadratic forms q and q′ over K are Witt equivalent, we will write q ∼ q′.

As we mentioned in the introduction, the a-fold Pfister forms generate I^{a}(K) as an abelian group. In other words, every q ∈ I^{a}(K) is Witt equivalent to ∑_{i=1}^{r} ±p_{i}, where each p_{i} is an a-fold Pfister form over K. We now define the a-Pfister number of q to be the smallest possible number r of Pfister forms appearing in any such sum. The (a, n)-Pfister number Pf_{k}(a, n) is the supremum of the a-Pfister number of q, taken over all field extensions K/k and all n-dimensional forms q ∈ I^{a}(K).

Proposition 4.1. Let k be a field of characteristic ≠ 2 and let n be a positive even integer. Then (a) Pf_{k}(1, n) ≤ n and (b) Pf_{k}(2, n) ≤ n − 2.

Proof. (a) Immediate from the identity

⟨a_{1}, a_{2}⟩ ∼ ⟨1, a_{1}⟩ − ⟨1, −a_{2}⟩ = ≪−a_{1} ≫ − ≪a_{2} ≫

in the Witt ring.

(b) Let q = ⟨a_{1}, …, a_{n}⟩ be an n-dimensional quadratic form over K. Recall that q ∈ I^{2}(K) iff n is even and d_{±}(q) = 1, modulo (K^{∗})^{2} [Lam73, Corollary II.2.2]. Here d_{±}(q) is the signed discriminant given by (−1)^{n(n−1)/2}d(q) where d(q) = ∏_{i=1}^{n} a_{n} is the discriminant of q; cf. [Lam73, p. 38].

To explain how to write q in terms of n − 2 Pfister forms, we will temporarily assume that \sqrt{−1} ∈ K. In this case, without loss of generality, a_{1} … a_{n} = 1. Since ⟨a, a⟩ is hyperbolic for every a ∈ K^{∗}, we see that q = ⟨a_{1}, …, a_{n}⟩ is Witt equivalent to

≪ a_{2}, a_{1} ≫ ⊕ ≪ a_{3}, a_{1}a_{2} ≫ ⊕ ⋯ ⊕ ≪ a_{n−1}, a_{1} … a_{n−2} ≫ .

By inserting appropriate powers of −1, we can modify this formula so that it remains valid even if we do not assume that \sqrt{−1} ∈ K, as follows:

q = ⟨a_{1}, …, a_{n}⟩ ∼ ∑_{i=2}^{n}(−1)^{i}≪(−1)^{i+1}a_{i}, (−1)^{i(i−1)/2+1}a_{1} … a_{i−1} ≫ ♠

We do not have an explicit upper bound on Pf_{k}(3, n); however, we do know that Pf_{k}(3, n) is finite for any k and any n. To explain this, let us recall that I^{3}(K) is the set of all classes q ∈ W(K) such that q has even dimension, trivial signed discriminant and trivial Hasse-Witt invariant [KMRT98]. The following result was suggested to us by Merkurjev and Totaro.

Proposition 4.2. Let k be a field of characteristic different from 2. Then Pf_{k}(3, n) is finite.

Sketch of proof. Let E be a versal torsor for Spin_{n} over a field extension L/k; cf. [GMS03, Section I.V]. Let q_{L} be the quadratic form over L corresponding to E under the map H^{1}(L, Spin_{n}) → H^{1}(L, O_{n}). The 3-Pfister number of q_{L} is then an upper bound for the 3-Pfister number of any ndimensional form in I^{3} over any field extension K/k. ♠

Remark 4.3. For a > 3 the finiteness of Pf_{k}(a, n) is an open problem.

5. Proof of Theorem 1.1

The goal of this section is to prove Theorem 1.1 stated in the introduction, which says, in particular, that

Pf_{k}(3, n) ≥ \frac{2^{(n+4)/4} − n − 2}{7}

for any field k of characteristic different from 2 and any positive even integer n. Clearly, replacing k by a larger field k′ strengthens the assertion of Theorem 1.1. Thus, we may assume without loss of generality that \sqrt{−1} ∈ k. This assumption will be in force for the remainder of this section.

