Lower bounds for essential dimensions via orthogonal representations

Vladimir Chernousov ^{∗}

Department of Mathematics University of Alberta Edmonton, Alberta T6G 2G1

Jean–Pierre Serre

Collège de France 3, rue d’Ulm 75231 Paris, Cedex 05

§ 1 Introduction

Let us first recall what is the essential dimension of a functor, cf. [BR 97] and [R 00]. Let k be a field, and let ℱ be a functor from the category of field extensions of k into the category of sets. Let F/k be an extension and let ξ be an element of ℱ(F). If E is a field with k ⊂ E ⊂ F we say that ξ comes from E if it belongs to the image of ℱ(E) → ℱ(F). The essential dimension ed (ξ) of ξ is the minimum of the transcendence degrees E/k, for all E with k ⊂ E ⊂ F such that ξ comes from E. One has ed(ξ) ≤ tr. deg. F. If there is equality, we say that ξ is incompressible. The essential dimension ed (ℱ) of ℱ is

ed (ℱ) = max {ed (ξ)},

the maximum being taken over all pairs (F, ξ) with k ⊂ F and ξ ∈ ℱ(F).

∗Supported by the Canada Research Chairs Program, and by NSERC’s Grant G121210944

Along similar lines, the essential dimension ed(ξ; p) of ξ ∈ ℱ(F) at a prime number p is

ed(ξ; p) = min {ed(ξ_{K})},

where ξ_{K} is the image of ξ in ℱ(K), and the minimum is taken over all extensions K/F with [K : F] finite and prime to p. The essential dimension of ℱ at p is

ed(ℱ; p) = max {ed(ξ; p)}

the maximum being taken over all pairs (F, ξ) with ξ ∈ ℱ(F). It is clear that ed(ℱ) ≥ ed(ℱ; p).

We will apply this to the functor ℱ defined by:

ℱ(F) = H^{1}(F, G) = { isomorphism classes of G-torsors over F },

where G is a smooth linear algebraic group over k. The essential dimension ed(G) of G (resp. the essential dimension ed(G; p) at p) is ed(ℱ) (resp. ed(ℱ; p)). If ξ is a versal G-torsor, in the sense of [GMS 03], p.13, one has

ed(G) = ed(ξ) and ed(G; p) = ed(ξ; p).

In case we feel the need to be precise about F, we write ed_{F} instead of just ed.

If char(k) = 0, Reichstein and Youssin have given a very efficient lower bound for ed(G; p), namely:

If G is connected and contains a finite abelian p-group A whose centralizer is finite, then one has ed(G; p) ≥ rk(A), where rk(A) denotes the minimum number of generators of A ([RY 00], th. 7.8).

The proof of Reichstein-Youssin uses resolution of singularities, hence does not apply (for the time being) when char(k) > 0. What we do in the present paper is to prove most of their results relative to p = 2 in arbitrary characteristic (except^{1} characteristic 2) by using orthogonal groups and quadratic forms (especially ”monomial” quadratic forms, cf. § 4). For instance :

(1.1) If G is semisimple of adjoint type, and −1 belongs to the Weyl group, then

ed(G; 2) ≥ rank(G) + 1.

1It seems likely that a similar method can also be applied in characteristic 2, but we have not checked all the necessary steps.

This is the case G = G^{∘} of th. 1 of § 2. Note that it implies

ed(E_{8}; 2) ≥ 9 and ed(E_{7}; 2) ≥ 8.

(1.2) ed (Spin_{n}; 2) ≥ [n/2] for n > 6, n ≠ 10 , the inequality being strict if n ≡ −1, 0 or 1 (mod 8), cf. th. 11 and th. 12.

(1.3) ed (HSpin_{n}; 2) > n/2 if n ≥ 8, n ≡ 0 (mod 8), cf. th. 13.

Of course, these results give lower bounds for ed(G) itself, for instance ed(E_{8}) ≥ 9.

§ 2 The Main Theorem

In what follows, we assume char(k) ≠ 2 and k algebraically closed.

Let G^{∘} be a simple algebraic group over k of adjoint type, and let T be a maximal torus of G^{∘}. Let c ∈ Aut(G^{∘}) be such that c^{2} = 1 and c(t) = t^{−1} for every t ∈ T (it is known that such an automorphism exists, see e.g. [DG 70], Exp. XXIV, Prop. 3.16.2, p. 355). This automorphism is inner (i.e. belongs to G^{∘}) if and only if −1 belongs to the Weyl group of (G, T). When this is the case, we put G = G^{∘}. If not, we define G to be the subgroup of Aut(G^{∘}) generated by G^{∘} and c. We have

• G = G^{∘} for types A_{1}, B_{r}, C_{r}, D_{r} (r even), G_{2}, F_{4}, E_{7}, E_{8} ;

• (G : G^{∘}) = 2 and G = Aut(G^{∘}) for types A_{r} (r ≥ 2), D_{r} (r odd), E_{6}.

Let r = dim (T) be the rank of G.

Theorem 1 If G is as above, we have ed (G; 2) ≥ r + 1.

The proof of Theorem 1 consists in:

a) construction of a G-torsor θ_{G} over a suitable extension K/k with tr. deg_{k}(K) = r + 1, see below;

b) proof that the image of θ_{G} in a suitable H^{1}(K, O_{N}) (cf. § 3) is incompressible (§§ 4–6); this implies that θ_{G} itself is incompressible, and th. 1 follows.

Let us start with part a). Let R be the root system of G with respect to T, and let R_{sh} be the (sub) root system formed by the short roots of R.

