ESSENTIAL DIMENSION OF QUADRATIC FORMS WITH TRIVIAL DISCRIMINANT AND CLIFFORD INVARIANT

VLADIMIR CHERNOUSOV AND ALEXANDER MERKURJEV

Abstract. We conclude the computation of the essential dimension of split spinor groups and give an application in algebraic theory of quadratic forms. We also compute essential dimension of quadratic forms with trivial discriminant and Clifford invariant.

1. Introduction

Let F be a field and let F : Fields/F → Sets be a functor from the category of field extensions over F to the category of sets. Let E ∈ Fields/F and K ⊂ E a subfield over F. We say that that K is a field of definition of an element α ∈ F(E) if α belongs to the image of the map F(K) → F(E). The essential dimension of α, denoted ed^{F}(α), is the least transcendence degree tr. deg_{F} (K) over all fields of definition K of α. The essential dimension of the functor F is

ed(F) = sup{ed^{F}(α)},

where the supremum is taken over all fields E ∈ Fields/F and all α ∈ F(E) (see [1, Def. 1.2] or [12, Sec.1]). Informally, the essential dimension of F is the smallest number of algebraically independent parameters required to define F and may be thought of as a measure of complexity of F.

Let p be a prime integer. The essential p-dimension of α, denoted ed_{p}^{F}(α), is defined as the minimum of ed^{F}(α_{E′}), where E′ ranges over all finite field extensions of E of degree prime to p. The essential p-dimension of F is

ed_{p}(F) = sup{ed_{p}^{F}(α)},

where the supremum ranges over all fields E ∈ Fields/F and all α ∈ F(E). By definition, ed(F) ≥ ed_{p}(F) for all p.

For convenience we write ed_{0}(F) = ed(F), so ed_{p}(F) is defined for p = 0 and all prime p.

Let G be an algebraic group scheme over F. Write F_{G} for the functor taking a field extension E/F to the set H_{et}^{1}(E, G) of isomorphism classes of principal homogeneous G-spaces (G-torsors) over E. The essential (p-)dimension of F_{G} is called the essential (p-)dimension of G and is denoted by ed(G) and ed_{p}(G) (see [14] and [15]).

2000 Mathematics Subject Classification. 11E04, 11E57, 11E72, 11E81, 14L35, 20G15.

Key words and phrases. Linear algebraic groups, Spinor groups, essential dimension, torsor, non-abelian cohomology, quadratic forms, Witt rings, the fundamental ideal.

The work of the first author has been supported in part by Canada Research Chairs Program and NSERC research grant.

The work of the second author has been supported by the NSF grant DMS #0652316.

In this paper we conclude computation of the essential dimension of the split spinor groups originated in [2] and [7] and continued in [12] (Theorem 2.2). We give an application in algebraic theory of quadratic forms (Theorem 4.2). We also compute essential dimension of quadratic forms with trivial discriminant and Clifford invariant (Theorem 7.1).

2. Essential dimension of Spin_{n}

Let G be an algebraic group over F and let C ⊂ G be a normal subgroup over F. For a torsor E → Spec(F) of the group H := G/C consider the stack [E/G] (see [16]). Recall that an object of the category [E/G](K) for a field extension K/F is a pair (E′, φ), where E′ is a G-torsor over K and φ : E′/C →^{∼} E_{K} is an isomorphism of H-torsors overK. The essential dimension ed[E/G] of the stack [E/G] is the essential dimension of the functor K ↦ set of isomorphism classes of objects in [E/G](K).

The following proposition was proven independently by R. Lötscher in [11, Ex. 3.4]:

Proposition 2.1. Let C be a normal subgroup of an algebraic group G over F and H = G/C. Then

ed(G) ≤ ed(H) + max ed[E/G],

where the maximum is taking over all field extensions L/F and all H-torsors E over L.

Proof. Let I′ be a G-torsor over a field extension K/F. Then I := I′/C is an H-torsor over K. There is a subextension K_{0}/F of K/F and an H-torsor E over K_{0} such that there is an isomorphism φ : I →^{∼} E_{K} of H-torsors and tr. deg(K_{0}/F) ≤ ed(H).

Consider the stack [E/G] over K_{0}. The pair (I′, φ) is an object of [E/G](K). There is a subextension K_{1}/K_{0} of K/K_{0} such that (I′, φ) is defined over K_{1} and tr. deg(K_{1}/K_{0}) ≤ ed[E/G]. It follows that I′ is defined over the field K_{1} with

tr. deg(K_{1}/F) = tr. deg(K_{0}/F) + tr. deg(K_{1}/K_{0}) ≤ ed(H) + ed[E/G]. □

The following theorem concludes computation of the essential dimension of the spinor groups initiated in [2] and [7] and continued in [12]. We write Spin_{n} for the split spinor group of a nondegenerate quadratic form of dimension n and maximal Witt index.

If char(F) ≠ 2, then the essential dimension of Spin_{n} has the following values for n ≤ 14 (see [7, §23]:

n  ≤ 6 7 8 9 10 11 12 13 14
ed_{2}(Spin_{n}) = ed(Spin_{n}) 0 4 5 5 4 5 6 6 7

In the following theorem we give the values of ed_{p}(Spin_{n}) for n ≥ 15 and p = 0 and 2. Note that ed_{p}(Spin_{n}) = 0 if p ≠ 0, 2 as every Spin_{n}-torsor over a field is split over an extension of degree a power of 2.

