ALGEBRAS WITH INVOLUTION THAT BECOME HYPERBOLIC OVER THE FUNCTION FIELD OF A CONIC

ANNE QU ÉGUINER-MATHIEU AND JEAN-PIERRE TIGNOL

Abstract. We study central simple algebras with involution of the first kind that become hyperbolic over the function field of the conic associated to a given quaternion algebra Q. We classify these algebras in degree 4 and give an example of such a division algebra with orthogonal involution of degree 8 that does not contain (Q, ), even though it contains Q and is totally decomposable into a tensor product of quaternion algebras.

Given two central simple algebras with involution (A, σ) and (B, τ ) over a field F , we say that (A, σ) contains (B, τ ) if A contains a σ-stable subalgebra isomorphic to B over which the involution induced by σ is conjugate to τ . By the double centralizer theorem [11, (1.5)], this is also equivalent to saying that (A, σ) is isomorphic to a tensor product (A, σ) ≃ (B, τ ) ⊗ (C, γ) for some central simple algebra with involution (C, γ) over F .

Let (Q, ) be a quaternion division algebra over F , endowed with its canonical involution. We denote by F Q the function field of the associated conic, which is the Severi-Brauer variety of Q. Since is of symplectic type, it becomes hyperbolic over any field that splits Q, hence in particular over F Q . From this, one may easily deduce that any (A, σ) that contains (Q, ) becomes hyperbolic over F Q . The main theme of this paper is to investigate the reverse implication. In the case where A is split and σ is anisotropic, it is an easy consequence of the Cassels-Pfister subform theorem in the algebraic theory of quadratic forms that the converse holds, see Proposition 2.1. When A is not split, the problem is much more delicate, and comparable to the characterization of quadratic forms that become isotropic over F Q , which was studied by Hoffmann, Lewis, and Van Geel [START_REF] Hoffmann | Function fields of quadratic forms[END_REF], [START_REF] Hoffmann | Minimal forms for function fields of conics, K-theory and algebraic geometry: connections with quadratic forms and division algebras[END_REF], [START_REF] Hoffmann | Minimal forms with respect to function fields of conics[END_REF]. Using an example from [START_REF] Hoffmann | Minimal forms with respect to function fields of conics[END_REF] of a 7-dimensional quadratic form over a suitable field F that becomes isotropic over F Q , we construct in §5 a division algebra A of degree 8 with an orthogonal involution σ such that (A, σ) becomes hyperbolic over F Q but does not contain (Q, ), even though A contains Q and (A, σ) decomposes into a tensor product of quaternion algebras with involution. This situation does not occur in lower degrees.

Algebras with involution that become split hyperbolic over F Q are considered in §2. It is shown in Proposition 2.3 that their anisotropic kernel contains (Q, ) (if it is not trivial). This applies in particular to algebras of degree 2m with m odd, see Corollary 2.4. The case of algebras of degree 4 is completely elucidated in §3, using Clifford algebras for orthogonal involutions and a relative cohomological invariant of degree 3 due to Knus-Lam-Shapiro-Tignol for symplectic involutions.

In the symplectic case, we classify the algebras of degree 4 that become hyperbolic

The second author gratefully acknowledges the hospitality of Université Paris 13, where part of the work leading to this paper was carried out. He was supported in part by the F.R.S.-FNRS.

over F Q but do not contain (Q, ), see Theorem 3.6. We show in §4 that our result is equivalent to the Hoffmann-Lewis-Van Geel classification of 5-dimensional quadratic forms that become isotropic over F Q without containing a Pfister neighbour of the norm form of Q, see Corollary 4.2. Orthogonal involutions on algebras of degree 8 are considered in § §5 and 6, using triality. In §5 we relate tensor products of quaternion algebras to quadratic forms of dimension 8 with trivial discriminant. The algebras with involution that do not decompose into tensor products of quaternion algebras with involution and become hyperbolic over F Q are determined in §6, see Theorem 6.3. They are isotropic, and their anisotropic kernel contains (Q, ). Finally, we use Laurent power series in §7 to construct algebras with involution of large degree that do not contain (Q, ).

Notations and preliminary observations

We work over a base field F of characteristic different from 2, and only consider algebras with involution of the first kind. We refer the reader to [START_REF] Knus | The book of involutions[END_REF] and to [START_REF] Lam | Introduction to quadratic forms over fields[END_REF] for background information on central simple algebras with involution and on quadratic forms. However, we depart from the notation in [START_REF] Lam | Introduction to quadratic forms over fields[END_REF] by using a 1 , . . . , a n to denote the n-fold Pfister form ⊗ n i=1 1, -a i . If h : V × V → D is a regular hermitian or skew-hermitian form on a finitedimensional vector space V over a division algebra D, we denote by ad h the involution on End D (V ) that is adjoint to h. In the particular case where D is split and h is the polar form of a quadratic form q, we also denote ad q for ad h . Following Becher [START_REF] Becher | A proof of the Pfister Factor Conjecture[END_REF], for any n-dimensional quadratic form q over F , we denote by Ad q the split orthogonal algebra with involution (M n (F ), ad q ).

Recall that a central simple F -algebra with involution (A, σ) is hyperbolic if and only if it is isomorphic to (End D V, ad h ) for some hyperbolic hermitian or skewhermitian form h on a vector space V over a division algebra D. In particular, A admits a hyperbolic involution if and only if the index of A divides 1 2 deg(A). If so, then A admits up to conjugation a unique orthogonal hyperbolic involution, and a unique symplectic one; see [START_REF] Bayer-Fluckiger | Hyperbolic involutions[END_REF] or [11, 6.B].

For any central simple F -algebra with involution (A, σ) and any field extension K/F , we let (A, σ) K = (A ⊗ F K, σ ⊗ Id). As pointed out in the introduction, we shall be mostly interested in the special case where K = F Q is the function field of the conic associated to a quaternion algebra Q, i.e. its Severi-Brauer variety. Throughout the paper, we fix the notation

Q = (a, b) F with a, b ∈ F × ,
and we assume Q is not split. We may then identify F Q with the quadratic extension

F ( √ at 2 + b) of F (t)
, where t is an indeterminate. For any central simple F -algebra A, we let

A(t) = A ⊗ F F (t) and A[t] = A ⊗ F F [t].
Moreover, the following conditions are equivalent:

(a') (A, σ) contains (Q, ); (b') A[t]
contains an element y satisfying (1).

Proof. The equivalence of (a) and (b) readily follows from the description of anisotropic involutions that become hyperbolic over a quadratic extension in [2, 3.3]. If y ∈ A(t) satisfies [START_REF] Kr | Cohomologische Invarianten quadratischer Formen[END_REF], then the version of the Cassels-Pfister theorem for algebras with involution in [START_REF]A Cassels-Pfister theorem for involutions on central simple algebras[END_REF] yields an element u ∈ A(t) such that σ(u)u = 1 and uy ∈ A[t].

Then y 0 = uy satisfies (2). If (a') holds, then A contains two skew-symmetric elements i, j such that i 2 = a, j 2 = b, and ji = -ij. Then y = it + j ∈ A[t] satisfies (1), so (b') holds. Conversely, suppose (b') holds and let y ∈ A[t] satisfy (1). We then have σ(y)y = -(at 2 + b), hence the degree of y is 1 since σ is anisotropic. Thus, y = λt+ µ for some λ, µ ∈ A. It follows from (1) that λ and µ are skew-symmetric and satisfy λ 2 = a, µ 2 = b, and µλ = -λµ, hence they generate a σ-stable subalgebra of (A, σ) isomorphic to (Q, ). Proposition 1.2. Let (A, σ) be a central simple F -algebra with involution. Assume A is division. If (A, σ) FQ is hyperbolic, then A contains Q and A[t] contains an element y 1 such that y 2 1 = at 2 + b. Proof. Since (A, σ) FQ is hyperbolic, the algebra A FQ is not division, hence Merkurjev's index reduction theorem [START_REF] Merkurjev | Simple algebras and quadratic forms[END_REF]Th. 1] shows that A contains Q, hence also two elements i, j such that i 2 = a, j 2 = b, and ji = -ij. Then

y 1 = it + b satisfies y 2 1 = at 2 + b.
Thus, the algebra with involution (A, σ) in Theorem 5.2 below is such that A(t) contains an element y satisfying (1), and A[t] contains elements y 0 , y 1 satisfying σ(y 0 )y 0 = -(at 2 + b), y 2 1 = at 2 + b, but no element satisfying both equations.

For the rest of this section, we focus on the case where σ is orthogonal. We then have a discriminant disc σ ∈ F × /F ×2 and a Clifford algebra C(A, σ), see [11, § §7, 8]. Recall from [11, (8.25)] that the center of C(A, σ) is the quadratic étale F -algebra obtained by adjoining a square root of disc σ. Therefore, if disc σ = 1 the algebra C(A, σ) decomposes into a direct product of two components, which are central simple F -algebras,

C(A, σ) ≃ C + (A, σ) × C -(A, σ).
If deg A ≡ 0 mod 4, then the canonical involution σ on C(A, σ) restricts to involutions σ + and σ -on C + (A, σ) and C -(A, σ). Moreover, the tensor product [11, (8.12), (9.14)].

C + (A, σ) ⊗ F C -(A, σ) is Brauer-equivalent to A, see
Proposition 1.3. Let (A, σ) be a central simple F -algebra with orthogonal involution and deg A ≡ 0 mod 4. If (A, σ) FQ is hyperbolic, then disc σ = 1 and at least one of (C + (A, σ), σ + ), (C -(A, σ), σ -) is split and isotropic.

Proof. Since σ FQ is hyperbolic, disc σ is a square in F Q , hence also in F since F is quadratically closed in F Q . By [11, (8.31)], the Clifford algebra of any hyperbolic involution has a split component; therefore C + (A, σ) FQ or C -(A, σ) is split. The corresponding involution σ ± is isotropic over F Q by the main theorem in [START_REF] Garibaldi | Clifford algebras of hyperbolic involutions[END_REF].

Since the Brauer group kernel of the scalar extension map from F to F Q is {0, [Q]}, the description of algebras (A, σ) with σ orthogonal and deg A ≡ 0 mod 4 such that (A, σ) FQ is hyperbolic falls into two cases:

Case 1: [C + (A, σ)], [C -(A, σ)] = 0, [A] ; Case 2: [C + (A, σ)], [C -(A, σ)] = [Q], [A ⊗ F Q] .
Following Garibaldi's definition in [START_REF]Orthogonal involutions on algebras of degree 16 and the Killing form of E 8 (with an appendix[END_REF], case 1 is when (A, σ) ∈ I 3 . Note that these two cases are not exclusive: they intersect if and only if A is Brauer-equivalent to Q; then (A, σ) becomes split hyperbolic over F Q . This case is easily dealt with in §2.

F to F Q is {0, [Q]}. Therefore, if (A, σ)
is a central simple F -algebra with involution that becomes split hyperbolic over F Q , then either A is split or A is Brauer-equivalent to Q. We consider each case separately.