Let Δ = {α_{1}, …, α_{r}} be a basis of R_{sh}. The root lattices of R and R_{sh} are the same; hence Δ is a basis of the character group X(T). This allows us to identify T with G_{m} × ⋯ × G_{m} using the basis Δ.

Call A_{0} the kernel of “multiplication by 2” on T. Let A = A_{0} × {1, c} be the subgroup of G generated by A_{0} and by the element c defined above. The group A is isomorphic to (±1)^{r+1}.

Take K = k(t_{1}, …, t_{r}, u) where t_{1}, …, t_{r} and u are independent indeterminates. We have H^{1}(K, A) = H^{1}(K, Z/2Z) × … × H^{1}(K, Z/2Z). Identify H^{1}(K, Z/2Z) with K^{×}/(K^{×})^{2} as usual. Then u and the t_{i}’s define elements (u) and (t_{i}) of H^{1}(K, Z/2Z). Let θ_{A} be the element of H^{1}(K, A) with components ((t_{1}), …, (t_{r}), (u)). Let θ_{G} be the image of θ_{A} in H^{1}(K, G). We will prove in § 6 :

Theorem 2 (K, θ_{G}) is incompressible, and remains so after any field extension of K of odd degree.

Note that Theorem 2 implies Theorem 1 since tr. deg. K = r + 1.

Remarks. (i) It would not be useful to take A_{0} instead of A. Indeed, A_{0} is a subgroup of T and H^{1}(K, T) = 1 by Hilbert Theorem 90. Hence the image in H^{1}(K, G) of any element of H^{1}(K, A_{0}) is trivial. In particular, the class θ_{G} defined above is killed by the quadratic extension K(\sqrt{u})/K. (ii) Suppose that G = G^{∘}, i.e. that c belongs to G^{∘}. The subgroup A constructed above is the same as the one described in [BS 53], p.139, for compact Lie groups. It is also the same one (with the same θ_{G}) as in Reichstein-Youssin theory [RY 00].

§ 3 An orthogonal representation

Proposition 3 There exists a quadratic space (V, q) over K, and an orthogonal irreducible linear representation

ρ : G ⟶ O (V, q)

with the following property :

(∗) the nonzero weights of T on V are the short roots and they have multiplicity 1.

Proof. Let B be a Borel subgroup containing T. This defines an order on the root system R. Let β be the highest root of R_{sh} with respect to that order. It is a dominant weight. We choose for V an irreducible representation L(β) of G^{∘} with highest weight β. By a well-known criterion ([St 67], Lemmas 78, 79, p. 226), L(β) is an orthogonal representation of G^{∘}. Since R_{sh} ∪ {0} is R-saturated in the sense of [Bo 75], VIII. § 7.2, the nonzero weights of L(β) belong to R_{sh}, hence are conjugate to β by the Weyl group. This implies that they have multiplicity 1, so that (∗) is fulfilled.

It remains to show that this orthogonal representation of G^{∘} extends to an orthogonal representation of Aut (G^{∘}), and hence of G. This can be done in the following way:

If Aut (G^{∘}) = G^{∘}, there is nothing to prove.

If Aut (G^{∘}) ≠ G^{∘}, the roots have the same length, so that β is the highest root of R, and V = L(β) is essentially the adjoint representation of G^{∘}. More precisely, if G̃^{∘} denotes the universal covering of G^{∘}, one can take for V the image of Lie (G̃^{∘}) in Lie (G^{∘}), with the obvious action of Aut(G^{∘}). One puts on V the ”normalized Killing form” q(x, y). That form is defined first over Z, in which case it is equal to Tr (ad(x) · ad(y))/2h where h is the Coxeter number (see [GN 04], [Sel 57], [SpSt 70]); it is then defined by base change for every simple group scheme, and the computation of its discriminant done in the references above shows that it is nondegenerate.

Examples. a) When the roots of R have the same length, we have V = Lie (G^{∘}), except for :

• type A_{n} when p divides n + 1;

• type E_{6} when p = 3.

In both cases, V has codimension 1 in Lie G^{∘}.

b) When the roots have different length, then:

• If G is of type G_{2}, then V = L(ω_{1}), where ω_{1} is the first fundamental weight (in Bourbaki’s notation); its dimension is 7.

• If G is of type F_{4}, then V = L(ω_{4}); its dimension is 26 if p ≠ 3 and 25 if p = 3.

• If G is of type B_{r}, r > 1, then V = L(ω_{1}) is the standard representation of G = SO_{2r+1} of dimension 2r + 1.

• If G is of type C_{r}, r > 1, then V = L(ω_{2}). When p ∤ r, one has V ⊕ 1 = ⋀^{2}(V_{1}) where V_{1} = L(ω_{1}) is the standard representation of G̃ = Sp_{2r}; one has dim V = 2r^{2} −r −1. When p | r, V is a subquotient of ⋀^{2}(V_{1}) of dimension 2r^{2} − r − 2.

§ 4 Monomial quadratic forms

Consider the following general situation. Let A be an abelian group of type (2, …, 2) and rank s and let λ : A → O(V, q) be an orthogonal representation of A. As above, take K = k(t_{1}, ..., t_{s}), where t_{1}, …, t_{s} are independent indeterminates, and define θ_{A} ∈ H^{1}(K, A) as in § 2. Let θ_{O} = λ(θ_{A}) be the image of θ_{A} in H^{1}(K, O (V, q)). Let X(A) = Hom (A, Z/2Z) be the character group of A. Let X_{λ} be the subset of X(A) made up of the characters whose multiplicity in λ is odd.

