ESSENTIAL DIMENSION AND ALGEBRAIC STACKS I

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

Abstract. Essential dimension is a numerical invariant of an algebraic group G which reflects the complexity of G-torsors over a field K. In the past 10 years it has been studied by many authors in a variety of contexts. In this paper we extend this notion to algebraic stacks. As an application of the resulting theory we obtain new results about the essential dimension of certain algebraic (and, in particular, finite) groups which occur as central extensions. In particular, we show that the essential dimension of the spinor group Spin_{n} grows exponentially with n, improving previous lower bounds of Chernousov-Serre and Reichstein-Youssin. In the last section we apply the result on spinor groups to show that quadratic forms with trivial discriminant and Hasse-Witt invariant are more complex, in high dimensions, than previously expected.

Contents

1. Introduction 1 2. The essential dimension of a stack: definition and first properties 5 3. Preliminaries on gerbes and canonical dimension 9 4. The essential dimension of a μ_{d}-gerbe 12 5. Proof of Theorem 1.4 17 6. Florence’s theorem 19 7. A theorem about the essential dimension of a finite group 21 8. Proof of Theorem 1.5 24 9. More on p-groups 27 10. Spinor groups 29 11. Pfister numbers 33 References 37

1. Introduction

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.

2000 Mathematics Subject Classification. Primary 14A20, 20G15, 11E04. †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.

Definition 1.1. Let a ∈ F (L), where L is an object of Fields_{k}. We say that a 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}.

These notions are relative to the base field k. To emphasize this, we will sometimes write ed_{k}(a), ed_{k}(a; p), ed_{k} F or ed_{k}(F; p) instead of ed(a), ed(a; p), ed F or ed(F; p), respectively.

If the functor F is limit-preserving, a condition that is satisfied in all cases that interest us, every element a ∈ F (L) has a field of definition K that is finitely generated over k, so ed(a) is finite; cf. Remark 2.5. On the other hand, ed F may be infinite even in cases of interest.

The following example describes a class of functors which will play a key role in the sequel.

Example 1.2. Let G be an algebraic group. Consider the Galois cohomology functor F = H^{1}(∗, G) sending K to the set H^{1}(K, G) 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. This number is usually denoted by ed_{k} G or (if k is fixed throughout) simply by ed G. Essential dimension was originally introduced and has since been extensively studied in this context; see [BR97, Rei00, RY00, Kor00, Led02, JLY02, BF03, Lem04, CS06, Gar]. The more general Definition 1.1 is due to A. Merkurjev; see [BF03, Proposition 1.17].

In this paper we define the notion of essential dimension for algebraic stacks and apply it in the “classical” setting of Example 1.2. The stacks that play a special role in these applications are gerbes banded by the multiplicative group μ_{n}; see §§3-4. Before proceeding to state out main results, we remark that the notion of essential dimension introduced in this paper also leads to interesting results for other types of stacks. In particular, in the forthcoming paper [BRV07] we will compute the essential dimension of the moduli stack M_{g,n} of smooth curves of genus g for every g ≥ 0. We also note that a related (but not equivalent) notion of arithmetic dimension has been studied by C. O’Neil [O’N05, O’N].

We will now describe the main results of this paper. Let

(1.3)  1 ⟶ Z ⟶ G ⟶ Q ⟶ 1

denote an exact sequence of algebraic groups over a field k with Z central and isomorphic to μ_{n} for some integer n > 1. For every field extension K/k the sequence (1.3) induces a connecting map ∂_{K}: H^{1}(K, Q) → H^{2}(K, Z). We define ind(G, Z) as the maximal value of ind(∂_{K}(t)), as K ranges over all field extensions of k and t ranges over all torsors in H^{1}(K, Q). (Note that ind(G, Z) does not depend on the choice of the isomorphism Z ≃ μ_{n}.) In §5 we will prove the following inequality.

Theorem 1.4. Let G be an extension as in (1.3). Assume that n a power of a prime integer p. Then ed(G; p) ≥ ind(G, Z) − dim G.

In §6 we will use Theorem 1.4 to give an alternative proof of a recent theorem of Florence about the essential dimension of a cyclic group (in a slightly strengthened form). We will then use Florence’s theorem to settle a particular case of a conjecture of Ledet [Led02, Section 3], relating the essential dimensions of the cyclic group C_{n} and the dihedral group D_{n} (n odd); see Corollary 6.2.

In §§7–9 we continue to study essential dimensions of finite p-groups. Let G be a finite abstract group. We write ed_{k} G (respectively, ed_{k}(G; p)) for the essential dimension (respectively, for the essential dimension at p) of the constant group G_{k} over the field k. Let exp(G) denote the exponent of G and let C(G) denote the center of G.

Theorem 1.5. Let G be a 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 which contains a primitive root of unity of degree exp(G).

Note that with the above hypotheses, |G/C(G)| is a complete square. In the case where G is abelian we recover the identity ed G = rank(G); see [BR97, Theorem6.1]. For most finite groups G the best previously known lower bounds on ed G were of the form

(1.6)  ed G ≥ rank(A) ,

where A was taken to be an abelian subgroup A of G of maximal rank. Theorem 1.5 represents a substantial improvement over these bounds. For example, if G is a non-abelian group of order p^{3} and k contains a primitive root of unity of degree p^{2} then Theorem 1.5 tells us that ed G = p, while (1.6) yields only ed G ≥ 2.

In §9 we record several interesting corollaries of Theorem 1.5. In particular, we show that ed(G; p) ≥ p for any non-abelian p-group G; see Corollary 9.3. We also answer a question of Jensen, Ledet and Yui [JLY02, p.204] by giving an example of a finite group G with a normal subgroup N such that ed(G/N) > ed G; see Corollary 9.6.

Another consequence of Theorem 1.5 is the following new bound on ed Spin_{n}. Here by Spin_{n} we will mean the totally split form of the spin group in dimension n over a field k.

Theorem 1.7. Suppose k is a field of characteristic ≠ 2, and that \sqrt{−1} ∈ k. If n is not divisible by 4 then

2^{⌊(n−1)/2⌋} − \frac{n(n − 1)}{2} ≤ ed(Spin_{n}; 2) ≤ ed(Spin_{n}) ≤ 2^{⌊(n−1)/2⌋} .

If n is divisible by 4 then

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

Theorem 1.7 is proved in §10, where we also prove similar estimates for the essential dimensions of pin and half-spin groups.

The lower bound of Theorem 1.7 was surprising to us because previously the best known lower bound, due of V. Chernousov and J.–P. Serre [CS06], was

(1.8) ed Spin_{n} ≥ \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].) Moreover, in low dimensions, M. Rost [Ros99] (cf. also [Gar]) computed the following table of exact values:

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.

Taken together these results seemed to suggest that ed Spin_{n} should be a slowly increasing function of n and gave no hint of its exponential growth.

The computation of ed Spin_{n} gives an example of a split, semisimple, connected linear algebraic group whose essential dimension exceeds its dimension. (Note that for a semisimple adjoint group G, ed G ≤ dim G; cf. Example 10.10.) Since ed SO_{n} = n − 1 for every n ≥ 3 (cf. [Rei00, Theorem 10.4]), it also gives an example of a split, semisimple, connected linear algebraic group G with a central subgroup Z such that ed G > ed G/Z.

In the last section we follow a suggestion of A. Merkurjev and B. Totaro to apply our results on ed Spin_{n} to a problem in the theory of quadratic forms. 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). It is well known that if q is a nondegenerate n-dimensional quadratic form whose class [q] in W(K) lies in I^{a}(K), then [q] can be expressed as the class a sum of a-fold Pfister forms. It is natural to ask how many Pfister form are needed. When a = 1 or 2 it is easy to see that n Pfister forms always suffice; see Proposition 11.1. We prove the following result, which shows that the situation is quite different when a = 3.

Theorem 1.9. Let k be a field of characteristic different from 2 and n an even positive integer. Then there is a field extension K/k and a class [q] ∈ I^{3}(K) represented by an n-dimensional quadratic form q/K such that [q] cannot be written as the sum of fewer than

\frac{2^{(n+4)/4} − n − 2}{7}

3-fold Pfister forms over K.

Finally we remark that the stack-theoretic approach to computing ed G developed in this paper has been recently extended and refined by Karpenko and Merkurjev [KM07]. Their main result is a general formula for the essential dimension of any finite p-group over any field k containing a primitive pth root of unity. This formula may be viewed as a common generalization of our Theorems 1.5 and 6.1.

Notation. Throughout this paper, a variety over a field k will be a geometrically integral separated scheme of finite type over k. Cohomology groups H^{i}(X, G), where X is a scheme and G is an abelian group scheme over X, will be taken with respect to the fppf topology unless otherwise specified. We will write μ_{n} for the group scheme of n-th roots of unity. If k is a field whose characteristic is prime to n, ζ_{n} will denote a primitive n-th root of unity in the algebraic closure of k. (Using the axiom of choice, we choose the ζ_{n} once and for all.) For typographical reasons, we sometimes write C_{n} for the cyclic group ℤ/n.

Acknowledgments. 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 K. Behrend, C.- L. Chai, D. Edidin, N. Fakhruddin, M. Florence, A. Merkurjev, B. Noohi, G. Pappas, D. Saltman and B. Totaro for helpful conversations.

2. The essential dimension of a stack: definition and first properties

We now return to the general setting of Definition 1.1. As we mentioned in Example 1.2 most of the existing theory is specific to the Galois cohomology functors F = H^{1}(∗, G) for various algebraic k-groups G. On the other hand, many naturally arising functors where the essential dimension is of interest are not of the form F (K) = H^{1}(K, G) for any algebraic group G. Two such examples are given below. In this section we identify a class of functors which is sufficiently broad to include most such examples, yet “geometric” enough to allow one to get a handle on their essential dimension. These functors are isomorphism classes of objects of an algebraic stack; see Definition 2.3 below. As we shall see in the sequel, these “new” functors turns out to be very useful even if one is only interested in the “classical” setting of Example 1.2.

Example 2.1. Let X/k be a scheme of finite type over a field k, and let F_{X}: Fields_{k} → Sets denote the functor given by K ↦ X(K). Then an easy argument due to Merkurjev shows that ed F_{X} = dim X; see [BF03, Proposition 1.17].

In fact, this equality remains true for any algebraic space X. Indeed, an algebraic space X has a stratification by schemes X_{i}. Any K-point η: Spec K → X must land in one of the X_{i}. Thus ed X = max ed X_{i} = dim X. ♠

Example 2.2. Let Curves_{g} be the functor that associates to a field K/k the set of isomorphism classes of smooth algebraic curves of genus g over K. The essential dimension of this functor is computed in [BRV07].

We are now ready to give the main definition of this section.

Definition 2.3. Suppose X is an algebraic stack over k. The essential dimension ed X of X (respectively, the essential dimension ed(X; p) of X at a prime integer p) is the essential dimension (respectively, the essential dimension at p) of the functor F_{X} : Fields_{k} → Sets which sends a field L/k to the set of isomorphism classes of objects in X (L).^{1}

As in Definition 1.1, we will write ed(X /k), or ed_{k} X when we need to be specific about the dependence on the base field k. Similarly for ed(ξ/k) or ed_{k} ξ, where ξ is an object of F_{X}, and for ed(X /k; p), ed_{k}(X; p), ed(ξ/k; p), ed_{k}(ξ; p).

All of the examples we have considered so far may be viewed as special cases of Definition 2.3. If X is a scheme of finite type (or an algebraic space), we recover Example 2.1. At the other extreme, if X = BG is the classifying stack of an agebraic group G defined over k (so that BG(T) is the category of G-torsors on T), we recover Example 1.2. If X = M_{g} is the stack of smooth algebraic curves of genus g, we recover Example 2.2.