Theorem 2.2. Let F be a field of characteristic zero. Then for every integer n ≥ 15 we have:

ed_{2}(Spin_{n}) = ed(Spin_{n}) = \{ \begin{matrix}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} + 2^{m} − \frac{n(n−1)}{2}, & if n ≡ 0 (mod 4), \end{matrix}

where 2^{m} is the largest power of 2 dividing n.

Proof. The case n ≥ 15 and n is not divisible by 4 has been considered in [2, Th. 3.3].

Now assume that n > 15 and n is divisible by 4. The inequality ”≥” was obtained in [12, Th. 4.9], so we just need to prove the inequality ”≤”. The case n = 16 was considered in [12, Cor. 4.10]. Assume that n ≥ 20 and n is divisible by 4.

Consider the following diagram with the exact rows:

1 ⟶ μ_{2} ⟶ Spin_{n} ⟶ Spin_{n}^{+} ⟶ 1
∥  ↓  ↓
1 ⟶ μ_{2} ⟶ O_{n}^{+} ⟶ PGO_{n}^{+} ⟶ 1,

where Spin_{n}^{+} is the semi-spinor group, O_{n}^{+} is the split special orthogonal group and PGO_{n}^{+} is the split special projective orthogonal group. We see from the diagram that the image of the connecting map

δ_{K} : H_{et}^{1}(K, Spin_{n}^{+}) → H_{et}^{2}(K, μ_{2}) ⊂ Br(K)

is contained in the image of the other connecting map

H_{et}^{1}(K, PGO_{n}^{+}) → H_{et}^{2}(K, μ_{2}) ⊂ Br(K)

for every field extension K/F. The image of the last map consists of the classes [A] of all central simple K-algebras A of degree n admitting orthogonal involutions (see [9, §31]). As ind(A) is a power of 2 dividing n, we have ind(A) ≤ 2^{m}, where 2^{m} is the largest power of 2 dividing n.

Let E be a Spin_{n}^{+}-torsor over K. We have shown that if δ_{K}([E]) = [A] for a central simple K-algebra A, then ind(A) ≤ 2^{m}. It follows from [3, Th. 4.1] that ed[E/ Spin_{n}] = ind(A) ≤ 2^{m}.

It is shown in [2, Rem. 3.10] that ed(Spin_{n}^{+}) = 2^{(n−2)/2} − \frac{n(n−1)}{2} for every integer n ≥ 20 divisible by 4. Finally, by Proposition 2.1,

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

3. The functors I_{n}^{k}

We use the following notation. Let F be a field of characteristic different from 2 and K/F a field extension. We define:

I_{n}^{1}(K) = Set of isomorphism classes of nondegenerate quadratic forms over K of dimension n

We have a natural bijection I_{n}^{1}(K) ≃ H_{et}^{1}(K, O_{n}) (see [9, §29.E]).

Recall that the discriminant disc(q) of a form q ∈ I_{n}^{1}(K) is equal to (−1)^{n(n−1)/2} det(q) ∈ K^{×}/K^{×2}. Set

I_{n}^{2}(K) = {q ∈ I_{n}^{1}(K) such that disc(q) = 1}.

We have a natural bijection I_{n}^{2}(K) ≃ H_{et}^{1}(K, O_{n}^{+}) (see [9, §29.E]).

The Clifford invariant c(q) of a form q ∈ I_{n}^{2}(K) is the class in the Brauer group Br(K) of the Clifford algebra of q if n is even and the class of the even Clifford algebra if n is odd [9, §8.B]. Define

I_{n}^{3}(K) = {q ∈ I_{n}^{2}(K) such that c(q) = 0}.

Remark 3.1. Our notation of the functors I_{n}^{k} for k = 1, 2, 3 is explained by the following property: I_{n}^{k}(K) consists of all classes of quadratic forms q ∈ W (K) of dimension n such that q ∈ I(K)^{k} if n is even and q ⊥ ⟨−1⟩ ∈ I(K)^{k} if n is odd, where I(K) is the fundamental ideal in the Witt ring W (K) of K.

The functor I_{n}^{3} is related to Spin_{n}-torsors as follows. The short exact sequence

1 → μ_{2} → Spin_{n} → O_{n}^{+} → 1

yields an exact sequence

(1)  H_{et}^{1}(K, μ_{2}) → H_{et}^{1}(K, Spin_{n}) → H_{et}^{1}(K, O_{n}^{+}) ⟶^{c} H_{et}^{2}(K, μ_{2}),

where c is the Clifford invariant. Thus Ker(c) = I_{n}^{3}(K).

The essential dimension of I_{n}^{1} and I_{n}^{2} was computed in [14, Th. 10.3 and 10.4]: we have ed(I_{n}^{1}) = n and ed(I_{n}^{2}) = n−1. In Section 7 we compute ed(I_{n}^{3}).

Lemma 3.2. We have: ed_{p}(I_{n}^{3}) ≤ ed_{p}(Spin_{n}) ≤ ed_{p}(I_{n}^{3}) + 1.