For each extension K of k, denote by T_{n}(K) the image of H^{1}(K, Spin_{n}) in H^{1}(K, SO_{n}). We will view T_{n} as a functor Fields_{k} → Sets. Note that T_{n}(K) is the set of isomorphism classes of n-dimensional quadratic forms q ∈ I^{3}(K).

Lemma 5.1. We have the following inequalities:

(a) ed Spin_{n} − 1 ≤ ed T_{n} ≤ ed Spin_{n},

(b) ed(Spin_{n}; 2) − 1 ≤ ed(T_{n}; 2) ≤ ed(Spin_{n}; 2).

Proof. In the language of [BF03, Definition 1.12], we have a fibration of functors

H^{1}(∗, μ_{2}) ⇝ H^{1}(∗, Spin_{n}) ⟶ T_{n}(∗).

The first inequality in part (a) follows from [BF03, Proposition 1.13] and the second from Proposition [BF03, Lemma 1.9]. The same argument proves part (b). ♠

Let K/k be a field extension. Let h_{K} = ⟨1, −1⟩ be the 2-dimensional hyperbolic form over K. (Note in §3 we wrote h in place of h_{k}; see (3.1).) For each n-dimensional quadratic form q ∈ I^{3}(K), let ed_{n}(q) denote the essential dimension of the class of q in T_{n}(K).

Lemma 5.2. Let q be an n-dimensional quadratic form in I^{3}(K). Then

ed_{n+2s}(h_{K}^{⊕s} ⊕ q) ≥ ed_{n}(q) − \frac{s(s + 2n − 1)}{2}

for any integer s ≥ 0.

Proof. Set m =^{def} ed_{n+2s}(h_{K}^{⊕s} ⊕ q). By definition, h_{K}^{⊕s} ⊕ q descends to an intermediate subfield k ⊂ F ⊂ K such that tr deg_{k}(F) = m. In other words, there is an (n + 2s)-dimensional quadratic form q̃ ∈ I^{3}(F) such that q̃_{K} is K-isomorphic to h_{K}^{⊕s} ⊕ q. Let X be the Grassmannian of sdimensional subspaces of F^{n+2s} which are totally isotropic with respect to q̃. The dimension of X over F is s(s + 2n − 1)/2.

The variety X has a rational point over K; hence there exists an intermediate extension F ⊆ E ⊆ K such that tr deg_{F} E ≤ s(s + 2n − 1)/2, with the property that q̃_{E} has a totally isotropic subspace of dimension s. Then q̃_{E} splits as h_{E}^{s} ⊕ q′, where q′ ∈ I^{3}(E). By Witt’s Cancellation Theorem, q′_{K} is K-isomorphic to q; hence

ed_{n}(q) ≤ tr deg_{k} E = tr deg_{k} F + tr deg_{F} E = m + s(s + 2n − 1)/2 ,

as claimed. ♠

We now proceed with the proof of Theorem 1.1. For n ≤ 10 the statement of the theorem is vacuous, because 2^{(n+4)/4} −n−2 ≤ 0. Thus we will assume from now on that n ≥ 12.

Lemma 5.1 implies, in particular, that ed(T_{n}; 2) is finite. Hence, there exist a field K/k and an n-dimensional form q ∈ I^{3}(K) such that ed_{n}(q) = ed(T_{n}; 2). We will show that this form has the properties asserted by Theorem 1.1. In fact, it suffices to prove that if q is Witt equivalent to

∑_{i=1}^{r} ≪a_{i}, b_{i}, c_{i} ≫.

over K then r ≥ \frac{2^{(n+4)/4} − n − 2}{7}. Indeed, by our choice of q, ed_{n}(q_{L}) = ed(T_{n}; 2) for any finite odd degree extension L/K. Thus if we can prove the above inequality for q, it will also be valid for q_{L}.