Remark 2.4. If G is an algebraic group, we will often write ed G for ed BG. That is, we will write ed G for the essential dimension of the stack BG and not the essential dimension of the scheme underlying G. We do this to conform to the, now standard, notation introduced at the beginning of this paper (and in Example 1.2). Of course, by Example 2.1, the essential dimension of the underlying scheme is dim G.

In fact, the reader may notice that we prefer to write ed BG earlier in the paper, where we are working in a general stack-theoretic setting and ed G later on, where we are primarily concerned with essential dimensions of algebraic groups.

1In the literature the functor F_{X} is sometimes denoted by X̂ or \overline{X}.

Remark 2.5. ed X takes values in the range {±∞} ∪ ℤ_{≥0} (with −∞ occurring if and only if X is empty). On the other hand, ed ξ is finite for every object ξ ∈ X (K) and every field extension K/k.

Indeed, any field K/k can be written as a filtered direct limit K = colim_{I} K_{i} of its subfields K_{i} of finite transcendence degree. Since X is limit preserving (cf. [LMB00, Proposition 4.18]), ξ lies in the essential image of X (K_{i}) → X (K) for some i ∈ I and thus ed ξ ≤ tr deg_{k}(K_{i}) < ∞.

We now recall Definitions (3.9) and (3.10) from [LMB00]. A morphism f: X → Y of algebraic stacks (over k) is said to be representable if, for every k-morphism T → Y, where T is an affine k-scheme, the fiber product X ×_{Y} T is representable by a scheme over T. A representable morphism f: X → Y is said to be locally of finite type and of fiber dimension ≤ d if the projection X ×_{Y} T → T is locally of finite type over T and every fiber has dimension ≤ d.

Proposition 2.6. Let d be an integer X → Y be a representable k-morphism of algebraic stacks which is locally of finite type and of fiber dimension at most d. Let L/k be a field, ξ ∈ X(L) and p be a prime integer. Then

(a) ed_{k} ξ ≤ ed_{k}(f(ξ)) + d,

(b) ed_{k}(ξ; p) ≤ ed_{k}(f(ξ); p) + d,

(c) ed_{k} X ≤ ed_{k} Y + d,

(d) ed_{k}(X; p) ≤ ed_{k}(Y; p) + d.

Proof. (a) By the definition of ed_{k}(f(ξ)) we can find an intermediate field k ⊂ K ⊂ L and a morphism η: Spec K → Y such that tr deg_{k} L ≤ ed f(ξ) and the following diagram commutes.

\xymatrix{Spec L \ar[r]^{ξ} \ar[d] & X \ar[d]^{f} \\ Spec K \ar[r]^{η} & Y}

Let X_{K} =^{def} X ×_{Y} Spec K. By the hypothesis, X_{K} is an algebraic space, locally of finite type over K and of relative dimension at most d. By the commutativity of the above diagram, the morphism ξ: Spec L → X factors through X_{L}:

\xymatrix{Spec L \ar[drr]^{ξ} \ar[dr]^{ξ_{0}} \ar[ddr] \\  & X_{K} \ar[r] \ar[d] & X \ar[d]^{f} \\  & Spec K \ar[r]^{η} & Y}

Let p denote the image of ξ_{0} in X_{K}. Since X_{K} has dimension at most d, we have tr deg_{k} k(p) ≤ d. Therefore tr deg_{k} k(p) ≤ ed(f(ξ)) + d. Since ξ factors through Spec k(p), part (a) follows.

(b) Let L′/L be a field extension. By part (a), ed_{k}(ξ_{L′}) ≤ ed_{k}(f(ξ_{L′}))+d. Taking the minimum over all prime-to p extensions L′/L, we obtain the desired inequality, ed_{k}(ξ; p) ≤ ed_{k}(f(ξ); p) + d.

(c) follow from (a) and (d) follows from (b) by taking the maximum on both sides over all L/k and all ξ ∈ X (L). ♠

Corollary 2.7. Let G be an algebraic group defined over k and H be a closed subgroup of G. Then

(a) ed H ≤ ed G + dim G − dim H and

(b) ed(H; p) ≤ ed(G; p) + dim G − dim H.

More generally, suppose G is acting on an algebraic space X (over k). Then

(c) ed [X/H] ≤ ed [X/G] + dim G − dim H and

(d) ed([X/H]; p) ≤ ed([X/G]; p) + dim G − dim H.

Here [X/G] and [X/H] denote the quotient stacks for the actions of G and H on X.

Recall that the objects in [X/G](K) are, by definition, diagrams of the form

\xymatrix{T \ar[r]^{f} \ar[d]^{π} & X \\ Spec(K),}

where π is a G-torsor and f is a G-equivariant morphism.

Proof. The natural morphism [X/H] → [X/G] of quotient stacks is easily seen to be representable, of finite type and with fibers of dimension d = dim G − dim H. Applying Proposition 2.6(c) and (d) to this morphism, we obtain the inequalities (c) and (d) respectively. Specializing X to a point (i.e., to Spec(k)) with trivial G-action, we obtain (a) and (b). ♠

In general, the inequality of Proposition 2.6 only goes in one direction. However, equality holds in the following important special case. We will say that a morphism f: X → Y of stacks is isotropic if for every field extension K of k and every object η of Y(K) there exists an object ξ of X (K) such that f(ξ) is isomorphic to η.

Proposition 2.8. Let f: X → Y be an isotropic morphism of stacks fibered over Points_{k}. Then ed X ≥ ed Y and ed(X; p) ≥ ed(Y; p) for every prime p.

Proof. Both inequalities are immediate consequences of the following observation.

Let K be an extension of k and η an object of Y(K). If ξ is an object of X (K) such that f(ξ) is isomorphic to η, then clearly a field of definition for ξ is also a field of definition for η. Thus ed ξ ≥ ed η. ♠

Proposition 2.8 can be used to deduce the following well-known inequality. See [BF03] for another proof.

Theorem 2.9. Let G be a linear algebraic group over a field k admitting a generically free representation on a vector space V. Then

ed BG ≤ dim V − dim G.

Proof. Let U denote a dense G-stable Zariski open subscheme of V on which G acts freely. Then [U/G] is an algebraic space of dimension dim V − dim G and the map [U/G] → BG is representable and isotropic. ♠

3. Preliminaries on gerbes and canonical dimension

Let X be a gerbe defined over a field K banded by an abelian K-group scheme G. For background material on gerbes we refer the reader to [Mil80, p. 144] and [Gir71, IV.3.1.1].

There is a natural notion of equivalence of gerbes banded by G; the set of equivalence classes is in a natural bijective correspondence with the group H^{2}(K, G). Given a gerbe X banded by G, we write [X] for its class in H^{2}(K, G). The identity is the class [B_{K}G] of the neutral gerbe B_{K}G.

We remark that it makes sense to talk about a gerbe banded by G, where G is not necessarily abelian, but we will not need to work in this more general setting, which makes the definition considerably more involved.

Let K be a field and let 𝔾_{m} denote the multiplicative group scheme over K. Recall that the group H^{2}(K, 𝔾_{m}) is canonically isomorphic to the Brauer group Br(K) of Brauer equivalence classes of central simple algebras (CSAs) over K. By Wedderburn’s structure theorem, any CSA over K isomorphic to the matrix algebra M_{n}(D) for D a division algebra over K which is unique up to isomorphism. Moreover, if A and B are two Brauer equivalent CSAs, the division algebras D and E corresponding to A and B respectively are isomorphic. For a class [A] ∈ Br(K), the index of A is \sqrt{dim_{K} D}.

Let α ∈ H^{2}(K, μ_{n}), where n is a positive integer prime to char K. We define the index ind α to be the index of the image on α under the composition

H^{2}(K, μ_{n}) ↪ H^{2}(K, 𝔾_{m}) ⟶^{≅} Br(K).

Note that the index of α is the smallest integer d such that α is in the image of the (injective) connecting homomorphism

(3.1)  ∂: H^{1}(K, PGL_{d}) ⟶ Br(K).

arising from the short exact sequence

(3.2)  1 ⟶ 𝔾_{m} ⟶ GL_{d} ⟶ PGL_{d} ⟶ 1.

The exponent ord([A]) of a class [A] ∈ Br K is defined to be its order in the Brauer group. The exponent ord([A]) always divides the index ind([A]) [Her68, Theorem 4.4.5].

In the next section, we address the problem of computing the essential dimension of a μ_{d}-gerbe over a field K. Our computation will rely, in a key way, on the notion of canonical dimension introduced in [BR05] and [KM06], which we shall now recall.

Let X be a smooth projective variety defined over a field K. We say that L/K is a splitting field for X if X(K) ≠ ∅. A splitting field L/K is called generic if for every splitting field L_{0}/K there exists a K-place L → L_{0}. The canonical dimension of X is defined as the minimal value of tr deg_{K}(L), where L/K ranges over all generic splitting fields. Note that the function field L = K(X) is a generic splitting field of X; see [KM06, Lemma 4.1]. In particular, generic splitting fields exist and cdim(X) is finite.

If X is a K-scheme and p is a prime integer, we also recall the relative variant of this notion from [KM06]. We say that L/K is a p-generic splitting field for X if for every splitting field L_{0}/K there is a prime-to-p extension L_{1}/L and a K-place L → L_{1}. The minimal value of tr deg_{K}(L′), as L′ ranges over all p-generic splitting fields, is called the p-canonical dimension of X and is denoted by cdim_{p}(X). (Here, as usual, by a prime-to-p extension we mean a finite field extension whose degree is prime to p.)

Lemma 3.3. If L/K is a p-generic extension for X then

(a) any intermediate extension M/K, where K ⊂ M ⊂ L is also pgeneric, and

(b) any finite prime-to-p extension L′/K is also p-generic.

Proof. (a) Let L_{0}/K be a splitting field for X. Since L is generic, there is a prime-to-p extension L_{1}/L_{0} and a K-place L → L_{1}. Restricting this place to M, we obtain a desired place M → L_{1}.

Part (b) is an immediate consequence of [KM06, Lemma 3.2]. ♠

The determination functor D_{X} : Fields_{K} → Sets is defined as follows: D associates to a field L/K the empty set, if X(L) = ∅, and a set consisting of one point, which we will denote by a(L), if X(L) ≠ ∅. The natural map D(L_{1}) → D(L_{2}) is then uniquely determined for any K ⊂ L_{1} ⊂ L_{2}.

Lemma 3.4. Let X be a complete regular K-variety then (a) cdim(X) = ed(D_{X}).

(b) cdim_{p}(X) = ed(D_{X}; p).

Proof. (a) Let L be a generic splitting field for X. By [KM06, Lemma 2.1], any subfield of L containing F is also a generic splitting field. Therefore ed D_{X} ≥ ed a(L) ≥ cdim(X).

To prove the opposite inequality, it suffices to show that ed a(F) ≤ tr deg_{K} L for any splitting field F/K and any generic splitting field L/K of X. In other words, given F and L as above, we want to construct an intermediate splitting field K ⊂ F_{0} ⊂ F with tr deg_{K}(F_{0}) ≤ tr deg_{K}(L). By the definition of a generic splitting field, there is a place L → F. That is, there is a valuation ring R with quotient field L and a local homomorphism f: R → F. We claim that the residue field F_{0} = R/m, where m is the maximal ideal of R, has the desired properties. Clearly K ⊂ F_{0} ⊂ F, and tr deg_{K}(F_{0}) ≤ tr deg_{K}(R) = tr deg_{K}(L), so we only need to check that F_{0} is a splitting field for X.

Since L is itself a splitting field, we have a diagram

\xymatrix{Spec L \ar[r] \ar[d] & X \ar[d] \\ Spec R \ar[r] & Spec K.}

By the valuative crition, we obtain a map Spec R → X and hence, a map Spec F_{0} = Spec R/m → X. Thus F_{0} is splitting field for X, as claimed.