Proof. Let K/F be a field extension. The group H_{et}^{1}(K, μ_{2}) = K^{×}/K^{×2} acts transitively on the fibers of the second map in the sequence (1). It follows that the natural map Spin_{n}-Torsors → I_{n}^{3} is a surjection with G_{m} acting surjectively on the fibers. The statement follows from [1, Prop. 1.13]. □

4. Subforms of forms in I_{n}^{3}

In this section we study the following problem in quadratic form theory which will be used in Section 7 in order to compute the essential dimension of I_{n}^{3}. Note that the problem is stated entirely in terms of quadratic forms, while in the solution we use the essential dimension. We don’t know how to solve the problem by means of quadratic form theory.

Problem 4.1. Given a field F, determine all integers n such that every form in I_{n}^{3}(K) contains a nontrivial subform in I^{2}(K) for any field extension K/F.

All forms in I_{n}^{3}(K) for n ≤ 14 are classified (see [7, Ex. 17,8, Th.17.13 and Th. 21.3]). Inspection shows that for such n the problem has positive solution.

In the following theorem we show that in the range n ≥ 15 the problem has negative solution (with possibly two exceptions).

Theorem 4.2. Let F be a field of characteristic zero, n ≥ 15 and let b be an even integer with 0 < b < n. Then there is a field extension K/F and a form in I_{n}^{3}(K) that does not contain a subform in I_{b}^{2}(K) (with possible exceptions: (n, b) = (15, 8) or (16, 8)).

Let a := n − b. Write H_{a,b} for the image of the natural homomorphism

(2)  Spin_{a} × Spin_{b} → Spin_{n} .

Note that the kernel of (2) is contained in

μ_{2} × μ_{2} = Ker(Spin_{a} × Spin_{b} → O_{a}^{+} × O_{b}^{+})

and therefore, is the cyclic group of order 2 generated by (−1, −1). Hence we have an exact sequence

1 → μ_{2} → H_{a,b} → O_{a}^{+} × O_{b}^{+} → 1

and therefore a map

H_{et}^{1}(R, H_{a,b}) → H_{et}^{1}(R, O_{a}^{+} × O_{b}^{+}) = H_{et}^{1}(R, O_{a}^{+}) × H_{et}^{1}(R, O_{b}^{+})

for a commutative F -algebra R.

We write q(η) := (q_{a}, q_{b}) for the image of an element η ∈ H_{et}^{1}(R, H_{a,b}) under this map, where q_{a} ∈ H_{et}^{1}(R, O_{a}^{+}) and q_{b} ∈ H_{et}^{1}(R, O_{b}^{+}).

Consider the commutative diagram with the exact rows

1 ⟶ μ_{2} ⟶ H_{a,b} ⟶ O_{a}^{+} × O_{b}^{+} ⟶ 1
∥  ↓  ^{τ}↓
1 ⟶ μ_{2} ⟶ Spin_{n} ⟶ O_{n}^{+} ⟶ 1.

The image of an element ξ ∈ H_{et}^{1}(R, Spin_{n}) in H_{et}^{1}(R, O_{n}^{+}) will be denoted by q(ξ).

If ξ ∈ H_{et}^{1}(R, Spin_{n}) is the image of an element η ∈ H_{et}^{1}(R, H_{a,b}), then q(ξ) = q_{a} ⊥ q_{b}, the image of (q_{a}, q_{b}) = q(η) under the map induced by τ. We can reverse this statement as follows:

Lemma 4.3. Let ξ ∈ H_{et}^{1}(R, Spin_{n}) with q(ξ) = q_{a} ⊥ q_{b}, where q_{a} ∈ H_{et}^{1}(R, O_{a}^{+}) and q_{b} ∈ H_{et}^{1}(R, O_{b}^{+}). Then ξ is the image of an element η under the map H_{et}^{1}(R, H_{a,b}) → H_{et}^{1}(R, Spin_{n}) such that q(η) = (q_{a}, q_{b}).

Proof. The diagram above yields a commutative diagram with the exact rows:

H_{et}^{1}(R, H_{a,b}) ⟶ H_{et}^{1}(R, O_{a}^{+}) × H_{et}^{1}(R, O_{b}^{+}) ⟶^{c′} H_{et}^{2}(R, μ_{2})
↓  ↓  ∥
H_{et}^{1}(R, Spin_{n}) ⟶  H_{et}^{1}(R, O_{n}^{+})  ⟶^{c} H_{et}^{2}(R, μ_{2}).

Moreover, the group H_{et}^{1}(R, μ_{2}) acts transitively on the fibers of the left maps in the two rows. The result follows. □

For non-negative integers a, b and a field extension K/F set

I_{a,b}^{3}(K) := {(q_{a}, q_{b}) ∈ I_{a}^{2}(K) × I_{b}^{2}(K) such that q_{a} ⊥ q_{b} ∈ I_{n}^{3}(K)}.

Corollary 4.4. For any η ∈ H_{et}^{1}(K, H_{a,b}) we have q(η) ∈ I_{a,b}^{3}(K). The morphism of functors q : H_{a,b}-Torsors → I_{a,b}^{3} is surjective. In particular, ed_{p}(I_{a,b}^{3}) ≤ ed_{p}(H_{a,b}) for every p ≥ 0.

