THE J-INVARIANT AND THE TITS ALGEBRAS OF A LINEAR ALGEBRAIC GROUP.

A. QUÉGUINER-MATHIEU, N. SEMENOV, K. ZAINOULLINE

Abstract. In the present paper we set up a connection between the indices of the Tits algebras of a simple linear algebraic group G and the degree one parameters of its J-invariant. Our main technical tool is the second Chern class map in the Riemann-Roch theorem without denominators.

As an application we recover some known results on the J-invariant of quadratic forms of small dimension; we describe all possible values of the J-invariant of an algebra with involution up to degree 8 and give explicit examples; we establish several relations between the J-invariant of an algebra A with involution and the J-invariant (of the quadratic form) over the function field of the Severi-Brauer variety of A.

Contents

Introduction  2
Notations  2
1. Characteristic maps and restriction maps  4
Characteristic map for Chow groups.  4
Restriction map for Chow groups.  5
Characteristic map for K_{0} and the Steinberg basis.  5
Restriction map for K_{0} and the Tits algebras.  6
Rational cycles versus rational bundles.  7
2. The J-invariant  7
3. The parameters of degree one and indices of Tits algebras.  10
Filtrations of K_{0} and the Riemann-Roch Theorem  12
Proof of Lemma 3.9  13
4. Applications to quadratic forms  14
5. J-invariant of an algebra with involution  16
Split case  17
Half-spin case  18
Witt-equivalent algebras with involution  18
Classification results in degree 6  19
Classification in degree 4  19
6. The trialitarian case  20
Explicit examples  22
7. Generic properties  25
8. Appendix  25
References  26

Introduction

The notion of a Tits algebra was introduced by J. Tits in his celebrated paper on irreducible representations [40]. This invariant of a linear algebraic group G plays a crucial role in the computation of the K-theory of twisted flag varieties by Panin [29] and in the statements and proofs of the index reduction formulas by Merkurjev, Panin and Wardsworth [27]. It has important applications to the classification of linear algebraic groups, and to the study of the associated homogeneous varieties.

Another invariant of a linear algebraic group, the J-invariant, has been recently defined in [34]. It can be viewed as an extension to arbitrary groups of the discrete motivic invariant of a quadratic form which was studied during the last decade, notably by Karpenko, Merkurjev, Rost and Vishik. This invariant describes the motivic behavior of the variety of Borel subgroups of G, and it is given by an r-tuple of integers J = (j_{1}, j_{2}, …, j_{r}).

The goal of the present paper is to set up a connection between the indices of the Tits algebras of a group G and the degree one parameters of its J-invariant (see Cor. 3.3 and Thm. 3.8 below). The paper is organized as follows. In § 1, we recall the definitions of the characteristic maps, both for Chow groups and K-theory, and their relation with the restriction maps. The definition of the J-invariant of a twisted form G = _{ξ}G_{0} of a split group G_{0} is recalled in § 2,where we also describe precisely how it depends on the choice of the cocycle ξ. We then state and prove the main results in § 3. As a crucial ingredient, we use Panin’s computation of K_{0}(𝔛), where 𝔛 is the variety of Borel subgroups of G [29]. The result is obtained using the so-called γ-filtration on K_{0}(𝔛), and relies on Lemma 3.11, which describes Chern classes of rational bundles of the first two layers of the γ-filtered group K_{0}(𝔛).

The rest of the paper is devoted to applications. First, we recover very easily some known results on the J-invariant of quadratic forms of small dimension using our main theorem. We then explain how one can compute the J-invariant of an algebra with involution up to degree 8. We describe the possible values and give explicit examples. As opposed to what happens for quadratic forms, we also show that some values, which were not excluded before, actually are impossible (see 6.1). Finally, we study the relations between the J-invariant of an algebra A with involution and the J-invariant (of the quadratic form) over the function field of the Severi-Brauer variety of A.

Notations.

Algebraic groups and Borel varieties. We work over a base field k of characteristic different from 2. Let G_{0} be a split simple linear algebraic group of rank n over k. We fix a split maximal torus T_{0} ⊂ G_{0}, and a Borel subgroup B_{0} ⊃ T_{0}, and we let T̂_{0} be the character group of T_{0}. We let Π = {α_{1}, α_{2}, …, α_{n}} be a set of simple roots with respect to B_{0}, and {ω_{1}, ω_{2}, …, ω_{n}} the respective set of fundamental weights, so that α_{i}^{∨}(ω_{j}) = δ_{ij}. The roots and weights are always numbered as in Bourbaki [1].

Recall that Λ_{r} ⊂ T̂_{0} ⊂ Λ_{ω}, where Λ_{r} and Λ_{ω} are the root and weight lattices, respectively. The lattice T̂_{0} coincides with Λ_{r} (respectively Λ_{ω}) if and only if G_{0} is adjoint (respectively simply connected).

0.1. Throughout the paper, G denotes a twisted form of G_{0}, and T ⊂ G is the corresponding maximal torus. We always assume that G is an inner twisted form of G_{0}, and even a little bit more, that is G = _{ξ}G_{0} for some cocycle ξ ∈ Z^{1}(k, G_{0}). Note that this hypothesis need not be satisfied by all groups isogeneous to G; for instance, it is valid for the simply connected cover G^{sc} of G if and only if G is a strongly inner twisted form of G_{0}.

0.2. We let 𝔛_{0} be the variety of Borel subgroups of G_{0}, or equivalently of its simply connected cover G_{0}^{sc}, and 𝔛 = _{ξ}𝔛_{0} the corresponding twisted variety. Both varieties are defined over k, and they are isomorphic over a separable closure k_{s} of k. The Picard group Pic(𝔛_{0}) can be computed as follows. Since any character λ ∈ T̂_{0} extends uniquely to B_{0}, it defines a line bundle L(λ) over 𝔛_{0}. Hence, there is a natural map T̂_{0} → Pic(𝔛_{0}), which is an isomorphism if G_{0} is simply connected by [28, Prop. 2.2]. So, we may identify the Picard group Pic(𝔛_{0}) with the weight lattice Λ_{ω}.

Algebras with involution. We refer to [23] for definitions and classical facts on algebras with involution. Throughout the paper, (A, σ) always stands for a central simple algebra of even degree 2n, endowed with an involution of orthogonal type with trivial discriminant. In particular, this implies that the Brauer class [A] of the algebra A is an element of order 2 of the Brauer group Br(k). Because of the discriminant hypothesis, the Clifford algebra of (A, σ), endowed with its canonical involution, is a direct product (C(A, σ), \underline{σ}) = (C_{+}, σ_{+})×(C_{−}, σ_{−}) of two central simple algebras. If moreover n is even, the involutions σ_{+} and σ_{−} also are of orthogonal type.

0.3. We refer to [23, § 6] for the definition of isotropic and hyperbolic involutions. In particular, recall that A has a hyperbolic involution if and only if it decomposes as A = M_{2}(B) for some central simple algebra B over k. When this occurs, A has a unique hyperbolic involution σ_{0} up to isomorphism. Moreover, σ_{0} has trivial discriminant, and if additionally the degree of A is divisible by 4, then its Clifford algebra has a split component by [23, (8.31)].

0.4. The connected component of the automorphism group of (A, σ) is denoted by PGO^{+}(A, σ). Since the involution has trivial discriminant, it is an inner twisted form of PGO_{2n}^{+}, and hence satisfies (0.1). Both groups are adjoint of type D_{n}. We recall from Bourbaki [1] the description of their cocenter Λ_{ω}/Λ_{r} in terms of the fundamental weights, for n ≥ 3:

If n = 2m is even, then Λ_{ω}/Λ_{r} ≃ ℤ/2ℤ×ℤ/2ℤ, and the three non-trivial elements are the classes of ω_{1}, ω_{2m−1} and ω_{2m}.

If n = 2m + 1 is odd, then Λ_{ω}/Λ_{r} ≃ ℤ/4ℤ, and the generators are the classes of ω_{2m} and ω_{2m+1}. Moreover, the element of order 2 is the class of ω_{1}.

Tits algebras. Consider the simply connected cover G_{0}^{sc} of G_{0} and the corresponding twisted group G^{sc}. We denote by Λ_{ω}^{+} ⊂ Λ_{ω} the cone of dominant weights. Since G is an inner twisted form of G_{0}, for any ω ∈ Λ_{ω}^{+} the corresponding irreducible representation G_{0}^{sc} → GL(V), viewed as a representation of G^{sc} × k_{s}, descends to an algebra representation G^{sc} → GL_{1}(A_{ω}), where A_{ω} is a central simple algebra over k, called a Tits algebra of G (cf. [40, §3,4] or [23, §27]). In particular, to any fundamental weight ω_{ℓ} corresponds a Tits algebra A_{ω_{ℓ}}

0.5. Taking Brauer classes, the assignment ω ∈ Λ_{ω}^{+} ↦ A_{ω} induces a homomorphism to the Brauer group of k (loc. cit.)

β : Λ_{ω}/T̂_{0} → Br(k).

0.6. Example. If G is adjoint of type D_{n}, that is G = PGO^{+}(A, σ), the Tits algebras A_{ω_{1}}, A_{ω_{2m−1}} and A_{ω_{2m}} are respectively the algebra A and the two components C_{+} and C_{−} of the Clifford algebra of (A, σ) (see [23, §27.B]). Applying Tits homomorphism, and taking into account the description of Λ_{ω}/T̂_{0} = Λ_{ω}/Λ_{r}, we get the so-called fundamental relations [23, (9.12)] relating their Brauer classes, namely:

If n = 2m is even, that is deg(A) ≡ 0 mod 4, then [C_{+}] and [C_{−}] are of order at most 2, and [A] + [C_{+}] + [C_{−}] = 0 ∈ Br(k). In other words, any of those three algebras is Brauer equivalent to the tensor product of the other two.

If n = 2m + 1 is odd, that is deg(A) ≡ 2 mod 4, then [C_{+}] and [C_{−}] are of order dividing 4, and [A] = 2[C_{+}] = 2[C_{−}] ∈ Br(k).

For any ω ∈ Λ_{ω}, we denote by i(ω) the index of the Brauer class β(ω̄), that is the degree of the underlying division algebra. For fundamental weights, i(ω_{ℓ}) is the index of the Tits algebra A_{ω_{ℓ}}.

1. Characteristic maps and restriction maps

Characteristic map for Chow groups. Let CH^{∗}(−) be the graded Chow ring of algebraic cycles modulo rational equivalence. Since 𝔛_{0} is smooth projective, the first Chern class induces an isomorphism between the Picard group Pic(𝔛_{0}) and CH^{1}(𝔛_{0}) [14, Cor. II.6.16]. Combining with the isomorphism Λ_{ω} ≃ Pic(𝔛_{0}) of 0.2, we get an isomorphism, which is the simply connected degree 1 characteristic map:

𝔠_{sc}^{(1)} : Λ_{ω} ˜→ CH^{1}(𝔛_{0}).

Hence, the cycles

h_{i} := c_{1}(L(ω_{i})), i = 1 … n,

form a ℤ-basis of the group CH^{1}(𝔛_{0}).

1.1. In general, the degree 1 characteristic map is the restriction of this isomorphism to the character group of T_{0},

𝔠^{(1)} : T̂_{0} ⊂ Λ_{ω} → CH^{1}(𝔛_{0}).

Hence, it maps λ = ∑_{i=1}^{n} a_{i}ω_{i} ∈ T̂_{0}, where a_{i} ∈ ℤ, to c_{1}(L(λ)) = ∑_{i=1}^{n} a_{i}h_{i}. For instance, in the adjoint case, the image of 𝔠^{(1)} is generated by linear combinations ∑_{j} c_{ij}h_{j}, where c_{ij} = α_{i}^{∨}(α_{j}) are the coefficients of the Cartan matrix.