(b) Let L be a p-generic splitting field for X. Lemma 3.3 tells us that if L′/L is a prime-to-p extension and K ⊂ M ⊂ L′ is an intermediate splitting field then M is again p-generic. Thus ed(a(L); p) = minimal value of tr deg_{K}(M) ≥ cdim_{p}(X).

To prove the opposite inequality, we need to show that ed(a(F); p) ≤ tr deg_{K}(L) for any splitting field F/K and any p-generic splitting field L/K. Indeed, after replacing F by a prime-to-p-extension F′, we obtain a Kplace L → F′. Now the same argument as in part (a) shows that there is an intermediate splitting subfield K ⊂ F_{0} ⊂ F′ such that tr deg_{K}(F_{0}) ≤ tr deg_{K}(L). Thus

ed(a(F); p) ≤ ed(a(F′)) ≤ tr deg_{K}(F_{0}) ≤ tr deg_{K}(L) ,

as claimed. ♠

Of particular interest to us will be the case where X is a Brauer-Severi variety over K. Let m be the index of X. If m = p^{a} is a prime power then

(3.5)  cdim(X) = cdim_{p}(X) = p^{a} − 1 ;

see [KM06, Example 3.10] or [BR05, Theorem 11.4]. If the highest power of p dividing m is p^{a} then

(3.6)  cdim_{p}(X) = p^{a} − 1 ;

see [KM06, Example 5.10]. On the other hand, if m is divisible by more than one prime, cdim(X) is not known in general. Suppose m = p_{1}^{a_{1}} … p_{r}^{a_{r}}. Then the class of P in Br L is the sum of classes α_{1}, …, α_{r} whose indices are p_{1}^{a_{1}}, …, p_{r}^{a_{r}}. Denote by X_{1}, …, X_{r} the Brauer–Severi varieties with classes α_{1}, …, α_{r}. It is easy to see that K(X_{1} × ⋯ × X_{r}) is a generic splitting field for X. Hence,

cdim(X) ≤ dim(X_{1} × ⋯ × X_{r}) = p_{1}^{a_{1}} + ⋯ + p_{r}^{a_{r}} − r .

In [CTKM06], Colliot-Thélène, Karpenko and Merkurjev conjectured that equality holds, i.e.,

(3.7)  cdim(X) = p_{1}^{a_{1}} + ⋯ + p_{r}^{a_{r}} − r

for all m ≥ 2. As we mentioned above, this in known to be true of m is a prime power (i.e., r = 1). Colliot-Thélène, Karpenko and Merkurjev also proved (3.7) for m = 6; see [CTKM06, Theorem 1.3].

4. The essential dimension of a μ_{d}-gerbe

We are now ready to proceed with our main theorem on gerbes.

Theorem 4.1. Let d be an integer with d > 1. Let K be a field and x ∈ H^{2}(K, μ_{d}), where char K does not divide d. Denote the image of x in H^{2}(K, 𝔾_{m}) by y, the μ_{d}-gerbe associated to x by X, the 𝔾_{m}-gerbe associated to y by Y, and the Brauer–Severi variety associated to y by P. Then

(a) ed(Y) = cdim P and (b) ed X = cdim P + 1. Moreover, if the index of x is a prime power p^{r} then

(c) ed(Y; p) = ed(Y) = p^{r} − 1 and (d) ed(X; p) = ed X = p^{r}.

Proof. The functor F_{Y} : Fields_{K} → Sets sends a field L/K to the empty set, if P (L) = ∅, and to a set consisting of one point, if P (L) ≠ ∅. In other words, F_{Y} is the determination functor D_{P} introduced in the previous section. Thus ed(Y) = ed(D_{P}) and ed(Y; p) = ed(D_{P}; p). On the other hand, by Lemma 3.4, ed(D_{P}) = cdim(P) and ed(D_{P}; p) = cdim_{p}(P). This, in combination with (3.5), proves parts (a) and (c).

(b) First note that the natural map X → Y is finite type and representable of relative dimension 1. By Proposition 2.6 this implies that ed X ≤ ed(Y)+ 1. By part (a) it remains to prove the opposite inequality, ed X ≥ ed(Y)+1. We will do this by constructing an object α of X whose essential dimension is ≥ ed(Y) + 1.

We will view X as a torsor for Bμ_{d} in the following sense: One has maps

X × Bμ_{d} → X
X × X → Bμ_{d}

satisfying various compatibilities, where the first map is the action of Bμ_{d} on X and the second map is the “difference” of two objects of X. For the definition and a discussion of the properties of these maps, see [Gir71, Chapter IV, Sections 2.3, 2.4 and 3.3]. (Note that, in the notation of Giraud’s book, X ∧ Bμ_{d} ≅ X and the action operation above arises from the map X × Bμ_{d} → X ∧ Bμ_{d} given in Chapter IV, Proposition 2.4.1. The “difference” operation — which we will not use here — arises similarly from the fact that, in Giraud’s notation, HOM(X , X) ≅ Bμ_{d}.)

Let L = K(P) be the function field of P. Since L splits P, we have a natural map a : Spec L → Y. Moreover since L is a generic splitting field for P,

(4.2) ed(a) = cdim(P) = ed(Y), (4.3) ed(a; p) = cdim_{p}(P) = ed(Y; p).

where we view a as an object in Y. Non-canonically lift a : Spec L → Y to a map Spec L → X using the fact that X → Y is isotropic. Let Spec L(t) → Bμ_{d} denote the map classified by (t) ∈ H^{1}(L(t), μ_{d}) = L(t)^{×}/L(t)^{×d}. Composing these two maps, we obtain an object

α : Spec L(t) → X × Bμ_{d} → X .

in X (L(t)). Our goal is to prove that ed(α) ≥ ed(Y) + 1. In other words, given a diagram of the form

(4.4)  \xymatrix{Spec L(t) \ar[r]^{α} \ar[d] & X \\ Spec M \ar[ur]^{β}}

where K ⊂ M ⊂ L is an intermediate field, we want to show that tr deg_{K}(M) ≥ ed(Y)+1. Assume the contrary: there is a diagram as above with tr deg_{K}(M) ≤ ed(Y). Let ν: L(t)^{∗} → ℤ be the usual discrete valuation corresponding to t and consider two cases.

Case 1. Suppose the restriction ν_{|M} of ν to M is non-trivial. Let M_{0} denote the residue field of ν and M_{≥0} denote the valuation ring. Since Spec M → X → Y, there exists an M-point of P. Then by the valuative criterion of properness for P, there exists an M_{≥0}-point and thus an M_{0}- point of P. Passing to residue fields, we obtain the diagram

\xymatrix{Spec L \ar[r]^{a} \ar[d] & Y \\ Spec M_{0} \ar[ur] & .}

which shows that ed(a) ≤ tr deg_{K} M_{0} = tr deg_{K} M − 1 ≤ ed(Y) − 1, contradicting (4.2).

Case 2. Now suppose the restriction of ν to M is trivial. The map Spec L → X sets up an isomorphism X_{L} ≅ B_{L}μ_{d}. The map Spec L(t) → X factors through X_{L} and thus induces a class in Bμ_{d}(L(t)) = H^{1}(L(t), μ_{d}). This class is (t). Tensoring the diagram (4.4) with L over K, we obtain

\xymatrix{Spec L(t) ⊗ L \ar[r]^{α} \ar[d] & X_{L} ≅ B_{L}μ_{d} \\ Spec M ⊗ L \ar@{-->}[ur]^{β}}

Recall that L = K(P) is the function field of P. Since P is absolutely irreducible, the tensor products L(t) ⊗ L and M ⊗ L are fields. The map Spec M ⊗ L → B_{L}μ_{d} is classified by some m ∈ (M ⊗ L)^{×}/(M ⊗ L)^{×d} = H^{1}(M ⊗ L, μ_{d}). The image of m in L(t) ⊗ L would have to be equal to t modulo d-th powers. We will now derive a contradiction by comparing the valuations of m and t.

To apply the valuation to m, we lift ν from L(t) to L(t) ⊗ L. That is, we define ν_{L} as the valuation on L(t) ⊗ L = (L ⊗ L)(t) corresponding to t. Since ν_{L}(t) = ν(t) = 1, we conclude that ν_{L}(m) ≡ 1 (mod d). This shows that ν_{L} is not trivial on M ⊗ L and thus ν is not trivial on M, contradicting our assumption. This contradiction completes the proof of part (b).

The proof of part (d) proceeds along similar lines, except for a small complication in prime characteristic, which is resolved by Lemma 4.7 below. For the sake of completeness we outline the argument.

Once again, applying Proposition 2.6 to the representable projection map X → Y of relative dimension 1, we obtain the inequality ed(X; p) ≤ ed(Y; p)+ 1. By part (c) it remains to show that ed(X; p) ≥ ed(Y; p) + 1. We will do this by showing that

(4.5)  ed(α; p) ≥ ed(Y; p) + 1 ,

where α: Spec L(t) → X is the same object in X (L(t)) we constructed in the proof of part (b). Here we continue to denote the function field of P by L = K(P). Once again, assume the contrary: there exists a finite field extension L′/L(t) of degree prime to p, such that ed(α_{L′}) ≤ ed(Y; p). Let L′_{s} be the separable closure of L(t) in L′. By Lemma 4.7 ed(α_{L′_{s}}) = ed(α_{L′}). Hence, after replacing L′ by L′_{s}, we may assume that L′ is a finite separable extension of L(t) of degree prime to p. The assumption that ed(α_{L′}) ≤ ed(Y; p) means, by definition, that there exists a commutative diagram

(4.6)  \xymatrix{Spec(L′) \ar[dr] \ar[dd]_{prime-to-p} \\  & Spec M \ar[dr]^{β} \\ Spec L(t) \ar[rr]^{α} &  & X}

where K ⊂ M ⊂ L′ is an intermediate field and tr deg_{K}(M) ≤ ed(Y; p).

Let ν: L(t)^{∗} → ℤ be the discrete valuation corresponding to t and ν′_{1}, …, ν′_{r} be the liftings of ν to L′. Since L′/L is separable, [Lan65, Proposition XII.18] tells us that

e_{1}f_{1} + ⋯ + e_{r}f_{r} = [L′ : L] ,

where e_{i} and f_{i} are, respectively, the ramification index and the residue class degree of ν′_{i}. Since [L′ : L] is prime to p, at least one of the valuations ν′_{1}, …, ν′_{r}, say ν′: (L′)^{∗} → \frac{1}{e}ℤ, has the property that its ramification index e and residue class degree f are both prime to p.

Once again we consider two cases. If the restriction of ν′ to M is nontrivial then after passing to residue fields and arguing as in Case 1 of the proof of part (b), we arrive at the diagram

\xymatrix{Spec(L′_{0}) \ar[dr] \ar[dd]_{degree f} \\  & Spec M_{0} \ar[dr] \\ Spec L \ar[rr] &  & Y}

which shows that ed(a; p) ≤ tr deg_{K} M_{0} = tr deg_{K} M − 1 ≤ ed(Y; p) − 1, contradicting (4.3).

If the restriction of ν′ to M is trivial then the induced valuation on M ⊗_{K} L′ is trivial, and we argue as in Case 2 in the proof of part (b). Tensoring the diagram 4.6 with L over K and identifying X_{L} with B_{L}μ_{d} via our chosen map ϕ: Spec(L) → X, we obtain the diagram

\xymatrix{Spec(L′ ⊗_{K} L) \ar[dr] \ar[dd]_{prime-to-p} \\  & Spec M ⊗_{K} L \ar[dr]^{β⊗_{K}ϕ} \\ Spec L(t) ⊗_{K} L \ar[rr]^{α⊗_{K}ϕ} &  & B_{L}μ_{d}.}

Since L is absolutely irreducible, L′ ⊗_{K} L, M ⊗_{K} L and L(t) ⊗_{K} L are fields. The map Spec M ⊗_{K} L → B_{L}μ_{d} gives rise to an m ∈ (M ⊗L)^{×}/(M ⊗L)^{×d} = H^{1}(M ⊗ L, μ_{d}) whose image in L′ ⊗_{K} L is equal to t, modulo d-th powers.