Proposition 2.1. Let (A, σ) be a split central simple F -algebra with involution.

(1) If σ is symplectic, then it is hyperbolic. In this case, (A, σ) contains (Q, ) if and only if deg A ≡ 0 mod 4. (2) If σ is orthogonal, then (A, σ) ≃ Ad q for some quadratic form q. We have (Ad q ) FQ hyperbolic if and only if the anisotropic kernel of q is a multiple of the norm form n Q . Assuming this condition holds, Ad q contains (Q, ) if and only if the Witt index of q is a multiple of 4.

Proof.

(1) Since every alternating form on an F -vector space is hyperbolic, it follows that every split algebra with hyperbolic involution is hyperbolic. If ρ is an orthogonal involution on Q, then (Q, ) ⊗ (Q, ρ) is a split algebra of degree 4 with symplectic involution. If deg A = 4m, we have

(A, σ) ≃ (Q, ) ⊗ (Q, ρ) ⊗ Ad q
for any quadratic form q of dimension m since all the hyperbolic involutions on A are conjugate. Conversely, if A contains Q, then the centralizer of Q in A is Brauer-equivalent to Q, hence of even degree. Therefore, deg A ≡ 0 mod 4.

(2) By definition, (Ad q ) FQ is hyperbolic if and only if q FQ is hyperbolic, hence the first statement follows from [12, X(4.11)]. Let n Q ⊗ q 0 be the anisotropic kernel of q. If the Witt index of q is 4m, then denoting by H the hyperbolic plane over F we have [11, (11.1)]) it follows that

q ≃ n Q ⊗ (q 0 ⊥ mH). Since Ad nQ ≃ (Q, ) ⊗ (Q, ) (see
Ad q ≃ (Q, ) ⊗ (Q, ) ⊗ Ad q0⊥mH ,
so Ad q contains (Q, ). Conversely, assume Ad q contains (Q, ) and let (A 1 , σ 1 ) be the centralizer of (Q, ) in (A, σ). Then

Ad q ≃ (Q, ) ⊗ (A 1 , σ 1 )
hence σ 1 is symplectic and A 1 is Brauer-equivalent to Q. Therefore, there is a hermitian Q-form h 1 such that σ 1 ≃ ad h1 . The anisotropic kernel of h 1 has a diagonalization a 1 , . . . , a r Q with a 1 , . . . , a r ∈ F × ; letting m be the Witt index of h 1 , we have h 1 ≃ a 1 , . . . , a r Q ⊥ mH Q . Consider the following quadratic form over F :

q 1 = a 1 , . . . , a r ⊥ mH.
Then (A 1 , σ 1 ) ≃ (Q, ) ⊗ Ad q1 , hence Ad q ≃ Ad nQ⊗q1 , and therefore q is isometric to n Q ⊗ q 1 up to a scalar factor. Now,

n Q ⊗ q 1 = n Q ⊗ a 1 , . . . , a r ⊥ m(n Q ⊗ H) = n Q ⊗ a 1 , . . . , a r ⊥ 4mH,
and n Q ⊗ a 1 , . . . , a r is anisotropic since a 1 , . . . , a r Q is anisotropic (see [START_REF] Scharlau | Quadratic and Hermitian forms[END_REF]Ch. 10,Th. 1.1]). Therefore, the Witt index of q is 4m.

For algebras that are Brauer-equivalent to Q, the result is as follows:

Proposition 2.2. Let (A, σ) be a central simple F -algebra with involution such that A is Brauer-equivalent to Q. ( 1
) If σ is symplectic, then (A, σ) contains (Q, ) and (A, σ) FQ is hyperbolic. (2) If σ is orthogonal and (A, σ) FQ is hyperbolic, then (A, σ) is hyperbolic and contains (Q, ).
Proof. (1) It was already observed in the proof of Proposition 2.1(2) that every central simple algebra with symplectic involution that is Brauer-equivalent to Q contains (Q, ). Therefore, every such algebra becomes hyperbolic over F Q .

(2) The first statement follows from the injectivity of the scalar extension map W -(Q, ) → W (F Q ) proved independently in [START_REF] Dejaiffe | Formes antihermitiennes devenant hyperboliques sur un corps de déploiement[END_REF] and [START_REF] Parimala | Hermitian analogue of a theorem of Springer[END_REF]. If (A, σ) is hyperbolic, then A = M r (F ) ⊗ Q for some even integer r, and σ is conjugate to τ ⊗ for any symplectic (hyperbolic) involution τ on M r (F ). In particular, (A, σ) contains (Q, ).

Focusing on the anisotropic case in the propositions above, we have: Corollary 2.3. Let (A, σ) be a central simple F -algebra with involution. Assume σ is anisotropic and (A, σ) FQ is split hyperbolic. Then either

• A is split, σ is orthogonal, and there is a quadratic form q such that

(A, σ) ≃ Ad nQ⊗q ≃ (Q, ) ⊗ (Q, ) ⊗ Ad q ; or • A is Brauer-equivalent to Q, σ is symplectic, and there is a quadratic form q such that (A, σ) ≃ (Q, ) ⊗ Ad q . In both cases, (A, σ) contains (Q, ).
Using the results in this section, we may describe the algebras with involution of degree 2 mod 4 that become hyperbolic over F Q : Corollary 2.4. Let (A, σ) be a central simple F -algebra with involution. If deg A ≡ 2 mod 4 and (A, σ) FQ is hyperbolic, then A FQ is split. If moreover σ is anisotropic, then it is symplectic and there is an odd-dimensional quadratic form q over F such that (A, σ) ≃ (Q, ) ⊗ Ad q .

Proof. Since deg A ≡ 2 mod 4, we have A ≃ M r (H) for some odd integer r and some quaternion algebra H over F . If H is division, A does not admit a hyperbolic involution. Therefore, the hypothesis that (A, σ) FQ is hyperbolic implies that H FQ is split, hence (A, σ) FQ is split hyperbolic. The last statement then readily follows from Corollary 2.3.

In view of this corollary, we only consider central simple algebras of degree divisible by 4 in the following sections.

Algebras of degree 4

Throughout this section, A is a central simple algebra of degree 4 over an arbitrary field F of characteristic different from 2. Involutions on A are classified by cohomological invariants (see [START_REF] Knus | The book of involutions[END_REF]Ch. 4]), which we use to give an explicit description of the involutions on A that become hyperbolic over F Q .

3.1. The orthogonal case. This case is easy to handle using Clifford algebras. Proposition 3.1. Let σ be an orthogonal involution on A.

(1) (A, σ) is hyperbolic if and only if disc σ = 1 and one of the components of

C(A, σ) is split; (2) (A, σ) contains (Q, ) if and only if disc σ = 1 and one of the components of C(A, σ) is isomorphic to Q; (3) A contains Q if and only if A ⊗ F Q is Brauer-equivalent to a quaternion algebra. Proof. (1) This readily follows from [2, 2.5]. (2) If (A, σ) contains (Q, ), then there is a quaternion F -algebra Q ′ with canon- ical involution such that (A, σ) ≃ (Q, ) ⊗ (Q ′ , ).
By [11, (15.12)], this relation holds if and only if C(A, σ) ≃ Q × Q ′ . This proves (2).

(3) If A contains (a copy of) Q, then the centralizer of Q in A is a quaternion F -algebra Q ′ , and we have

A ≃ Q ⊗ F Q ′ .

This relation holds if and only if

A ⊗ F Q is Brauer-equivalent to Q ′ , hence (3) follows.
Corollary 3.2. Let σ be an orthogonal involution on A. Then (A, σ) FQ is hyperbolic if and only if (A, σ) is hyperbolic or contains (Q, ).

Proof. As noticed in the introduction, any (A, σ) containing (Q, ) is hyperbolic over F Q , so we only have to prove the converse. Assume (A, σ) FQ hyperbolic. Since F is quadratically closed in F Q , the discriminant of σ is trivial, so C(A, σ) is a direct product of two quaternion algebras. At least one of these quaternion algebras splits over F Q by Proposition 3.1(1), hence that component must be either split or isomorphic to Q. It follows that (A, σ) is either hyperbolic or contains (Q, ), by Proposition 3.1. [11, (15.12)]. Therefore, Proposition 3.1 [START_REF] Bayer-Fluckiger | Hyperbolic involutions[END_REF] shows that (A, σ) does not contain (Q, ), even though Proposition 3.1 [START_REF] Bayer-Fluckiger | Pfister involutions[END_REF] shows that A contains Q. This situation does not occur in the symplectic case, in view of Proposition 3.3 below.

Note that a central simple algebra of degree 4 with hyperbolic involution does not necessarily contain

(Q, ), even if it contains Q: if Q ′ is a quaternion F -algebra such that Q⊗ F Q ′ has index 2, then (A, σ) = (M 2 (F ), )⊗(Q ′ , ) is hyperbolic over F and satisfies C(A, σ) ≃ M 2 (F )×Q ′ by
3.2. The symplectic case. Symplectic involutions on A are classified up to conjugation by a relative invariant ∆ with values in the Galois cohomology group H 3 (F, µ 2 ), see1 [11, (16.9)]. We denote by [A] the Brauer class of A, viewed as an element of H 2 (F, µ 2 ), and for λ ∈ F × we denote by (λ) the square class of λ, viewed as an element in H 1 (F, µ 2 ). Using the invariant ∆, we show:

Proposition 3.3. Let σ be a symplectic involution on A. If A contains Q and (A, σ) FQ is hyperbolic, then (A, σ) contains (Q, ).
Proof. Since A contains Q, it decomposes as A = Q⊗Q ′ for some quaternion algebra

Q ′ = (a ′ , b ′ ) F . Let τ ′ be an orthogonal involution on Q ′ of discriminant a ′ . The involution τ = ⊗ τ ′ on A = Q ⊗ Q ′ is
of symplectic type, and clearly hyperbolic over F Q since (A, τ ) contains (Q, ). Similarly, for every invertible y ∈ Sym(Q ′ , τ ′ ), the involution

τ y = Int(1 ⊗ y) • τ = ⊗ (Int(y) • τ ) is such that (A, τ y ) contains (Q, )
. We prove below that σ is conjugate to τ y for a suitable y. By [11, (16.18)], the relative discriminant ∆ τ (τ ′ ) is given by

∆ τ (τ ′ ) = (Nrp τ (1 ⊗ y)) ∪ [A] ∈ H 3 (F, µ 2 ),
where Nrp τ is the Pfaffian norm, as defined in [11, (2.9)]. Since this relative discriminant classifies symplectic involutions on A up to conjugation, it suffices to show: Lemma 3.4. There exists an invertible element y ∈ Sym(Q ′ , τ ′ ) such that

∆ τ (σ) = (Nrp τ (1 ⊗ y)) ∪ [A].
Proof. The involution σ can be written as σ = Int(x) • τ for some x ∈ Sym(A, σ), hence we already have