1.2. Example. We let p = 2 and consider the Chow group with coefficients in 𝔽_{2} Ch^{1}(𝔛_{0}) = CH^{1}(𝔛_{0}) ⊗_{ℤ} 𝔽_{2}. Assume G_{0} is of type D_{4}. Using the simply connected characteristic map, we may identify the degree 1 Chow group modulo 2 with the 𝔽_{2}-lattice

Ch^{1}(𝔛_{0}) = 𝔽_{2}h_{1} ⊕ 𝔽_{2}h_{2} ⊕ 𝔽_{2}h_{3} ⊕ 𝔽_{2}h_{4}

Moreover, numbering roots as in [1], the Cartan matrix modulo 2 is given by

( \begin{matrix}0 & 1 & 0 & 0 \\ 1 & 0 & 1 & 1 \\ 0 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{matrix} )

Therefore, in the adjoint case the image of the characteristic map 𝔠_{ad}^{(1)} with 𝔽_{2}- coefficients is the subgroup

im(𝔠_{ad}^{(1)}) = 𝔽_{2}h_{2} ⊕ 𝔽_{2}(h_{1} + h_{3} + h_{4}) ⊂ Ch^{1}(𝔛_{0})

In the half-spin case, that is when one of the two weights ω_{3}, ω_{4} is in T̂_{0}, say ω_{3} ∈ T̂_{0}, we get

im(𝔠_{hs}^{(1)}) = 𝔽_{2}h_{2} ⊕ 𝔽_{2}h_{3} ⊕ 𝔽_{2}(h_{1} + h_{4}) ⊂ Ch^{1}(𝔛_{0})

1.3. The degree 1 characteristic map extends to a characteristic map

𝔠: S^{∗}(T̂_{0}) → CH^{∗}(𝔛_{0}),

where S^{∗}(T̂_{0}) is the symmetric algebra of T̂_{0} (see [13, §4], [3, §1.5]). Its image im(𝔠) is generated by the elements of codimension one, that is by the image of 𝔠^{(1)}.

Restriction map for Chow groups.

1.4. Let G and ξ ∈ Z^{1}(k, G_{0}) be as in 0.1, so that G = _{ξ}G_{0}. The cocycle ξ induces an identification 𝔛 ×_{k} k_{s} ≃ 𝔛_{0} ×_{k} k_{s}. Moreover, since 𝔛_{0} is split, CH^{∗}(𝔛_{0} ×_{k} k_{s}) = CH^{∗}(𝔛_{0}). Hence the restriction map can be viewed as a map

res_{CH} : CH^{∗}(𝔛) → CH^{∗}(𝔛 ×_{k} k_{s}) ≃ CH^{∗}(𝔛_{0}).

A cycle of CH^{∗}(𝔛_{0}) is called rational if it belongs to the image of the restriction.

In [22, Thm.6.4(1)], it is proven that, under the hypothesis (0.1), any cycle in the image of the characteristic map 𝔠 is rational, i.e.

im(𝔠) ⊂ im(res_{CH}).

(See [22, §7] to compare their φ̄_{G} with our characteristic map.)

1.5. Remark. Note that the image of the restriction map does not depend on the choice of G in its isogeny class, while the image of the characteristic map does. But the inclusion holds only if G can be obtained from G_{0} by twisting by a cocycle with values in G_{0}. Such a cocycle cannot always be lifted to a cocycle with values in a larger group isogeneous to G_{0}. For instance, if G is not a strongly inner form of G_{0}, then there might be some non rational cycles in the image of the simply connected characteristic map.

For a split group G_{0}, the restriction map is an isomorphism, and this inclusion is strict, except if H^{1}(k, G_{0}) is trivial. On the other hand, generic torsors are defined as the torsors for which it is an equality:

1.6. Definition. A cocycle ξ ∈ Z^{1}(k, G_{0}) defining the twisted group G = _{ξ}G_{0} is said to be generic if any rational cycle is in im(𝔠), so that

im(𝔠) = im(res_{CH}).

Observe that a generic cocycle always exists over some field extension of k by [22, Thm.6.4(2)].

Characteristic map for K_{0} and the Steinberg basis.

1.7. Using the identification between Λ_{ω} and Pic(𝔛_{0}) of 0.2, one also gets a characteristic map for K_{0} (see [3, §2.8]),

𝔠_{K}: ℤ[T̂_{0}] → K_{0}(𝔛_{0}),

where ℤ[T̂_{0}] ⊂ ℤ[Λ_{ω}] denotes the integral group ring of the character group T̂_{0}. Any generator e^{λ}, λ ∈ T̂_{0}, maps to the class of the associated line bundle [L(λ)] ∈ K_{0}(𝔛_{0}).

Combining a theorem of Pittie [31] (see also [29, §0]), and Chevalley’s description of the representation rings of the simply connected cover G_{0}^{sc} of G_{0} and its Borel subgroup B_{0}^{sc}, one may check that K_{0}(𝔛_{0}) is isomorphic to the tensor product ℤ[Λ_{ω}] ⊗_{ℤ[Λ_{ω}]^{W}} ℤ. That is, the simply-connected characteristic map 𝔠_{K,sc} : ℤ[Λ_{ω}] → K_{0}(𝔛_{0}) is surjective, and its kernel is generated by the elements of the augmentation ideal that are invariant under the action of the Weyl group W.

1.8. Moreover, Steinberg described in [39, §2] (see also [29, §12.5]) an explicit basis of ℤ[Λ_{ω}] as a free module over ℤ[Λ_{ω}]^{W}. It consists of the weights ρ_{w} defined for any w in the Weyl group W by

ρ_{w} = ∑_{{α_{k}∈Π, w^{−1}(α_{k})∈Φ^{−}}} w^{−1}(ω_{k}),

where Φ^{−} denotes the set of negative roots with respect to Π. Hence, we get that the elements

g_{w} := 𝔠_{K,sc}(e^{ρ_{w}}) = [L(ρ_{w})], w ∈ W,

form a ℤ-basis of K_{0}(𝔛_{0}), called the Steinberg basis. Note that if w is the reflection w = s_{i}, 1 ≤ i ≤ n, associated to the root α_{i}, we get

ρ_{s_{i}} = ∑_{{α_{k}∈Π, s_{i}(α_{k})∈Φ^{−}}} s_{i}(ω_{k}) = s_{i}(ω_{i}) = ω_{i} − α_{i}.

1.9. Definition. The elements of the Steinberg basis

g_{i} = [L(ρ_{s_{i}})], i = 1 … n

are called special.

Restriction map for K_{0} and the Tits algebras.

1.10. As we did for Chow groups, we use the identification 𝔛 ×_{k} k_{s} ≃ 𝔛_{0} ×_{k} k_{s} to view the restriction map for K_{0} as a morphism

res_{K_{0}} : K_{0}(𝔛) → K_{0}(𝔛_{0}) = ⨁_{w∈W} ℤ · g_{w}.

By Panin’s theorem [29, Thm. 4.1], the image of the restriction map, whose elements are called rational bundles, is the sublattice with basis

{i(ρ_{w}) · g_{w}, w ∈ W },

where i(ρ_{w}) is the index of the Brauer class β(ρ̄_{w}), that is the index of any corresponding Tits algebra (see 0.5).

1.11. Note that since the Weyl group acts trivially on Λ_{ω}/T̂_{0}, we have

ρ̄_{w} = ∑_{{α_{k}∈Π|w^{−1}(α_{k})∈Φ^{−}}} ω̄_{k}.

Therefore, the corresponding Brauer class is given by

β(ρ̄_{w}) = Σ_{{α_{k}∈Π|w^{−1}(α_{k})∈Φ^{−}}}β(ω̄_{k}).

For special elements, we get β(ρ̄_{s_{i}}) = β(ω̄_{i}), so that i(ρ_{s_{i}}) is the index of the Tits algebra A_{ω_{i}}.

Rational cycles versus rational bundles. Since the total Chern class of a rational bundle is a rational cycle, the graded-subring 𝔅^{∗} of CH^{∗}(𝔛_{0}) generated by Chern classes of rational bundles consists of rational cycles. We use Panin’s description of rational bundles to compute 𝔅^{∗}. The total Chern class of i(ρ_{w}) · g_{w} is given by

c(i(ρ_{w}) · g_{w}) = (1 + c_{1}(L(ρ_{w})))^{i(ρ_{w})} = ∑_{k=1}^{i(ρ_{w})} \binom{i(ρ_{w})}{k}c_{1}(L(ρ_{w}))^{k}

Therefore, 𝔅^{∗} is generated as a subring by the homogeneous elements

\binom{i(ρ_{w})}{k}c_{1}(L(ρ_{w}))^{k}, for w ∈ W, 1 ≤ k ≤ i(ρ_{w}).

Let p be a prime number, and denote by i_{w} the p-adic valuation of i(ρ_{w}), so that i(ρ_{w}) = p^{i_{w}}q for some prime-to-p integer q. By Luca’s theorem [4, p. 271] the binomial coefficient \binom{i(ρ_{w})}{p^{iw}} is congruent to q modulo p. Hence its image in 𝔽_{p} is invertible. Considering the image in the Chow group modulo p of the rational cycle \binom{i(ρ_{w})}{p^{iw}}c_{1}(L(ρ_{w}))^{p^{iw}}, we get:

1.12. Lemma. Let p be a prime number. For any w in the Weyl group, the cycle

c_{1}(L(ρ_{w}))^{p^{iw}} ∈ Ch(𝔛_{0}) = CH(𝔛_{0}) ⊗_{ℤ} 𝔽_{p}

is rational.

2. The J-invariant

In this section, we recall briefly the definition and the key properties of the Jinvariant of an algebraic group, following [34]. The definition involves the choice of a cocycle ξ ∈ Z^{1}(k, G_{0}), such that G is the inner twisted form _{ξ}G_{0} of the split group G_{0}. For adjoint groups of type D_{4}, as opposed to all other types, we show that the J-invariant of G does depend on this cocycle, and should actually be considered as an invariant of the cohomology class of ξ, or of the underlying algebra with involution.

2.1. Let us denote by π : CH^{∗}(𝔛_{0}) → CH^{∗}(G_{0}) the pull-back induced by the natural projection G_{0} → 𝔛_{0}, where 𝔛_{0} is the variety of Borel subgroups of G_{0}. By [13, §4, Rem. 2], π is surjective and its kernel is the ideal I(𝔠) ⊂ CH^{∗}(𝔛_{0}) generated by non constant elements in the image of the characteristic map (see 1.3). Therefore, there is an isomorphism of graded rings

CH^{∗}(𝔛_{0})/I(𝔠) ≃ CH^{∗}(G_{0})

In particular, in degree 1, we get

(1)  CH^{1}(G_{0}) ≃ CH^{1}(𝔛_{0})/(im 𝔠^{(1)}) ≃ Λ_{ω}/T̂_{0}.

Using this fact, V. Kac computed in [17, Thm. 3] the Chow group of G_{0} with 𝔽_{p}-coefficients. Namely, he proved it is isomorphic as an 𝔽_{p}-algebra (and even as a Hopf algebra) to

Ch^{∗}(G_{0}) ≃ 𝔽_{p}[x_{1}, …, x_{r}]/(x_{1}^{p^{k_{1}}} , …, x_{r}^{p^{kr}})

for some integers r and k_{i}, for 1 ≤ i ≤ r, which depend on the group G_{0}. For each i, we let d_{i} be the degree of the generator x_{i}. Note in particular that the number of generators of degree 1 is the dimension over 𝔽_{p} of the vector space Λ_{ω}/T̂_{0} ⊗_{ℤ} 𝔽_{p}.