Lifting μ from L′ to L′ ⊗_{K} L (so that ν′_{L} is trivial on 1 ⊗_{K} L), we see that

ν′_{L}(m) − ν′_{L}(t) = ν′_{L}(m) − 1 ∈ d\frac{1}{e}ℤ .

Since d and e are relatively prime, this shows that ν′_{L}(m) ≠ 0. We conclude that ν′_{L} is non-trivial on M ⊗_{K} L. Consequently, ν′ is non-trivial on M, contradicting our assumption. This contradiction completes the proof of part (d) (modulo Lemma 4.7 below). ♠

Lemma 4.7. Let F be a functor from the category of field extensions of K to the category of sets.

(a) Suppose that the natural map F (A) → F (B) is bijective for every algebraic inseparable field extension B/A. Then ed(a) = ed(a_{B}) for every algebraic inseparable field extension B/A and every a ∈ F (A).

(b) Let X be a gerbe over a field K banded by μ_{d}, A a field over K, and B/A an algebraic inseparable field extension. Suppose d is not divisible by char K. Then ed(a) = ed(a_{B}) for every a ∈ X (A).

Proof. (a) The inequality ed(a) ≥ ed(a_{B}) follows immediately from Definition 1.1. To prove the opposite inequality it suffices to show that if a_{B} descends to an intermediate field M, where K ⊂ M ⊂ B, then a descends to an intermediate field M_{0}, where K ⊂ M_{0} ⊂ A and tr deg_{K}(M_{0}) ≤ tr deg_{K}(M). To prove this assertion, set M_{0} = M ∩ A and p = char K. Then for every m ∈ M there is a power of p, say e = p^{r} such that m^{e} lies in both A and M (and hence, in M_{0}). This shows that M is algebraic and inseparable over M_{0}; cf. [Lan65, VII.7]. Since M_{0} is a subfield of M, the inequality tr deg_{K}(M_{0}) ≤ tr deg_{K}(M) is obvious. (In fact, equality holds, since M is algebraic over M_{0}.) The commutative diagram

\xymatrix{F (M) \ar[r] & F (B) ∋ a_{B} \\ F (M_{0}) \ar[u]^{≃} \ar[r] & F (A) ∈ a \ar[u]^{≃}}

shows that a descends to M_{0}.

(b) By part (a) it suffices to show that the natural map X (A) → X (B) is an isomorphism for every algebraic inseparable extension B/A. If A doesn’t split the class [X] ∈ H^{1}(K, μ_{d}) then neither does B, so, X (A) = X (B) = ∅, and there is nothing to prove. If A splits [X], i.e., there is a map Spec(A) → X then we can use this map to trivialize X. The map X (A) → X (B) can then be identified with the natural map H^{1}(A, μ_{d}) → H^{1}(B, μ_{d}) or equivalently,

A^{∗}/(A^{∗})^{d} → B^{∗}/(B^{∗})^{d} .

We want to show that this map is an isomorphism. To do this, we recall that for every y ∈ B^{∗} there exists an r ≥ 0 such that y^{p^{r}} lies in A. Since p and d are relatively prime, both injectivity and surjectivity of the above map follow. ♠

In view of Theorem 4.1(a), Conjecture (3.7) of Colliot-Thélène, Karpenko and Merkurjev be restated in the language of essential dimension of gerbes as follows.

Conjecture 4.8. If X is a gerbe banded by μ_{n} over a field K, let p_{1}^{a_{1}} … p_{r}^{a_{r}} be the decomposition into prime factors of the index of the class of X in the Brauer group of K. Then

ed X = p_{1}^{a_{1}} + ⋯ + p_{r}^{a_{r}} − r + 1.

When the index is 6 this follows from [CTKM06, Theorem 1.3].

In view of the fact that the conjecture holds for r = 1, it can also be rephrased as follows: if m and n are relatively prime positive integers, X and Y are gerbes banded by μ_{m} and μ_{n}, then

ed(X × Y) = ed X + ed Y − 1.

Back to the language of canonical dimension, one could ask the following more general question. Let X and Y be smooth projective varieties over a field K. Assume that there are no rational functions X −−→ Y or Y −−→ X. Then is it true that cdim(X × Y) = cdim(X) + cdim(Y)? A positive answer to this question would imply Conjecture 4.8.

5. Proof of Theorem 1.4

In this section k is a field and p is a prime number not equal to char k. We begin with some preliminary facts.

Proposition 5.1. Let U be an integral algebraic space locally of finite type over k with function field K =^{def} k(U), and let f: X → U be a stack over U. Let X_{K} denote the pullback of X to Spec K. Then,

ed X ≥ ed(X_{K}/K) + dim U;
ed(X; p) ≥ ed(X_{K}/K; p) + dim U.

Proof. Let j : X_{K} → X denote the inclusion. If ξ : Spec L → X_{K} be a morphism, then it is easy to see that ed(ξ/K) = ed(j ∘ ξ/k) + dim U. This implies the first inequality of the proposition directly. The second inequality then follows by taking the infinum of ed(ξ_{M}/K) for all prime-to-p extensions M/L. ♠

Let X be a locally noetherian stack over a field k with presentation P: X → X. Recall that the dimension of X at a point ξ: Spec K → X is given by dim_{x}(X) − dim_{x} P where x is an arbitrary point of X lying over ξ [LMB00, (11.14)]. Let Y be stack-theoretic closure of the image of ξ; that is, the intersection of all the closed substacks Y_{i} such that ξ^{−1}(Y_{i}) = Spec K. The morphism ξ factors uniquely through Y ⊆ X. We defined the dimension of the point ξ to be the dimension of the stack Y at the point Spec K → Y.

Proposition 5.2. Let X → Y be a morphism of algebraic stacks over a field k. Let K/k be a field extension and let y: Spec K → Y be a point of dimension d ∈ ℤ. Let X_{K} =^{def} X ×_{Y} Spec K. Then

ed(X_{K}/K) ≤ ed(X /k) − d;
ed(X_{K}/K; p) ≤ ed(X /k; p) − d.

Proof. By [LMB00, Theorem 11.5], Y is the disjoint union of a finite family of locally closed, reduced substacks Y_{i} such that each Y_{i} is an fppf gerbe over an algebraic space Y_{i} with structural morphism A_{i}: Y_{i} → Y_{i}. We can therefore replace Y by one of the Y_{i} and assume that Y is an fppf gerbe over an algebraic space Y. Without loss of generality, we can assume that Y is an integral affine scheme of finite type over k.

Let p be the image of y in Y. Since Y is limit-preserving, we can find an integral affine scheme U equipped with a morphism i: U → Y and a dominant morphism j: Spec K → U such that y is equivalent to i ∘ j. We can also assume that the composition U → Y → Y is dominant.

Since Y is a gerbe over Y, it follows that U → Y is representable of fiber dimension at most dim U − d. Now, form the following diagram with Cartesian squares.

\xymatrix{X_{K} \ar[r] \ar[d] & Spec K \ar[d] \\ X_{U} \ar[r] \ar[d] & U \ar[d] \\ X \ar[r] & Y}

Since the vertical maps in the lower square are finite type and representable of fiber dimension at most dim U −d, it follows from Propositions 2.6 and 5.1 that

ed(X_{K}/K) ≤ ed(X_{k(U)}/k(U))
≤ ed X_{U} − dim U
≤ ed X + dim U − d + dim U
≤ ed X − d.  ♠

Similarly, ed(X_{K}/K; p) ≤ ed(X; p) − d,

We now proceed with the proof of Theorem 1.4, which we restate for the convenience of the reader. Let

(5.3)  1 ⟶ Z ⟶ G ⟶ Q ⟶ 1

denote an extension of groups over a field k, with Z central and isomorphic to μ_{n} for some integer n > 1. Recall from the introduction, that we defined ind(G, Z) as the maximal value of the index ind(∂_{K}(t)), as K ranges over all field extensions of k and t ranges over all torsors in H^{1}(K, Q).

Theorem 5.4. Let G be an extension as in (5.3). Assume that n = p^{r} for some non-negative integer r. Then ed_{k}(G; p) ≥ ind(G, Z) − dim G.

Proof. Let K/k be a field extension and let t : Spec K → BQ be a Q-torsor over Spec K. The dimension of BQ at the point t is − dim Q = − dim G. Let X denote the pull-back in the following diagram.

\xymatrix{X \ar[r] \ar[d] & Spec K \ar[d]^{t} \\ BG \ar[r] & BQ}

By Proposition 5.2, ed(X /K; p) ≤ ed(BG/k; p)+dim G. On the other hand, since BG is a gerbe banded by Z over BQ, X is a gerbe banded by Z over Spec K. Therefore, by Theorem 4.1(d), ed(X /K; p) = ind ∂_{K}(t). By substitution, ind ∂_{K}(t)−dim G ≤ ed_{k}(BG; p). Since this inequality holds for all field extensions K/k and all Q-torsors t over K, the theorem follows. ♠ Remark 5.5. Suppose G is a simple algebraic group whose center Z is cyclic. It is tempting to apply Theorem 1.4 to the natural sequence

1 ⟶ Z ⟶ G ⟶ G^{ad} ⟶ 1

where the adjoint group G^{ad} is G/Z. Given a torsor t ∈ H^{1}(K, G^{ad}), the central simple algebra representing ∂_{K}(t) ∈ H^{2}(K, Z) is called the Tits algebra of t. The values of the index of the Tits algebra were studied in [Tit92], where this index is denoted by b(X) (for a group of type X) and its possible values are listed on p. 1133. A quick look at this table reveals that for most types these indices are smaller than dim G, so that the bound of Theorem 1.4 becomes vacuous. The only exception are groups of types B and D, in which case Theorem 1.4 does indeed, give interesting bounds; cf. Remark 10.7.

Remark 5.6. If n ≥ 2 is not necessarily a prime power, then the above argument shows that

ed_{k} G ≥ cdim(∂_{K}(t)) − dim G .

Here by cdim(∂_{K}(t)) we mean the canonical dimension of the Brauer-Severi variety representing the class of ∂_{K}(t) in H^{2}(K, μ_{n}). Assuming that Conjecture 4.8 holds, we obtain the following (conjectural) inequality: if ind(G, Z) = p_{1}^{a_{1}} … p_{r}^{a_{r}} is the prime factorization of ind(G, Z) then

ed_{k} G ≥ p_{1}^{a_{i}} + ⋯ + p_{r}^{a_{r}} − r + 1 − dim G .

As we remarked at the end of §4, Conjecture 4.8 (and thus the above inequality) is known to be true if ind(G, Z) is a prime power (i.e., r = 1) or ind(G, Z) = 6.

6. Florence’s theorem

In this section we give an alternative proof of a recent theorem of Florence (in a slightly strengthened form) by combining the lower bound of Theorem 1.4 with the Brauer–Rowen Theorem. The idea to use the Brauer– Rowen theorem in this context is due to Florence. Thus while our proof is different from the one in [Flo], it is not entirely independent.

Theorem 6.1 (M. Florence [Flo]). Let p be a prime, k a field of characteristic ≠ p. Suppose ζ_{p^{n}} ∈ k but ζ_{p^{n+1}} ∉ k for some integer n ≥ 1. Moreover, if p = 2 and n = 1, assume also that k(ζ_{4}) ≠ k(ζ_{8}). Then

ed_{k}(C_{p^{m}}; p) = ed_{k} C_{p^{m}} = \begin{cases}p^{m−n} & if n < m, \\ 1 & if n ≥ m. \end{cases}

Before proceeding with the proof, we remark that, in general, the value of ed_{k} C_{p^{m}} is not known if ζ_{p} ∉ k. On the other hand, for the sake of computing ed_{k}(C_{p^{m}}; p), we may replace k by k(ζ_{p}) and then apply Theorem 6.1. Indeed, since [k(ζ_{p}) : k] is prime to p, ed_{k}(C_{p^{m}}; p) = ed_{k(ζ_{p})}(C_{p^{m}}; p).