Theorem 4 The integers ed(θ_{O}) and ed(θ_{O}; 2) are both equal to the rank r_{λ} of the subgroup of X(A) generated by X_{λ}.

Note that θ_{O} ∈ H^{1}(K, O (V, q)) may be interpreted as a quadratic form (namely, the twist of q by θ_{O}); we will denote this form by q_{O}; it is well defined up to K-isomorphism. To prove Theorem 4, we first need to compute explicitly q_{O}.

Computation of q_{O}

If α ∈ X(A), let V_{α} be the corresponding weight subspace of V. We have an orthogonal decomposition V = ⊕_{α} V_{α}; put m_{α} = dim V_{α}.

Let α_{1}, …, α_{s} be the canonical basis of X(A) corresponding to the projections A = Z/2Z × ⋯ × Z/2Z → Z/2Z. Any element a ∈ A acts by multiplication by α(a) on V_{α}. Hence twisting q|_{V_{α}} by θ_{O} we obtain a quadratic form ⟨t^{α}, …, t^{α}⟩ of dimension m_{α}, where t^{α} = α(t) = t_{1}^{n_{1}} ⋯ t_{s}^{n_{s}} ∈ K^{×} for α = n_{1}α_{1} + ⋯ + n_{s}α_{s}. Hence q_{O} can be written as

q_{O} = ⊕ m_{α} ⟨t^{α}⟩,

where m_{α} ⟨t^{α}⟩ means the direct sum of m_{α} copies of the 1-dimension form ⟨t^{α}⟩. Note that, because −1 is a square, we have ⟨t^{α}, t^{α}⟩ = 0 in the Witt group W (K), so that the formula can also be written as

q_{O} = ⊕_{α∈X_{λ}} ⟨t^{α}⟩ in W (K),

where the sum is over the set X_{λ} defined above.

Examples. Let ρ : G → O (V, q) be as in Proposition 3 and let λ denote the composition A → G →^{ρ} O (V, q).

a) If G = G_{2} and V is as defined in § 3, then

q_{O} = ⟨ u, t_{1}, ut_{1}, t_{2}, ut_{2}, t_{1}t_{2}, ut_{1}t_{2} ⟩ = ⟨⟨ u, t_{1}, t_{2} ⟩⟩ − ⟨1⟩,

where ⟨⟨ u, t_{1}, t_{2} ⟩⟩ is the generic 3-Pfister form.

b) Similarly, in the case of G = F_{4} (and p ≠ 3), q_{O} is

q_{O} = q_{3} ⊗ (q_{2} − ⟨1⟩) ⊕ ⟨1, 1⟩,

where q_{2} (resp. q_{3}) is a generic 2-Pfister form (resp. 3-Pfister form). When p = 3, the term ⟨1, 1⟩ is replaced by ⟨1⟩.

c) In case G = E_{8}, q_{0} can also be computed. One finds:

q_{O} = 8 ⟨1⟩ ⊕ ⟨1, u⟩ ⊗ ( ⊕_{(m)} ⟨t_{1}^{m_{1}} ⋯ t_{8}^{m_{8}}⟩)

where m = (m_{1}, …, m_{8}) runs through the 120 octuples of 0, 1’s such that

m_{1}m_{2} + m_{3}m_{4} + m_{5}m_{6} + m_{7}m_{8} ≡ 1 (mod 2).

Monomial quadratic forms

A rank n quadratic form f(X_{1}, …, X_{n}) over K = k(t_{1}, …, t_{s}) is called monomial if it is a diagonal form

f(X) = ∑ a_{i}X_{i}^{2},

with a_{i} ∈ K and if each coefficient a_{i} is a monomial in t_{1}, …, t_{s} with exponents in Z (“Laurent monomial”). As usual, we write such a form as f = ⟨a_{1}, …, a_{n}⟩.

Examples. a) The generic 2-Pfister form ⟨1, t_{1}, t_{2}, t_{1}t_{2}⟩ is monomial over k(t_{1}, t_{2}).

b) The form q_{O} = ⊕_{α} ⟨ t^{α}⟩ defined above is monomial.

Let f = ⟨a_{1}, …, a_{n}⟩ be monomial. After dividing the a_{i}’s by squares, we may assume that they are ”square-free”, i.e. for every i and j, the exponent of t_{j} in the monomial a_{i} is 0 or 1. We can then write f as:

f = ⊕ m_{f}(μ) ⟨t^{μ}⟩,

where the exponent μ = (μ_{1}, …, μ_{s}) belongs to {0, 1}^{s} = (F_{2})^{s}, t^{μ} means t_{1}^{μ_{1}} ⋯ t_{s}^{μ_{s}}, and m_{f}(μ) is ≥ 0. We say that f is multiplicity free if it is squarefree and m_{f}(μ) = 0 or 1 for every μ.

Proposition 5 A multiplicity free monomial quadratic form f over K = k(t_{1}, …, t_{s}) is anisotropic.

Proof. Let v be the valuation of K with value group Z^{s} (with lexicographic order) which is trivial on k and such that

v(t_{1}) = (1, 0, …, 0), …, v(t_{s}) = (0, …, 0, 1)

(see [Bo 64], Chap. 6, § 10). If f represents 0 we get an equation

∑ t^{μ}ϕ_{μ}(t)^{2} = 0,

where the non-zero terms have different v-valuations (and even different valuations in Γ/2Γ). This is only possible if all the terms are 0. Alternate proof: use the fact that f is a subform of a generic s-Pfister form, and that such a form is anisotropic, cf. [Pf 95], p. 111.