The J-invariant of G is related to the Chow-motif M(𝔛) of the variety of Borel subgroups of G. Namely, the main result (Thm. 5.13) in [34] asserts that the motif M(𝔛) splits as a direct sum of twisted copies of some indecomposable motif R_{p}(G). Moreover, the Poincaré polynomial of R_{p}(G) over a separable closure of k (see [34, §1.3]) is given by

(2)  P (R_{p}(G) ×_{k} k_{s}, t) = ∏_{i=1}^{r} \frac{1 − t^{d_{i}p^{j_{i}}}}{1 − t^{d_{i}}} , where 0 ≤ j_{i} ≤ k_{i}.

As opposed to r, d_{i} and k_{i} for i ∈ 1 … r, which only depend on G_{0}, the parameters j_{i} depend on G, and reflect its splitting properties.

From the values given in the table [17, Table II] (see also [34, §4]), one may check that, except if p = 2 and G is adjoint of type D_{n} with n even, the degrees d_{i} are pairwise distinct. If so, we may assume d_{1} < d_{2} < ⋯ < d_{r} and we get a well defined r-tuple J_{p}(G) = (j_{1}, j_{2}, …, j_{r}). By (2), this tuple is an invariant of G, and does not depend on the choice of the cocycle ξ.

2.2. We now recall from [34, Def. 4.6] the computation of the indices j_{i} corresponding to the degree 1 generator(s). Let us fix a cocycle ξ ∈ Z^{1}(k, G_{0}) such that G = _{ξ}G_{0}. Let R_{ξ} be the pull-back in Ch^{∗}(G_{0}) of the rational cycles of Ch^{∗}(𝔛_{0}), that is the image of the composition

R_{ξ} = Im(Ch^{∗}(𝔛) →^{res_{Ch}} Ch^{∗}(𝔛_{0}) →^{π} Ch^{∗}(G_{0})),

where the restriction map is as defined in 1.4, and π denotes as above the pull-back with respect to the natural projection G_{0} ↦ 𝔛_{0}.

Let us denote by s the dimension of Ch^{1}(G_{0}) over 𝔽_{p}, and assume first that s = 1. We pick a generator x_{1} ∈ Ch^{1}(G_{0}) ≃ 𝔽_{p}. Then the parameter j_{1} is the smallest non negative integer a such that x_{1}^{p^{a}} belongs to R_{ξ} (see [34, §4.4]). Observe in particular that j_{1} = 0 if and only if x_{1} ∈ R_{ξ}.

2.3. We now assume that p = 2 and G_{0} is adjoint of type D_{2m}, with m ≥ 2, that is G_{0} = PGO_{4m}^{+}. As mentioned earlier, this is the only case when s > 1. Moreover, in view of (1), choosing two degree 1 generators for Ch^{∗}(G_{0}) amounts to the choice of two generators of the cocenter (Λ_{ω}/Λ_{r}) = ℤ/2ℤ × ℤ/2ℤ of the group. We pick the classes of the fundamental weights ω_{1} and ω_{2m} (see 0.4), and denote by x_{1} and x_{2} the corresponding elements in Ch^{1}(G_{0}),

(3)  x_{1} = π(h_{1}), x_{2} = π(h_{2m}).

By definition [34, §4.4] , j_{1} is the smallest non negative integer a such that x_{1}^{2^{a}} belongs to R_{ξ}, and j_{2} is the smallest integer b such that

x_{2}^{2^{b}} + ∑_{0<i≤2^{b}} a_{i}x_{1}^{i}x_{2}^{2^{b}−i} ∈ R_{ξ} for some a_{i} ∈ 𝔽_{2}.

Note that, since ω_{1} + ω_{n−1} + ω_{n} ∈ Λ_{r} = T̂_{0} (cf. 0.4), replacing ω_{2m} by ω_{2m−1} amounts to computing (x_{1} + x_{2})^{2^{b}} instead of x_{2}^{2^{b}}, and does not affect the value of j_{2}.

2.4. It follows from the definition that the values of j_{1} and j_{2} might decrease under field extension. This also applies to the indices corresponding to generators of higher degrees, as explained in [34, 4.7]. Hence, we have

J_{p}(G_{E}) ≤ J_{p}(G), for any extension E of the base field F.

2.5. We now describe how j_{1} and j_{2} depend on the choice of a cocycle ξ ∈ Z^{1}(k, G_{0}) such that G = _{ξ}G_{0}, with G_{0} = PGO_{4m}^{+}. The exact sequence

1 ↦ G_{0} → Aut(G_{0}) → Aut(Δ) ↦ 1,

where Δ denotes the Dynkin diagram of G_{0}, induces a map

H^{1}(k, G_{0}) ↦ H^{1}(k, Aut(G_{0})).

By [23, §29.F], a cohomology class in H^{1}(k, G_{0}) corresponds to the k-isomorphism class of a triple (A, σ, ε) where A is a degree 4m central simple algebra over k, σ is an orthogonal involution on A with trivial discriminant, and ε is an isomorphism between the center of the Clifford algebra of (A, σ) and k × k. Moreover, the image in H^{1}(k, Aut(G_{0})) of (A, σ, ε) is the connected component PGO^{+}(A, σ) of its automorphism group.

Assume first that n ≠ 4, so that Aut(Δ) = ℤ/2. This group acts on H^{1}(k, G_{0}), by sending (A, σ, ε) to (A, σ, ε′), where ε′ is the composite of ε with the non trivial automorphism of k × k. By [37, §5, Prop.39(ii)], we may replace ξ by a cocycle ξ′ representing (A, σ, ε′). To prove that this does not affect j_{1} and j_{2}, consider the following commutative diagram:

\xymatrix{ & Ch^{∗}(𝔛_{0}) \ar[r] & Ch^{∗}(G_{0}) \\ Ch^{∗}(𝔛) \ar[r] & Ch^{∗}(𝔛 ×_{k} k_{s}) \ar[r] \ar[u]_{≃} & Ch^{∗}(G ×_{k} k_{s}). \ar[u]_{≃}}

The vertical arrows, and only those, depend on the choice of a cocycle ξ. We claim that the antecedent in Ch^{1}(G×_{k} k_{s}) of the generator x_{1} ∈ Ch^{1}(G_{0}) does not depend on ξ. Hence the parameter j_{1} is a well-defined invariant of the group G.

2.6. Indeed, since we numbered roots as in [1], the generator x_{1}, which corresponds to the class of ω_{1} in (Λ_{ω}/Λ_{r}) ⊗ 𝔽_{2} ≃ Ch^{1}(G_{0}) can be characterized as follows: it is the unique element of Ch^{1}(G_{0}) killed by the pull-back map Ch^{1}(G_{0}) → Ch^{1}(O_{4m}^{+}) induced by taking the quotient map modulo the center. Hence, its image in Ch^{1}(G ×_{k} k_{s}) is the unique class killed by the pull-back map Ch^{1}(G ×_{k} k_{s}) → Ch^{1}(O^{+}(A, σ) ×_{k} k_{s}). For a given group G = PGO^{+}(A, σ), this defines the antecedent of x_{1} in Ch^{1}(G ×_{k} k_{s}) uniquely, independently of the choice of ξ.

As opposed to this, the antecedent of the second generator x_{2} does depend on ξ. Hence changing the cocycle amounts to changing the second generator x_{2}, and we already observed that this does not affect the value of j_{2}.

2.7. The situation is quite different in the trialitarian case, since the automorphism group of Δ then is the symmetric group S_{3}. Let us assume now that G is adjoint of type D_{4}, that is G_{0} = PGO_{8}^{+}. The J-invariant then is a triple (j_{1}, j_{2}, j_{3}), where the first two parameters correspond to generators of degree 1, and the last one to a generator of degree 3. In this case, there are three possibly non isomorphic algebras with involution (A, σ_{A}), (B, σ_{B}) and (C, σ_{C}) such that

G = PGO^{+}(A, σ_{A}) = PGO^{+}(B, σ_{B}) = PGO^{+}(C, σ_{C}),

which leads to 6 cohomology classes in H^{1}(k, PGO_{8}^{+}) by [23, §42.A]. The very same argument as before now shows that the value of j_{1} does depend on the choice of a cocycle, or more precisely on the choice of one of the three possible algebras with involution. Indeed, if the cocycle ξ ∈ Z^{1}(k, PGO_{4m}^{+}) corresponds to one of the triples (A, σ, ε) and (A, σ, ε′), then the antecedent of x_{1} in Ch^{1}(G ×_{k} k_{s}) is the unique element killed by the pull back to Ch^{1}(O^{+}(A, σ)). We denote by J(A, σ) the triple (j_{1}, j_{2}, j_{3}), where j_{1} and j_{2} are computed as in 2.3, and for such a choice of ξ. It is a well-defined invariant of (A, σ).

2.8. By (2), if J(A, σ_{A}) = (j_{1}, j_{2}, j_{3}), and G = PGO^{+}(A, σ_{A}) = PGO^{+}(B, σ_{B}), then J(B, σ_{B}) ∈ {(j_{1}, j_{2}, j_{3}), (j_{2}, j_{1}, j_{3})}. In 6.3 and 6.8 below, we give a more precise statement, and provide explicit examples of algebras with involution having isomorphic automorphism groups and different J-invariant.

3. The parameters of degree one and indices of Tits algebras.

In this section, we prove the main results of the paper, which give connections between the indices of the J-invariant corresponding to generators of degree 1 and indices of Tits algebras of the group G (cf. [12, §3]).

3.1. From now on, we let s be the dimension over 𝔽_{p} of Λ_{ω}/T̂_{0} ⊗𝔽_{p} ≃ Ch^{1}(G_{0}), and we fix G_{0} and p so that s ≥ 1. If s = 1, we fix an index i_{1} ∈ {1, …, n} such that the class of ω_{i_{1}} generates Λ_{ω}/T̂_{0} ⊗ 𝔽_{p}, and we let x_{1} ∈ Ch^{1}(G_{0}) be the corresponding generator. If p = 2 and G_{0} is adjoint of type D_{2m}, so that s = 2, we let i_{1} = 1, i_{2} = 2m, and define x_{1} and x_{2} as in (3). If moreover m = 2, that is G_{0} is adjoint of type D_{4}, the J-invariant in this section always refers to the J-invariant of (A, σ), where A is the Tits algebra associated to the weight ω_{1}.

3.2. Consider the special elements g_{i}, i = 1 … n of the Steinberg basis of K_{0}(𝔛_{0}) (see Definition 1.9). Since c_{1}(g_{i}) = h_{i} − c_{1}(L(α_{i})) ∈ Ch^{1}(𝔛_{0}), we have

π(c_{1}(g_{i})) = π(h_{i}) − π(c_{1}(L(α_{i}))) = π(h_{i}) ∈ Ch^{1}(G_{0}).

Hence the generators x_{ℓ}, 1 ≤ ℓ ≤ s may also be defined by x_{ℓ} = π(c_{1}(g_{i_{ℓ}})). In view of the isomorphism (1), it follows that for any g ∈ Pic(𝔛_{0}) its Chern class modulo p can be written as

(4)  c_{1}(g) = ∑_{ℓ=1}^{s} a_{ℓ}c_{1}(g_{i_{ℓ}}) mod im 𝔠^{(1)} ∈ Ch^{1}(𝔛_{0})

As an immediate consequence of rationality of cycles introduced in Lemma 1.12 we obtain another proof of the first part of [32, Prop. 4.2]:

3.3. Corollary. The first entry j_{1} of the J-invariant is bounded

j_{1} ≤ i_{i_{1}},

by the p-adic valuation i_{i_{1}} of the index of the Tits algebra A_{ω_{i_{1}}} associated to ω_{i_{1}}.