Proof. If m ≤ n, then C_{p^{m}} = μ_{p^{m}}. Therefore ed C_{p^{m}} = 1 (see [BF03, Example 2.3]) and hence, ed(C_{p^{m}}; p) = 1 as well. We can therefore restrict our attention to the case where n < m.

We first show that ed C_{p^{m}} ≤ p^{m−n}. To do this, pick a faithful character χ: C_{p^{n}} → 𝔾_{m} defined over K and set V =^{def} ind_{C_{p^{n}}}^{C_{p^{m}}} χ. A simple calculation shows that V is faithful, thus, V is generically free since C_{p^{m}} is finite. By Theorem 2.9, it follows that ed C_{p^{m}} ≤ dim V = p^{m−n}.

It remains to show that ed(C_{p^{m}}; p) ≥ p^{m−n}. By Theorem 1.4 it suffices to show that ind(C_{p^{m}}, C_{p^{n}}) ≥ p^{m−n}. To establish this inequality, we will view the representation V as a homomorphism ρ: C_{p^{m}} → GL(V) of algebraic groups. Let π : GL(V) → PGL(V) denote the obvious projection and note that the kernel of π ∘ρ is exactly C_{p^{n}}. It follows that we have a commutative diagram

\xymatrix{0 \ar[r] & C_{p^{n}} \ar[r] \ar[d] & C_{p^{m}} \ar[r]^{ρ} \ar[d] & C_{p^{m−n}} \ar[r] \ar[d]^{ι} & 1 \\ 1 \ar[r] & 𝔾_{m} \ar[r] & GL(V) \ar[r]^{π} & PGL(V) \ar[r] & 1}

where the rows are exact and the columns are injective.

Let K/k be a field extension and let t ∈ H^{1}(K, C_{p^{m}}) be a torsor. Let ι_{∗}: H^{1}(K, C_{p^{m−n}}) → H^{1}(K, PGL(V)) denote the map induced by ι. Then, from the commutativity of the above diagram (and the injectivity of the columns), it follows that ind_{K}(t) is the index of the CSA ι_{∗}(t).

We claim that there is a field extension K/k and a t ∈ H^{1}(K, C_{p^{m}}) such that ι_{∗}(t) is a division algebra. From this it will easily follow that ind_{K}(t) = dim V = p^{m−n}. In fact, we will take t ∈ H^{1}(K, C_{p^{m}}) to be a versal element, constructed as follows. Let L = K(x_{1}, …, x_{p^{m−n}}) denote the field obtained by adjoining p^{m−n} independent variables to K, and let C_{p^{m−n}} act on L by cyclically permuting the variables, i.e., a · x_{i} = x_{i+a} (mod p^{m−n}) for any a ∈ C_{p^{m−n}}. Let K = L^{C_{p^{m−n}}}. Then L/K defines a C_{p^{m−n}}-torsor t over K.

The CSA corresponding to ι_{∗}(t) ∈ H^{1}(K, PGL_{n}) is the Brauer-Rowen algebra R_{p^{n},p^{m},p^{m}}; cf. [Row88, §7.3]. The fact that this algebra has index n is a variant of a theorem of Brauer; for a proof see [Flo, Theorem 2.17] or (for k = ℚ(ζ_{p}) only) [Row88, Theorem 7.3.8]. ♠

Let D_{n} be the dihedral group of order 2n. Ledet [Led02, Section 3] conjectured that if n is odd then ed_{k} C_{n} = ed_{k} D_{n} over any field k of characteristic zero. As an application of Theorem 6.1 we will now prove Ledet’s conjecture in the case where n = p^{r} is a prime power and k contains a primitive pth root of unity.

Corollary 6.2. Let p be an odd prime, m ≥ 1 be an integer, and k be a field containing a primitive pth root of unity. Then ed_{k} D_{p^{m}} = ed_{k} C_{p^{m}}.

Proof. If ζ_{p^{m}} ∈ k then we know that

ed_{k} C_{p^{m}} = ed_{k} D_{p^{m}} = 1 ;

see the proof of [BR97, Theorem 6.2]. Thus we may assume ζ_{p^{m}} ∉ k. Let n be the largest integer such that ζ_{p^{n}} ∈ k. By our assumption 1 ≤ n ≤ m − 1. Since C_{p^{m}} ⊂ D_{p^{m}}, we have ed_{k} D_{p^{m}} ≥ ed_{k} C_{p^{m}}.

To prove the opposite inequality, note that D_{p^{m}} ≃ C_{p^{m}} ⋊ C_{2} has a subgroup isomorphic to D_{p^{n}} = C_{p^{n}} ⋊ C_{2} of index p^{m−n}. As we pointed out above, ζ_{p^{n}} ∈ k implies ed_{k} D_{p^{n}} = 1. Thus

ed_{k} D_{p^{m}} ≤ (ed_{k} D_{p^{n}}) · [D_{p^{m}} : D_{p^{n}}] = 1 · p^{m−n} = ed C_{p^{m}} .

where the inequality is given by [Led02, Section 3] and the last equality by Theorem 6.1. ♠

7. A theorem about the essential dimension of a finite group

The following proposition, extending [BR97, Theorem 5.3] and [Kan06, Theorem 4.5], will be used in the proof of Theorem 1.5 in the next section.

Theorem 7.1. Let G be a finite group, H be a central cyclic subgroup of G, and χ: G → k^{∗} be a character of G whose restriction to H is faithful. Assume char k does not divide |G|. If H is maximal among central cyclic subgroups of G then

(a) ed_{k} G = ed_{k}(G/H) + 1.

(b) Moreover, if G is a p-group then ed_{k}(G; p) = ed_{k}(G/H; p) + 1.

Proof. Let ϕ: G → G/H ↪ GL(V) be a faithful representation of G/H. Then ϕ ⊕ χ: G → GL(V × 𝔸^{1}) is a faithful representation of G.

(a) To prove the inequality ed G ≤ ed(G/H) + 1, recall that ed(G/H) is the minimal dimension of a faithful G/H-variety X which admits a dominant G/H-equivariant rational map f: V −−→ X; see [BR97]. Then

f × id: V × 𝔸^{1} −−→ X × 𝔸^{1}

is a dominant rational map of G-varieties. Thus

ed G ≤ dim(X × 𝔸^{1}) = ed(G/H) + 1 .

We will now prove the opposite inequality, ed G ≥ ed(G/H) + 1. Choose a dominant rational map

V × 𝔸^{1} −−→ Y

of faithful G-varieties such that dim Y = ed G. Equivalently, if K = k(V), we choose G-invariant subfield F ⊂ K(t) such that the G-action on F is faithful and tr deg_{k}(F) = ed G. (Here, of course, F = k(Y).)

Let ν: K(t) → ℤ be the natural valuation associated to t (and trivial on K). Restricting ν to F we obtain a discrete valuation ν_{|F} of F. Denote the residue field for this valuation by F_{0}.

Claim: (i) ν_{|F} is non-trivial, and (ii) the inertia group of ν_{|F} (i.e., the subgroup of G that acts trivially on F_{0}) is precisely H.

To prove (i), assume the contrary. Then there is a natural G-equivariant embedding of F in the residue field K(t)_{0} = K. Since H acts faithfully on F and trivially on K, this is impossible.

To prove (ii), let H_{F} ⊂ G be the inertia group for ν_{|F}. By our construction H is the inertia group for the valuation ν on K(t). Hence, H ⊂ H_{F}. On the other hand, since char k is prime to |G|, H_{F} is cyclic (see [Ser79, Corollaries 2 and 3 to Proposition IV.7]) and central (see [Ser79, Propositioni IV.10]). By our assumption H is maximal among central cyclic subgroups of G; thus H = H_{F}, proving (ii).

We are now ready to complete the proof of the inequality ed G ≥ ed(G/H)+ 1 (and thus of part (a)). By (i) the G-equivariant inclusion F ↪ K(t) induces a G/H-equivariant inclusion F_{0} ↪ K(t)_{0} = K of residue fields. In other words, we have a rational dominant G/H-equivariant map V −−→ Y_{0} where k(Y_{0}) = F_{0}. Moreover, by (ii) G/H acts faithfully on F_{0} (or equivalently, on Y_{0}). Since V is a faithful linear representation of G/H, we have ed(G/H) ≤ dim Y_{0}. On the other hand,

dim Y_{0} = tr deg_{k}(F_{0}) = tr deg_{k}(F) − 1 = dim Y − 1 ≤ ed G − 1 .

and the desired inequality ed(G/H) ≤ ed G − 1 follows.

(b) To prove the inequality ed(G; p) ≤ ed(G/H; p)+1, recall that ed(G/H) is the minimal dimension of a faithful G/H-variety X which admits a diagram

\xymatrix{V′ \ar@{-->}[d]_{prime-to-p} \ar@{-->}[dr] \\ V & X ,}

of dominant rational G/H-equivariant maps, where X is a generically free G-variety, dim V′ = dim V and [k(V′) : k(V)] is prime to p. (Note that, in the definition of ed(G; p) we require that G should transitively permute the connected components of V′ (and of X). However, if G is a p-group and the degree of the cover V′ → V is prime to p, a simple counting argument shows that G has to fix an irreducible component of V′. Consequently, in this situation V′ and X are necessarily irreducible.) Multiplying every variety in the above diagram by 𝔸^{1} (on which G acts via the character χ), we obtain the diagram

\xymatrix{V′ × 𝔸^{1} \ar@{-->}[d]_{prime-to-p} \ar@{-->}[dr] \\ V × 𝔸^{1} & X × 𝔸^{1} ,}

of dominant rational map of faithful G-varieties, which shows that

ed(G; p) ≤ dim(X × 𝔸^{1}) = ed(G/H; p) + 1 .

To prove the opposite inequality, ed(G; p) ≤ ed(G/H; p) + 1, choose a diagram

\xymatrix{W \ar@{-->}[d]_{prime-to-p} \ar@{-->}[dr] \\ V × 𝔸^{1} & Y ,}

of rational maps of faithful G-varieties, where [k(W) : k(V × 𝔸^{1})] is finite and prime to p and dim Y = ed_{k}(G; p). Equivalently, we have a diagram of field extensions

[figure: L; prime-to-p; K(t); F ,]

where K, K(t), L and F are the function fields of V, V × 𝔸^{1}, W and Y, respectively.

First observe that after replacing L by the separable closure L_{s} of K(t) in L and F by F ∩ L_{s}, we may assume without loss of generality that L is separable over K(t). Indeed, the G-action on L and F descends to (and remains faithful on) L_{s} and F ∩ L_{s}. Moreover, some power of every x ∈ F lies in F ∩ L_{s}; hence, F is algebraic over F ∩ L_{s}, i.e., the two have the same transcendence degree over k.

From now on we will assume that L is separable over K(t) (or equivalently, W is separable over V × 𝔸^{1}). Let ν be the natural discrete valuation on K(t), by the degree in t (the same valuation we considered in part (a)). We claim that there exists a G-invariant lifting η of ν to L whose residue degree (i.e., the degree of the induced extension L_{0}/K of the residue fields) in prime to p.

Clearly the inertia group H_{L} ⊂ G of L (with respect to η) is contained in the inertia group of K(t), which is equal to H by our construction. On the other hand, the cyclic p-group H/H_{L} faithfully acts on the extension L_{0}/K whose degree is prime to p. This is only possible if H/H_{L} = {1}, i.e., H = H_{L}. Thus if the above claim is established, we can use the same argument as in part (a) (with K(t), ν replaced by L, η) to complete the proof of the inequality ed(G; p) ≤ ed(G/H; p) + 1.