Proof. Note that the map c′ in the proof of Lemma 4.3 when R = K takes a pair (q_{a}, q_{b}) to the Clifford invariant of q_{a} ⊥ q_{b} in Br(K). The pair (q_{a}, q_{b}) ∈ I_{a}^{2}(K) × I_{b}^{2}(K) comes from H_{et}^{1}(K, H_{a,b}) if and only if the Clifford invariant of q_{a} ⊥ q_{b} is split, i.e., q_{a} ⊥ q_{b} ∈ I_{n}^{3}(K). □

Lemma 4.5. For an even a and any b,

ed_{p}(I_{a,b}^{3}) ≤ ed_{p}(I_{a−1,b}^{3}) + 1

for every p ≥ 0.

Proof. Consider the morphism of functors

α : G_{m} ×I_{a−1,b}^{3} → I_{a,b}^{3}, (λ; f, g) ↦ (λ(f ⊥ ⟨−1⟩), g).

Every form h in I_{a}^{2}(K) can be written in the form h = λ(f ⊥ ⟨−1⟩) for a value λ of h and a form f ∈ I_{a−1}^{2}(K), i.e., α is a surjection, whence the result. □

Write V_{n} (respectively W_{n}) for the (semi-)spinor (respectively regular) representation of the group Spin_{n}. We have

dim(V_{n}) = \{ \begin{matrix}2^{(n−1)/2}, & if n is odd; \\ 2^{(n−2)/2}, & if n is even, \end{matrix}

and dim(W_{n}) = n. We consider the tensor product V_{a,b} := V_{a} ⊗ V_{b} as the representation of the group H_{a,b}. We also view W_{a} (respectively W_{b}) as a H_{a,b}-representation via the natural homomorphism H_{a,b} → O_{a}^{+} (respectively H_{a,b} → O_{b}^{+}).

A representation V of an algebraic group H is generically free if the stabilizer of a generic vector in V is trivial. In this case by [15],

ed(H) ≤ dim(V) − dim(H).

Lemma 4.6. Let a be odd and b even. Suppose that V_{a,b} is a generically free representation of the image of the homomorphism H_{a,b} → GL(V_{a,b}). Then V_{a,b} ⊕ W_{b} is a generically free representation of H_{a,b}. In particular,

ed(H_{a,b}) ≤ dim(V_{a,b}) + dim(W_{b}) − dim(H_{a,b}).

Proof. Write C_{n} for the kernel of Spin_{n} → PGO_{n}^{+} and C′_{n} for the kernel of Spin_{n} → O_{n}^{+} so C′_{n} = {±1} ⊂ C_{n}. By assumption, the generic stabilizer H of the action of Spin_{a} × Spin_{b} on V_{a,b} is contained in the center C_{a} × C_{b}. Since C_{b}/C′_{b} = μ_{2} acts on W_{b} by multiplication by −1 we have H ⊂ C_{a} × C′_{b} ≃ μ_{2} × μ_{2}. Note that μ_{2} × 1 and 1 × μ_{2} act by multiplication by −1 on V_{a,b}, hence H is generated by (−1, −1). It follows that H_{a,b} = (Spin_{a} × Spin_{b})/H acts generically freely on V_{a,b} ⊕ W_{b}. □

Proposition 4.7. Let char(F) = 0. If n = a + b ≥ 15 with a ≤ b, then V_{a,b} is a generically free representation of the image of H_{a,b} → GL(V_{a,b}) if and only if (a, b) ≠ (3, 12), (4, 11), (4, 12), (6, 10) and (8, 8).

Proof. All the cases of infinite generic stabilizers H are listed in [5, §3] (row 7 of Table 6): H is infinite if and only if (a, b) = (3, 12) and (4, 12).

If H is finite, by [13, Th. 1] (rows 1, 12 and 13 of Table 1) H is nontrivial if and only if (a, b) = (4, 11), (6, 10) and (8, 8). □

Proof of Theorem 4.2. Note that the case (n, b) with n even implies the case (n − 1, b). Indeed suppose that every form in I_{n−1}^{3} for an even n contains a subform from I_{b}^{2}. Take any form q ∈ I_{n}^{3}(K) for a field extension K/F and write q = λ(f ⊥ ⟨−1⟩) for a λ ∈ K^{×} and f ∈ I_{n−1}^{3}(K). If f contains a subform h ∈ I_{b}^{2}(K), then q contains λh.

We need to show that the natural morphism of functors I_{a,b}^{3} → I_{n}^{3} is not surjective. It suffices to prove that ed(I_{a,b}^{3}) < ed(I_{n}^{3}). We may assume that n (and hence also a) is even. Moreover, we may assume that a ≤ b.

Suppose that n ≥ 18. By Proposition 4.7, Lemma 4.5, Lemma 4.6 and Corollary 4.4,

ed(I_{a,b}^{3}) ≤ ed(I_{a−1,b}^{3}) + 1
≤ ed(H_{a−1,b}) + 1
≤ dim(V_{a−1,b}) + dim(W_{b}) − dim(H_{a−1,b}) + 1
= 2^{n/2−2} + b − (a − 1)(a − 2)/2 − b(b − 1)/2 + 1
= 2^{n/2−2} − (a^{2} + b^{2} − 3a − 3b)/2
≤ 2^{n/2−2} − (n^{2} − 6n)/4

as a^{2} + b^{2} ≥ n^{2}/2. The last integer is strictly less than

2^{n/2−1} − n(n − 1)/2 − 1 ≤ ed(Spin_{n}) − 1 ≤ ed(I_{n}^{3})

by Theorem 2.2 and Lemma 3.2.