Proof. We apply lemma 1.12 to the weight ρ_{s_{i_{1}}} = ω_{i_{1}} − α_{i_{1}}. As noticed in 1.11, the index i(ρ_{s_{i_{1}}} ) is equal to the index i(ω_{i_{1}}) of the Tits agebra A_{ω_{i_{1}}} . Hence, the cycle

c_{1}(g_{i_{1}})^{p^{i_{i_{1}}}} is rational, and its image x_{1}^{p^{i_{i_{1}}}} ∈ Ch^{∗}(G_{0}) belongs to R_{ξ}. The inequality then follows from the definition of j_{1} (see 2.2). □

If s = 2, the very same argument also applies to ρ_{s_{ℓ}} for ℓ = 2m − 1 and ℓ = 2m. Combining with the definition of j_{2} given in 2.3, we get 3.4. Corollary. If p = 2 and G is adjoint of type D_{2m}, so that s = 2, then the second entry j_{2} of the J-invariant is bounded

j_{2} ≤ min{i_{2m−1}, i_{2m}},

where i_{ℓ} is the 2-adic valuation of the index of the Tits algebra A_{ω_{ℓ}}.

The next result, which gives an inequality in the other direction, uses the notion of common index, which we introduce now.

3.5. Definition. Consider the Tits algebras A_{ω_{i_{ℓ}}} associated to the fundamental weights ω_{i_{ℓ}}, for 1 ≤ ℓ ≤ s, where i_{ℓ} are as in 3.1. We define their common index i_{J} to be the p-adic valuation of the greatest common divisor of all the indices ind(A_{ω_{i_{1}}}^{⊗a_{1}} ⊗ … ⊗ A_{ω_{is}}^{⊗a_{s}}), where at least one of the a_{i} is coprime to p.

3.6. Example. If s = 1, then i_{J} is the p-adic valuation i_{i_{1}} of the index of the Tits algebra A_{ω_{i_{1}}} . Assume for instance that G is adjoint of type D_{2m+1}. As recalled in 0.4, we may take i_{1} = 2m or i_{1} = 2m + 1, so that i_{J} is the 2-adic valuation of any component C_{+} or C_{−} of the Clifford algebra of (A, σ). From the fundamental relations 0.6, we know that the two components have the same index.

3.7. Example. If s = 2, we have p = 2 and G_{0} is adjoint of type D_{2m}. Using 0.6, one may check that i_{J} is the p-adic valuation of the greatest common divisor of the indices of A_{ω_{1}}, A_{ω_{2m−1}} and A_{ω_{2m}}, that is

i_{J} = min{i_{1}, i_{2m−1}, i_{2m}}.

We will prove:

3.8. Theorem. Let i_{J} be the common index of the Tits-algebras A_{ω_{i_{ℓ}}}, for 1 ≤ ℓ ≤ s. If i_{J} > 0, then j_{ℓ} > 0 for any ℓ, 1 ≤ ℓ ≤ s. If i_{J} > 1 and p = 2, then for any ℓ such that k_{ℓ} > 1, the corresponding index

also satisfies j_{ℓ} > 1.

Consider the ideal I(res_{Ch}) of Ch^{∗}(𝔛_{0}) generated by the non constant rational elements. For any integer i, we let I(res_{Ch})^{(i)} ⊂ Ch^{i}(𝔛_{0}) be the homogeneous part of degree i. Since the image of the characteristic map consists of rational elements, we have I(𝔠) ⊂ I(res_{Ch}). The theorem follows easily from the following lemma:

3.9. Lemma. If i_{J} > 0, then I(res_{Ch})^{(1)} = I(𝔠)^{(1)} ⊂ Ch^{1}(𝔛_{0}).

If i_{J} > 1 and p = 2, then I(res_{Ch})^{(2)} = I(𝔠)^{(2)} ⊂ Ch^{2}(𝔛_{0})

Indeed, let us assume first that i_{J} > 0. By the lemma, any element in im(res_{Ch}^{(1)}) = I(res_{Ch})^{(1)} belongs to I(𝔠)^{(1)}, which is in the kernel of π. Therefore, the image of the composition

R_{ξ}^{(1)} = im(Ch^{1}(𝔛) →^{res_{Ch}^{(1)}} Ch^{1}(𝔛_{0}) →^{π} Ch^{1}(G_{0}))

is trivial, R_{ξ}^{(1)} = {0}. From the definition 2.3 of j_{1} and j_{2}, this implies that they are both strictly postive.

The proof of the second part follows the same lines. We write it in details for s = 2 and k_{1}, k_{2} > 1. Assume that i_{J} > 1. Since the image im(res_{Ch})^{(2)} is contained in I(res_{Ch})^{(2)}, the lemma again implies that R_{ξ}^{(2)} = {0}. On the other hand, the hypothesis on k_{1} and k_{2} guarantees that in the truncated polynomial algebra 𝔽_{2}[x_{1}, x_{2}]/(x_{1}^{2^{k_{1}}} , x_{2}^{2^{k_{2}}} ) ⊂ Ch^{∗}(G_{0}), the elements x_{1}^{2} and x_{2}^{2} + a_{1}x_{1}x_{2} + a_{2}x_{1}^{2} are all non trivial. Hence they do not belong to R_{ξ}, and we get j_{1}, j_{2} > 1.

The rest of the section is devoted to the proof of Lemma 3.9. The main tool is the Riemann-Roch theorem, which we now recall.

Filtrations of K_{0} and the Riemann-Roch Theorem.

3.10. Let X be a smooth projective variety over k. Consider the topological filtration on K_{0}(X) given by

τ^{i}K_{0}(X) = ⟨[O_{V}], codimV ≥ i⟩,

where O_{V} is the structure sheaf of the closed subvariety V in X. There is an obvious surjection

p : CH^{i}(X) → τ^{i/i+1}K_{0}(X) = τ^{i}K_{0}(X)/τ^{i+1}K_{0}(X),

given by V ↦ [O_{V}]. By the Riemann-Roch theorem without denominators [6, §15], the i-th Chern class induces a map in the opposite direction

c_{i} : τ^{i/i+1}K_{0}(X) → CH^{i}(X)

and the composite c_{i} ∘ p is the multiplication by (−1)^{i−1}(i − 1)!. In particular, it is an isomorphism for i ≤ 2 (see [6, Ex. 15.3.6 ]).

The topological filtration can be approximated by the so-called γ-filtration. Let c_{i}^{K_{0}} be the i-th Chern class with values in K_{0} (see [6, Ex. 3.2.7(b)], or [19, §2]). We use the convention c_{1}^{K_{0}}([L]) = 1 − [L^{v}] for any line bundle L, where L^{v} is the dual of L, so that in the Chow group,

c_{1}(c_{1}^{K_{0}}([L])) = c_{1}(L).

Similarly, one may compute the second Chern class

(5)  c_{2}(c_{1}^{K_{0}}([L_{1}])c_{1}^{K_{0}}([L_{2}])) = −c_{1}(L_{1})c_{1}(L_{2}).

The γ-filtration on K_{0}(X) is given by the subgroups (cf. [11, §1])

γ^{i}K_{0}(X) = ⟨c_{n_{1}}^{K_{0}}(b_{1}) · . . . · c_{n_{m}}^{K_{0}}(b_{m}) | n_{1} + … + n_{m} ≥ i, b_{l} ∈ K_{0}(X)⟩,

(see [6, Ex.15.3.6], [7, Ch.3,5]). We let γ^{i/i+1}(K_{0}(X)) = γ^{i}K_{0}(X)/γ^{i+1}K_{0}(X) be the respective quotients, and γ^{∗}(K_{0}(X)) = ⊕_{i≥0}γ^{i/i+1}(K_{0}(X)) the associated graded ring.

By [19, prop 2.14], γ^{i}(K_{0}(X)) is contained in τ^{i}(K_{0}(X)), and they coincide for i ≤ 2. Hence, by Riemann-Roch theorem, the Chern class c_{i} with values in CH^{i}(X) vanishes on γ^{(i+1)}K_{0}(X), and induces a map

c_{i} : γ^{i/i+1}(K_{0}(X)) → CH^{i}(X).

In codimension 1 we get an isomorphism

c_{1} : γ^{1/2}(K_{0}(X)) →^{≃} CH^{1}(X)

which sends for a line bundle L the class c_{1}^{K_{0}}(L) to c_{1}(L). In codimension 2 the map

c_{2} : γ^{2/3}(K_{0}(X)) ↠ CH^{2}(X),

is surjective and has torsion kernel [19, Cor. 2.15].

Let us now apply this to the varieties 𝔛_{0} and 𝔛 of Borel subgroups of G_{0} and G respectively. Since K_{0}(𝔛_{0}) is generated by the line bundles g_{w} = [L(ρ_{w})] for w ∈ W, one may check that γ^{i/i+1}(𝔛_{0}) is generated by the products

{c_{1}^{K_{0}}(g_{w_{1}}) … c_{1}^{K_{0}}(g_{w_{i}}), w_{1}, …, w_{i} ∈ W }.

Moreover, the restriction map commutes with Chern classes, so it induces

res_{γ} : γ^{∗}(𝔛) → γ^{∗}(𝔛_{0}).

Using Panin’s description of the image of the restriction map res_{K_{0}} recalled in 1.10, we get that the image of res_{γ}^{(1)} : γ^{1/2}(𝔛) → γ^{1/2}(𝔛_{0}) is generated by the elements c_{1}^{K_{0}}(i(ρ_{w})g_{w}) = i(ρ_{w})c_{1}^{K_{0}}(g_{w}), for any w ∈ W, while the image of res_{γ}^{(2)} is generated by

i(ρ_{w_{1}})i(ρ_{w_{2}})c_{1}^{K_{0}}(g_{w_{1}})c_{1}^{K_{0}}(g_{w_{2}}) and c_{2}^{K_{0}}(i(ρ_{w})g_{w}) for w_{1}, w_{2}, w ∈ W.

If the index i(ρ_{w}) is 1, then c_{2}^{K_{0}}(i(ρ_{w})g_{w}) = 0. Otherwise, the Whitney sum formula gives

c_{2}^{K_{0}}(i(ρ_{w})g_{w}) = \binom{i(ρ_{w})}{2}c_{1}^{K_{0}}(g_{w})^{2}.

Applying the morphisms c_{1} and c_{2}, and using (5), we now get

3.11. Lemma. The subgroup c_{1}(im(res_{γ}^{(1)})) ∈ CH^{1}(𝔛_{0}) is generated by i(ρ_{w})c_{1}(g_{w}), for all w ∈ W. The subgroup c_{2}(im(res_{γ}^{(2)})) ∈ CH^{2}(𝔛_{0}) is generated by the elements i(ρ_{w_{1}})i(ρ_{w_{2}})c_{1}(g_{w_{1}})c_{1}(g_{w_{2}}) and \binom{i(ρ_{w})}{2}c_{1}(g_{w})^{2} for all w_{1}, w_{2}, w ∈ W.

Proof of Lemma 3.9. Since the image of the characteristic map consists of rational elements (see 1.4), we already know that I(𝔠) ⊂ I(res_{Ch}). We now prove the reverse inclusions for the homogeneous parts of degree 1 and 2 under the relevant hypothesis on the common index i_{J}. Note that since c_{1} and c_{2} are both surjective, and commute with restriction maps, one has

im(res_{Ch}^{(k)}) = c_{k}(im(res_{γ}^{(k)})), for k = 1, 2.

In degree 1, we have I(res_{Ch})^{(1)} = im(res_{Ch}^{(1)}), so to prove the first part of the lemma, we have to prove that if i_{J} > 0, then for any w ∈ W, the element i(ρ_{w})c_{1}(g_{w}) belongs, after tensoring with 𝔽_{p}, to I(𝔠)^{(1)} = im 𝔠^{(1)}. Let us write

c_{1}(g_{w}) = ∑_{ℓ=1}^{s} a_{ℓ}c_{1}(g_{i_{ℓ}}) mod im 𝔠^{(1)},

as in (4). If all the a_{ℓ} ∈ 𝔽_{p} are trivial, we are done, so we may assume at least one of them is invertible in 𝔽_{p}. The weights ρ_{w} and ρ_{i_{ℓ}} satisfy the same relation

ρ_{w} = ∑_{ℓ=1}^{s} a_{ℓ}ρ_{i_{ℓ}} mod T̂_{0} ⊗_{ℤ} 𝔽_{p}.