∆ τ (σ) = (Nrp τ (x)) ∪ [A],
(3) and we want to prove we can substitute for x some element 1 ⊗ y with y ∈ Sym(Q ′ , τ ′ ). Since both σ and τ become hyperbolic over

F Q , the relative discrim- inant ∆ τ (σ) is killed by F Q , hence (Nrp τ (x)) ∪ [Q ′ ] = (λ) ∪ [Q] for some λ ∈ F × .
By the common slot lemma [1, Lemma 1.7], we may even assume

(Nrp τ (x)) ∪ [Q ′ ] = (λ) ∪ [Q ′ ] = (λ) ∪ [Q], (4)
from which we deduce

(λ) ∪ ([Q] + [Q ′ ]) = 0, and (λ Nrp τ (x)) ∪ [Q ′ ] = 0. ( 5
)
Hence the relative discriminant of σ is

∆ τ (σ) = (Nrp τ (x)) ∪ ([Q] + [Q ′ ]) = (λ Nrp τ (x)) ∪ ([Q] + [Q ′ ]) = (λ Nrp τ (x)) ∪ [Q].
Moreover, the quadratic space (Sym(A, σ), Nrp τ ) is an Albert quadratic space for the biquaternion algebra A by [11, (16.8)]. Hence, its Clifford invariant is

e 2 (Nrp τ ) = [A] = [Q]+[Q ′ ].
We deduce from (5) that the quadratic form λ ⊗Nrp τ has trivial Arason invariant e 3 , and hence is hyperbolic by the Arason-Pfister

Hauptsatz. So there exists an element x ′ ∈ Sym(A, σ) such that λ Nrp τ (x) = Nrp τ (x ′ ), and we have proven

∆ τ (σ) = (Nrp τ (x ′ )) ∪ [Q] and (Nrp τ (x ′ )) ∪ [Q ′ ] = 0 (6)
for some x ′ ∈ Sym(A, σ).

Let us now pick a pure quaternion i ′ ∈ Q ′ such that i ′2 = a ′ and τ ′ = Int(i ′ ) • . Denoting by Q 0 the vector space of pure quaternions in Q, we have

Sym(A, τ ) = (1 ⊗ Sym(Q ′ , τ ′ )) ⊕ (Q 0 ⊗ i ′ ).
Hence x ′ can be written as

x ′ = x 0 + 1 ⊗ ξ 1 + ξ 2 ⊗ i ′ for some x 0 ∈ F , some pure quaternion ξ 1 ∈ Q ′0 ∩ Sym(Q ′ , τ ′
) and some ξ 2 ∈ Q 0 . The pure quaternion ξ 1 being τ ′ -symmetric, it anticommutes with i ′ , and changing of quaternionic basis if necessary, we may assume that ξ 2 1 = b ′ x 2 1 for some x 1 ∈ F . Similarly, we may assume that ξ 2 2 = ax 2 2 for some x 2 ∈ F . Note that we allow x i = 0 for i = 0, 1, and 2 since some term might be zero in the decomposition of x ′ . We then have

Nrp τ (x ′ ) = x 2 0 -b ′ x 2 1 -aa ′ x 2 2 , (7)
and the following lemma finishes the proof:

Lemma 3.5. There exists z 0 , z 1 , y 0 , y 1 and

y 2 such that Nrp τ (x ′ )(z 2 0 -az 2 1 ) = y 2 0 -b ′ y 2 1 + a ′ b ′ y 2 2 = 0. Indeed, if we let y = y 0 + y 1 j ′ + y 2 i ′ j ′ , where j ′ is a pure quaternion in Q ′ such that j ′2 = b ′ and i ′ j ′ = -j ′ i ′ , we have Nrp τ (1 ⊗ y) = N rd Q ′ (y) = y 2 0 -b ′ y 2 1 + a ′ b ′ y 2 2
(see [11, (2.11)]). Hence,

∆ τ (σ) = (Nrp τ (x ′ )) ∪ [(a, b) F ] = (Nrp τ (x ′ )(z 2 0 -az 2 1 )) ∪ [(a, b) F ] = (Nrd Q ′ (y)) ∪ [A] = (Nrp τ (1 ⊗ y)) ∪ [A], with y ∈ Sym(Q ′ , τ ′ ).
Proof of Lemma 3.

is hyperbolic. On the other hand, by [START_REF]Orthogonal involutions on algebras of degree 16 and the Killing form of E 8 (with an appendix[END_REF], the quadratic form 1, -b ′ , -aa ′ , -µ is isotropic. Multiplying by (-ab ′ µ), we get that -a ′ b ′ µ is represented by the form µa, ab ′ , -µab ′ . Hence, in view of (8), the quadratic form 1, -µ, -a, µb ′ , -a ′ b ′ µ , which is a 5-dimensional subform of µ, a, b ′ , is necessarily isotropic. Since the quadratic forms 1, -a and 1, -b ′ , a ′ b ′ are anisotropic, this completes the proof.

Using Proposition 3.3, we may characterize the symplectic involutions on A that become hyperbolic over F Q : Theorem 3.6. Let σ be a symplectic involution on A.

If (A, σ) FQ is hyperbolic, then either (a) (A, σ) ≃ (Q, ) ⊗ (Q ′ , ρ)
for some quaternion F -algebra with orthogonal involution (Q ′ , ρ), or (b) (A, σ) ≃ Ad λ ⊗(Q ′ , ) for some quaternion F -algebra Q ′ and some λ ∈ F × with the following properties:

Q ⊗ F Q ′ is a division algebra and the norm forms n Q , n Q ′ satisfy λ • n Q ≃ λ • n Q ′ .
Conversely, if (A, σ) is of either type above, then (A, σ) FQ is hyperbolic.

Note that (a) and (b) are mutually exclusive since in the first case A ⊗ F Q has Schur index 1 or 2, whereas it has index 4 in case (b).

Proof. Suppose (A, σ) is not as in case (a) and (A, σ) FQ is hyperbolic. Then Proposition 3.3 shows that A does not contain Q. In particular, A is not division, by Proposition 1.2, so A ≃ M 2 (Q ′ ) for some quaternion F -algebra Q ′ , and Q ⊗ F Q ′ is a division algebra by Proposition 3.1 [START_REF] Bayer-Fluckiger | Pfister involutions[END_REF]. As observed in the proof of Proposition 2.1(2), (A, σ) contains (Q ′ , ), hence

(A, σ) ≃ Ad λ ⊗(Q ′ , )
for some λ ∈ F × .

The algebra A carries a hyperbolic symplectic involution τ , and we have by [11, (16.21)] [12, X(4.11)] and Arason's common slot lemma we may assume as in the proof of Lemma 3.4

∆ τ (σ) = (λ) ∪ [Q ′ ]. Since (A, σ) FQ is hyperbolic, this invariant vanishes over F Q . Hence, by
that (λ) ∪ [Q ′ ] = (λ) ∪ [Q]. Therefore, λ • n Q ′ ≃ λ • n Q , which shows that (A, σ) is as in case (b).
It is clear that (A, σ) FQ is hyperbolic in case (a). In case (b) the algebra A carries a hyperbolic symplectic involution τ and ∆ τ (σ) = (λ) ∪ [Q ′ ] vanishes over F Q . Therefore, (A, σ) FQ is hyperbolic.

F Q -minimal quadratic forms of dimension 5

A quadratic form ϕ over F is called F Q -minimal if ϕ FQ is isotropic and ψ FQ is anisotropic for every proper subform ψ ⊂ ϕ. In this section, we show that Theorem 3.6 can be used to recover (and is in fact equivalent to) the description of F Q -minimal forms of dimension 5 due to Hoffmann, Lewis, and Van Geel [START_REF] Hoffmann | Minimal forms for function fields of conics, K-theory and algebraic geometry: connections with quadratic forms and division algebras[END_REF]Prop. 4.1].

A general procedure to construct central simple algebras with involution that become hyperbolic over F Q uses Clifford algebras. Recall from [12, V(1.9), V(2.4)] that for any quadratic form ϕ of odd dimension 2m + 1 over F the even Clifford algebra C 0 (ϕ) is central simple over F of degree 2 m . It carries a canonical involution τ 0 , which is the restriction of the involution on the full Clifford algebra that leaves invariant every vector in the underlying vector space of ϕ. The involution τ 0 is orthogonal if m ≡ 0 or 3 mod 4 and symplectic otherwise, see [11, (8.4)].

Proposition 4.1. Let ϕ be a quadratic form of odd dimension over F .

(1) If ϕ FQ is isotropic, then (C 0 (ϕ), τ 0 ) FQ is hyperbolic. The converse holds if dim ϕ = 5. (2) If ϕ contains a subform similar to 1, -a, -b , then (C 0 (ϕ), τ 0 ) contains (Q, ). The converse holds if dim ϕ = 5.

Proof.

(1) The first statement readily follows from [11, (8.5)] and the second from [11, (15.21)].

(2) Suppose the underlying vector space of ϕ contains orthogonal vectors e 0 , e 1 , e 2 satisfying for some λ ∈ F × ϕ(e 0 ) = λ, ϕ(e 1 ) = -λa, ϕ(e 2 ) = -λb.

Then the products e 0 e 1 and e 0 e 2 generate a τ 0 -stable subalgebra of C 0 (ϕ) isomorphic to (Q, ).

For the rest of the proof, suppose dim ϕ = 5 and (C 0 (ϕ), τ 0 ) contains (Q, ). The centralizer of (Q, ) is a quaternion algebra with orthogonal involution (Q ′ , ρ) such that

(C 0 (ϕ), τ 0 ) = (Q, ) ⊗ (Q ′ , ρ).
Let V ⊂ Sym(C 0 (ϕ), τ 0 ) be the vector space of τ 0 -symmetric elements of trace 0. The map x → x 2 defines a quadratic form s : V → F that is similar to ϕ by the equivalence B 2 ≡ C 2 , see [11, (15.16)]. Let y ∈ Q ′ be a ρ-skew-symmetric unit and let Q 0 ⊂ Q be the vector space of pure quaternions. The restriction of s to the subspace Q 0 ⊗ y ⊂ V is similar to 1, -a, -b , hence the proof is complete.

C 0 (ϕ) ≃ M 2 (Q ′ ) for some quaternion F -algebra Q ′ such that Q ⊗ F Q ′ is a division algebra.
Proof. Proposition 4.1 shows that ϕ is F Q -minimal if and only if (C 0 (ϕ), τ 0 ) is hyperbolic but (C 0 (ϕ), τ 0 ) does not contain (Q, ). By Theorem 3.6, this condition is equivalent to

(C 0 (ϕ), τ 0 ) ≃ Ad λ ⊗(Q ′ , ) (9)
for some quaternion F -algebra

Q ′ = (a ′ , b ′ ) F and some λ ∈ F × with Q ⊗ Q ′ a division algebra
hold if ϕ is F Q -minimal. Conversely, if C 0 (ϕ) ≃ M 2 (Q ′ ) for some quaternion F -algebra Q ′ = (a ′ , b ′ ) F ,

Totally decomposable orthogonal involutions of degree 8

In this section, A denotes a central simple F -algebra of degree 8 and σ is an orthogonal involution on A. The algebra with involution (A, σ) is called totally decomposable if there are σ-stable quaternion subalgebras

Q 1 , Q 2 , Q 3 in A such that A = Q 1 ⊗ Q 2 ⊗ Q 3 . Denoting by σ i the restriction of σ to Q i for i = 1, 2, 3, we then have (A, σ) = (Q 1 , σ 1 ) ⊗ (Q 2 , σ 2 ) ⊗ (Q 3 , σ 3 ).
The totally decomposable algebras of degree 8 are characterized by the property that disc σ = 1 and one of the components of the Clifford algebra C(A, σ) is split, see [11, (42.11)]. Proposition 5.1 below shows how to use this criterion to relate totally decomposable algebras to quadratic forms.