It remains to consider the case n = 16. Note that by Theorem 2.2 and Lemma 3.2,

(3)  ed(I_{16}^{3}) ≥ ed(Spin_{16}) − 1 = 23.

We shall prove that ed(I_{a,b}^{3}) < 23. All possible values of b are 8, 10, 12 and 14.

Case (n, b) = (16, 10): Consider the representation V := W_{6} ⊕ V_{6,10} ⊕ W_{10} of H_{6,10}. We claim that V is generically free. The stabilizer in Spin_{6} of a point in general position in W_{6} is Spin_{5}. Hence the stabilizer in H_{6,10} of a point in general position in W_{6} is H_{5,10}. Note that the restriction of V_{6,10} to H_{5,10} is isomorphic to V_{5,10}. Finally, the H_{5,10}-representation V_{5,10} ⊕ W_{10} is generically free by Proposition 4.7.

It follows from (3) and Corollary 4.4 that

ed(I_{6,10}^{3}) ≤ ed(H_{6,10}) ≤ dim(V) − dim(H_{6,10}) = 80 − 60 = 20.

Case (n, b) = (16, 12): Consider the representation V := W_{3} ⊕ W_{3} ⊕ V_{3,12} ⊕ W_{12} of H_{3,12}. We claim that V is generically free as the representation of H_{3,12}. Indeed, the stabilizer in H_{3,12} of a generic vector in W_{12} is H_{3,11}. We are reduced to showing that W_{3} ⊕ W_{3} ⊕ V_{3,11} is a generically free representation of H_{3,11}. By [13, §5, p. 246] the generic stabilizer S of H_{3,11} in V_{3,11} is finite (isomorphic to μ_{2} × μ_{2}), and the restriction to S of the natural projection H_{3,11} → O_{3}^{+} is injective. It remains to notice that the representation W_{3} ⊕ W_{3} of O_{3}^{+} = PGL_{2} is generically free.

It follows from Lemmas 4.5 and 4.6 and Corollary 4.4 that

ed(I_{4,12}^{3}) ≤ ed(I_{3,12}^{3}) + 1 ≤ ed(H_{3,12}) + 1 ≤ dim(V) − dim(H_{3,12}) + 1 = 82 − 69 + 1 = 14.

Case (n, b) = (16, 14): As every form in I_{2}^{3} is hyperbolic, we have I_{2,14}^{3} = I_{14}^{3} and ed(I_{14}^{3}) = 7 by Theorem 2.2. □

5. Unramified principal homogeneous spaces

Let G be an algebraic group over F and let K/F be a field extension with a discrete valuation v trivial on F. Write O for the valuation ring of v. It is a local F -algebra. We say that a class ξ ∈ H_{et}^{1}(K, G) is unramified (with respect to v) if ξ belongs to the image of the map H_{et}^{1}(O, G) → H_{et}^{1}(K, G).

Let \overline{K} be the residue field of v. The ring homomorphism O → \overline{K} yields a map H_{et}^{1}(O, G) → H_{et}^{1}(\overline{K}, G). This map is a bijection if K is complete (see [4, Exp. XXIV. Prop. 8.1]). Hence we have the map

(4)  H_{et}^{1}(\overline{K}, G) →^{∼} H_{et}^{1}(O, G) → H_{et}^{1}(K, G).

Example 5.1. Let char(F) ≠ 2 and G = O_{n}. Then H_{et}^{1}(K, G) is the set of isomorphism classes of nondegenerate quadratic forms of dimension n over K. A quadratic form q over a field K with a discrete valuation is unramified if and only if q ≃ ⟨a_{1}, a_{2}, …, a_{n}⟩, where a_{i} are units in the valuation ring O in K. In general, every q can be written q = q_{1} ⊥ πq_{2} ⊥ h, where π is a prime element, q_{1} and q_{2} are unramified anisotropic quadratic forms and h is a hyperbolic form. The form q is unramified if and only if q_{2} = 0. It follows that if two forms q and πq are both unramified, then q is hyperbolic. If K is complete, then the map (4) takes f = ⟨ā_{1}, ā_{2}, …, ā_{n}⟩ over \overline{K}, where a_{i} are units in O, to f_{K} := ⟨a_{1}, a_{2}, …, a_{n}⟩.

6. Essential dimension of P I_{n}^{3}

Two quadratic forms f and g over a field K are called similar if f = λg for some λ ∈ K^{×}. If n is even, we write P I_{n}^{3}(K) for the set of similarity classes of forms in I_{n}^{3}(K). The group K^{×} acts transitively on the fibers of the the natural surjective map I_{n}^{3}(K) → P I_{n}^{3}(K). Hence

ed_{p}(P I_{n}^{3}) ≤ ed_{p}(I_{n}^{3}) ≤ ed_{p}(P I_{n}^{3}) + 1

for any p ≥ 0 by [1, Prop. 1.13].