Let f be a monomial square-free quadratic form over K, and let X_{f} be the subset of (F_{2})^{s} made up of the μ’s such that m_{f}(μ) is odd. Let e = e_{f} be the rank of X_{f}, i.e. the dimension of the F_{2}-subspace of (F_{2})^{s} generated by X_{f}.

Proposition 6 The integers ed(f) and ed(f; 2) are both equal to e.

Note that for f = q_{O} the rank of X_{q_{O}} is obviously equal to that of X_{λ}, hence Theorem 4 follows from Proposition 6.

Remark. One may wonder whether the equality ed(f) = ed(f; 2) remains true for an arbitrary quadratic form f. It is not hard to see that it does if ed(f; 2) ≤ 2, but we do not know what happens for larger values of ed(f; 2).

§ 5 Proof of Proposition 6

We use induction on the number s of the indeterminates t_{1}, …, t_{s}, the case s = 0 being obvious. Since −1 is a square in k, each pair ⟨t^{μ}, t^{μ}⟩ is hyperbolic, and can be replaced by ⟨1, −1⟩ = ⟨1, 1⟩. Hence every monomial quadratic form f can be written as f = ⟨1, …, 1⟩ ⊕ q, with q multiplicity free. Since X_{f} = X_{q} or X_{q} ∪ {0} , we have e = rank(X_{f}) = rank(X_{q}).

We now make a further reduction on f. In order to state it, let us say that q is e-reduced if the set X_{q} contains the first e basic vectors

x_{1} = (1, 0, …, 0); x_{2} = (0, 1, …, 0); …; x_{e} = (0, …, 1, …, 0).

This amounts to saying that

q = ⟨t_{1}, …, t_{e}, a_{e+1}, …, a_{n}⟩,

where the a_{i}, for i > e, are pairwise distinct square-free monomials in t_{1}, …, t_{e} of total degree ≠ 1.

Lemma 7 There is an automorphism of the extension K/k which transforms q into an e-reduced form.

Proof. Note that GL_{s}(Z) acts in a natural way on the set of monomials with exponents in Z. This gives a natural embedding of GL_{s}(Z) into Aut(K/k). Moreover, the natural map

GL_{s}(Z) ⟶ GL_{s}(F_{2})

is surjective, since GL_{s}(F_{2}) = SL_{s}(F_{2}) and

SL_{s}(Z) → SL_{s}(Z/mZ)

is well known to be surjective for any m. By the very definition of e, the set X_{q} contains e elements z_{1}, …, z_{e} which are linearly independent over F_{2}. Hence there exists ϕ ∈ GL_{s}(Z) ⊂ Aut(K/k) whose reduction mod 2 transforms the z_{i} into the first e basic vectors x_{1}, …, x_{e}. It is clear that ϕ(q) is e-reduced.

We now use the “residue operators” of the local theory of quadratic forms (see e.g. [L 73], Chap. VI, § 1.5). Recall that, if v is a discrete valuation of a field K, with residue field K̃, one may write any quadratic form q over K in the form

q = ⟨ u_{1}, …, u_{m}, πu_{m+1}, …, πu_{n} ⟩,

where the u’s are units, π is a uniformizing element and m is an integer with 0 ≤ m ≤ n. One defines the first residue ∂_{1}(q) of q as the class in the Witt group W (K̃) of the quadratic form ⟨ũ_{1}, …, ũ_{m} ⟩, where ũ_{i} denotes the image of u_{i} in K̃; similarly, the second residue ∂_{2}(q) of q is the class in W (K̃) of ⟨ ũ_{m+1}, …, ũ_{n} ⟩. It is known (loc. cit.) that the class of ∂_{1}(q) does not depend on the choice of π, nor on the choice of the diagonalization of q; as for the class of ∂_{2}(q), it is only defined up to similarity (i.e. up to multiplication by a 1-dimensional quadratic form).

Proposition 8 Let v be a discrete valuation on an extension L of k trivial on k, let L̃ be its residue field, and let ϕ be a quadratic form over L. Let e be a positive integer. Assume :

a) ∂_{2}(ϕ) ≠ 0 in W (L̃).

b) edL̃(ψ; 2) ≥ e − 1 for every quadratic form ψover L̃ belonging to the Witt class of ∂_{1}(ϕ).

Then ed_{L}(ϕ; 2) ≥ e. (Both ed’s are relative to k, viewed as a subfield of L and of L̃.)

Proposition 8 implies Proposition 6

We apply induction on e. The case e = 0 or 1 is trivial. Let us assume e > 1. We may suppose that f is of the form f = ⟨1, …, 1⟩⊕q, where q is e-reduced and multiplicity free. Since the exponents μ appearing in X_{f} are sums of the x_{i} (1 ≤ i ≤ e), the t^{μ} appearing in q belong to the subfield k(t_{1}, ..., t_{e}) of K. This shows that ed(f) ≤ e.

It remains to show that ed(f; 2) ≥ e. To do so, consider the valuation v on K associated to the indeterminate t_{1}. Such a valuation is characterized by the properties:

v(t_{1}) = 1;
v(x) = 0 if x ∈ k(t_{2}, ..., t_{s})^{×}.

Moreover, we have K̃ = k(t_{2}, ..., t_{s}).