To prove the claim, denote the liftings of ν on K(t) to L by η_{1}, …, η_{r}. As usual, we will denote the ramification index of η_{i} by e_{i} and the residue degree by f_{i}. By [Lan65, Proposition XII.18]

e_{1}f_{1} + ⋯ + e_{r}f_{r} = [L : K(t)]

or, equivalently,

∑_{e,f≥1} |V_{e,f}|ef = [L : K(t)]

where V_{e,f} be the subset of {η_{1}, …, η_{n}} consisting of those valuations η_{i} with e_{i} = e and f_{i} = f. (Note that V_{e,f} = ∅ for all but finitely many pairs (e, f).)

Thus there exist integers e, f ≥ 1 such that e, f, and |V_{e,f}| are all prime to p. The p-group G permutes the elements of V_{e,f}; since |V_{e,f}| is prime to p, G must fixed a valuation η ∈ V_{e,f}. This is the valuation we were looking for. (Note that the residue degree f of η is prime to p by our construction.) This completes the proof of the claim and thus of part (b). ♠

Corollary 7.2. Let G be a finite p-group and S be a direct summand of C(G) such that [G, G] ∩ S = {1}. If ζ_{e} ∈ k, where e is the exponent of S then ed_{k} G = ed_{k}(G/S) + rank(S).

Remark 7.3. The following elementary observation will be useful in the sequel.

Let G be a finite group and S be a normal subgroup such that [G, G]∩S = {1}. Then C(G/S) = C(G)/S.

To prove this assertion, suppose g ∈ G projects to a central element of G/S. Then for every x ∈ G the commutator c = gxg^{−1}x^{−1} lies in both [G, G] and S. Hence, c = 1, and thus g ∈ C(G), as desired. ♠

Proof of Corollary 7.2. Suppose C(G) as S ⊕ T. Write S = H_{1} ⊕ ⋯ ⊕ H_{r} as a direct sum of cyclic groups, where r = rank(S). We argue by induction on r. When r = 0, i.e., S = {1}, there is nothing to prove. For the induction step, note that by Remark 7.3,

S/H_{r} ≃ H_{1} ⊕ ⋯ ⊕ H_{r−1}

is a direct summand of C(G/H_{r}) = S/H_{r} ⊕ T. Thus by the induction assumption ed(G/H_{r}) = ed(G/S)+r−1. On the other hand, by Theorem 7.1 ed G = ed(G/H_{r}) + 1, and the corollary follows. ♠

8. Proof of Theorem 1.5

Let Z be a maximal cyclic subgroup of C(G) containing [G, G]. Then C(G) = Z ⊕ S for some central subgroup W of G. By Corollary 7.2,

ed G = ed(G/S) + rank(S) = ed(G/S) + rank C(G) − 1 .

By Remark 7.3, |C(G/S)| = |C(G)|/|S|. Thus the quantity |G/C(G)| does not change when we replace G by G/S. We conclude that it suffices to prove Theorem 1.5 for G/S. In other words, we may assume without loss of generality that the center Z = C(G) of G is cyclic. In this case the theorem reduces to the identity

ed(G; p) = ed G = \sqrt{|G/Z|} .

To prove this identity it suffices to establish Lemma 8.1 below. Indeed, part (a) of this lemma tells us that ed(G; p) ≥ \sqrt{|G/Z|} (see Theorem 1.4), and part (b) implies the inequality, ed G ≤ \sqrt{|G/Z|} (see Theorem 2.9), and Theorem 1.5 follows.

Lemma 8.1. Let G be a p-group such that the center Z = C(G) is cyclic and the quotient group A =^{def} G/Z is abelian. (or equivalently, [G, G] ⊂ Z). Then

(a) ind(G, Z) ≥ \sqrt{|A|}, and

(b) G has a faithful linear representation of dimension \sqrt{|A|}.

Proof. We will use additive notation for the groups Z and A, multiplicative for G. In this situation we can define a skew-symmetric bilinear form ω: A× A → Z by

ω(a_{1}, a_{2}) = g_{1}g_{2}g_{1}^{−1}g_{2}^{−1} ,

where a_{i} = g_{i}, modulo Z, for i = 1, 2. (Note that ω(a_{1}, a_{2}) is independent of the choice of g_{1} and g_{2}.) Clearly g lies in C(G) if and only if its image a lies in the kernel of ω; i.e., ω(a, b) = 0 for every b ∈ A. Since we are assuming that C(G) = Z, we conclude that the kernel of ω is trivial; i.e., ω is a symplectic form on A. It is well known (see for example [TA86, §3.1]) that the order of A, which equals the order of G/C(G), is a complete square.

Fix a generator z of Z. We recall the basic result on the structure of a symplectic form ω on a finite abelian group A (the proof is easy; it can be found, e.g., in [Wal63, §3.1] or [TA86, §7.1]). There exist elements a_{1}, …, a_{2r} in A and positive integers d_{1}, …, d_{r} with the following properties.

(a) d_{i} divides d_{i−1} for each i = 2, …, r, and d_{r} > 1.

(b) Let i be an integer between 1 and r. If A_{i} denotes the subgroup of A generated by a_{i} and a_{r+i}, then there exists an isomorphism A_{i} ≃ (ℤ/d_{i}ℤ)^{2} such that a_{i} corresponds to (1,0) and a_{r+i} to (0, 1).

(c) The subgroups A_{i} are pairwise orthogonal with respect to ω.

(d) ω(a_{i}, a_{r+i}) = z^{n/d_{i}} ∈ Z.

(e) A = A_{1} ⊕ ⋯ ⊕ A_{r}.

Then the order of A is d_{1}^{2} … d_{r}^{2}, hence \sqrt{|A|} = d_{1} … d_{r}.

Let G_{i} be the inverse image of A_{i} in G; note that G_{i} commutes with G_{j} for any i ≠ j.

Let u_{1}, …, u_{2r} be indeterminates, and set K =^{def} k(u_{1}, …, u_{2r}). Identify Z with μ_{n} by sending z into ζ_{n}. Consider the boundary map

∂_{i}: H^{1}(K, A_{i}) ⟶ H^{2}(K, Z)

obtained from the exact sequence

1 ⟶ Z ⟶ G_{i} ⟶ A_{i} ⟶ 1.

Claim. There exists a class ξ_{i} ∈ H^{1}(K, A_{i}) such that ∂_{i}ξ_{i} is the class of the cyclic algebra (u_{i}, u_{r+i})_{d_{i}} in Br K.

To see that part (a) follows from the claim, consider the commutative diagram

\xymatrix{1 \ar[r] & Z^{r} \ar[r] \ar[d]^{m} & ∏_{i} G_{i} \ar[r] \ar[d] & ∏_{i} A_{i} \ar[r] \ar[d] & 1 \\ 1 \ar[r] & Z \ar[r] & G \ar[r] & A \ar[r] & 1}

in which m is defined by the formula m(z_{1}, …, z_{r}) = z_{1} … z_{r}, and the homomorphism ∏_{i} G_{i} → G is induced by the inclusions G_{i} ⊆ G. This yields a commutative diagram

\xymatrix{∏_{i} H^{1}(K, A_{i}) \ar[r]^{∏_{i} ∂_{i}} \ar[d] \ar[r] & H^{2}(K, Z)^{r} \ar[d]^{m_{∗}} \\ H^{1}(K, A) \ar[r]^{∂} & H^{2}(K, Z)}

in which the map m_{∗} is given by m_{∗}(α_{1}, …, α_{r}) = α_{1} … α_{r}. So, if ξ ∈ H^{1}(K, A) is the image of (ξ_{1}, …, ξ_{r}) then ∂ξ is the class of the product

(u_{1}, u_{r+1})_{d_{1}} ⊗_{K} (u_{2}, u_{r+2})_{d_{2}} ⊗_{K} ⋯ ⊗_{K} (u_{r}, u_{2r})_{d_{r}},

whose index is d_{1} … d_{r}. Hence ind(G, Z) ≥ d_{1} … d_{r} = \sqrt{|A|}, as needed.

We now proceed with the proof of the claim. Choose a power of p, call it d, that is divisible by the order of Z and by the order of each a_{i}. Consider the group Λ(d) defined by the presentation

⟨x_{1}, x_{2}, y | x_{1}^{d} = x_{2}^{d} = y^{d} = 1, x_{1}x_{2} = yx_{2}x_{1}, x_{1}y = yx_{1}, x_{2}y = yx_{2}⟩.

Call ρ_{i}: Λ(d) → G_{i} the homomorphism obtained by sending x_{1} to a_{i}, x_{2} to a_{r+i}, and y to z^{n/d_{i}} = ω(a_{i}, a_{r+i}).

Let ζ_{d} be a primitive d-th root of 1 in k such that ζ_{n} = ζ_{d}^{n/d}. The subgroup ⟨y⟩ in Λ(d) is cyclic of order d; we fix the isomorphism ⟨y⟩ ≃ μ_{d} so that y corresponds to ζ_{d}. The restriction of ρ_{i} to ⟨y⟩ → Z corresponds to the homomorphism μ_{d} → μ_{n} defined by α ↦ α^{d/d_{i}}. We have a commutative diagram

\xymatrix{1 \ar[r] & μ_{d} \ar[r] \ar[d]_{\begin{matrix}α \\ ↓ \\ α^{d/d_{i}} \end{matrix}} & Λ(d) \ar[r] \ar[d]^{ρ_{i}} & (ℤ/dℤ)^{2} \ar[d] \ar[r] & 1 \\ 1 \ar[r] & μ_{n} \ar[r] & G_{i} \ar[r] & A_{i} \ar[r] & 1.}

We have H^{1}(K, (ℤ/dℤ)^{2}) = (K^{∗}/K^{∗}^{d})^{2}. According to [Vel00, Example 7.2], the image of the element (u_{i}, u_{r+i}) ∈ H^{1}(K, (ℤ/dℤ)^{2}) is the cyclic algebra (u_{i}, u_{r+i})_{d}; hence, if ξ_{i} is the image in H^{1}(K, A_{i}) of (u_{i}, u_{r+i}), the image of ξ_{i} in H^{2}(K, μ_{d}) is the algebra (u_{i}, u_{r+i})_{d}^{⊗d/d_{i}}, which is equivalent to (u_{i}, u_{r+i})_{d_{i}}. This concludes the proof of Lemma 8.1(a).

We now turn to the proof of Lemma 8.1(b). Suppose |C(G)| = p^{h} and |A| = p^{2m}; we want to construct a faithful representation of G of dimension p^{m}. By [TA86, §3.1] A contains a Lagrangian subgroup L of order p^{m}. Denote by H the inverse image of L in G; then H is an abelian subgroup of G of order p^{h+m}. Since ζ_{p^{e}} ∈ k we can embed Z in k^{∗} and extend this embedding to a homomorphism χ: H → k^{∗}. We claim that the representation ρ: G → GL_{p^{m}} induced by χ is faithful.

It is enough to show that ρ(g) ≠ id for any g ∈ G of order p, or, equivalently, that ρ |_{⟨g⟩} is non-trivial for any such g. If s ∈ G consider the subgroup H_{s} =^{def} s⟨g⟩s^{−1} ∩ H of H, which is embedded in ⟨g⟩ via the homomorphism x ↦ s^{−1}xs. By Mackey’s formula ([Ser77, §7.3]), ρ |_{⟨g⟩} contains all the representations of ⟨g⟩ induced by the restrictions χ |_{H_{s}} via the embedding above.

If g ∉ H then H_{1} = ⟨g⟩ ∩ H = {1}: we take s = 1, and we see that ρ |_{⟨g⟩} contains a copy of the regular representation of ⟨g⟩, which is obviously non-trivial.