Let us write a 3-fold Pfister form ≪a, b, c≫ as ⟨1⟩ ⊕ ≪a, b, c≫_{0}, where

≪a, b, c≫_{0} =^{def} ⟨a_{i}, b_{i}, c_{i}, a_{i}b_{i}, a_{i}c_{i}, b_{i}c_{i}, a_{i}b_{i}c_{i}⟩.

Set

ϕ =^{def} \begin{cases}∑_{1=1}^{r} ≪a_{i}, b_{i}, c_{i} ≫_{0}, if r is even, and \\ ⟨1⟩ ⊕ ∑_{1=1}^{r} ≪a_{i}, b_{i}, c_{i} ≫_{0}, if r is odd. \end{cases}

Then q is Witt equivalent to ϕ over K; in particular, ϕ ∈ I^{3}(K). The dimension of ϕ is 7r or 7r + 1, depending on the parity of r.

We claim that n < 7r. Indeed, assume the contrary. Then dim(q) ≤ dim(ϕ), so that q is isomorphic to a form of type h_{K}^{s} ⊕ ϕ over K. Thus

\frac{3n}{7} ≥ 3r ≥ ed_{n}(q) = ed(T_{n}; 2) ≥^{by Lemma 5.1} ed(Spin_{n}; 2) − 1 .

The resulting inequality fails for every even n ≥ 12 because for such n

ed(Spin_{n}; 2) ≥ n/2;

see (3.2).

So, we may assume that 7r > n, i.e., ϕ is isomorphic to h_{K}^{⊕s}⊕q over K, for some s ≥ 1. By comparing dimensions we get the equality 7r = n+2s when r is even, and 7r + 1 = n + 2s when r is odd. The essential dimension of the form ϕ, as an element of T_{7r}(K) or T_{7r+1}(K) is at most 3r, while Lemma 5.2 tells us that this essential dimension is at least ed_{n}(q) − s(s + 2n − 1)/2.

From this, Lemma 5.1 and Theorem 3.3(a) we obtain the following chain of inequalities

3r ≥ ed_{n}(q) − \frac{s(s + 2n − 1)}{2} = ed(T_{n}; 2) − \frac{s(s + 2n − 1)}{2}

≥ ed(Spin_{n}; 2) − 1 − \frac{s(s + 2n − 1)}{2} (5.3)

≥ 2^{(n−2)/2} − \frac{n(n − 1)}{2} − 1 − \frac{s(s + 2n − 1)}{2}.

Now suppose r is even. Substituting s = (7r − n)/2 into inequality (5.3), we obtain

\frac{49r^{2} + (14n + 10)r − 2^{(n+4)/2} − n^{2} + 2n − 8}{8} ≥ 0.

We interpret the left hand side as a quadratic polynomial in r. The constant term of this polynomial is negative for all n ≥ 8; hence this polynomial has one positive real root and one negative real root. Denote the positive root by r_{+}. The above inequality is then equivalent to r ≥ r_{+}. By the quadratic formula

r_{+} = \frac{\sqrt{49 · 2^{(n+4)/2} + 168n − 367} − (7n + 5)}{49} ≥ \frac{2^{(n+4)/4} − n − 2}{7} .

This completes the proof of Theorem 1.1 when r is even. If r is odd then substituting s = (7r + 1 − n)/2 into (5.3), we obtain an analogous quadratic inequality whose positive root is

r_{+} = \frac{\sqrt{49 · 2^{(n+4)/2} + 168n − 199} − (7n + 12)}{49} ≥ \frac{2^{(n+4)/4} − n − 2}{7} ,

and Theorem 1.1 follows. ♠

References

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(Brosnan, Reichstein) Department of Mathematics, The University of British Columbia, 1984 Mathematics Road, Vancouver, B.C., Canada V6T 1Z2

(Vistoli) Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy E-mail address, Brosnan: brosnan@math.ubc.ca E-mail address, Reichstein: reichst@math.ubc.ca E-mail address, Vistoli: angelo.vistoli@sns.it