Applying the morphism β, we get that the p-primary part of the Brauer class β(ρ̄_{w}) coincides with the p-primary part of the Brauer class of ⊗_{ℓ=1}^{s}A_{ω_{i_{ℓ}}}^{a_{ℓ}} (see 1.11). The hypothesis on i_{J} guarantees that this index of this algebra is divisible by p. Hence i(ρ_{w}), which is the index of β(ρ̄_{w}) also is divisible by p, so that i(ρ_{w})c_{1}(g_{w}) = 0 in the Chow group Ch^{1}(𝔛_{0}) modulo p, and we are done.

Let us now assume that p = 2 and i_{J} > 1. The homogeneous part I(res_{Ch})^{(2)} decomposes as

I(res_{Ch})^{(2)} = im(res_{Ch}^{(1)}) Ch^{1}(𝔛_{0}) + im(res_{Ch}^{(2)}).

By the first part of the Lemma, we already know that

im(res_{Ch}^{(1)}) Ch^{1}(𝔛_{0}) ⊂ I(𝔠).

Hence it remains to prove that im(res_{Ch}^{(2)}) = c_{2}(im res_{γ}^{(2)}) ⊂ I(𝔠)^{(2)}. The proof for the degree 1 part already shows that i(ρ_{w_{1}})i(ρ_{w_{2}})c_{1}(g_{w_{1}})c_{1}(g_{w_{2}}) belongs to I(𝔠)^{(2)}. The same argument extends to \binom{i(ρ_{w})}{2}c_{1}(g_{w})^{2}. Indeed, if the coefficients a_{ℓ} are not all trivial modulo 2, the condition on the common index now implies that 4 divides i(ρ_{w}), so that \binom{i(ρ_{w})}{2} is zero modulo 2.

4. Applications to quadratic forms

The purpose of this section is to apply our main theorem 3.8 to quadratic forms. The results presented here are not new, but some of them can be recovered very easily from our inequalities. Let φ be a quadratic form of even dimension 2n. We always assume that φ has trivial discriminant, so that its special orthogonal group O^{+}(φ) satisfies condition 0.1. We define the J-invariant of φ as follows:

4.1. Definition. Let φ be a 2n dimensional quadratic form over F, with trivial discriminant. Its J-invariant is

J(φ) = J_{2}(O^{+}(φ)).

4.2. Remark. (i) The J-invariant of a quadratic form was initially defined by Vishik in [41, Def 5.11]. The invariant considered here is closely related to Vishik’s J-invariant, and also to the dual version given in [5, §88]. We refer the reader to[34, §4.8] for a more precise statement.

(ii) Let φ_{0} be any non degenerate subform of φ of codimension 1. The maximal orthogonal grassmanian of φ_{0} is isomorphic to any connected component of the maximal orthogonal grassmanian of φ (see [5, 85.2]). So by (2), the forms φ_{0} and φ have the same J-invariant (see also [5, §88]). Since any odd-dimensional form can be embedded in an even dimensional form with trivial discriminant, we only consider the even-dimensional case.

The J-invariant is an invariant of the Witt-class of a quadratic form, in the following sense:

4.3. Proposition. Let φ and φ′ be two Witt-equivalent even-dimensional quadratic forms with trivial discriminant. All the non trivial indices in their J-invariant are equal.

Moreover, J(φ) = (0, …, 0) if and only if φ is hyperbolic.

Proof. In general, the forms φ and φ′ have different dimensions, so strictly speaking, their J-invariants are different. Nevertheless, we claim that deleting all the zero indices on both sides, we get the same tuple of integers. In view of (2), this is a direct consequence of [34, 5.18(iii)], since the maximal orthogonal grassmanians of φ and φ′ clearly have the same splitting fields.

From [34, 6.7], a quadratic form with trivial J-invariant is hyperbolic over some odd-degree extension of the base field. Hence the second statement follows from Springer’s theorem [5, 18.5]. □

In small dimension, the J-invariant of a quadratic form φ can be explicitly computed in terms of the index of its full Clifford algebra C(φ). More precisely, we let i_{S} be the 2-adic valuation of the index of C(φ). Since φ has trivial discriminant, its even Clifford algebra has center F × F and splits as C_{0}(φ) = C × C for some central simple algebra C which is Brauer-equivalent to C(φ), that is C(φ) ≃ M_{2}(C). In particular, it follows that i_{S} ≤ n − 1. In this setting, we have s = 1, and C is the Tits algebra associated to a generator of Λ_{ω}/T̂_{0} ⊗ 𝔽_{2}. So the common index i_{J} is given by i_{J} = i_{S}. With this notation, the inequalities 3.3 and 3.8 can be translated as follows:

4.4. Corollary. Let φ be a 2n dimensional quadratic form with trivial discriminant. The 2-adic valuation i_{S} of its Clifford algebra and the first index j_{1} of its J-invariant are related as follows:

(1) j_{1} ≤ i_{S}

(2) If n ≥ 2, and i_{S} > 0, then j_{1} > 0.

(3) If n ≥ 3 and i_{S} > 1, then j_{1} > 1.

Assume now that φ has dimension 4 or 6, that is n = 2 or 3. By table [34, 4.13], the J invariant of φ consists of a single integer, J(φ) = (j_{1}), which is bounded by 1 (respectively 2) if φ has dimension 4 (respectively 6). Therefore, by Corollary 4.4, we have:

4.5. Proposition. Let φ be a quadratic form of dimension 4 or 6 with trivial discriminant. Its J-invariant is J(φ) = (i_{S}).

We can give a more precise description of the quadratic form φ in each case. In dimension 4, the Clifford algebra of φ is Brauer-equivalent to a quaternion algebra Q over F, and φ is similar to the norm form n_{Q} of Q, which is a 2-fold Pfister form. It is hyperbolic if Q is split and anisotropic otherwise.

If φ has dimension 6, its Clifford algebra is a biquaternion algebra B over F, and φ is an Albert form of B. If J(φ) = (2), or equivalently B is division, then φ is anisotropic. If J(φ) = (1), or equivalently B is Brauer equivalent to a non split quaternion algebra Q, then φ is similar to n_{Q} ⊕ ℍ. If J(φ) = (0), or equivalently B is split, then φ is hyperbolic. Hence we get:

4.6. Corollary. A quadratic form φ of dimension 6 and trivial discriminant is anisotropic if and only if its J-invariant is J(φ) = (2); it is isotropic and nonhyperbolic if and only if J(φ) = (1).

Let us now consider quadratic forms of dimension 8 with trivial discriminant. Their special orthogonal group have type D_{4}, and table [34, 4.13] now says J(φ) = (j_{1}, j_{2}) with 0 ≤ j_{1} ≤ 2 and 0 ≤ j_{2} ≤ 1. We have:

4.7. Proposition. Let φ be a quadratic form of dimension 8 with trivial discriminant, and consider its J-invariant J(φ) = (j_{1}, j_{2}). The first index j_{1} is given by j_{1} = min{i_{S}, 2}. Moreover, j_{2} is 0 if φ is isotropic and 1 if φ is anisotropic.

Proof. Again, the first assertion follows instantly from Corollary 4.4. To prove the second, let us first assume that φ is isotropic. If it is hyperbolic, we already know that J(φ) = (0, 0), so in particular j_{2} = 0. Otherwise, φ splits as φ = φ_{0} ⊕ ℍ for some 6-dimensional non hyperbolic form φ_{0} with trivial discriminant. By 4.3, this implies that one of the two indices j_{1} and j_{2} of J(φ) is zero. Since the Clifford algebra of φ is Brauer-equivalent to the Clifford algebra of φ_{0}, which is non split by 4.6, j_{1} = i_{S} ≠ 0. Therefore, we get j_{2} = 0 as expected.

To prove the converse, we distinguish two cases, depending on the value of i_{S}. Let us first assume that φ is anisotropic and i_{S} ≤ 2. Consider a generic splitting F_{C} of the Clifford algebra of φ. By a theorem of Laghribi [25, Thm. 4], the form φ remains anisotropic after scalar extension to F_{C}. Hence the J-invariant of φ_{F_{C}} is non-trivial. On the other hand, since its Clifford algebra is split, the first index is zero. Therefore the second index j_{2} is 1, and this is a fortiori the case over the base field.

Finally, let us assume i_{S} = 3, which implies in particular that φ is anisotropic. Any field extension over which the quadratic form φ becomes hyperbolic splits its Clifford algebra. Therefore the index of such a field extension has to be at least 8 = 2^{3}. Hence, by [34, 6.6], we have 3 ≤ j_{1} + j_{2}, so that J(φ) = (2, 1). □

In his paper [15], Detlev Hoffmann classified quadratic forms of small dimension in terms of their splitting pattern. Using his classification, one can give a precise description of quadratic forms of dimension 8 with trivial discriminant, depending on the value of their J-invariant. The results are summarized in the table below. The notation J_{v}(φ) stands for Vishik’s J-invariant, as defined in [5, §88]. The index i is the 2-adic valuation of the greatest common divisor of the degrees of the splitting fields of φ. In the explicit description, P f_{k} stands for a k-fold Pfister form, s_{l/k}(P f_{2}) for the Scharlau transfer of a 2-fold Pfister form with respect to a quadratic field extension, and Al_{6} for an Albert form.

J(φ) J_{v}(φ)  i_{S}, i  Splitting Pattern  Description
(0)  ∅  i_{S} = i = 0  (4)  hyperbolic
(1,0)  {1}  i_{S} = i = 1  (2,4)  P f_{2} ⊥ 2ℍ
(2,0) {1, 2}  i_{S} = i = 2  (1,2,4)  Al_{6} ⊥ ℍ
(0,1)  {3}  i_{S} = 0; i = 1  (0,4)  P f_{3}
(1,1) {1, 3}  i_{S} = i = 1  (0,2,4)  q = ⟨1, −a⟩ ⊗ q′
(2,1) {1, 2, 3} i_{S} = i = 2  (0,1,2,4)  P f_{2} ⊥ P f_{2} or s_{l/k}(P f_{2})
i_{S} = i = 3  “  generic

In particular, all possible values for the J-invariant do occur in this setting. Moreover, it follows from the table that the J-invariant and the splitting pattern uniquely determine each other. This is not true anymore in higher degree. Indeed, consider a 10-dimensional quadratic form φ over F with splitting pattern {0, 2, 3, 5}. By Hoffmann’s classification theorem [15, Thm 5.1], φ has trivial discriminant, its Clifford algebra has index 4, so that i_{S} = 2 and it is a Pfister neighbor. We claim that its J-invariant is J(φ) = (2, 0). To prove this, one may use the relation between the splitting pattern and Vishik’s J invariant of a quadratic form as described in [5, 88.8]. Once translated in terms of J(φ) following [34, 4.8], we get that J(φ) is (j_{1}, 0) for some integer j_{1}, 1 ≤ j_{1} ≤ 3. Since i_{S} = 2, Corollary 4.4 give j_{1} = 2. From the classification in degree 8, the value (2, 0) also is the J invariant of π + 2ℍ for any Albert form π. On the other hand, such a form has splitting pattern {2, 3, 5}.

5. J-invariant of an algebra with involution

Recall that (A, σ) is a degree 2n central simple algebra over k, endowed with an involution of orthogonal type and trivial discriminant. In particular, this implies that A has exponent 2, so that it has index 2^{i_{A}} for some integer i_{A}. The connected component PGO^{+}(A, σ) of the automorphism group of (A, σ) is an adjoint group of type D_{n}. Because of the discriminant hypothesis, it is an inner twisted form of PGO_{2n}^{+}. If the degree of A is different from 8, we define

J(A, σ) = J_{2}(PGO^{+}(A, σ)).