Assume g ∈ H. Then H_{s} = ⟨sgs^{−1}⟩ for any s ∈ G; it is enough to prove that χ(sgs^{−1}) ≠ 1 for some s ∈ G. If χ(g) ≠ 1 then we take s = 1. Otherwise χ(g) = 1; in this case g ∉ C(G), because χ |_{C(G)} : C(G) → k^{∗} is injective. Hence the image \overline{g} of g in A is different from 0, and we can find s ∈ G such that ω(\overline{s}, \overline{g}) ≠ 1. Then

χ(sgs^{−1}) = χ(ω(\overline{s}, \overline{g})g) = χ(ω(\overline{s}, \overline{g}))χ(g) = χ(ω(\overline{s}, \overline{g})) ≠ 1.

This concludes the of Lemma 8.1 and thus of Theorem 1.5. ♠

9. More on p-groups

In this section we will discuss some consequences of Theorem 1.5.

Example 9.1. Recall that a p-group G is called extra-special if its center Z is cyclic of order p, and the quotient G/Z is elementary abelian. The order of an extra special p-group G is an odd power of p; the exponent of G is either p or p^{2}; cf. [Rob96, pp. 145–146]. Note that every non-abelian group of order p^{3} is extra-special. For extra-special p-groups Theorem 1.5 reduces to the following.

Let G be an extra-special p-group of order p^{2m+1}. Assume that the characteristic of k is different from p, that ζ_{p} ∈ k, and ζ_{p^{2}} ∈ k if the exponent of G is p^{2}. Then ed(G; p) = ed G = p^{m}.

Example 9.2. Let p be an odd prime and G = C_{p^{r}} ⋉ C_{p^{s}} be the natural semidirect product of cyclic groups of order p^{r} and p^{s} (in other words, C_{p^{s}} is identified with the unique subgroup of C_{p^{r}}^{∗} of order p^{s}). If s ≤ r/2 then

ed_{k}(C_{p^{r}} ⋉ C_{p^{s}}; p) = ed_{k}(C_{p^{r}} ⋉ C_{p^{s}}) = p^{s} ,

for any field k containing a primitive pth root of unity ζ_{p}.

Proof. The center C(G) of G is the (unique) subgroup of C_{p^{r}} of order p^{s}. Since s ≤ r/2, this subgroup is central. Thus, if ζ_{p^{r}} ∈ k, the equality ed_{k} G = p^{s} is an immediate consequence of Theorem 1.5. But we are only assuming that ζ_{p} ∈ k, so Theorem 1.5 only tells us that ed_{k} G ≥ p^{s}. To prove the opposite inequality, we argue as follows. Let F be the prime subfield of k. By [Led02, Corollary to Proposition 2], ed_{F (ζ_{p})}(G) ≤ p^{s}. Since we are assuming that F (ζ_{p}) ⊂ k, we conclude that ed_{k} G ≤ p^{s} as well. ♠

Corollary 9.3. Suppose k is a base field of characteristic ≠ p. If G is a non-abelian finite p-group then ed(G; p) ≥ p.

Proof. Assume the contrary: let G be a non-abelian p-group of smallest possible order such that ed G < p. Since G has a non-trivial center, there exists a cyclic central subgroup Z ⊂ G. The short exact sequence

(9.4)  1 ⟶ Z ⟶ G ⟶ G/Z ⟶ 1

gives rise to the exact sequence of pointed sets

H^{1}(K, G) ⟶ H^{1}(K, G/Z) ⟶^{∂_{K}} H^{2}(K, Z)

for any field extension K of our base field k. We will now consider two cases.

Case 1. Suppose the map H^{1}(K, G) → H^{1}(K, G/Z) is not surjective for some K/k. Then ∂_{K} is non-trivial, and Theorem 1.4 tells us that ed(G; p) ≥ p, a contradiction.

Case 2. Suppose the map H^{1}(K, G) → H^{1}(K, G/Z) is surjective for every K/k. Then the morphism BG → B(G/Z) is isotropic, and Proposition 2.8 implies that p > ed(G; p) ≥ ed(G/Z; p). By the minimality of G, the group G/Z has to be abelian. Consequently, [G, G] ⊂ Z is cyclic and central in G. Since G is non-abelian, |G/C(G)| ≥ p^{2}. Theorem 1.5 now tells us that

ed(G; p) = \sqrt{|G|/|C(G)|} + rank C(G) − 1 ≥ p ,

a contradiction. ♠

We will conclude this section by answering the following question of Jensen, Ledet and Yui [JLY02, p. 204].

Question 9.5. Let G be a finite group and N be a normal subgroup. Is it true that ed G ≥ ed(G/N)?

The inequality ed G ≥ ed(G/N) is known to hold in many cases; cf., e.g., Theorem 7.1. We will now show that it does not hold in general, even if H is assumed to be central.

Corollary 9.6. For every real number λ > 0 there exists a finite p-group G, with a central subgroup H ⊂ G such that ed(G/H) > λ ed G.

Proof. Let Γ be a non-abelian group of order p^{3}. The center of Γ has order p; denote it by C. The center of Γ^{n} = Γ × ⋯ × Γ (n times) is then C^{n}.

Let H_{n} be the subgroup of C^{n} consisting of n-tuples (c_{1}, …, c_{n}) such that c_{1} … c_{n} = 1. Clearly

ed Γ^{n} ≤ n · ed Γ = np;

see Example 9.1.

On the other hand, Γ^{n}/H_{n}, is easily seen to be extra-special of order p^{2n+1}, so ed(Γ^{n}/H_{n}) = p^{n}, again by Example 9.1. Setting G = Γ^{n} and H = H_{n}, we see that the desired inequality ed(G/H) > λ ed G, holds for suitably large n. ♠

10. Spinor groups

In this section we will prove Theorem 1.7 stated in the introduction. As usual, we will write ⟨a_{1}, …, a_{n}⟩ for the rank n-quadratic form q given

by q(x_{1}, …, x_{n}) = ∑_{i=1}^{n} a_{i}x_{i}^{2}. Set h to be the standard hyperbolic quadratic form given by h(x, y) = xy. (Thus h ≅ ⟨1, −1⟩). For each n ≥ 0 define

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

Set Spin_{n} = Spin(h_{n}); this is the totally split spin group which appears in the statement of Theorem 1.7. We also denote the totally split orthogonal and special orthogonal groups by O_{n} =^{def} O(h_{n}) and SO_{n} =^{def} SO(h_{n}).

Now, one of the hypotheses of Theorem 1.7 is that ζ_{4} ∈ k. Therefore we can write Spin_{n} as Spin(q), where

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

Consider the subgroup Γ_{n} ⊆ SO_{n} consisting of diagonal matrices, which is isomorphic to μ_{2}^{n−1}. Call G_{n} the inverse image of Γ_{n} in Spin_{n}. It is a constant group scheme over k. Denote by μ_{2} the kernel of the homomorphism Spin_{n} → SO_{n}.

Lemma 10.2. Every Spin_{n}-torsor over an extension K of k 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 any field extension K/k.

Proof. Let P → Spec K be a Spin_{n}-torsor: we are claiming that the Kscheme P/G_{n} has a rational point. We have P/G_{n} = (P/μ_{2})/Γ_{n}. However P/μ_{2} → Spec K is the SO_{n} torsor associated with P → Spec K, and every SO_{n}-torsor admits reduction of structure group to Γ_{n}. ♠

This means that the natural morphism BG_{n} → BSpin_{n} is isotropic; so from Corollary 2.7 and Proposition 2.8 we get the bounds

(10.3)  ed G_{n} − dim Spin_{n} ≤ ed Spin_{n} ≤ ed G_{n};

cf. also [BF03, Lemma 1.9]. Of course dim Spin_{n} = n(n − 1)/2; we need to compute ed G_{n}. The structure of G_{n} is well understood; in particular, it is very clearly described in [Woo89]. Recall that Spin_{n} is a subgroup of the group of units in the Clifford algebra A_{n} of the quadratic form −(x_{1}^{2} + ⋯ + x_{n}^{2}). The algebra A_{n} is generated by elements e_{1}, …, e_{n}, with relations e_{i}^{2} = −1 and e_{i}e_{j} + e_{j}e_{i} = 0 for all i ≠ j. The element e_{i} is in Pin_{n}, and image of e_{i} in O_{n} is the diagonal matrix with −1 as the i-th diagonal entry, and 1 as all the other diagonal entries. The kernel of the homomorphism Pin_{n} → O_{n} is {±1}. (For background material on the theory of Clifford algebras and spin modules, we refer the reader to [Che54], [FH91, §20.2] or [Ada96]).

For any I ⊆ {1, …, n} write I = {i_{1}, …, i_{r}} with i_{1} < i_{2} < ⋯ < i_{r} and set e_{I} =^{def} e_{i_{1}} … e_{i_{r}}. The group G_{n} consists of the elements of A_{n} of the form ±e_{I}, where I ⊆ {1, …, n} has an even number of elements. 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 [G_{n}, G_{n}] = {±1} is cyclic, and the center C(G) is given by

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}

Theorem 1.5 now tells us that

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}

Substituting this into (10.3), we obtain the bounds of Theorem 1.7. ♠

Remark 10.4. The same argument can be applied to Pin groups. (For the definition of Pin groups, see, e.g., [Ada96].) Here we replace G_{n} the inverse image G′_{n} of the diagonal subgroup of O_{n} in Pin_{n}. One easily checks that

C(G′_{n}) ≃ \begin{cases}ℤ/2ℤ, if n is even, \\ ℤ/4ℤ, if n ≡ 1 (mod 4), \\ ℤ/2ℤ × ℤ/2ℤ, if n ≡ 3 (mod 4). \end{cases}

This yields the following bounds on the essential dimensions of Pin_{n}:

2^{⌊n/2⌋} − \frac{n(n − 1)}{2} ≤ ed_{k} Pin_{n} ≤ 2^{⌊n/2⌋}, if n ≢ 1 (mod 4),
2^{⌊n/2⌋} − \frac{n(n − 1)}{2} + 1 ≤ ed_{k} Pin_{n} ≤ 2^{⌊n/2⌋} + 1, if n ≡ 1 (mod 4).

for any field k of characteristic ≠ 2 containing ζ_{4}.

Remark 10.5. When n ≤ 14 the lower bound of Theorem 1.7 is negative and the upper bound is much larger than the true value of ed Spin_{n}. For n = 15 and 16 our inequalities yield

23 ≤ ed Spin_{15} ≤ 128

and

9 ≤ ed Spin_{16} ≤ 129.

When n = 16 our lower bound coincides with the lower bound (1.8) of Reichstein–Youssin and Chernousov–Serre, while for n = 15 it is substantially larger. When n ≥ 17 the exponential part of the lower bound takes over, the growth becomes fast and the gap between the lower bound and the upper bound proportionally small. For values of n close to 15 our estimates are quite imprecise; it would be interesting to improve them.

Remark 10.6. The lower bounds in Theorem 1.7 hold over any field of characteristic different from 2 (and for any form of the Spin group).

On the other hand, if we do not assume that ζ_{4} ∈ k, we have the following slightly weaker upper bound

ed Spin_{n} ≤ 2^{⌊(n−1)/2⌋} + n − 1

for the totally split form of the spin group in dimension n, over k. To prove this inequality, we observe that a generically free representation of Spin_{n} can be constructed by taking a spin, or half-spin, representation V of Spin_{n} of dimension 2^{⌊(n−1)/2⌋}, and adding a generically free representation W of SO_{n}. Since the essential dimension of SO_{n−1} is n − 1 over any field of characteristic different from 2, there is an SO_{n}-equivariant dominant rational map f: W −−→ X, where dim X = dim SO_{n} + n − 1. Now id × f: V × W −−→ V × X is a Spin_{n}-equivariant dominant rational map V × W. Consequently,

ed Spin_{n} ≥ dim(V × X) − dim Spin_{n}
= 2^{⌊(n−1)/2⌋} + dim SO_{n} + n − 1 − dim Spin_{n}
= 2^{⌊(n−1)/2⌋} + n − 1 ,

as claimed.