Recall that for any quadratic form ϕ of dimension 8 with disc ϕ = 1 the even Clifford algebra decomposes into a direct product of central simple F -algebras of degree 8,

C 0 (ϕ) ≃ C + (ϕ) × C -(ϕ).
The canonical involution τ 0 on C 0 (ϕ) restricts to orthogonal involutions τ + , τ -on C + (ϕ) and C -(ϕ), and we have

(C + (ϕ), τ + ) ≃ (C -(ϕ), τ -).
It is easily checked that (C + (ϕ), τ + ) is totally decomposable.

Proposition 5.1. For every central simple algebra of degree 8 with totally decomposable orthogonal involution (A, σ), there is a quadratic form ϕ with dim ϕ = 8 and disc σ = 1 such that (A, σ) ≃ (C + (ϕ), τ + ). The form ϕ is uniquely determined by (A, σ) up to similarity. Moreover, -the algebra A is split if and only if ϕ is a multiple of a 3-fold Pfister form; in that case (A, σ) ≃ Ad ϕ ; -the algebra (A, σ) contains (Q, ) if and only if ϕ contains a subform similar to 1, -a, -b . Furthermore, the following conditions are equivalent:

(a) (A, σ) is isotropic; (b) (A, σ) is hyperbolic; (c) ϕ is isotropic.
Proof. Since (A, σ) is totally decomposable, it follows from [11, (42.11)] that one of the components C + (A, σ) of the Clifford algebra is split. The canonical involution σ + on C + (A, σ) is orthogonal, hence there is a quadratic form ϕ of dimension 8 such that (C + (A, σ), σ + ) ≃ Ad ϕ . By triality (see [11, (42.3)]) we have

(C 0 (ϕ), τ 0 ) ≃ (A, σ) × (A, σ).
Therefore, disc ϕ = 1 and (A, σ) ≃ (C + (ϕ), τ + ). Conversely, triality also shows that if (A, σ) ≃ (C + (ϕ), τ + ), then the canonical involution σ on C(A, σ) satisfies (C(A, σ), σ) ≃ Ad ϕ ×(A, σ), hence the form ϕ is uniquely determined up to similarity.

The Clifford algebra of ϕ splits if and only if ϕ is a multiple of a 3-fold Pfister form, by [16, Ch. 2, Th. 14.4] and [12, X(5.6)]. When that condition holds we have (A, σ) ≃ Ad ϕ by [11, (35.1)].

If ϕ contains a multiple of 1, -a, -b , then the same argument as in the proof of Proposition 4.1 [START_REF] Bayer-Fluckiger | Hyperbolic involutions[END_REF] shows that (C 0 (ϕ), τ 0 ) contains (Q, ). Projecting on each component, it follows that (C

+ (ϕ), τ + ) contains (Q, ). Conversely, if (A, σ) contains (Q, ), then we have (A, σ) = (Q, ) ⊗ (A 1 , σ 1 ) (10)
for some central simple algebra with symplectic involution (A 1 , σ 1 ) of degree 4. By [11, (15.19)] there is a 5-dimensional quadratic form ψ such that disc ψ = 1 and (A 1 , σ 1 ) ≃ (C 0 (ψ), τ 0 ). Letting τ ′ 0 be the canonical involution on C 0 ( ab, -a, -b ), we may rewrite [START_REF] Hoffmann | Minimal forms with respect to function fields of conics[END_REF] as

(A, σ) ≃ (C 0 ( ab, -a, -b ), τ ′ 0 ) ⊗ (C 0 (ψ), τ 0 ). (11)
In view of the canonical embedding

C 0 ( ab, -a, -b ) ⊗ F C 0 (ψ) ֒→ C 0 ( ab, -a, -b ⊥ ψ),
modifying it by a square if necessary, we may write it as λ 0 , -aλ 0 , -bλ 0 or abλ 0 for some λ 0 ∈ F × 1 . The same argument as above then proves the claim. Thus, to prove that ϕ does not contain any subform similar to 1, -a, -b , we have to show that no three of the quadratic forms α, β, γ, and δ have any common value over F 1 . This can be checked after some scalar extension. For instance, by [10, Lemma(4.4)(iii)], the only common value of α and β over F 0 (t)((u)) is the square class of u. On the other hand, applying [START_REF] Lam | Introduction to quadratic forms over fields[END_REF]VI(1.3)], one may check that γ and δ are both isomorphic over F 0 ((t))((u)) to 1, t , which does not represent u. On the other hand, over F 0 (t)((t + u)) the form γ only represents the square classes of 1 and t + u, whereas δ only represents the square classes of t and t(1 + t)(t + u). Therefore, γ and δ have no common value in F 1 .

To complete the proof, we show A is a division algebra. Taking the Clifford invariant of each side of ( 12) and applying [12, V(3.13)], we obtain the following equality in the Brauer group of F :

[A] + [C 0 ( t(1 + t), -at, -btu(t + u), ab, abu )] = [Q].
The even Clifford algebra of the 5-dimensional form is easily computed:

C 0 ( t(1 + t), -at, -btu(t + u), ab, abu ) ≃ (-u, bt) F ⊗ (ab(t + u), -au(1 + t)) F , hence A ≃ (a, b) F ⊗ (-u, bt) F ⊗ (ab(t + u), -au(1 + t)) F ≃ (-u, t) F ⊗ (a(t + u), u(1 + t)(t + u)) F ⊗ (b, 1 + t) F .
Since b is a uniformizing parameter for the b-adic valuation on F , it follows from [15, §19.6, Prop.] that the right side is a division algebra if (and only if) the algebra

B = (-u, t) F1((a)) ⊗ (a(t + u), u(1 + t)(t + u)) F1((a)) ⊗ F 1 √ 1 + t ((a))
is division. (Alternatively, one may view A as a ring of twisted Laurent series over B in an indeterminate whose square is b.) Now, a(t+u) is a uniformizing parameter for the a-adic valuation on F 1 √ 1 + t ((a)), hence the same argument shows that B is a division algebra if (and only if) the algebra

C = (-u, t) F1 ⊗ F 1 √ 1 + t, u(1 + t)(t + u)
is division. Since 1 + t and u(t + u) are squares in F 0 (u)((t)), we may embed C in the quaternion algebra (-u, t) F0(u)((t)) , which is clearly division. Therefore, A is a division algebra.

Non-totally decomposable orthogonal involutions of degree 8

In this section, we consider the case of central simple algebras with orthogonal involution (A, σ) of degree 8 that are not totally decomposable. These algebras do not contain any quaternion algebra with canonical involution (H, ), since the centralizer of H would be an algebra of degree 4 with symplectic involution, hence decomposable by [11, (16.16)]; the algebra (A, σ) would then be totally decomposable.

We start with a couple of lemmas of independent interest related to triality. Let

Q 1 , Q 2 , Q 3 be quaternion F -algebras such that Q 1 ⊗ F Q 2 ⊗ F Q 3 is split.
By a well-known result due to Albert and to Pfister, this condition implies that Q 1 , Q 2 , and Q 3 have a common maximal subfield, see [11, (16.30)]. Therefore, we may write

Q 1 = (c, d 1 ) F , Q 2 = (c, d 2 ) F , Q 3 = (c, d 3 ) F for some c, d 1 , d 2 , d 3 ∈ F × such that the quaternion algebra (c, d 1 d 2 d 3 ) F is split.
For α = 1, 2, 3, let ρ α be the orthogonal involution on Q α with disc ρ α = c. The involution ρ α is uniquely determined up to conjugation by [11, (7.4)]. Lemma 6.1. For {α, β, γ} = {1, 2, 3} we have

(Q α , ) ⊗ (Q β , ) ≃ Ad dα ⊗(Q γ , ρ γ ) ≃ Ad d β ⊗(Q γ , ρ γ ).
Proof. By Tao's computation of the Clifford algebra of a decomposable involution [START_REF] Tao | A variety associated to an algebra with involution[END_REF] we have

C (Q α , ) ⊗ (Q β , ) ≃ Q α × Q β ≃ (c, d α ) F × (c, d β ) F , C Ad dα ⊗(Q γ , ρ γ ) ≃ (c, d α ) F × (c, d α d γ ) F ≃ (c, d α ) F × (c, d β ) F , C Ad d β ⊗(Q γ , ρ γ ) ≃ (c, d β ) F × (c, d β d γ ) F ≃ (c, d β ) F × (c, d α ) F .
Since central simple algebras with orthogonal involutions of degree 4 are classified by their Clifford algebra (see [11, (15.7)]), the lemma follows. Now, for α = 1, 2, 3, let (A α , σ α ) be a central simple F -algebra with orthogonal involution of degree 8 such that for β, γ with {α, β, γ} = {1, 2, 3},

(A α , σ α ) ≃ Ad 1,-1,1,-d β ⊗(Q α , ρ α ) ≃ Ad 1,-1,1,-dγ ⊗(Q α , ρ α ). Thus, (A α , σ α ) is Witt-equivalent to (Q β , ) ⊗ (Q γ , ) by Lemma 6.1.
Lemma 6.2. The triple (A 1 , σ 1 ), (A 2 , σ 2 ), (A 3 , σ 3 ) is trialitarian, in the sense that for {α, β, γ} = {1, 2, 3} we have

C(A α , σ α ) ≃ (A β , σ β ) ⊗ (A γ , σ γ ).
(See [11, p. 548].) Proof. By triality, it suffices to prove the isomorphism for α = 1, β = 2, and γ = 3. By definition, (A 1 , σ 1 ) is an orthogonal sum of the algebra M 2 (Q 1 ) with a hyperbolic involution and of Ad d2 ⊗(Q 1 , ρ 1 ), so by Lemma 6.1

(A 1 , σ 1 ) ≃ (M 2 (F ), ) ⊗ (Q 1 , ) ⊞ (Q 2 , ) ⊗ (Q 3 , ) .
By [11, (15.12)] we have

C (M 2 (F ), ) ⊗ (Q 1 , ) ≃ (M 2 (F ), ) × (Q 1 , ) and C (Q 2 , ) ⊗ (Q 3 , ) ≃ (Q 2 , ) × (Q 3 ,
). Arguing as in Garibaldi's "Orthogonal Sum Lemma" [6, Lemma 3.2], we get

(C(A 1 , σ 1 ), σ 1 ) ≃ (C + (A 1 , σ 1 ), σ + ) × (C -(A 1 , σ 1 ), σ -) with (C + (A 1 , σ 1 ), σ + ) ≃ (M 2 (F ), ) ⊗ (Q 2 , ) ⊞ (Q 1 , ) ⊗ (Q 3 , ) and (C -(A 1 , σ 1 ), σ -) ≃ (M 2 (F ), ) ⊗ (Q 3 , ) ⊞ (Q 1 , ) ⊗ (Q 2 , ) .