In degree 8, J(A, σ) was defined in 2.7 as the J-invariant of PGO^{+}(A, σ), computed with a suitable cocycle. Therefore, from the table [34, 4.13], one may check that J(A, σ) is an r-tuple J(A, σ) = (j_{1}, j_{2}, …, j_{r}), with r = m + 1 if n = 2m and r = m if n = 2m + 1. Note that our notation slightly differs from the notation in the table, where in the n-odd case, they have an additional index, but which is bounded by k_{1} = 0. So, for n odd, our (j_{1}, …, j_{r}) coincides with (j_{2}, …, j_{r+1}) in [34]. In particular, the indices corresponding to generators of degree 1 are j_{1} if n is odd and j_{1} and j_{2} if n is even.

Since σ has trivial discriminant, its Clifford algebra splits as a direct product C(A, σ) = C_{+} × C_{−} of two central simple algebras over k. We let i_{A} (respectively i_{+}, i_{−}) be the 2-adic valuation of the index of A (respectively C_{+}, C_{−}). From Examples 3.6 and 3.7, the common index i_{J} is

i_{J} = \{ \begin{matrix}i_{+} = i_{−} & if n is odd, \\ min\{i_{A}, i_{+}, i_{−}\} & if n is even. \end{matrix}

Hence, Corollaries 3.3 and 3.4 and Theorem 3.8 translate as follows:

5.1. Corollary. Assume that n is odd, so that deg(A) ≡ 2[4], and let i_{S} = i_{+} = i_{−}. We have:

(1) j_{1} ≤ i_{S};

(2) If i_{S} > 0, then j_{1} > 0;

(3) If deg(A) ≥ 6 and i_{S} > 1, then j_{1} > 1.

5.2. Corollary. Assume now that n is even, that is deg(A) ≡ 0[4], and let i_{J} = min{i_{A}, i_{+}, i_{−}}. We have:

(1) j_{1} ≤ i_{A};

(2) j_{2} ≤ min{i_{+}, i_{−}};

(3) If i_{J} > 0, then j_{1} > 0 and j_{2} > 0.

(4) If deg(A) ≡ 0[8] and i_{J} > 1, then j_{1} > 1.

(5) If deg(A) ≥ 8 and i_{J} > 1, then j_{2} > 1.

The additional conditions on the degrees are obtained from the table [34, 4.13], and guarantee that k_{1} > 1 or k_{2} > 1.

Split case. If A is split, the involution σ is adjoint to a quadratic form φ over k. We then have:

5.3. Proposition. If A is split and σ is adjoint to the quadratic form φ, the Jinvariants of (A, σ) and φ are related as follows:

J(A, σ) = \{ \begin{matrix}J(φ) & if deg(A) ≡ 2[4] \\ (0, J(φ)) & if deg(A) ≡ 0[4] \end{matrix}

Proof. Assume A is split, and σ is adjoint to the quadratic form φ. Since d(φ) = d(σ) = 1, the J invariant of φ is defined,

J(φ) = J_{2}(O^{+}(φ)).

Moreover, the groups O^{+}(φ) = O^{+}(A, σ) and PGO^{+}(A, σ) = PGO^{+}(φ) are isogeneous. Therefore, the corresponding varieties of Borel subgroups are the same, and by (2), the non trivial indices in the J-invariants of φ and (A, σ) are the same. If n is odd, this is enough to conclude that the J-invariants are equal. If n is even, the only difference comes from the presentations of Ch^{∗}(O_{2n}^{+}) and Ch^{∗}(PGO_{2n}^{+}): the second group has two generators of degree 1, while the first has only one. Since we precisely defined x_{1} to be the generator of Ch^{1}(PGO_{2n}^{+}) killed by pull-back to Ch^{1}(O_{2n}^{+}) (see § 2.6), we get that the index j_{1} of J(A, σ) is trivial, and this proves the proposition. □

Half-spin case. We now assume that the Clifford algebra C(A, σ) = C_{+} × C_{−} has a split component. If deg(A) ≡ 2[4], by the fundamental relations 0.6, the algebra A is split, so that J(A, σ) = J(φ) for a suitable quadratic form φ. So we may assume deg(A) ≡ 0[4]. Let us pick one of the two (isomorphic) half-spin groups Spin_{2n}^{+} ⊂ Spin_{2n}. As explained in [10, Lem. 4.1], C(A, σ) has a split component if and only if at least one of the cocycles ξ and ξ′ ∈ Z^{1}(k, PGO_{2n}^{+}) corresponding to the classes of (A, σ, ε) and (A, σ, ε′) lifts to a cocycle η ∈ Z^{1}(k, Spin_{2n}^{+}). Let Spin^{+}(A, σ) be the corresponding half-spin group, Spin^{+}(A, σ) = _{η}Spin_{2n}^{+}. It satisfies condition 0.1, and its J-invariant is well defined. For the same reason as before, the non trivial indices of this J-invariant are equal to the non trivial indices of J(A, σ), and again the only difference comes from the presentations of Ch^{∗}(Spin_{2n}^{+}) and Ch^{∗}(PGO_{2n}^{+}): one of the two generators x_{2} and x_{1}+x_{2} is killed by the pull-back map Ch^{1}(PGO_{2n}^{+}) ↦ Ch^{1}(Spin_{2n}^{+}) by 1.2. Hence, we have proven:

5.4. Proposition. Assume that deg(A) ≡ 0[4]. The Clifford algebra C(A, σ) = C_{+} × C_{−} has a split component if and only if one of the two half-spin groups, say Spin^{+}(A, σ), satisfies condition 0.1. If so, the algebra with involution (A, σ) is said to be half-spin, and its J-invariant satisfies

J(A, σ) = (j_{1}, 0, j_{3}, …, j_{r}), where (j_{1}, j_{3}, …, j_{r}) = J_{2}(Spin^{+}(A, σ)).

If (A, σ) is half-spin, we can refine the inequalities given in 5.2 by applying Theorem 3.8 to the half-spin group Spin^{+}(A, σ). We get the following:

5.5. Corollary. Assume deg(A) ≡ 0[4] and (A, σ) is half-spin, that is its Clifford algebra has a split component. The following hold:

(1) If i_{A} > 0, then j_{1} > 0.

(2) If i_{A} > 1, then j_{1} > 1.

Indeed, in degree 1, the Chow group modulo 2 of a half-spin group is

Ch^{1}(Spin_{2n}^{+}) = Ch^{1}(𝔛_{0})/ im(𝔠_{hs}^{(1)}) = Λ_{ω}/T̂_{0} ⊗_{ℤ} 𝔽_{2}.

It is generated by x_{1} = π(h_{1}) = π(c_{1}(L(ω_{1})). Hence the common index in this case is i_{J} = i_{A}. Moreover, if i_{A} > 1, then 4| deg(A) and k_{1} > 1.

Witt-equivalent algebras with involution. The arguments of 4.3 also apply to algebras with involution. Consider two Brauer-equivalent algebras A and B, respectively endowed with the orthogonal involutions σ and τ. They can be represented as (End_{D}(M), ad_{h}) and (End_{D}(M′), ad_{h′}), for some hermitian modules (M, h) and (M′, h′) over a division algebra with orthogonal involution (D, ), Brauerequivalent to A. The algebras (A, σ) and (B, τ) are said to be Witt-equivalent if the hermitian modules (M, h) and (M′, h′) are Witt-equivalent. If so, (A, σ) and (B, τ) are split hyperbolic over the same fields. Hence, the corresponding twisted Borel varieties 𝔛_{A} and 𝔛_{B} are split over the function field of each other. So [34, 5.18(iii)] applies and we get:

5.6. Proposition. Let (A, σ) and (B, τ) be two Witt-equivalent algebras with involution. All the non trivial indices in their J-invariant are equal.

This proposition will prove useful to complete the classification in degree 8. Before, we study the degree 6 and degree 4 cases.

Classification results in degree 6. Assume that deg(A) ≡ 2[4]. From the table [34, 4.13], 6 is the smallest value for which the J-invariant may be non trivial. This is not surprising, since in degree 2, any quaternion algebra endowed with an orthogonal involution with trivial discriminant is split hyperbolic [23, (7.4)]. In degree 6, the J-invariant is given by J(A, σ) = (j_{1}), with 0 ≤ j_{1} ≤ 2. It can be computed as follows:

5.7. Theorem. Let A be a degree 6 algebra endowed with an orthogonal involution σ with trivial discriminant. Its J-invariant is given by

J(A, σ) = (i_{S}),

where, as before, i_{S} = i_{+} = i_{−} is the 2-adic valuation of any component of the Clifford algebra C(A, σ). So, we have:

(1) J = (0) ⟺ (A, σ) is split hyperbolic.

(2) J = (1) ⟺ (A, σ) is split isotropic and non hyperbolic.

(3) J = (2) ⟺ (A, σ) is anisotropic.

Note that the algebra can be split or non split in the last case.

Proof. For degree reasons, the index i_{s} is bounded by 2. Hence the equality J(A, σ) = (i_{S}) is a direct consequence of corollary 5.1. Moreover, the fundamental relations 0.6 show that if i_{S} ≤ 1, the algebra A is split. If so, σ is adjoint to a quadratic form φ, and the J invariant of (A, σ) is J(A, σ) = J(φ) (see 5.3). By 4.6, this proves (1) and (2), and also (3) in the split case. To finish the proof, it is enough to check that if A is non split, then σ is anisotropic. For the sake of contradiction, assume A is non split, that is A = M_{3}(Q) for some division quaternion algebra Q over k, and σ is isotropic. Since A is the endomorphism ring of a 3-dimensional Q-module, (A, σ) is a hyperbolic extension, in the sense of [9, 3.1], of (Q, σ_{an}), for some orthogonal involution σ_{an} of Q. Moreover, by [23, (7.5)], we have d(σ_{an}) = d(σ) = 1. This is impossible if Q is non split (see [23, (7.4)]). □

Classification in degree 4. Assume the degree of A satisfies deg(A) ≡ 0[4]. The first two parameters of J(A, σ) = (j_{1}, j_{2}, …, j_{r}) correspond to generators of degree 1, and we now have:

5.8. Lemma. Assume the algebra A has degree deg(A) ≡ 0[4], and consider the first indices j_{1} and j_{2} of its J-invariant. We have:

(1) j_{1} = 0 ⟺ A is split;

(2) j_{2} = 0 ⟺ (A, σ) is half-spin, i.e. its Clifford algebra C(A, σ) has a split component.

Proof. We already know from 5.2 that j_{1} = 0 if A is split and j_{2} = 0 if (A, σ) is half-spin. To prove the converse, assume first that A is non split, that is i_{A} > 0. If min{i_{+}, i_{−}} > 0, Corollary 5.2(3) shows that j_{1} > 0. Otherwise, we are in the half spin case, so we can apply Corollary 5.5, which also gives j_{1} > 0. Similarly, assume that i_{+} > 0, and i_{−} > 0. If i_{A} > 0, Corollary 5.2(3) gives j_{2} > 0. If A is split, Corollary 4.4 gives the conclusion since, by Proposition 5.3, j_{2} is the first index of the J-invariant of the underlying quadratic form.

□

For algebras of degree 4, the J-invariant is given by J(A, σ) = (j_{1}, j_{2}) with 0 ≤ j_{1}, j_{2} ≤ 1. Hence the previous lemma suffices to determine J(A, σ). Moreover, it is well-known that (A, σ) is hyperbolic if and only if one component of C(A, σ) is split (see [23, (15.14)]). So we can rephrase the result as follows:

5.9. Lemma. Let A be a degree 4 algebra endowed with an orthogonal involution σ with trivial discriminant. Its J-invariant J(A, σ) = (j_{1}, j_{2}) can be computed as follows:

(1) j_{1} is 0 if A is split and 1 otherwise;

(2) j_{2} is 0 if σ is hyperbolic, and 1 otherwise.