Let us write f as f = ϕ ⊕ ⟨t_{1}⟩ ⊗ ϕ′, where ϕ, ϕ′ are monomial quadratic forms over k(t_{2}, …, t_{s}). The second residue of f with respect to v is given by ∂_{2}(f) = ∂_{2}(q) = ϕ′. Since q is multiplicity free, so is ϕ′. It is clear that ϕ′ ≠ 0, and hence ϕ′ is anisotropic, by Proposition 5. Since the Witt class of ϕ′ is ∂_{2}(f), we have checked condition a) of Proposition 8.

Let us look at condition b). Of course ϕ is a representative of ∂_{1}(f). Moreover, it is clear that ϕ can be written as ϕ = m⟨1, 1⟩ ⊕ ψ, where m is an integer ≥ 0 and ψ is a multiplicity free (e − 1)-reduced monomial quadratic form over K̃, hence is anisotropic, by Proposition 5. Since ⟨1, 1⟩ = ⟨1, −1⟩, this shows that any quadratic form ψ′ over K̃ which belongs to the Witt class ∂_{1}(f) is isomorphic to m′⟨1, 1⟩ ⊕ ψ, hence is (e − 1)-reduced. We may thus apply the induction assumption to ψ′, and deduce that edK̃(ψ′; 2) ≥ e − 1. By Proposition 8, we get ed_{K}(f; 2) ≥ e, as required.

Proof of Proposition 8

Let L′ be an odd-degree extension of L, and let F be a subfield of L′, containing k, and such that ϕ is L′-isomorphic to a quadratic form ϕ_{F} over F. We have to show that tr. deg_{k}(F) ≥ e. We distinguish two cases:

i) The case L′ = L. Let w be the restriction of v to the subfield F. There are three possibilities:

i_{1}) w is trivial on F (i.e. v(x) = 0 for every x ∈ F^{×}). In that case, the coefficients of ϕ_{F} are v-units, and this implies that ∂_{2}(ϕ) = 0, which we assumed is not true.

i_{2}) The value group v(F^{×}) is a subgroup of even index of v(L^{×}) = Z. The same argument as for i_{1}) shows that ∂_{2}(ϕ) = 0.

i_{3}) The index of v(F^{×}) in v(L^{×}) is odd. In that case, ∂_{1}(ϕ) ∈ W (L̃) is the image of ∂_{1}(ϕ_{F}) ∈ W (F̃) under the natural map W (F̃) → W (L̃). Here F̃ is the residue field of F with respect to w. Choose any representative ψF̃ of ∂_{1}(ϕ_{F}); it gives a representative ψL̃ of ∂_{1}(ϕ), hence we have

edF̃(ψF̃; 2) ≥ edL̃(ψL̃; 2) ≥ e − 1

by hypothesis b). This implies that tr. deg_{k}(F̃) ≥ e−1, hence tr. deg_{k}(F) ≥ e by a standard result of valuation theory, cf. [Bo 64], Chap. 6, § 10, no. 3.

ii) The general case. Let S be the set of extensions w of v to L′. For each w ∈ S, let e(w/v) and f(w/v) be the ramification index and the residue degree of w with respect to v.

Lemma 9 There exists w ∈ S such that both e(w/v) and f(w/v) are odd.

Proof. By dévissage, it is enough to prove this in the following two cases.

a) The extension L′/L is separable. In that case, we have the standard formula (cf. [Bo 64], Chap. 6, § 8, no. 5)

∑_{w∈S} e(w/v)f(w/v) = [L′ : L].

Since [L′ : L] is odd, there is at least one w ∈ S such that e(w/v)f(w/v) is odd.

b) We have char(L) = p > 0 and L′/L is purely inseparable. In that case, S is reduced to one element w, and one checks that e(w/v) and f(w/v) are powers of p, hence are odd.

End of proof of ii). Select w as in Lemma 9. We are going to apply case i) to (L′, ϕ, w). Note first that the w-residues of ϕ are the images of its v-residues by the base change L̃ → L̃′. Since [L̃′ : L̃] is odd, the map W (L̃) → W (L̃′) is injective. This shows that ∂_{2}(ϕ) ≠ 0 in W (L̃′), so that condition a) is satisfied by (L′, ϕ, w).

It remains to check condition b). Let ψ_{0} be the unique anisotropic representative of ∂_{1}(ϕ); by a classical theorem of Springer (cf. [L 73], p.198), it remains anisotropic in L̃′. Hence the representatives ψ of ∂_{1}(ϕ) over L̃′ are the sums of ψ_{0} and some hyperbolic forms; in particular they come from L̃. Since an odd degree extension does not change ed( ; 2) we have edL̃′(ψ; 2) ≥ e − 1. We have thus checked conditions a) and b) over L′, and we may apply part i) of the proof.

This concludes the proof of Proposition 8 and hence of Proposition 6 and of Theorem 4. □

Remark. Let K/k be a field extension, with k algebraically closed. Let q and q′ be quadratic forms over K which belong to the same Witt class. Is it true that ed(q) = ed(q′) and ed(q; 2) = ed(q′; 2)? It is so when K = k(t_{1}, …, t_{e}) and one of the forms q or q′ is monomial. We do not know what happens in general.

§ 6 Proof of Theorem 2

Let ρ : G → O (V, q) be as in Proposition 3, and let θ_{O} = ρ(θ_{G}) be the image of θ_{G} in H^{1}(K, O (V, q)). If ρ_{A} denotes the composition A → G →^{ρ} O (V, q), we have θ_{O} = ρ_{A}(θ_{A}). By Theorem 4, it suffices to show that the rank of ⟨X_{ρ_{A}}⟩ is r + 1. We need the following.