Thus, (C

+ (A 1 , σ 1 ), σ + ) is Witt-equivalent to (Q 1 , ) ⊗ (Q 3 , ), hence it is isomor- phic to (A 2 , σ 2 ). Likewise, (C -(A 1 , σ 1 ), σ -) is isomorphic to (A 3 , σ 3 ).
Theorem 6.3. Let (A, σ) be a central simple F -algebra with orthogonal involution of degree 8. Assume (A, σ) is not totally decomposable. Then (A, σ) FQ is hyperbolic if and only if there is a quaternion F -algebra Q ′ with the following properties:

-

ind(Q ⊗ F Q ′ ) ≤ 2, and -(A, σ) is Witt-equivalent to (Q, ) ⊗ (Q ′ , ). When these equivalent properties hold, we can find c, d, d ′ ∈ F × such that Q ≃ (c, d) F , Q ′ ≃ (c, d ′ ) F , and
(A, σ) ≃ Ad 1,-1,1,-d ⊗(Q ′′ , ρ ′′ )
where

Q ′′ = (c, dd ′ ) F and ρ ′′ is an orthogonal involution on Q ′′ with disc ρ ′′ = c. Proof. Clearly, (A, σ) FQ is hyperbolic if (A, σ) is Witt-equivalent to an algebra containing (Q, ). Conversely, suppose (A, σ) FQ is hyperbolic. Since (A, σ) is not totally decomposable, it is not hyperbolic. If A is split, Proposition 2.1 shows that the anisotropic kernel of (A, σ) is Ad nQ ≃ (Q, ) ⊗ (Q, ), hence (A, σ) ≃ Ad 1,-1,1,-a ⊗ Ad b .
For the rest of the proof, we may thus assume A is not split. By Proposition 1.

(A, σ), σ + ) is Witt-equivalent to a product (Q ′ , ) ⊗ (Q ′′ , ) for some quaternion F -algebras Q ′ , Q ′′ such that Q ′ ⊗ Q ′′ is Brauer-equivalent to Q. We may therefore find c, d, d ′ , d ′′ ∈ F × such that Q ≃ (c, d) F , Q ′ ≃ (c, d ′ ) F , Q ′′ ≃ (c, d ′′ ) F .
Letting ρ (resp. ρ ′ , resp. ρ ′′ ) be an orthogonal involution on Q (resp. Q ′ , resp. Q ′′ ) with discriminant c, we have by Lemma 6.1

(C + (A, σ), σ + ) ≃ Ad 1,-1,1,-d ′ ⊗(Q, ρ) ≃ Ad 1,-1,1,-d ′′ ⊗(Q, ρ).
By Lemma 6.2, it follows that (A, σ) is isomorphic to

Ad 1,-1,1,-d ⊗(Q ′ , ρ ′ ) or Ad 1,-1,1,-d ⊗(Q ′′ , ρ ′′ ), hence it is Witt-equivalent to (Q, ) ⊗ (Q ′ , ) or (Q, ) ⊗ (Q ′′ , ).
Interchanging Q ′ and Q ′′ if necessary, we thus obtain the stated description of (A, σ).

Examples of arbitrarily large degree

Let (A, σ) be a central simple F -algebra with involution of orthogonal or symplectic type. Consider the (iterated) Laurent series fields F 1 = F ((x)), F 2 = F ((x))((y)), and the quaternion F 2 -algebra H = (x, y) F2 . Let ρ be any involution of orthogonal or symplectic type on H, and let

(A 1 , σ 1 ) = (A, σ) ⊗ F Ad x , (A 2 , σ 2 ) = (A, σ) ⊗ F (H, ρ).
If (A, σ) FQ is hyperbolic, then (A 1 , σ 1 ) FQ and (A 2 , σ 2 ) FQ also are hyperbolic, since they contain a hyperbolic factor.

Theorem 7.1. Assume (A, σ) is anisotropic. Then (A 1 , σ 1 ) and (A 2 , σ 2 ) are anisotropic. Moreover, the following conditions are equivalent:

(i) (A, σ) contains (Q, ); (ii) (A 1 , σ 1 ) contains (Q, ); (iii) (A 2 , σ 2 ) contains (Q, ). Proof. Let ξ 1 = 1 0 0 -1 , η 1 = 0 x 1 0 ∈ M 2 (F 1 ), so ad x (ξ 1 ) = ξ 1 and ad x (η 1 ) = -η 1 .
Let also (a i ) i∈I be an F -basis of A, so (a i ⊗ 1,

a i ⊗ ξ 1 , a i ⊗ η 1 , a i ⊗ ξ 1 η 1 ) i∈I is an F 1 -basis of A 1 .
We extend the x-adic valuation v 1 on F 1 to a map

g 1 : A 1 → 1 2 Z ∪ {∞} defined by g 1 i∈I a i ⊗ (α i + β i ξ 1 + γ i η 1 + δ i ξ 1 η 1 ) = min i∈I v 1 (α i ), v 1 (β i ), v 1 (γ i ) + 1 2 , v 1 (δ i ) + 1 2 for α i , β i , γ i , δ i ∈ F 1 .
It is readily verified that the map g 1 satisfies the following conditions for s, t ∈ A 1 and α ∈ F 1 :

• g 1 (1) = 0 and g 1 (s) = ∞ if and only if s = 0;

• g 1 (s + t) ≥ min g 1 (s), g 1 (t) and g 1 (sα) = g 1 (s) + v 1 (α);

• g 1 (st) ≥ g 1 (s) + g 1 (t). (It suffices to prove the last inequality for s, t in the above F 1 -base of A 1 , see [START_REF] Tignol | Value functions and associated graded rings for semisimple algebras[END_REF]Lemma 1.2].) The map g 1 defines a filtration of A 1 , and the associated graded ring gr(A 1 ) is gr(A 1 ) ≃ A ⊗ F M 2 (F [x, x -1 ]) with the grading defined by gr(A 1 ) λ = A ⊗ x λ 0 0 x λ for λ ∈ Z,

gr(A 1 ) λ = A ⊗ 0 x λ+ 1 2 x λ-1 2 0 for λ ∈ 1 2 Z \ Z.
Therefore, gr(A 1 ) is a graded simple algebra, and g 1 is a v 1 -gauge in the sense of [START_REF] Tignol | Value functions and associated graded rings for semisimple algebras[END_REF]. The involution σ 1 preserves g 1 . On gr(A 1 ) 0 , the induced involution σ 1 is σ ⊗ Id, hence it is anisotropic. Therefore, σ 1 is anisotropic by [21, Cor. 2.3], g 1 is the unique v 1 -gauge that is preserved by σ 1 by [21, Th. 2.2], and we have

g 1 (σ 1 (s)s) = 2g 1 (s) for all s ∈ A 1 . (14)
Now, suppose (A 1 , σ 1 ) contains (Q, ); it then contains elements i, j such that i 2 = a, j 2 = b, ji = -ij, σ 1 (i) = -i, σ 1 (j) = -j.

Then by ( 14) we have g 1 (i) = 1 2 g 1 (-a) = 0 and, similarly, g 1 (j) = 0. The images i, j of i, j in gr 1 (A 1 ) 0 satisfy conditions similar to [START_REF] Pierce | Associative algebras[END_REF]. Since gr(A 1 ) 0 ≃ A × A we may consider a projection gr(A) 0 → A, which is a homomorphism of algebras with involution π : (gr(A 1 ) 0 , σ 1 ) → (A, σ). The images π( i), π( j) generate a copy of (Q, ) in (A, σ). Thus, (A, σ) contains (Q, ) if (A 1 , σ 1 ) contains (Q, ). The converse is clear.

The argument for (A 2 , σ 2 ) follows the same lines. Let ξ 2 , η 2 ∈ H be such that

ξ 2 2 = x, η 2 2 = y, η 2 ξ 2 = -ξ 2 η 2 .
Note that if ρ is orthogonal its discriminant is represented by the quadratic form x, y, -xy , hence it is the square class of x, y, or -xy. Therefore, we may assume ρ = Int(ξ 2 )• , Int(η 2 )• , or Int(ξ 2 η 2 )• . In each case (and also if ρ is symplectic) we have ρ(ξ 2 ) = ±ξ 2 and ρ(η 2 ) = ±η 2 . Let v 2 : F 2 → Z 2 ∪ {∞} be the (x, y)-adic valuation such that v 2 (x λ y µ ) = (λ, µ) for λ, µ ∈ Z, where Z 2 is endowed with the right-to-left lexicographic ordering. Considering again an F -basis (a i ) i∈I of A, we extend v 2 to a map

g 2 : A 2 → 1 2 Z 2 ∪ {∞} defined by g 2 i∈I a i ⊗ (α i + β i ξ 2 + γ i η 2 + δ i ξ 2 η 2 ) = min i∈I v 2 (α i ), v 2 (β i ) + ( 1 2 , 0), v 2 (γ i ) + (0, 1 2 ), v 2 (δ i ) + ( 1 2 , 1 2 )
for α i , β i , γ i , δ i ∈ F 2 . The map g 2 is a v 2 -gauge on A 2 with associated graded ring gr(A 2 ) = A ⊗ (x, y) F [x,x -1 ,y,y -1 ] .

The involution σ 2 preserves g 2 and the induced involution σ 2 on gr(A 2 ) 0 = A is σ. Therefore, the same arguments as for (A 1 , σ 1 ) show that (A 2 , σ 2 ) is anisotropic, and that (A 2 , σ 2 ) contains (Q, ) if and only if (A, σ) contains (Q, ).

Theorem 7.1 applies in particular to the division algebra with orthogonal involution (A, σ) of Theorem 5.2, and yields central simple algebras with anisotropic involution (A 1 , σ 1 ) and (A 2 , σ 2 ) of degree 16 that do not contain (Q, ), even though they are hyperbolic over F Q . The involution σ 1 is orthogonal and ind A 1 = 8, while the involution σ 2 may be of orthogonal or symplectic type and A 2 is division. Of course, these constructions can be iterated to obtain examples of algebras with anisotropic involution of arbitrarily large degree that become hyperbolic over F Q and do not contain (Q, ). Such examples can also be derived from the central simple algebras of degree 4 with symplectic involution in case (b) of Theorem 3.6, although no division algebra can be obtained in this way since the algebras in case (b) of Theorem 3.6 have index 2.