We can give a precise description of (A, σ) is each case. As explained in [23, (15.14)], since A has degree 4 and σ has trivial discriminant, (A, σ) decomposes as (A, σ) = (Q_{1}, ) ⊗_{k} (Q_{2}, ), where the quaternion algebras Q_{1} and Q_{2} are the two components of the Clifford algebra C(A, σ), each endowed with its canonical involution. The algebra A is split if and only if Q_{1} and Q_{2} are isomorphic, in which case σ is adjoint to the norm form of Q_{1} = Q_{2}, which is a 2-fold Pfister form. So J = (0, 0) if and only if Q_{1} = Q_{2} is split, and σ is hyperbolic. Otherwise, Q_{1} = Q_{2} is division, its norm form is anisotropic, and J = (0, 1). If A is non split, then Q_{1} and Q_{2} are not isomorphic. If one of them, say Q_{1} is split, then A has index 2, A = M_{2}(Q_{2}), σ is hyperbolic, and J = (1, 0). Otherwise, J = (1, 1), the involution is anisotropic, and A has index 2 or 4. Using this description, one may easily construct explicit examples for each possible value of the J-invariant.

6. The trialitarian case

From now on, we assume that (A, σ) has degree 8. The J-invariant of (A, σ) is a triple J(A, σ) = (j_{1}, j_{2}, j_{3}) with 0 ≤ j_{1}, j_{2} ≤ 2 and 0 ≤ j_{3} ≤ 1. In this section, we will explain how to compute J(A, σ). As a consequence of our results, we will prove:

6.1. Corollary. (i) There is no algebra of degree 8 with orthogonal involution with trivial discriminant having J-invariant equal to (1, 2, 0), (2, 1, 0) or (2, 2, 0).

(ii) All other possible values do occur.

In particular, this shows that the restrictions described in the table [34, 4.13] (see also § 8), which were obtained by applying the Steenrod operations on Ch^{∗}(G_{0}) (loc. cit. 4.12) are not the only ones.

Recall that the group PGO^{+}(A, σ) is of type D_{4}. To complete the classification in this case, we need to understand the action of the symmetric group S_{3} on the J-invariant (see 2.8). Let (B, τ) and (C, γ) be the two components of the Clifford algebra C(A, σ), each endowed with its canonical involution. It follows from the structure theorems [23, (8.10) and(8.12)] that both are degree 8 algebras with orthogonal involutions. The triple ((A, σ), (B, τ), (C, γ)) is a trialitarian triple in the sense of loc.cit. § 42.A, and in particular, the Clifford algebra of any of those three algebras with involution is the direct product of the other 2. Hence, if one of them, say (A, σ) is split, then the other two are half-spin.

6.2. Definition. The trialitarian triple ((A, σ), (B, τ), (C, γ)) is said to be ordered by indices if the indices of the algebras A, B and C satisfy

ind(A) ≤ ind(B) ≤ ind(C).

The J invariant of such a triple can be computed as follows:

6.3. Theorem. Let ((A, σ), (B, τ), (C, γ)) be a trialitarian triple ordered by indices, so that i_{A} ≤ i_{B} ≤ i_{C}. The J-invariants are given by

J(A, σ) = (j, j′, j_{3}) and J(B, τ) = J(C, γ) = (j′, j, j_{3}),

where j = min{i_{A}, 2} and j′ = min{i_{B}, i_{C}, 2} = min{i_{B}, 2}.

Moreover, the third index j_{3} is 0 if the involution is isotropic and 1 otherwise.

6.4. Remark. (i) The first index of the J-invariant of an algebra with involution (D, ρ) is min{i_{D}, 2} if D is not of maximal index in its triple. But it might be strictly smaller in general. In 6.9 below, we will give an explicit example where j_{1} < i_{D} = 2.

(ii) By 2.8, we already know that j_{3} does not depend on the choice of an element of the triple. On the other hand, as explained in [8], the involutions σ, τ and γ are either all isotropic or all anisotropic. The triple is said to be isotropic or anisotropic accordingly.

Proof. To start with, let us compute the first two indices j_{1} and j_{2} of the J-invariant of (A, σ). Since we are in degree 8, they are both bounded by 2. Moreover, the triple being ordered by indices, the common index is given by i_{J} = i_{A}. So the equality j_{1} = j follows directly from the inequalities of Corollary 5.2. If additionally j′ = j, the very same argument gives j_{2} = j′. Assume now that j and j′ are different, that is j < j′. If so, j = 0 or j = 1. In the first case, we have i_{A} = 0 so that the algebra A is split, and the result follows from 4.7 and 5.3. The only remaining case is j = i_{A} = 1 and i_{B} ≥ 2, so that j′ = 2. Consider the function field F_{A} of the Severi-Brauer variety of A, which is a generic splitting field of A. By the fundamental relations 0.6, the algebra C is Brauer equivalent to A ⊗ B. Hence Merkurjev’s index reduction formula [26] says

ind(B_{F_{A}}) = min{ind(B), ind(B ⊗ A)} = ind(B).

So the values of i_{B} and j′ are the same over F and F_{A}. We know the result hold over F_{A} by reduction to the split case. Since the index j_{2} can only decrease under scalar extension, we get j_{2} ≥ j′ = 2, which concludes the proof in this case.

So the J-invariant of (A, σ) is given by J(A, σ) = (j, j′, j_{3}) for some integer j_{3}. It remains to compute the J-invariant of (B, τ) and (C, γ). Recall from 2.8 that (j, j′, j_{3}) and (j′, j, j_{3}) are the only possible values. So, if j = j′, there is no choice and we are done. Again, there are two remaining cases. Assume first that j = i_{A} = 0 and j′ ≥ 1, so that J(A, σ) = (0, j′, j_{3}). Since A is split, (B, τ) and (C, γ) are half-spin, so they have trivial j_{2} and this gives the result. Assume now that j = 1 and j′ = 2, so that J(A, σ) = (1, 2, j_{3}). By the previous case, over the field F_{A}, both (B, τ) and (C, γ) have J-invariant (2, 0, j_{3}). So the value over F has to be (2, 1, j_{3}).

To conclude the proof, it only remains to compute j_{3}. If A is split, this was done in 4.7. In the anisotropic case, we can reduce to the split case by generic splitting.

Indeed, by [18] in the division case, [38, Prop. 3] in index 4, and [30, Cor. 3.4] in index 2 (see also [20]) the triple remains anisotropic after scalar extension to a generic splitting field F_{A} of the algebra A. Hence j_{3} is equal to 1 over F_{A}, and this implies j_{3} = 1. In the isotropic case, if i_{C} ≥ 2, then we actually are in the split case. Indeed, if ind(C) ≥ 4 and γ is isotropic, then C = M_{2}(D) for some degree 4 division algebra D, and γ has to be hyperbolic. So by 0.3, (C, γ) is half-spin, that is A is split. The only remaining case is i_{A} = i_{B} = i_{C} = 1 and all three involutions are isotropic. In this case, (A, σ) is Witt-equivalent to a non-split algebra of degree 4 with anisotropic involution, which has J-invariant (1, 1) by 5.9. Hence, in view of 5.6, the J-invariant of (A, σ), which already has j_{1} = j_{2} = 1 must have j_{3} = 0.

□

The first part of Corollary 6.1 follows easily from Theorem 6.3. Indeed, if one of j_{1}, j_{2} is 2 and the other one is ≥ 1, then the algebras A, B and C are all three non split, and B and C have index ≥ 4. By 0.3, since A and B are non split, the involution γ on C is not hyperbolic, so it is anisotropic, and the theorem gives j_{3} = 1.

Explicit examples. We now prove the second part of Corollary 6.1. Recall from 5.3 that if A is split, and σ is adjoint to a quadratic form φ, then J(A, σ) = (0, J(φ)). Hence any triple with j_{1} = 0 is obtained for a suitable choice of φ by 4.7. Considering the components of the even Clifford algebra of those quadratic forms, we also obtain all triples with j_{2} = 0 by Theorem 6.3. The maximal value (2, 2, 1) is obtained from a generic cocycle; such a cocycle exists by [22, Thm. 6.4(ii)]. Hence, it only remains to prove that the values (1, 1, 0), (1, 1, 1), (1, 2, 1) and (2, 1, 1) occur. For any of those, we will produce an explicit example, inspired by the trialitarian triple constructed in [35, Lemma 6.2]

Our construction uses the notion of direct sum for algebras with involution, which was introduced by Dejaiffe [2]. Consider two algebras with involution (E_{1}, θ_{1}) and (E_{2}, θ_{2}) which are Morita-equivalent, that is E_{1} and E_{2} are Brauer equivalent and the involutions θ_{1} and θ_{2} are of the same type. Dejaiffe defined a notion of Morita equivalence data, and explains how to associate to any such data an algebra with involution (A, σ), which is called a direct sum of (E_{1}, θ_{1}) and (E_{2}, θ_{2}). In the split orthogonal case, if θ_{1} and θ_{2} are respectively adjoint to the quadratic forms φ_{1} and φ_{2}, any direct sum of (E_{1}, θ_{1}) and (E_{2}, θ_{2}) is adjoint to φ_{1} ⊕ ⟨λ⟩φ_{2} for some λ ∈ F^{×}, and the choice of a Morita-equivalence data precisely amounts to the choice of a scalar λ. In general, there exist non isomorphic direct sums of two given algebras with involution. We will use the following characterization of direct sums [35, Lemma 6.3] :

6.5. Lemma. The algebra with involution (A, σ) is a direct sum of (E_{1}, θ_{1}) and (E_{2}, θ_{2}) if and only if there is an embedding of the direct product (E_{1}, θ_{1}) × (E_{2}, θ_{2}) in (A, σ) and deg(A) = deg(E_{1}) + deg(E_{2}).

Slightly extending Garibaldi’s ‘orthogonal sum lemma’ [9, Lemma 3.2], we get:

6.6. Proposition. Let Q_{1}, Q_{2}, Q_{3} and Q_{4} be quaternion algebras such that Q_{1} ⊗Q_{2} and Q_{3} ⊗ Q_{4} are Brauer equivalent. If (A, σ) is a direct sum of (Q_{1}, ) ⊗ (Q_{2}, ) and (Q_{3}, ) ⊗ (Q_{4}, ) then one of the two components of the Clifford algebra of (A, σ) is a direct sum of (Q_{1}, ) ⊗ (Q_{3}, ) and (Q_{2}, ) ⊗ (Q_{4}, ), while the other is a direct sum of (Q_{1}, ) ⊗ (Q_{4}, ) and (Q_{2}, ) ⊗ (Q_{3}, ).

6.7. Remark. If one of the four quaternion algebras is split, as we assumed in [35], then all three direct sums have a hyperbolic component. Hence they are uniquely defined. This is not the case anymore in the more general setting considered here. The algebra with involution (A, σ) does depend on the choice of an equivalence data. Nevertheless, once such a choice is made, its Clifford algebra is well defined. So the equivalence data defining the other two direct sums are determined by the one we have chosen.

Proof. Denote (E_{1}, θ_{1}) = (Q_{1}, )⊗(Q_{2}, ) and (E_{2}, θ_{2}) = (Q_{3}, )⊗(Q_{4}, ). By [23, (15.12)], their Clifford algebras with canonical involution are (Q_{1}, )×(Q_{2}, ), and (Q_{3}, ) × (Q_{4}, ) respectively. The embedding of the direct product (E_{1}, θ_{1}) × (E_{2}, θ_{2}) in (A, σ) induces an embedding of the tensor product of their Clifford algebras in the Clifford algebra of (A, σ):

((Q_{1}, ) × (Q_{2}, )) ⊗ ((Q_{3}, ) × (Q_{4}, )) ↪ (C(A, σ), \underline{σ}).