Lemma 10 Let R be an irreducible root system, and R_{sh} the set of short roots. Let Q(R) be the root lattice of R. If α and β are elements of R_{sh}, we have :

α = β (mod 2Q(R)) ⟺ α = ±β.

Proof. This can be checked by inspection of all possible root systems.

Let us compute the weights of ρ_{A} and their multiplicities. For a short root α ∈ R_{sh}^{+} we denote by V_{α} the corresponding weight subspace of V for T. By construction, dim V_{α} = 1 and we have an orthogonal decomposition

V = V_{0} ⊕ {⊕_{α} (V_{α} ⊕ V_{−α})} ,

where the sum is taken over all positive short roots.

Any element a ∈ A^{∘} acts by multiplication by α(a) on W_{α} = V_{α}⊕V_{−α}, and acts trivially on V_{0}. The automorphism c of § 2 preserves V_{0} and permutes V_{α} and V_{−α}. Since k is algebraically closed, there is a basis {u_{α}, v_{α}} of W_{α} such that c(u_{α}) = u_{α}, and c(v_{α}) = −v_{α}. It follows that the weight subspaces for ρ_{A} belonging to ⊕_{α} (V_{α} ⊕ V_{−α}) correspond to characters α ∈ R_{sh}^{+} and αγ, where

γ ∈ X(A) = Hom(A, ±1) is given by A^{∘} ↦ 1 and c ↦ −1. Furthermore, all these weights of ρ_{A} have multiplicity 1, by Lemma 10. Depending on the action of c on V_{0} the set X_{ρ_{A}} may contain additionally 0 and γ. In all cases the rank of ⟨X_{ρ_{A}}⟩ is r + 1, as required. □

§ 7 Spin groups

We keep the notation of the previous §§. In particular, the ground field k is algebraically closed of characteristic ≠ 2. If n > 2, we denote by Spin_{n} the universal covering of the group SO_{n} (relative to the unit quadratic form ⟨1, ..., 1⟩). For n ≤ 6, this group is “special”, which implies that ed(Spin_{n}) = 0, cf. [R 00]. The situation is different for n > 6. In order to state it precisely, let us define an integer e(n) by :

e(10) = 4;
e(n) = [n/2] = rank Spin_{n} if n > 6 and n ≠ 10.

Theorem 11 We have ed(Spin_{n}; 2) ≥ e(n) for every n > 6.

Proof. Let us write e = e(n), and put K = k(t_{1}, ..., t_{e}), where t_{1}, …, t_{e} are independent indeterminates. We are going to construct a monomial quadratic form f_{n} of rank n over K with the following properties:

(i) the Stiefel-Whitney classes w_{1}(f_{n}) and w_{2}(f_{n}) are both zero. (For the definitions of the Stiefel-Whitney classes, see e.g. [GMS 03], § 17.)

(ii) rank(X_{f_{n}}) = e, with the notation of the lines preceding Proposition 6.

Such a form f_{n} corresponds to an element [f_{n}] of H^{1}(K, O_{n}) which belongs to the image of H^{1}(K, Spin_{n}) → H^{1}(K, O_{n}) (because of (i)) and is such that ed([f_{n}]; 2) = e (because of (ii), cf. Proposition 6). This shows that H^{1}(K, Spin_{n}) contains an element ξ_{n} with ed(ξ_{n}; 2) ≥ e; hence the theorem.

Here is the construction of f_{n}. There are four cases, depending on the value of n modulo 4:

a) n ≡ 0 ( mod 4), n ≥ 8. We have e = n/2, which is even. We define f_{n} by:

f_{n} = ⟨t_{1}, …, t_{e}⟩ ⊗ ⟨1, t_{1} ⋯ t_{e}⟩.

Condition (ii) is obvious (but would not be true in the excluded case n = 4). As for condition (i), it follows from the general formulae:

w_{1}(f ⊗ f′) = 0 and w_{2}(f ⊗ f′) = w_{1}(f) · w_{1}(f′)

if rank(f) and rank(f′) are even. Indeed this shows that w_{1}(f_{n}) = 0 and that w_{2}(f_{n}) = (t_{1} ⋯ t_{e}) · (t_{1} ⋯ t_{e}) = (−1) · (t_{1} ⋯ t_{e}) = 0 since −1 is a square in k.

b) n ≡ −1 ( mod 4 ), n ≥ 7. Here e = (n − 1)/2, which is odd. We put:

f_{n} = ⟨t_{1}, …, t_{e}⟩⊗⟨1, t_{1} ⋯ t_{e}⟩ ⊕ ⟨t_{1} ⋯ t_{e}⟩.

Conditions (i) and (ii) are checked as in case a).

c) n ≡ 1 ( mod 4 ), n ≥ 9. Here e = (n − 1)/2, which is even. We put:

f_{n} = f_{n−1} ⊕ ⟨1⟩ = ⟨t_{1}, …, t_{e}⟩⊗⟨1, t_{1} ⋯ t_{e}⟩ ⊕ ⟨1⟩.