Proposition 6.1. Let char(F) ≠ 2. For an even n ≥ 8, and p = 0 or 2, we have

ed_{p}(P I_{n}^{3}) = ed_{p}(I_{n}^{3}) − 1.

Proof. Let K/F be a field extension and let q ∈ I_{n}^{3}(K) be a non-hyperbolic form. Consider the form tq over the field K((t)). It suffices to show that

ed_{p}^{I_{n}^{3}}(tq) ≥ ed_{p}^{P I_{n}^{3}}(q) + 1.

Let M/K((t)) be a finite field extension of degree prime to p (i.e., M = K((t)) if p = 0 and [M : K((t))] is odd if p = 2), let L/F be a subextension of M/F and let f ∈ I_{n}^{3}(L) be such that tr. deg(L/F) = ed_{p}^{I_{n}^{3}}(tq) and tq_{M} ≃ f_{M}.

Let v be the (unique) extension on M of the discrete valuation of K((t)) and let w be the restriction of v on L. The residue field \overline{M} is a finite extension of K of degree prime to p. As the form q is not hyperbolic, q_{M} is not hyperbolic and therefore, the form tq_{M} ≃ f_{M} is ramified by Example 5.1. It follows that w is nontrivial, i.e., w is a discrete valuation on L.

Let L̂ be the completion of L. Note that as M is complete, we can identify L̂ with a subfield of M. Write fL̂ ≃ (f_{1})L̂ ⊥ π(f_{2})L̂, where f_{1} and f_{2} are quadratic forms over the residue field \overline{L} and π ∈ L is a prime element (see Example 5.1). Note that f_{1}, f_{2} ∈ I^{2}(\overline{L}) by [6, Lemma 19.4]. If the ramification index e of M/L is even, then π is a unit in the valuation ring O of M modulo squares in M^{×}, hence f_{M} is unramified, a contradiction. It follows that e is odd. Writing π = ut^{e} with a unit u ∈ O^{×} we have

tq_{M} ≃ f_{M} ≃ (f_{1})_{M} ⊥ π(f_{2})_{M} ≃ (f_{1})_{M} ⊥ ut(f_{2})_{M},

hence (f_{1})_{M} = 0 and q_{M} = u(f_{2})_{M} in W (M). It follows that (f_{1})\overline{M} = 0 and q\overline{M} = ū(f_{2})\overline{M} in W (\overline{M}) and therefore,

(5)  q\overline{M} = ū(f_{2})\overline{M} = ūg\overline{M},

where g := f_{1} ⊥ f_{2} is the form over \overline{L} of dimension n. Note that fL̂ − gL̂ = ⟨π, −1⟩(f_{2})L̂ ∈ I^{3}(L̂), hence gL̂ ∈ I^{3}(L̂) and g ∈ I^{3}(\overline{L}).

It follows from (5) that q\overline{M} is similar to g\overline{M}, i.e., the form q is p-defined over \overline{L} for the functor P I_{n}^{3} (see [12, §1.1]) and therefore

ed_{p}^{I_{n}^{3}}(tq) = tr. deg(L/F) ≥ tr. deg(\overline{L}/F) + 1 ≥ ed_{p}^{P I_{n}^{3}}(q) + 1.  □

7. Essential dimension of I_{n}^{3}

In this section we compute the essential dimension of I_{n}^{3}.

Theorem 7.1. Let F be a field of characteristic zero. Then for every integer n ≥ 15 we have:

ed_{2}(I_{n}^{3}) = ed(I_{n}^{3}) = \{ \begin{matrix}2^{(n−1)/2} − 1 − \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} + 2^{m} − 1 − \frac{n(n−1)}{2}, & if n ≡ 0 (mod 4), \end{matrix}

where 2^{m} is the largest power of 2 dividing n.

If char(F) ≠ 2, then the essential dimension of I_{n}^{3} has the following values for n ≤ 14:

\begin{matrix}n & ≤ 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 \\ ed_{2}(I_{n}^{3}) = ed(I_{n}^{3}) & 0 & 3 & 4 & 4 & 4 & 5 & 6 & 6 & 7 \end{matrix}

Proof. We will prove the theorem case by case.

7.1. Case n ≡ 2 (mod 4) and n ≥ 10. The exact sequence

1 → μ_{4} → Spin_{n} → PGO_{n}^{+} → 1

yields a surjective map Spin_{n}-Torsors(K) → P I_{n}^{3}(K) for any K/F with the group K^{×} acting transitively on the fibers of this map. It follows from Theorem 2.2, Proposition 6.1 and Lemma 3.2 that

ed_{2}(I_{n}^{3}) = ed_{2}(P I_{n}^{3}) + 1 ≥ ed_{2}(Spin_{n}) = ed(Spin_{n}) ≥ ed(I_{n}^{3}) ≥ ed_{2}(I_{n}^{3}).

Hence ed_{2}(I_{n}^{3}) = ed(I_{n}^{3}) = ed(Spin_{n}). The latter value is known by Theorem 2.2.