Proposition 1 . 1 . 2 = 1 )

 1121 Let (A, σ) be a central simple F -algebra with involution. Assume σ is anisotropic. The following conditions are equivalent: (a) (A, σ) FQ is hyperbolic; (b) A(t) contains an element y satisfying σ(y) = -y and y at 2 + b. (If conditions (a) and (b) hold, then A[t] contains an element y 0 such that σ(y 0 )y 0 = -(at 2 + b).

5 .

 5 If 1, -b ′ , a ′ b ′ is isotropic, then it is universal, hence represents Nrp τ (x ′ ) and we are done. Otherwise, denote µ = Nrp τ (x ′ ); by (7), we have (µ) ∪ [(aa ′ , b ′ ) F ] = 0, and combining with (6), we get (µ) ∪ [(a, b ′ )] = 0. In terms of quadratic forms, this means that the 3-fold Pfister form µ, a, b ′ = 1, -µ, -a, µb ′ ⊥ µa, ab ′ , -µab ′ ⊥ -b ′

Corollary 4 . 2 (

 42 Hoffmann-Lewis-Van Geel[START_REF] Hoffmann | Minimal forms for function fields of conics, K-theory and algebraic geometry: connections with quadratic forms and division algebras[END_REF] Prop. 4.1]). A 5-dimensional quadratic form ϕ over F is F Q -minimal if and only if the following conditions hold:(a) ϕ is similar to a Pfister neighbour of the 3-fold Pfister form a, b, λ for some λ ∈ F × , and (b)

  and a, b, λ ≃ a ′ , b ′ , λ . It follows from the isomorphism (9) that ϕ is similar to a Pfister neighbour of a ′ , b ′ , λ , by[11, p. 271]. Thus, (a) and (b)

  then as observed in the proof of Proposition 2.1(2) we have (C 0 (ϕ), τ 0 ) ≃ Ad µ ⊗(Q ′ , ) for some µ ∈ F × . It then follows from [11, p. 271] that ϕ is similar to a Pfister neighbour of a ′ , b ′ , µ . If (a) holds, then a, b, λ ≃ a ′ , b ′ , µ and by the common slot lemma we may assume λ = µ. Thus, ϕ is F Q -minimal if (a) and (b) hold.

  Since F Q is the function field of the Severi-Brauer variety of Q, the Brauer group kernel of the scalar extension from

	If deg A = 4, the first case arises if and only if (A, σ) is hyperbolic, and the
	second case if and only if (A, σ) contains (Q, ), see §3.1. For algebras of degree 8, case 1 is when (A, σ) is totally decomposable, see §5; case 2 is considered in §6.
	2. Algebras with involution that become split hyperbolic over F Q

  3, one of the components of the Clifford algebra, C + (A, σ) say, is split by F Q . However, C

+ (A, σ) is not split since (A, σ) is not totally decomposable, hence C + (A, σ) is Brauer-equivalent to Q. As was observed in Proposition 1.3, (C + (A, σ), σ + ) FQ is isotropic. If it is hyperbolic, then (A, σ

) is hyperbolic by the main theorem of

[START_REF] Garibaldi | Clifford algebras of hyperbolic involutions[END_REF]

, a contradiction. Therefore, the anisotropic kernel of (C + (A, σ), σ + ) has degree 4. It has discriminant 1 by triality, hence (C +

The discussion in[START_REF] Knus | The book of involutions[END_REF] is in terms of the 3-fold Pfister form j whose Arason invariant is ∆.

which is compatible with the canonical involutions, [START_REF] Knus | The book of involutions[END_REF] yields (A, σ) ≃ (C + ( ab, -a, -b ⊥ ψ), τ + ).

Uniqueness of ϕ shows that ϕ is similar to ab, -a, -b ⊥ ψ, hence it contains a subform similar to 1, -a, -b .

The equivalence of (a), (b), (c) is clear if A is split, since then (A, σ) ≃ Ad ϕ and ϕ is a 3-fold Pfister form, hence it is isotropic if and only if it is hyperbolic. For the rest of the proof, we may thus assume A is not split. If (A, σ) is hyperbolic, then it follows from [START_REF] Garibaldi | Clifford algebras of hyperbolic involutions[END_REF] that the split component of (C(A, σ), σ) is isotropic, hence (b)⇒(c). Conversely, if (c) holds, then [11, (8.5)] shows that (C 0 (ϕ), τ 0 ) is hyperbolic, hence (C + (ϕ), τ + ) also is hyperbolic, proving (c)⇒(b). The equivalence of (a) and (b) readily follows from [3, Prop. 2.10].

To give an example of a division F -algebra of degree 8 with a totally decomposable orthogonal involution that is hyperbolic over F Q but does not contain (Q, ), we use an example of F Q -minimal quadratic form of dimension 7 due to Hoffmann and Van Geel [START_REF] Hoffmann | Minimal forms with respect to function fields of conics[END_REF]. For the rest of this section, we fix the following notation: [10, p. 43]) and ϕ = ϕ 0 ⊥ ab t(1 + t)(t + u) , so dim ϕ = 8 and disc ϕ = 1. Let also

a central simple F -algebra of degree 8 with a totally decomposable involution.

Theorem 5.2. The algebra with involution (A, σ) does not contain (Q, ), yet (A, σ) FQ is hyperbolic. Moreover, A is a division algebra.

Proof. Since 1, t ≃ 1 + t, t(1 + t) and t, u ≃ t + u, tu(t + u) , we have

Since the right side is hyperbolic over F Q , it follows that (ϕ 0 ) FQ is isotropic, and therefore (A, σ) FQ is hyperbolic by Proposition 4.1(1) or 5.1.

To show (A, σ) does not contain (Q, ), we prove ϕ does not contain any subform similar to 1, -a, -b . As in [10, p. 43], we decompose ϕ as

where α = 1 + t, u , β = 1, u , γ = 1, t + u , and δ = t, t(1 + t)(t + u) . By Springer's theorem, the isometry classes of α, β, γ, and δ over F 1 are uniquely determined by ϕ. If three of those quadratic forms represent a common value λ, then ϕ contains a 3-dimensional subform of λ a, b , hence a subform similar to 1, -a, -b . We claim that the converse also holds. Indeed, assume first that ϕ contains λ 1, -a, -b for some λ ∈ F × 1 . Writing ϕ = λ 1, -a, -b ⊥ ϕ ′ as in [START_REF] Merkurjev | Simple algebras and quadratic forms[END_REF], we get, by uniqueness of the forms α, β and γ up to isometry, that all three represent λ. Consider now the general situation, where λ ∈ F × need not be in F 1 ;


Géraldine Ducos
email: geraldine.ducos@rennes.inra.fr

Pierre Dupraz

CONSERVATION CONTRACT ADOPTION UNDER FIXED AND VARIABLE COSTS

Keywords: agro-environmental contract, willingness to accept, fixed compliance costs, JEL Classification: D23, Q12, Q28

Farmers' conservation contract adoption is usually explained through the technology lens, i.e. with operational costs depending on the subscribed area and environmental efforts. In their 2003 article, Dupraz et al. analysed this issue by introducing utility farmers directly derive from contract environmental outcomes. They specified farmers' willingness to accept, which is the minimum compensation payment the farmer accepts to enrol and produce environmental services on a given area. Here, we deepen this analysis at two levels. First, at the theoretical one, by introducing fixed compliance costs in the willingness to accept approach. Secondly, we empirically show that among fixed compliance costs, fixed transaction costs are a significant contracting barrier. It explains why smallest farms have a reduced probability to contract.

I) INTRODUCTION

The degradation of environmental quality and rural landscape has induced a strong demand of rural amenities in European countries. This led to a significant increase of agro-environmental schemes in Europe from the 90's. The trend was confirmed with the 1999-Common Agricultural Policy (CAP) reform and the implementation of "agro-environmental contracts". In this policy framework, farmer's voluntary participation is compensated if he adopts more environmental friendly practices than what is mandatory. First appraisals show the adoption rate remains globally low (about 25% of EU farms) and smallest farms are not involved. From this situation, displayed environmental objectives are expected not to be reached. The overall aim of this study is therefore to improve the understanding of farmers' adoption behaviour.

Several studies have examined factors influencing farmers' adoption. Four main determinants have been identified, namely (i) farmer and farm household characteristics, (ii) farm biophysical characteristics, (iii) farm financial/management characteristics, and (iv) exogenous factors such as information availability, sources of information, society social capital [START_REF] Knowler | Farmers' adoption of conservation agriculture: A review and synthesis of recent research[END_REF]. Even if they give an important insight on the overall adoption behaviour, a sharper analysis is obviously required. In particular, a first distinction between fixed and variable compliance costs should provide some useful explanations. Indeed, while compensation payment are on a per-hectare basis calculation, technology related fixed costs (a hedge cutter or an adapted seeder for instance) and fixed transaction costs (the costs of gathering information on contracts or writing the contract) amount to total significant levels.

This paper thus first seeks to know if fixed compliance costs are a significant adoption barrier. This should explain why smallest farms or farms having a small eligible area are not involved in agro-environmental contracts. The second research question is whether, among fixed compliance costs, transaction costs are a significant adoption barrier. We particularly think of transaction costs as an adoption barrier since we suspect agro-environmental contract to require specific investments. Indeed, contract requirements lead the farmer to entirely modify his production system, which ends up to a non negligible total investment. Most of the time, these investments being non redeployable outside the agro-environmental contract, they are threfore transaction specific and produce transaction costs.

As [START_REF] Dupraz | The Environmental Supply of Farm Households, a Flexible Willingness to Accept Model[END_REF], we base our analysis on a farm household model so that we incorporate the farmer's producer and consumer behavior. The household willingness to accept (WTA) is the minimum per-hectare compensation payment the farmer accepts to adopt an agro-environmental contract on a given area instead of not to adopt while his initial utility level remains the same (eq. Hicksian optimization). Besides, considering the presence of fixed compliance costs, we work the minimal per-hectare payment out from the average WTA and get a minimum compensation payment and a minimum necessary area. Finally, when the perunit payment is higher, farmers with a higher eligible area will be able to choose over the area to subscribe, this choice being derived from his Marshallian behavior. Therefore, as long as his marginal WTA is lower than the per-hectare payment, they will accept to subscribe an additional hectare.

The empirical analysis is based on a European sample made of 2262 farmers interviewed in 2005. It aims at highlighting the impact of fixed compliance costs on adoption behavior, and transaction cost one in particular. These last costs being not included in the calculation of compensation payments, it is therefore statistically possible to distinguish their effect from other costs. However, transaction costs being difficult to measure and endogenous to the decision to adopt, their effect is thus indirectly identified through their determinants. It is important to note that some of these determinants impact on transaction costs only whereas others have more complex effects, i.e. they impact on transaction costs, farmer's utility or operational costs too. These multiple effects will complicate the estimation interpretation.