This tensor product splits as a direct product of four tensor products of quaternion algebras with canonical involution; for degree reasons, two of them embed in each component of C(A, σ). To identify them, it is enough to look at their Brauer classes. From the hypothesis, we have Brauer equivalences Q_{1} ⊗ Q_{3} ∼ Q_{2} ⊗ Q_{4} and Q_{1} ⊗ Q_{4} ∼ Q_{2} ⊗ Q_{3}. If Q_{1} ⊗ Q_{3} and Q_{1} ⊗ Q_{4} are not Brauer equivalent, that is if A is non split, this concludes the proof. Otherwise, all four tensor products are isomorphic, and the result is still valid. □

With this in hand, we now give explicit examples of algebras with involution having J-invariant (1, 2, 1), (2, 1, 1), (1, 1, 1) and (1, 1, 0).

6.8. Example. Let F = k(x, y, z, t) be a function field in 4 variables over k, and consider the following quaternion algebras over F:

Q_{1} = (x, zt), Q_{2} = (y, zt), Q_{3} = (xy, z) and Q_{4} = (xy, t).

We let (A, σ) be a direct sum of (Q_{1}, ) ⊗ (Q_{2}, ) and (Q_{3}, ) ⊗ (Q_{4}, ) as in 6.6, and denote by (B, τ), and respectively (C, γ), the component of C(A, σ) Brauer equivalent to Q_{1} ⊗ Q_{3} ∼ (x, t) ⊗ (y, z) and Q_{1} ⊗ Q_{4} ∼ (x, z) ⊗ (y, t). The algebras A, B and C have index 2, 4 and 4, so that ((A, σ), (B, τ), (C, γ)) is a trialitarian triple ordered by indices. By Theorem 6.3, we get J(A, σ) = (1, 2, j_{3}) and J(B, τ) = J(C, γ) = (2, 1, j_{3}) for some j_{3}. Finally, assertion (i) of Corollary 6.1 implies j_{3} = 1; in other words, this triple is anisotropic.

6.9. Example. This example is obtained from the previous one by scalar extension. Consider the Albert form φ = ⟨x, t, −xt, −y, −z, yz⟩ associated to the biquaternion algebra Q_{1} ⊗Q_{3}. We let F′ be its function field, F′ = F (φ), and denote by (A′, σ′), (B′, τ′) and (C′, γ′) the extensions of (A, σ), (B, τ) and (C, γ) to F′. Since B is Brauer equivalent to Q_{1} ⊗ Q_{3}, the algebra B′ has index 2. On the other hand, it follows from Merkurjev’s index reduction formula [26, Thm. 3] that the indices of A and C are preserved by scalar extension to F′, so that A′ and C′ have indices 2 and 4 respectively. Hence ((A′, σ′), (B′, τ′), (C′, γ′)) again is a trialitarian triple ordered by indices and Theorem 6.3 now gives J(A′, σ′) = J(B′, τ′) = J(C′, γ′) = (1, 1, j_{3}) for some j_{3}. The same argument as in the proof of the first assertion of Corollary 6.1 applies here: since A′ and B′ are non split and C′ has index 4, the involutions are anisotropic and Theorem 6.3 gives j_{3} = 1. Note that, in particular, we have J(C′, γ′) = (1, 1, 1), even though C′ has index 4 = 2^{2}.

6.10. Example. We now produce another example of an anisotropic trialitarian triple having J-invariant (1, 1, 1) in which all three algebras have index 2. Namely, consider the F -quaternion algebras

Q_{1} = (x, y), Q_{2} = (x, z), Q_{3} = (x, t) and Q_{4} = (x, yzt).

Pick an arbitrary orthogonal involution ρ on H = (x, yz) over F. Since Q_{1} ⊗ Q_{2} is isomorphic to 2 by 2 matrices over H, the tensor product of the canonical involutions of Q_{1} and Q_{2} is adjoint to a 2-dimensional hermitian form h_{12} over (H, ρ). Similarly, (Q_{3}, )⊗(Q_{4}, ) is isomorphic to M_{2}(H) endowed with the adjoint involution with respect to some hermitian form h_{34}. Since h_{12} and h_{34} are both anisotropic, the hermitian form h = h_{12} ⊕ ⟨u⟩h_{34} over H′′ = H ⊗ F (u), for some indeterminate u, also is anisotropic. We define

(A, σ) = (M_{4}(H′′), ad_{h}).

It is clear from the definition that (A, σ) is a direct sum of (Q_{1}, ) ⊗ (Q_{2}, ) and (Q_{3}, ) ⊗ (Q_{4}, ). Hence, by 6.6, the two components (B, τ) and (C, γ) of its Clifford algebra are Brauer equivalent to (x, yt) and (x, zt). This shows that all three algebras have index 2. Since the involutions are anisotropic, by Theorem 6.3, their J-invariant is (1, 1, 1).

6.11. Remark. Note that there are many other examples, and not all of them can be described as in 6.6. In particular, any triple which includes a division algebra cannot be obtained from this proposition. Consider for instance the algebra with involution (A, σ) described in [36, Exple 3.6], and let (B, τ) and (C, γ) be the two components of its Clifford algebra. As explained there, A is a division indecomposable algebra, and one component of its Clifford algebra, say B, has index 2. Since A is Brauer equivalent to B ⊗ C, its indecomposability guarantees that C is division, and we get J(A, σ) = J(C, γ) = (2, 1, 1) and J(B, τ) = (1, 2, 1).

To produce examples of algebras with involution having J-invariant (1, 1, 1), we now construct examples of isotropic non split and non half-spin triples. As opposed to the previous examples, they can always be described using Proposition 6.6, as we now prove:

6.12. Proposition. If ((A, σ), (B, τ), (C, γ)) is an isotropic trialitarian triple with A, B and C non split, then there exists division quaternion algebras Q_{1}, Q_{2} and Q_{3} such that Q_{1}⊗Q_{2}⊗Q_{3} is split and the triple is described as in 6.6 with Q_{4} = M_{2}(k).

Proof. Since B and C are non split, the involution σ is not hyperbolic by 0.3. Hence A has index 2, A = M_{4}(Q_{1}) for some quaternion algebra Q_{1} over k. Fix an orthogonal involution ρ_{1} on Q_{1}; the involution σ is adjoint to a hermitian form h = h_{0} ⊕h_{1} over (Q_{1}, ρ_{1}), with h_{0} hyperbolic, h_{1} anisotropic and both of dimension 2 and trivial discriminant. Therefore, (A, σ) is a direct sum of (M_{2}(Q_{1}), ad_{h_{0}}) and (M_{2}(Q_{1}), ad_{h_{1}}). Since the first summand is hyperbolic, it is isomorphic to (M_{2}(k), ) ⊗ (Q_{1}, ). The second is (Q_{2}, ) ⊗ (Q_{3}, ), where Q_{2} and Q_{3} are the two components of the Clifford algebra ad_{h_{1}}, and this concludes the proof. □

We refer the reader to [35, §6] for a more precise description of those triples. They are the only ones for which the J-invariant is (1, 1, 1).

7. Generic properties

In the present section we investigate the relationship between the values of the Jinvariant of an algebra with involution (A, σ) and the J-invariant of the respective adjoint quadratic form φ_{σ} over the function field F_{A} of the Severi-Brauer variety of A, which is a generic splitting field of A.

7.1. Definition. We say (A, σ) is generically Pfister iff φ_{σ} is a Pfister form. Observe that in this case deg A is always a power of 2 and the J-invariant over F_{A} has the form:

J((A, σ)_{F_{A}}) = (0, …, 0, ∗)

(all zeros except possibly the last entry which is 0 or 1).

We say (A, σ) is in I^{s}, s > 2, if and only if φ_{σ} belongs to the s-th power I^{s}(F_{A}) of the fundamental ideal I(F_{A}) ⊂ W (F_{A}) of the Witt ring of F_{A}.

7.2. Theorem. Let (A, σ) be an algebra with orthogonal involution with trivial discriminant.

(a) If (A, σ) is in I^{s}, s > 2, then

J(A, σ) = (j_{1}, \underbrace{0, …, 0}_{2^{s−2}−1 times}, ∗, …, ∗)

(b) In particular, if (A, σ) is generically Pfister, then J(A, σ) = (∗, 0, …, 0, ∗).

Proof. (a) Let X = D_{n}/P_{i} be the variety of maximal parabolic subgroups of type i := 2 · [\frac{n+1}{2}] − 2^{s−1} + 1. Since i is odd, A_{k(X)} splits, and therefore the quadratic form φ_{σ} is defined over k(X). By assumption φ_{σ} ∈ I^{s}(k(X)). The Witt index of φ_{σ} is at least i. Therefore the anisotropic part of φ_{σ} has dimension at most 2(n − i) < 2^{s}. Thus, by the Arason-Pfister theorem φ_{σ} is hyperbolic. In particular, the variety X is generically split. Therefore by [33, Theorem 2.3] we obtain the desired expression for the J-invariant.

(b) Finally, if (A, σ) is generically Pfister, then φ_{σ} ∈ I^{s}(k(X)), where 2^{s} = 2n and (b) follows from (a). □

7.3. Remark. Let (j_{2}, …, j_{r}) be the J-invariant of φ_{σ} over F_{A}, r = [\frac{n+2}{2}]. In view of the theorem one may conjecture that the J-invariant of (A, σ) is obtained from J(φ_{σ}) just by adding an arbitrary left term, i.e.

J(A, σ) = (∗, j_{2}, …, j_{r}).

For example, if φ_{σ} is excellent, then the J-invariant has to be equal

J(A, σ) = (∗, 0, …, 0, ∗, 0, …, 0),

where the second ∗ has degree 2^{s} − 1 for some s and equals either 0 or 1. Observe that this holds for algebras of degree 8 (see §6).

8. Appendix

The following table provides the values of the parameters of the J-invariant for all orthogonal groups (here p = 2).

G_{0}  r  d_{i}  k_{i}  j_{i}
O_{n}^{+}  [\frac{n+1}{4}] 2i − 1  [log_{2} \frac{n−1}{d_{i}}] if d_{v} + l = 2^{s}d_{u} and 2 ∤ \binom{d_{v}}{l},
then j_{u} ≤ j_{v} + s
P GO_{4m}^{+} m + 1 1, i = 1  2^{k_{1}−1} ∥ m if d_{v} + l = 2^{s}d_{u} and 2 ∤ \binom{d_{v}}{l},
2i − 3, i ≥ 2 [log_{2} \frac{4m−1}{d_{i}}] for u, v > 1, then j_{u} ≤ j_{v} + s

The Spin_{n}-case is obtained from the O_{n}^{+}-case by removing the first entry, i.e. replacing r by r−1 and i by i+1. The P GO_{4m+2}^{+}-case is the same as the O_{4m+2}^{+}-case. The Spin_{4m}^{±}-case is the same as the O_{4m}^{+}-case.

Note that this table coincides with [34, Table 4.13] except of the last column which in our case contains more restrictive conditions. For the groups O_{n}^{+} the conditions of the last column coincide with the ones of [41, Prop. 5.12].

All values of the J-invariant which satisfy the restrictions given in the table will be called admissible.

Acknowledgments. The last author was supported by NSERC Discovery 385795-2010 and Accelerator Supplement 396100-2010 grants

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(Quéguiner-Mathieu) LAGA - UMR 7539 - Institut Galilée - Universitée Paris 13 - 93430 Villetaneuse - France

   URL: http://www.math.univ-paris13/~queguin/

(Semenov) Institut für Mathematik - Johannes Gutenberg-Universität - Staudingerweg 9 - 55099 Mainz - Germany

   URL: http://www.staff.uni-mainz.de/semenov/

(Zainoulline) Department of Mathematics and Statistics - University of Ottawa - 585 King Edward - Ottawa ON K1N6N5 - Canada

   URL: http://www.mathstat.uottawa.ca/~kirill/