Remark 10.7. It is natural to ask whether the inequality

ed Spin_{n} ≥ 2^{⌊(n−1)/2⌋} − \frac{n(n − 1)}{2}

can be proved by a direct application of Theorem 1.7 to the exact sequence

(10.8)  1 ⟶ μ_{2} ⟶ Spin_{n} ⟶ SO_{n} ⟶ 1

without considering the finite subgroup G_{n} of Spin_{n}. The answer is “yes.” Indeed, consider the associated coboundary map

\xymatrix{H^{1}(K, SO_{m}) \ar[r]^{∂_{K}} & H^{2}(K, μ_{2}).}

A class in H^{1}(K, SO_{m}) is represented by a m-dimensional quadratic form q of discriminant 1 defined over K. The class of ∂_{K}(q) ∈ H^{2}(K, μ_{2}) is then the Hasse-Witt invariant of q; following Lam [Lam73], we will denote it by c(q). (Note that since we are assuming that −1 is a square in k, the Hasse invariant and the Witt invariant coincide; see [Lam73, Proposition V.3.20].)

Our goal is thus to show that for every n ≥ 1 there exists a quadratic form q_{n} of dimension n and discriminant 1 such that c(q_{n}) has index 2^{⌊(n−1)/2⌋}.

If n is even this is proved in [Mer91, Lemma 5]. (Note that in this case c(q) ∈ H^{2}(K, μ_{2}) is the class of the Clifford algebra of q.) If n = 2r + 1 is odd, set K = k(a_{1}, b_{1}, a_{2}, b_{2}, …), where a_{1}, b_{1}, a_{2}, b_{2}, … are independent variables, and define q_{2r+1} recursively by

q_{1} = ⟨1⟩ and q_{2r+1} = ⟨a_{r}, b_{r}⟩ ⊕ a_{r}b_{r}⟩ ⊗ q_{2r−1}⟩.

One easily sees by induction on r that every q_{2r+1} has discriminant 1. Moreover, a direct computation using basic properties of the Hasse-Witt invariant (see, e.g., [Lam73, V.3.15 and 3.16]) shows that c(q_{2r+1}) is the class of the tensor product (a_{1}, b_{1})_{2}⊗_{K} ⋯⊗_{K} (a_{r}, b_{r})_{2} of quaternion algebras. This class has index 2^{r}, as claimed. ♠

In summary, this approach recovers the lower bound of Theorem 1.7 in the case where n is not divisible by 4. In the case where n is divisible by 4 Theorem 1.7 gives a slightly stronger lower bound.

To conclude this section, we will now prove similar bounds on the essential dimensions on half-spin groups. We begin with the following simple corollary of [CGR06, Theorem 1.1], which appears to have been previously overlooked.

Lemma 10.9. Let G be a closed (but not necessarily connected) subgroup of GL_{n} defined over a field k. Assume that char k = 0 and either k is algebraically closed or G is connected. Then ed G ≤ n.

Proof. By [CGR06, Theorem 1.1], there exists a finite k-subgroup S ⊆ G such that every G-torsor over Spec K, admits reduction of structure to S, for any field extension K/k. This means that the morphism BS → BG is isotropic and thus

ed G ≤ ed S;

cf. also [BF03, Lemma 1.9]. The restriction of the representation G ⊆ GL_{n} to S is faithful; since S is a finite group over a field of characteristic zero, any faithful representation of S is necessarily generically free. By Theorem 2.9, ed S ≤ n, and hence, ed G ≤ n, as claimed. ♠

Example 10.10. Suppose G/k satisfies one of the conditions of Lemma 10.9 and the centralizer C_{G}(G^{0}) of the connected component of G is trivial. Then the adjoint representation of G is faithful and Lemma 10.9 tells us that ed G ≤ dim G. In particular, this inequality is valid for every connected semisimple adjoint group G. (In the case of simple adjoint groups, a stronger bound is given by [Lem04, Theorem 1.3].)

We are now ready to proceed with our bounds on the essential dimension of half-spin groups. Recall that the half-spin group HSpin_{n} is defined, for every n divisible by 4, as Spin_{n}/⟨η⟩, where η is an element of the center of Spin_{n} different from −1. (There are two such elements, but the resulting quotients are isomorphic.)

Theorem 10.11. (a) Suppose k is a field of characteristic 0 and ζ_{4} ∈ k. Then

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

for any positive integer n divisible by 4.

The conditions that char k = 0 and ζ_{4} ∈ k are used only in the proof of the upper bound. The lower bound of Theorem 10.11 remains valid for any base field k of characteristic ≠ 2.

Proof. The group HSpin_{n} contains G_{n}/⟨η⟩ ≃ G_{n−1}, which is an extra-special group of order 2^{n−1}. By Example 9.1 ed(G_{n}/⟨η⟩) = 2^{(n−2)/2} and thus

ed HSpin_{n} ≥ ed(G_{n}/⟨η⟩) − dim HSpin_{n} = 2^{(n−2)/2} − \frac{n(n − 1)}{2} ,

as in the proof of Theorem 1.7.

For the upper bound notice that one of the two half-spin representations of Spin_{n} descends to HSpin_{n}, and is a faithful representation of HSpin_{n} of dimension 2^{(n−2)/2}. The upper bound now follows from Lemma 10.9 ♠

11. Pfister numbers

Let k be a field of characteristic not equal to 2 and write W(k) for the Witt ring of k; see [Lam73, Chapter 2]. Let I = I(k) denote the ideal of all even dimensional forms in the Witt ring. Then, for any integer a > 0, I^{a} is generated as an abelian group by the a-fold Pfister forms [Lam73, Proposition 1.2].

Let q be a quadratic form of rank n > 0 whose class [q] in W(k) lies in I^{a} for a > 0. Define the a-Pfister number of q to be the minimum number r appearing in a representation

q = ∑_{i=1}^{r} ±p_{i}

with the p_{i} being a-fold Pfister forms. 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 such that [q] ∈ I^{a}(K).

We have the following easy (and probably well-known) result.

Proposition 11.1. Let k be a field of characteristic not equal to 2 and let n be a positive even integer.

(a) Pf_{k}(1, n) ≤ n.

(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 determinant given by (−1)^{n(n−1)/2}d(q) where d(q) = ∏_{i=1}^{n} a_{n} is the determinant [Lam73, p. 38].

To explain how to write q as a sum of n − 2 Pfister forms, we will temporarily assume that ζ_{4} ∈ K. In this case we may assume that 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 ζ_{4} ∈ 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 determinant and trivial Hasse-Witt invariant [KMRT98].

Let n be a positive integer. Let q be a non-degenerate n-dimensional quadratic form over K whose whose signed determinant is 1. The class of q in H^{1}(K, O_{n}) lies in H^{1}(K, SO_{n}). We say that q admits a spin structure if its class is in the image of H^{1}(K, Spin_{n}) into H^{1}(K, SO_{n}). As pointed out in Remark 10.7, the obstruction to admitting a spin structure is the Hasse-Witt invariant c(q). Thus, the forms in I^{3} are exactly the even dimensional forms admitting a spin structure. The following result was suggested to us by Merkurjev and Totaro.

Proposition 11.2. 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 n-dimensional form in I^{3} over any field extension K/k. ♠

Remark 11.3. For a > 3 the finiteness of Pf_{k}(a, n) is an open problem.

The goal of this section is to prove Theorem 1.9 stated in the introduction, which says that

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

for any field k be a field of characteristic different from 2 and any positive even integer n. Since this is a lower bound on Pf_{k}(3, n), we may assume, without loss of generality that k contains ζ_{4} in the proof. To simplify matters, 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. The essential dimension of this functor is closely related to the essential dimension of Spin_{n}.

Lemma 11.4. ed Spin_{n} − 1 ≤ ed T_{n} ≤ ed Spin_{n}.

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 then follows from [BF03, Proposition 1.13] and the second follows from Proposition 2.8 (or [BF03, Lemma 1.9]). ♠

Lemma 11.5. Let q and q′ be non-degenerate even-dimensional quadratic forms over K. Suppose that q admits a spin structure. Then q ⊕ q′ admits a spin structure if and only if q′ admits a spin structure.

Proof. Immediate from the fact that I^{3}(K) is an ideal of W (K). ♠

Let h_{K} be the standard 2-dimensional hyperbolic form h_{K}(x, y) = xy over an extension K of k. 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 11.6. Let q be an n-dimensional quadratic form over K whose class in W (K) lies in I^{3}(K). Then for any positive integer s

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

Proof. Set m =^{def} ed_{n+2s}(h_{K}^{⊕s} ⊕ q). Let F be a field of definition of h_{K}^{⊕s} ⊕ q of transcendence degree m, and let q̃ be an (n+2s)-dimensional quadratic form with a spin structure over F such that q̃_{K} is K-isomorphic to h_{K}^{⊕s} ⊕q. Let X be the Grassmannian of s-dimensional subspaces of F^{n+2s} which are totally isotropic with respect to q̃; the dimension of X is precisely 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′. By Witt’s Cancellation Theorem, q′_{K} is K-isomorphic to q; hence ed_{n}(q) ≤ m + s(s + 2n − 1)/2, as claimed. ♠

Proof of Theorem 1.9. If n ≤ 10 then the statement is vacuous, because then 2^{(n+4)/4} − n − 2 ≤ 0, so we assume that n ≥ 12. We may also assume without loss of generality that ζ_{4} ∈ k. In this case W(K) is a ℤ/2-vector space; it follows that the 3-Pfister number of a form q is the smallest r appearing in an expression

q = ∑_{i=1}^{r} ≪a_{i}, b_{i}, c_{i} ≫.

in W(K). Choose an n-dimensional form q such that [q] ∈ I^{3}(K) and ed_{n}(q) = ed T_{n}. (Here we view q as an object in T_{n}(K).) Suppose that q is equivalent in the Witt ring to ∑_{1=1}^{r} ≪a_{i}, b_{i}, c_{i} ≫.

Let us write a Pfister form ≪a, b, c≫ as

≪a, b, c≫ = ⟨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} ∑_{1=1}^{r} ≪a_{i}, b_{i}, c_{i} ≫_{0}

if r is even, and

ϕ =^{def} ⟨1⟩ ⊕ ∑_{1=1}^{r} ≪a_{i}, b_{i}, c_{i} ≫_{0}

if r is odd. Then q is equivalent to ϕ in the Witt ring, and hence, 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 the dimension of q is less than of equal to the dimension of ϕ, so q is isomorphic to a form of type h_{K}^{s} ⊕ ϕ. Thus

\frac{3n}{7} ≥ 3r ≥ ed_{n}(q) = ed T_{n} ≥^{by Lemma 11.4} ed Spin_{n} − 1 .

The resulting inequality fails for every even n ≥ 12 because, for such n, ed Spin_{n} ≥ n/2; see (1.8).

So we may assume that 7r ≥ n; then there is an isomorphism between the quadratic forms ϕ and a form of the type h_{K}^{⊕s} ⊕q. 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 11.6 tells us that this essential dimension is at least ed(q) − s(s + 2n − 1)/2. From this, Lemma 11.4 and Theorem 1.7 we obtain the chain of inequalities

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

= ed T_{n} − \frac{s(s + 2n − 1)}{2} (11.7)

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

≥ 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 the inequality (11.7), 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 inquality 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 1.9 when r is even. If r is odd then substituting s = (7r+1−n)/2 into (11.7), 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 1.9 follows. ♠

b

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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