We use the two-stage Heckman method to shed light on whether and to what extent variations in fixed compliance costs are responsible for observed adoption rates. In the first step, the most influential factors on participation are the type of farming system, past experience in agro-environmental contracts and farmer's trust in the implementation process. In the second step, the area under contract is the dependant variable. The disappearance of significant transaction cost determinants from the first step in the second step reveals the presence of fixed transaction costs and their significant effect on adoption behavior. As regard to fixed compliance costs, given that none of variables describing the production system disappear in the second step, we may say that, whether the compensation payment do include fixed operational costs, or, they have no significant effect on adoption behavior. Concerning variables with more complex effects and that are present in both estimations, the interpretation is harder. Whether variable transaction costs exist and their effect is mixed with other variable operational costs, or, they do not exist and we may conclude fixed transaction cost effect on adoption is lower than utility or operational cost effects. Finally, variables appearing in the second step purely highlight the effect of the production system characteristics on the farmer's choice over area to subscribe. This paper has a double contribution. First, by introducing fixed compliance costs in the usual adoption model, we improve the understanding of farmers' conservation practice adoption behavior. Second, by empirically showing that among fixed compliance costs, fixed transaction costs are a significant contracting barrier, we support the idea fixed compliance costs and transaction costs in particular should be taken into account in participation constraints.

The paper is organized as follows. Section 2 presents the conceptual framework from which we derived propositions. Section 3 describes the transaction cost issue in AEC and the data set on which the empirical analysis is based on. Section 4 provides estimation results and section 5 concludes.

II) SPECIFYING FARMERS' WILLINGNESS TO ACCEPT

II.1) Model settings

As [START_REF] Dupraz | The Environmental Supply of Farm Households, a Flexible Willingness to Accept Model[END_REF], we consider a utility maximizing farmer with an income constraint depending on farm profit. The restricted profit function, or short-term profit function, R π , enables to derive the income from the on-farm activity with an agro-environmental contract that is attached to an area v. .

0 . ) , , ( ) , , ( 0 ,
≥ + + = v e v Z v p m Z v m U R v m Max ρ π (1)
The variables of the utility function are the farmer's income , expressed in monetary value, the quantity of an environmental service co-produced by farming activities (the number of hectares submitted to environmentally friendly practices for instance) and other relevant utility characteristics m v Z . This function is supposed to be increasing, concave and differentiable in and . m v The farmer's restricted profit function R π represents the agro-environmental technology using the dual approach by a short-term optimization of products and production factor variables of prices p . Z represents other relevant profit characteristics. R π is linearly homogenous in prices . It is assumed to be non increasing and convex in v . The scalar represents offfarm incomes that are assumed to be exogenous.

p 0 e
The contract payment is based on a positive per-unit premium ρ . The budget constraint always binds. In this model, the farmer is assumed to enroll when the premium is higher than 0 ρ , the farmer's marginal willingness to accept (WTA) for the first unit of co-produced environmental service. It is defined in (2), which provides the corner solution of (1), noted , when the positivity constraint regarding binds, meaning the contract is rejected.

*) *, ( v m v v Z p m Z m U v Z m U v R ∂ ∂ - ∂ ∂ ∂ ∂ - = ≤ ⇔ = / ) , 0 , ( / ) , 0 *, ( / ) , 0 *, ( 0 * 0 π ρ ρ (2) When 0 ρ ρ > , the marginal WTA, noted , is then defined by (3). ) , , ( Z v p m m ρ m R v m v v e Z v p m Z v m U Max ≥ + = 0 , ) , , ( ) , , ( π (3)
First order conditions bring:

v Z v p m Z v m U v Z v m U R m ∂ ∂ - ∂ ∂ ∂ ∂ - = / ) *
ρ m v m ρ m v m v m ρ 0 ρ .
Let's go back to program (1) and distinguish a fixed cost function associated with strictly positive v . Fixed costs may come from physical or immaterial investments which are needed to conclude or implement the contract. The optimization programme becomes:

(.) F 0 . ) , , ( ) , , ( ) , , ( 0 ,
≥ + + - = v e v Z v p F Z v p m Z v m U Max R v m ρ π (5)
It depends on Z describing the technology and the transaction environment such as the governance structure type or the asset specificity level. Following is a detailed specification of (.) F . It does not depend on v for strictly positive values. We also assume that this fixed cost deters smallest contracted areas (6).

ρ > ⇒ ≤ < > ∃ = ∂ ∂ = ) , , ( 0 ; 0 ~0 ) , , ( 0 ) , 0 , ( Z v p F v v v v Z v p F and Z p F (6)
Finally, we specify the cost function of contracting:

(.) C ) , , ( ) , , ( ) , 0 , ( ) , , ( Z v p F Z v p Z p Z v p C R R + - = π π (7)
It is positive, continuous, increasing and concave with all strictly positive v . Under the preceding assumptions, the average cost function is U-shaped. As usual, the profit maximizing farmer will sign a contract if the per-unit payment exceeds the minimal average cost.

Next section is the analysis of the willingness to accept function that we associate to 0 ρ .

II.2) Specifying the farmer's willingness to accept

Under the preceding assumptions, especially those regarding the fixed compliance costs, the farmer will not produce the environmental service without payment. In this case, the solution of ( 5) is:

0 , ) , 0 , ( ), , 0 , ( 0 0 0 0 = + = = v e Z p m Z m U U R π (8)
The farmer's willingness to accept (WTA) an agro-environmental contract is the minimum compensation payment 0 ρ the farmer accepts to enroll and produce a fixed amount of environmental services instead of not to produce those services (eq. ) while his initial utility level, , remains the same. It is specified in two steps [START_REF] Dupraz | The Environmental Supply of Farm Households, a Flexible Willingness to Accept Model[END_REF].

v 0 = v 0 U First, the restricted expenditure function, called , is minimized in m.
is continuous, increasing and concave with all strictly positive v. We get the Hicksian optimal consumption :

R e R e ) , , , ( * 0 U Z v p m ( ) 0 ) , , ( ) , , ( ) , , ( ) , , , ( 0 0 ≥
≤ + - = v Z v m U U Z v p F Z v p m Min U Z v p e R m R π (9)
The corner solution described in ( 8) is also a solution for (9). Then, having assumed a utility maximizing farmer, the utility constraint always binds.

Second, the farmer's WTA is obtained by deriving the surplus variation:

{ } ) , , , ( ) , , ( ) , , , ( * ) , , 0 , ( * ) , , ( ) , , ( ) , 0 , ( ) , , 0 , ( ) , , , ( ) , , , ( 0 0 0 0 0 0 U Z v p WTP Z v p C U Z v p m U Z p m Z v p F Z v p Z p U Z p e U Z v p e U Z v p WTA R R R R - = - - + - = - = π π (10)
The first term is the farm profit loss or the cost function of contracting (cf. equation 7). The second term is the farmer's willingness to pay (WTP) for an increase in the environmental service on his farm site. Therefore, when the farmer's utility from environmental services he produces is negligible, his WTA equals the profit loss. Otherwise, the profit loss overestimates the farmer's WTA. With convex preferences, the WTP is positive, increasing and concave with v. Under our preceding assumptions, there is no production of environmental service without strictly positive payment. It means that the WTP is always lower than the cost of contracting for strictly positive v. Hence, the WTA function is positive, continuous, increasing and concave with all strictly positive v. Next section considers the actual policy context in which the farmer chooses the subscribed area.

II.3) Introducing the choice over the subscribed area

II.3.1) Specifying the minimum per-unit payment

The cost function including fixed compliance costs, the minimum per-unit payment * ρ that triggers the contract is therefore the minimum of the average WTA:

v v WTA v WTA Min v M ) ( *) ( * = = ρ (11)
The first order condition provides (12) and the minimum contractible area:

) , *, , ( * 0 U Z m p v 0 ) , *, , ( * ) *, , ( * 1 ² * *) ( 0 = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ + ∂ ∂ - + - v U Z v p m v Z v p v v v WTA R π (12)
Since the WTA refers to a minimal utility, we set ) *, *, ( 9) with . The total differentiation of the utility function thus gives:

0 Z v m U U = from (
) , *, , ( * 0 U Z v p m m Z v m U v Z v m U v U Z v p m ∂ ∂ ∂ ∂ - = ∂ ∂ / ) *, *, ( / ) *, *, ( ) , *, , ( * 0
The right part being the marginal rate of substitution between m and v, i.e. the farmer's WTP for environmental services.

We finally get (13) for specifying the minimal per-unit payment:

m Z v m U v Z v m U v Z v p v WTA R M ∂ ∂ ∂ ∂ - ∂ ∂ - = = / ) *, *, ( / ) *, *, ( ) *, , ( *) ( * π ρ (13)
And, we get ( 14) from ( 10) and ( 11

): 0 * * ) *, , ( ) *, , ( * e v Z v p F Z v p m R + + - = ρ π (14)

II.3.1) Specifying the optimal subscribed area

The farmer's choice over the area to enroll is derived from his Marshallian behavior with an unchanged exogenous income . His Marshallian optimal contracted area is given by program (5).

II.4) Utility maximizing behaviour with fixed costs

In order to show the contracting decision based on the minimal average WTA is compatible with a utility maximizing behaviour, we shall now prove that ) *, *, (

0 Z v m U U =
is also a solution of program ( 5 First order conditions of (5) bring:

) , ( ) , ( ) ( * v m U v m U v m v R v - - = π ρ (17)
The budget constraint, which is binding, brings :

0 . * ) , , ( ) , , ( e v Z v p F Z v p m R + + - = ρ π (18)
i) First we suppose that * v v > . As the marginal cost of contracting is increasing with v, (13) and ( 17) imply that the marginal rate of substitution (MRS) for is lower than the MRS for

*) *, ( v m ) , ( v m . ) , ( ) , ( *) *, ( *) *, ( ) 17 ( ) 13 ( ) , , ( ) *, , ( * v m U v m U v m U v m U and v Z v p v Z v p v v m v m v R R < ⇒ ∂ ∂ - < ∂ ∂ - ⇒ > π π (19)
As the MRS is decreasing with v when m is constant, it implies that * m m > must hold, and is compatible with a maximized utility. However, by using ( 14) and ( 18) this inequality contradicts our basic assumption of a positive MRS:

( ) 0 * ) , , ( 0 * ) , , ( ) *, , ( ) *, , ( 0 *) ( * ) , , ( ) *, , ( ) *, , ( ) , , ( 0 * * > + ⇒ > + ⇒ = > - + - + - ⇒ > - ∫ ρ π ρ π ρ π π Z v p dv Z v p Z v p F Z v p F v v Z v p F Z v p F Z v p Z v p m m R v v v R v R R (20)
This result contradicts both ( 13) and ( 17) where the marginal willingness to pay is positive. Therefore * v v > can not hold.

ii) Secondly, we suppose that * v v < . As the marginal cost of contracting is increasing with v, (13) and ( 17) imply that the MRS for is higher than the MRS for

*) *, ( v m ) , ( v m . As the MRS is decreasing with v when m is constant, it implies that * m m < must hold. It implies 0 ) *, *, ( ) , , ( U Z v m U Z v m U = <
. This inequality contradicts the optimization program (5). Indeed, the relationships ( 13) and ( 14) shows that respects the necessary conditions of this program, as stated in ( 17) and ( 18), and provides an higher utility than

*) *, ( v m ) , ( v m . Therefore ) , ( v m
can not be a solution of (5

) if * v v < . From i) and ii) we can conclude *) *, ( ) , ( v m v m = .