7.2. Case n / ≡ 2 (mod 4) and n ≥ 15. Let n = a + b with even b ≠ 2. Let Z be the trivial group if b = 0 and the image of the center C_{b} of Spin_{b} in H_{a,b} if b ≥ 4. Then Z is central in H_{a,b}, hence the group H_{et}^{1}(K, Z) acts on H_{et}^{1}(K, H_{a,b}).

Lemma 7.2. Let ξ, η ∈ H_{et}^{1}(K, H_{a,b}) with even b ≠ 2. Suppose that q(ξ) = q_{a} ⊥ q_{b} and q(η) = q_{a} ⊥ λq_{b} with the forms q_{a} ∈ I_{a}^{2}(K) and q_{b} ∈ I_{b}^{2}(K) and λ ∈ K^{×}. Then η = αξ for some α ∈ H_{et}^{1}(K, Z).

Proof. The statement is trivial if b = 0, so assume that b ≥ 4. The restriction of the natural homomorphism H_{a,b} → O_{b}^{+} to the subgroup Z yields a surjection φ : Z → μ_{2} = Center(O_{b}^{+}). The kernel of φ coincides with the kernel C of the canonical homomorphism H_{a,b} → O_{a}^{+} × O_{b}^{+}.

As Z is isomorphic to μ_{2} × μ_{2} or μ_{4}, the homomorphism φ^{∗} : H_{et}^{1}(K, Z) → H_{et}^{1}(K, μ_{2}) = K^{×}/K^{×2} is surjective. Let γ ∈ H_{et}^{1}(K, Z) be such that φ^{∗}(γ) = λK^{×2}. Then q(γξ) = q_{a} ⊥ λq_{b} = q(η). Then there is β ∈ H_{et}^{1}(K, C) such that η = β(γξ). Hence η = αξ, where α = β′γ with β′ the image of β under the map H_{et}^{1}(K, C) → H_{et}^{1}(K, Z) induced by the inclusion of C into Z. □

Let ξ ∈ H_{et}^{1}(K, Spin_{n}) be such that the form q = q(ξ) ∈ I_{n}^{3}(K) is generic for the functor I_{n}^{3} (see [12, §2.2]). In particular, ed^{I_{n}^{3}}(q) = ed(I_{n}^{3}). Note that q is anisotropic.

Identifying μ_{2} with the kernel of Spin_{n} → O_{n}^{+}, we have an action of H_{et}^{1}(E, μ_{2}) = E^{×}/E^{×2} on H_{et}^{1}(E, Spin_{n}) where E = K((t)). Consider the element tξ_{E} ∈ H_{et}^{1}(E, Spin_{n}) over E. We claim that tξ_{E} is ramified. Suppose not, i.e., tξ_{E} comes from an element ρ ∈ H_{et}^{1}(O, Spin_{n}) where O = K[[t]]. Let q′ ∈ H_{et}^{1}(O, O_{n}^{+}) be the image of ρ, viewed as a quadratic form over O. We have

q′_{E} = q(tξ_{E}) = q(ξ_{E}) = q_{E},

hence q′ = q_{O}. Then ρ and ξ_{O} belong to the same fiber of the map

H_{et}^{1}(O, Spin_{n}) → H_{et}^{1}(O, O_{n}^{+}).

As the group H_{et}^{1}(O, μ_{2}) = O^{×}/O^{×2} acts transitively on the fiber, there is a unit u ∈ O^{×} satisfying tξ_{E} = uξ_{E}. It follows from [9, Prop. 28.11] that tu^{−1} is in the image spinor norm map

O^{+}(q_{E}) → H_{et}^{1}(E, μ_{2}) = E^{×}/E^{×2}

for the form q_{E}, hence q is isotropic by [6, Th. 18.3], a contradiction. The claim is proven.

Let L/F be a subextension of E/F and let η ∈ H_{et}^{1}(L, Spin_{n}) be such that tr. deg(L/F) = ed^{Spin^{n}}(tξ) and η_{E} ≃ tξ_{E}. We have q(η)_{E} = q(tξ) = q(ξ_{E}) = q_{E}, hence the form q(η)_{E} is anisotropic.

Let v be the restriction on L of the discrete valuation of E. As tξ is ramified, v is nontrivial, hence v is a discrete valuation. Let π ∈ L be a prime element.

Consider the completion L̂ of L. As E is complete, we can view L̂ as a subfield of E. Write q(ηL̂) = (q_{a})L̂ ⊥ π(q_{b})L̂, where q_{a} and q_{b} are anisotropic quadratic forms over the residue field \overline{L} of dimension a and b respectively. As q(η) ∈ I^{3}(L̂) we have q_{b} ∈ I^{2}(\overline{L}) and therefore, b is even and b ≠ 2. By Lemma 4.3, there is η′ ∈ H_{et}^{1}(L̂, H_{a,b}) that maps to η with q(η′) = ((q_{a})L̂, π(q_{b})L̂).

We claim that the ramification index e of the extension E/L̂ is odd. Suppose e is even. Note that q_{a} ⊥ q_{b} ∈ I_{n}^{3}(\overline{L}). Lemma 4.3 allows us to choose an unramified element ν ∈ H_{et}^{1}(L̂, H_{a,b}) with q(ν) = ((q_{a})L̂, (q_{b})L̂). By Lemma 7.2, there is α ∈ H_{et}^{1}(L̂, Z) such that η′ = αν. If b is divisible by 4, we have Z ≃ μ_{2} × μ_{2}. As e is even, α is unramified over E, hence η′_{E} is unramified. It follows that η_{E} ≃ tξ is also unramified, a contradiction.