Conditions (i) and (ii) follow from case a).

d) n ≡ 2 ( mod 4 ). This case splits into four subcases: d_{1}) n = 10. Here e = 4 and we put

f_{10} = f_{8} ⊕ ⟨1, 1⟩ = ⟨t_{1}, …, t_{4}⟩⊗⟨1, t_{1}...t_{4}⟩ ⊕ ⟨1, 1⟩.

d_{2}) n = 14. Here e = 7. We put

f_{14} = ⟨t_{7}⟩⊗(⟨⟨ t_{1}, t_{2}, t_{3} ⟩⟩_{0} ⊕ ⟨⟨ t_{4}, t_{5}, t_{6} ⟩⟩_{0})

where ⟨⟨a, b, c⟩⟩_{0} means ⟨⟨a, b, c⟩⟩ − ⟨1⟩, i.e. ⟨a, b, c, ab, bc, ac, abc⟩. The simplest way to check condition (i) is to rewrite f_{14} in the Witt ring W (K) as

f_{14} = ⟨t_{7}⟩ · (⟨⟨ t_{1}, t_{2}, t_{3} ⟩⟩ + ⟨⟨ t_{4}, t_{5}, t_{6} ⟩⟩).

This shows that f_{14} belongs to the cube I^{3} of the augmentation ideal I of W (K), and that implies condition (i). Condition (ii) is easy to check.

d_{3}) n = 18. Here e = 9. We put:

f_{18} = ⟨t_{1}, t_{2}, t_{1}t_{2}⟩⊗⟨t_{7}, t_{8}⟩ ⊕ ⟨t_{3}, t_{4}, t_{3}t_{4}⟩⊗⟨t_{8}, t_{9}⟩ ⊕ ⟨t_{5}, t_{6}, t_{5}t_{6}⟩⊗⟨t_{7}, t_{9}⟩.

In the Witt ring W (K), one has:

f_{18} = ⟨⟨ t_{1}, t_{2} ⟩⟩ · ⟨t_{7}, t_{8}⟩ + ⟨⟨ t_{3}, t_{4} ⟩⟩ · ⟨t_{8}, t_{9}⟩ + ⟨⟨ t_{5}, t_{6} ⟩⟩ · ⟨t_{7}, t_{9}⟩,

and this shows that f_{18} ∈ I^{3}, hence condition (i). As for condition (ii), one checks that, if one makes the change of variables:

T_{1} = t_{1}t_{7}, T_{2} = t_{2}t_{7}, T_{3} = t_{1}t_{2}t_{7}, T_{4} = t_{3}t_{8}, T_{5} = t_{4}t_{8},
T_{6} = t_{3}t_{4}t_{8}, T_{7} = t_{5}t_{9}, T_{8} = t_{6}t_{9}, T_{9} = t_{5}t_{6}t_{9},

then f_{18} becomes e-reduced (as a monomial quadratic form in the T_{i}). This implies (ii).

d_{4}) n ≡ 2 ( mod 4 ), n > 18. We define f_{n} by induction on n, as the sum of f_{n−8} and f_{8} (with independent variables):

f_{n} = f_{n−8} ⊕ ⟨t_{e−3}, t_{e−2}, t_{e−1}, t_{e}⟩⊗⟨1, t_{e−3}t_{e−2}t_{e−1}t_{e}⟩.

Conditions (i) and (ii) are proved by induction on n. This concludes the proof.

Remarks. 1) The reader may wonder whether the quadratic form f_{n} used above could have been defined via an abelian finite subgroup of Spin_{n}(k) whose image in SO_{n}(k) is of type (2, …, 2). The answer is “yes”; this follows from the well known construction of abelian 2-subgroups of Spin_{n} from binary linear codes (see e.g. [RY 00], pp. 1043–1044). Indeed, this is how we first obtained case d_{3} above (n = 18).

2) When n ≡ −1, 0 or 1 ( mod 8 ), the bound given by Theorem 11 can be slightly improved. This is due (in characteristic 0, at least) to Reichstein-Youssin ([RY 00], Theorem 8.16). More precisely:

Theorem 12 Assume n ≡ −1, 0 or 1 ( mod 8 ), n ≥ 7. Then :

ed(Spin_{n}; 2) ≥ [n/2] + 1.

Proof. We define a (2, ..., 2)-subgroup A of Spin_{n}(k) as in § 2, namely as A_{0} × {1, c̃}, where A_{0} is the 2-division subgroup of the maximal torus T, and c̃ is a lifting in Spin_{n}(k) of the element c of the adjoint group. The congruence condition on n implies that c̃ is of order 2. We have rank A = r+1 = [n/2]+1. Let us suppose first that n ≡ ±1 ( mod 8 ). The spin representation is then orthogonal, and it gives a homomorphism

ρ : Spin_{n} → O_{N}, with N = 2^{r} = 2^{(n−1)/2}.

If K = k(t_{1}, …, t_{r}, u) we define θ_{A} ∈ H^{1}(K, A) as in § 2. The image of θ_{A} by ρ corresponds to a rank N quadratic form q, which is easily shown to be isomorphic (up to a change of variables) to ⟨u⟩ ⊗ ⟨⟨ t_{1}, …, t_{r} ⟩⟩. By Proposition 6, we have ed(q; 2) ≥ r + 1. This shows that the image θ of θ_{A} in H^{1}(K, Spin_{n}) is such that ed(θ; 2) ≥ r+1, and the theorem follows. The case where n ≡ 0 ( mod 8 ) is analogous : one takes for ρ the direct sum of the two half-spin representations (which are orthogonal, because n ≡ 0 ( mod 8 )).

§ 8 Other examples

Theorem 13 (i) ed(HSpin_{n}; 2) ≥ n/2 + 1 if n > 0, n ≡ 0 (mod 8). (ii) ed(PSO_{n}; 2) ≥ n − 2 if n is even ≥ 4. (iii) ed(2.E_{7}; 2) ≥ 7. (iv) ed(PGL_{n}) ≥ v_{2}(n) if n > 0.