Suppose that b ≡ 2 (mod 4). Note that 0 < b < n since n / ≡ 2 (mod 4). Write π = ut^{k} with a unit u ∈ O^{×} and even k. Then

(q_{a} ⊥ uq_{b})_{E} ≃ (q_{a} ⊥ πq_{b})_{E} ≃ q(η_{E}) ≃ q(tξ_{E}) = q(ξ_{E}) = q_{E}.

It follows that q ≃ (q_{a})_{K} ⊥ (ūq_{b})_{K}, i.e., q contains the subform (ūq_{b})_{K} in I^{2}(K) of dimension b. This contradicts Theorem 4.2. The claim is proven.

Thus e is odd. We have

(q_{a} ⊥ utq_{b})_{E} ≃ (q_{a} ⊥ πq_{b})_{E} ≃ q(η_{E}) ≃ q(tξ_{E}) = q(ξ_{E}) = q_{E}.

It follows that (q_{b})_{K} is hyperbolic and hence (q_{a} ⊥ q_{b})_{K} = (q_{a})_{K} = q in W (K), i.e., (q_{a} ⊥ q_{b})_{K} ≃ q.

Note that (q_{a})L̂ = (q_{a})L̂ + π(q_{b})L̂ = q(ηL̂) ∈ I^{3}(L̂), hence q_{a} ∈ I^{3}(\overline{L}) and q_{a} ⊥ q_{b} ∈ I_{n}^{3}(\overline{L}). Therefore, q is defined over \overline{L} for the functor I_{n}^{3}, hence

ed^{Spin^{n}}(tξ) = tr. deg(L/F) ≥ tr. deg(\overline{L}/F) + 1 ≥ ed^{I_{n}^{3}}(q) + 1 = ed(I_{n}^{3}) + 1.

It follows that ed(Spin_{n}) ≥ ed(I_{n}^{3}) + 1, hence ed(I_{n}^{3}) = ed(Spin_{n}) − 1 by Lemma 3.2. The value of ed(Spin_{n}) is given in Theorem 2.2.

In what follows we use the following observation (see [1]): if a functor F admits a nontrivial cohomological invariant of degree d with values in ℤ/2ℤ, then ed_{2}(F) ≥ d.

7.3. Case n = 7. Every form q in I_{7}^{3}(K) is the pure subform of a 3-fold Pfister form ⟨⟨a, b, c⟩⟩, hence ed(I_{7}^{3}) ≤ 3. On the other hand, the Arason invariant e_{3}(q ⊥ ⟨−1⟩) = (a) ∪ (b) ∪ (c) ∈ H^{3}(K, ℤ/2ℤ) is nontrivial (see [7, §18.6]), hence ed_{2}(I_{7}^{3}) ≥ 3.

7.4. Case n = 8. Every form q in I_{8}^{3}(K) is a multiple e⟨⟨a, b, c⟩⟩ of a 3-fold Pfister form, hence ed(I_{8}^{3}) ≤ 4. The invariant a_{4}(q) = (e) ∪ (a) ∪ (b) ∪ (c) ∈ H^{4}(K, ℤ/2ℤ) is nontrivial, hence ed_{2}(I_{8}^{3}) ≥ 4.

7.5. Case n = 9 and 10. Every form q in I_{9}^{3}(K) (respectively in I_{10}^{3}(K)) is equal to f ⊥ ⟨1⟩ (respectively, f ⊥ ⟨1, −1⟩), where f is a 3-fold Pfister form over K, by [10, XII.2.8]. Hence I_{8}^{3} ≃ I_{9}^{3} ≃ I_{10}^{3}.

7.6. Case n = 11. The degree 5 cohomological invariant a_{5} of Spin_{11} defined in [7, §20.8] factors through a nontrivial invariant of I_{11}^{3}, hence ed_{2}(I_{11}^{3}) ≥ 5. On the other hand, ed(I_{11}^{3}) ≤ ed(Spin_{11}) = 5.

7.7. Case n = 12. The degree 6 cohomological invariant a_{6} of Spin_{12} defined in [7, §20.13] factors through a nontrivial invariant of I_{12}^{3}, hence ed_{2}(I_{12}^{3}) ≥ 6. On the other hand, ed(I_{12}^{3}) ≤ ed(Spin_{12}) = 6.

7.8. Case n = 13 and 14. By Case 7.1 and Theorem 2.2, ed_{2}(I_{14}^{3}) = ed(I_{14}^{3}) = ed(Spin_{14}) = 7. By Lemma, 4.5, ed_{2}(I_{13}^{3}) = ed_{2}(I_{13,0}^{3}) ≥ ed_{2}(I_{14,0}^{3}) − 1 = 6. On the other hand, ed(I_{13}^{3}) ≤ ed(Spin_{13}) = 6. □

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Department of Mathematical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1

E-mail address: chernous@math.ualberta.ca

Department of Mathematics, University of California, Los Angeles, CA 90095-1555, USA

E-mail address: merkurev@math.ucla.edu