(Undefined notation will be explained below.)

Proof (sketch). Let G be the group HSpin_{n} (resp. PSO_{n}, resp. 2.E_{7}, resp. PGL_{n}) mentioned in the theorem. We apply the method of the previous sections to a suitable abelian subgroup A of G(k), of rank e = n/2 + 1 (resp. n − 2, resp. 7, resp. v_{2}(n)) and to a suitable orthogonal representation ρ : G → GL(V). We thus get a monomial quadratic form q over K = k(t_{1}, …, t_{e}), and a routine computation, based on Theorem 4, shows that ed(q; 2) = e, hence the result.

Here are the definitions of A and ρ in each case (the “routine computation” is left to the reader):

Case (i). The group G = HSpin_{n} is the half-spin group, i.e. the quotient of Spin_{n} by a central subgroup of order 2 distinct from the kernel of Spin_{n} → SO_{n}. This is well defined whenever n ≡ 0 (mod 4), with a slight ambiguity for n = 8, since in that case HSpin_{8} is isomorphic to SO_{8} (and hence ed(HSpin_{8}; 2) = 7). The group G acts faithfully on the corresponding half-spin representation S. Since n ≡ 0 (mod 8), this is an orthogonal representation. Let T be a maximal torus of G. As in § 2 we define A to be the subgroup of G(k) generated by the elements of order 2 of T and by an element c of order 2 of N(T) such that ctc = t^{−1} for every t ∈ T (such an element exists because n is divisible by 8). The group A is an elementary abelian (2, …, 2)-group of rank e = n/2+1. We choose for V the direct sum S ⊕ Lie(G).

Case (ii). The group G = PSO_{n} is the quotient SO_{n}/μ_{2}, i.e. an adjoint group of type D_{n/2}. The group A is the image in G(k) of the diagonal matrices of square 1 in SO_{n}. It is a (2, …, 2)-abelian group of rank e = n−2. One takes for V the Lie algebra of G, with the quadratic form defined by Tr(x · y).

Case (iii). The group G = 2.E_{7} is a simply connected group of type E_{7}. Choose a maximal torus T of G, and let c ∈ N(T) be such that ctc^{−1} = t^{−1} for every t ∈ T. We have c^{2} = z, where z is the non trivial element of the center of G. Let A_{0} be the kernel of t ↦ t^{2}; it is an elementary group of type (2, …, 2) and of rank 7; it contains z. The subgroup A of G generated by A_{0} and c is an abelian group of type (4, 2, …, 2) and of rank 7. The image of A in the adjoint group G′ = G/{1, z} is A′ = A/{1, z}; it is elementary abelian of rank 7. If K = k(t_{1}, …, t_{7}), we have a canonical element θ_{A′} in H^{1}(K, A′); since −1 is a square in K, there exists an element θ_{A} ∈ H^{1}(K, A) whose image in H^{1}(K, A′) is θ_{A′}. We choose for orthogonal representation of G the adjoint representation. The action of A on this representation factors through A′, hence gives a monomial quadratic form q over k(t_{1}, …, t_{7}) and one checks that q is 7-reduced.

Case (iv). Here G = PGL_{n} and e = 2m, where m is the 2-adic valuation of n. If we write n as 2^{m}N, with N odd, there is a natural injection of PGL_{2} ×…×PGL_{2} (m factors) in G. Let A_{1} be a (2, 2)-subgroup of PGL_{2}, and let A = A_{1} × … × A_{1} (m factors). We have an embedding

A ⟶ PGL_{2} × … × PGL_{2} ⟶ PGL_{n} = G,

and A is a (2, …, 2)-group of rank e. We select for V the space M_{n} of n × n matrices, with the scalar product Tr(x · y). The group G acts by conjugation on M_{n}. (Here the monomial quadratic form q is the tensor product of a generic e-Pfister form by the unit form ⟨1, …, 1⟩ of rank N^{2}; since N is odd, Theorem 4 shows that the essential dimension of q at 2 is indeed equal to e.) □

Remarks. 1) We do not know how good are the lower bounds of Theorems 1,11,12 and 13. Some are rather weak: for instance, th. 1 applied to type B_{n} gives roughly half the true value of ed(G; 2). What about those on Spin_{n}, HSpin_{n}, and E_{8}? These questions are related: an upper bound for HSpin_{16} would give one for E_{8}.

2) Applying Proposition 6 to the generic quadratic form q = ⟨t_{1}, …, t_{n}⟩ and the generic quadratic form q′ = ⟨t_{1}, …, t_{n−1}, t_{1} ⋯ t_{n−1}⟩ of discriminant 1 one recovers the well known facts that ed(O_{n}; 2) ≥ n and ed(SO_{n}; 2) ≥ n−1 (if n ≥ 2), cf. e.g. [R 00], Theorems 10.3 and 10.4.

3) There are cases where the method “A → G → O(V, q)” fails to give any result. For instance, let G be a group of type E_{6} (adjoint, or simply connected, it does not matter). By using the relations of this group with G_{2} (cf. [GMS 03], Exercise 22.9) it is not hard to see that ed(G; 2) is equal to 3. One can show that there is no way to prove this by the A → G → O(V, q) method: every orthogonal representation G → O(V, q) gives a map H^{1}(K, G) → H^{1}(K, O(V, q)) which is trivial, hence gives no information on ed(G).

Acknowledgement

Work on the present paper started in 2002, at the Centre Bernoulli, E.P.F.L., Lausanne. We want to thank the Centre Bernoulli for its hospitality and its stimulating atmosphere.

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