II.5) Synthesis

We can observe from graph (1), the marginal WTA curve ( ) where, in the absence of fixed compliance costs, the minimum WTA is m WTA 0 ρ .

In the presence of fixed compliance costs, we can observe the U-shaped average WTA curve (

) whose minimum is located in and the associated minimum per-unit payment for contracting is

M WTA * v * ρ . v € M WTA m WTA 1 ρ * v * * v + v - v * ρ 0 ρ
Graph 1: Average WTA and marginal WTA with the number of hectare under contract When the per-unit payment is higher than * ρ , 1 ρ for instance, as long as , farmers respecting the area condition enroll. This corresponds to the hachured area between , the minimal necessary area to enrol, and , the maximal possible area to enrol, without utility loss. Farmers with an eligible area higher than will prefer to subscribe the optimal area . If the eligible area of their farm is lower than , they will enter 100% of this available suboptimal area.

1 ρ ≤ M WTA - v + v * v * * v * * v

III) FIXED AND VARIABLE COMPLIANCE COSTS: APPLICATION TO AGRO-ENVIRONMENTAL CONTRACTS

We here seek to observe the impact of fixed compliance costs on adoption behavior, and fixed transaction costs in particular. The empirical analysis is based on a European sample made of 2262 farmers interviewed in 2005. They come from 10 different regions spread among 9 EU countries, namely France, the Netherlands, Belgium, England, Germany, Italy (two regions: Emilia Romagna and Veneto), Ireland, Finland and Czech Republic. 55% of interviewed farmers have subscribed an agro-environmental contract and the proportion of contractors is quite similar in every region except for Finland where only 32% of farmers are considered as contractors. Next table presents an overview of the origin population. This table highlights an over-coverage of contractors. Indeed, 60% of surveyed farms are contractors while they represent only 25% of farms on average. It also shows an overcoverage of some types of farming systems, namely breeding systems (82%) and dairy systems (38%) . This makes a high proportion of grassland in total farmland (52%).

The questionnaire addresses issues concerning farmers' socio-demographic and cultural characteristics, their farm, their professional environment, their perceptions on agroenvironmental contract and their income. The sample is quite representative although contracting farmers are over represented on purpose in order to get better information on contracts.

III.1) The empirical methodology

Agro-environmental contracts (AEC) involve various compliance costs we can present as follows. Fixed compliance costs do not vary with the enrolled area. Fixed transaction costs are information gathering before contracting, contract writing, or bureaucratic costs for the contract follow-up. Variable compliance costs are varying with the enrolled area. Variable transaction costs are maladaptation costs and the costs of contract renegotiations. Indeed, we assume that the more the farmer has hectares under AEC, the more hazards may appear and lead him to be trapped in contracts commitments.

From this observation, taking back our theoretical model, fixed compliance costs encompasses fixed costs of production and fixed transaction costs, whereas the variable compliance cost function gathers variable production costs and variable transaction costs. Equations (?) give the new specification, where Z

T T T T R T R Z v p T Z v p Y Z Z v p F z v p t z v p y z z v p z z p + = + = - π π π π π π π
It is important to remind that farmer's characteristics, Z U , determine neo-classical preferences, variables describing the technology, π Z , determine the profit according to the duality theory, and, variables describing the transaction environment, Z T , determine transaction costs according to the transaction cost theory. Some of these determinants being common to the utility function, the profit function and the transaction cost function, we clarify these relationships in table 2 in order to simplify our estimation interpretations. This table is based on the previous work of [START_REF] Ducos | The asset specificity issue in the private provision of environmental services: Evidence from agro-environmental contracts[END_REF] analyzing determinants of the choice over asset specificity level in agro-environmental contracts. It provides determinants expected effects on each component of farmers' WTA, namely the production cost variation, the transaction cost variation and the WTP, other things being equal. Among these five determinants, three have an impact on the transaction cost function only and two impact on other components of the WTA too. Before providing propositions about the relationship between these determinants and the choice over contracting and the area to subscribe, some assumptions are to be set. First, agro-environmental contracts are centrally designed, i.e. the European commission has specified agro-environmental contract provisions (duration, flexibility1 , eligibility and compensation payment calculation). Consequently, contract design is assumed to be the same whatever the EU state. Second, we assume agroenvironmental contracts involve non negligible investments and their magnitude is similar across contractors. Third, we assume agro-environmental contract involve specific investments, such investments being non redeployable without sacrifice of productive value [START_REF] Ducos | The asset specificity issue in the private provision of environmental services: Evidence from agro-environmental contracts[END_REF]. More precisely, we assume contracting farmers got engaged in the same specificity level of investments. This is a reasonable assumption since they subscribed for the least specific suggested environmental friendly practices such as "extensive management of grassland, "extensive management of arable land" or "landscape maintenance". We thus consider the asset specificity level does not vary from a contractor to an other.

In other respects, in line with the 1999-CAP regulation, the implemented compensation payment is a per-unit payment based on average operational costs and income foregone in each region and do not include transaction costs. We therefore expect transaction costs originating from involved investment specificity to affect agro-environmental contract uptake rate, ceteris paribus.

From this last observation, we set the following testable propositions2 on the relationship between transaction characteristics and farmers' choice to enroll an agro-environmental contract: Proposition 1: The more the farmer trusts in the State, the lower the transaction cost variation, the lower his WTA and the higher the probability he enrolls, ceteris paribus. We argue the same effects for less uncertainty and less bounded rationality.

Proposition 2: The more the agro-environmental transaction is similar to ones the farmer is already engaged, the lower the transaction cost variation and the profit loss. The probability to enroll is thus higher and because of the compensation payment calculation, we may argue this is due to transaction cost effect, ceteris paribus. Proposition 3: The more utility the farmer gets from environmental services he produces through the agro-environmental contract, the lower the transaction cost variation and the greater his WTP. The probability to enroll is thus higher but it is not possible to distinguish both effects, ceteris paribus.

By introducing the prospect of estimating the structural relationships underlying the adoption decision, the application of the two-stage Heckman method stands to shed light on the issue of whether and to what extent variations in fixed costs (fixed transaction costs) are responsible for observed adoption rate.

The econometric specification is based on the optimal area the farmer would contract if he were obliged to contract: this is the notional area v n where the marginal WTA equals the per hectare premium (). It does not depend on fixed costs. So contracting this area may entail a utility loss under a threshold area because of fixed costs. Or this optimal area is negative if the premium do not reach the marginal WTA for any positive contracted area. In fact, the farmer refuses the contract in these both cases.

) , , , ( ) , , , (

T n n T n m z z p v v z z v p π π ρ ρ ρ = ⇒ =
The first step of the Heckman method analyses the decision to contract or not, using a probit model. It is based on the probability that the premium exceeds the minimal average WTA, which depends on fixed costs according to ( 14), or 0 ρ in the absence of fixed costs.

Alternatively this is the probability that the notional area v n exceeds v* or zero. The latent variable of the probit model is the difference b between v n and v*. This first step enable the calculation of the inverse Mills ratio to be used in the second step.

) , , , ( *

T n Z Z p b b v v b π ρ = ⇒ - =
The second step analyses the contracted area with a least square regression. The contracted area is the optimal area, knowing that the contract is accepted. Hence the regression is augmented with the inverse Mills ratio to take into account the outcomes of the first step.

Accordingly, the econometric specification assumes that the pair (v n ,b) has a bivariate normal distribution: The lowest payments per hectare are in France, Netherlands, Germany, Ireland, Finland, and Czech Republic. The highest are in Belgium, England, Italy Emilia Romania and Italy Veneto.

2 1 2 2 2 1 . . ) , ( ) ( ' ) / ( ) ( ' ) / ( σ σ θ σ β σ α = = = = = b v Cov b Var X X b E v Var x x v E n n n

IV) RESULTS

This section provides estimation results. Probit estimates allow to distinguish factors of transaction cost variation from those of the profit ones. Heckman estimates allow to distinguish factors of fixed transaction cost from those of variable ones;

IV.1) Probit estimations

The reference choice is not to enroll. The reference farmer is high agricultural education, intermediate general education, no experience in previous AEC, no children under 6 yearsold, his production system is conventional and dairy farming oriented, and, for other variables, average values of continuous explanatory variables have been taken. Country specificities, including characteristics of sampling in each case study, are taken into account through country dummies, which are introduced as control variables. France is the reference. The different parameter estimations are gathered in table 7. Significant variables are presented only. The model has kept every observations. Looking at variable signs, we observe seven expected factors, namely trust factors ("to trust in the implementation process of AEC", "to trust in institutions"), bounded rationality factor ("to have a high general education"), similarity factors ("environmental awareness", "to have children under 6 years-old"), and similarity factors ("organic production system", "grassland share"). Trust and bounded rationality factors thus support the proposition 1 whereas similarity and utility factors support propositions 2 and 3. Non expected variables are "to have no or a low general education" and "Free time dedicated to nature related hobbies.

Instead of looking at estimation coefficient magnitude, we observe marginal effects on the probability to enrol an AEC in table 6. They are calculated for the reference farmer whose probability to participate is 0.41. Only significant effects are indicated.

IV.2) Heckman estimations

Table 7 provides the estimation with the Heckman procedure. The inverse Mills ratio, λ , is calculated in the first step and used in the second one to correct data. This specific parameter accounts for differences between participants and non-participants that are captured by the error term. λ being non-significantly different from zero, it may be considered that omitted factors do not account for differences between the two categories. In other words, explanatory variables are enough to discriminate farmers and to estimate the share of area entering AECs. Three additional explanatory variables appear and describe the similarity of transactions, namely "to have a dairy production system", "farm land area", and "square farm land share". An other one appear and is a control variable "changes in the production system in the last five years". On the other hand, variables describing trust and bounded rationality disappeared and a utility variable ("to have children under six years-old") too. From this observation, and concentrating on trust and bounded rationality variables, which impact on transaction cost only, we can argue the presence of fixed transaction costs as a barrier of contracting AEC.

Two variables describing the similarity of transaction ("production system type", "grassland share") and the utility in the transaction ("environmental awareness") positively influence the relative area under AEC. They support the presence of variable costs but, since they impact on the profit loss and the WTP respectively, it is not possible to disentangle their effect on transaction costs. An additional explanatory variable ("changes in the production system in the last five years") whose effect is positive is also considered. As expected, farmers who made significant changes over the last five years enter a larger area than the others.

In addition, there is a significant effect of farm size on the contracted surface. This effect has been modelled using two components, a linear one and a quadratic one. Based on the estimation of the coefficient of the surface and square surface, a threshold (around 2 600 ha) has been calculated. There is a negative effect under the threshold and a positive one over, this threshold.

Finally there is a specific country effect which is mainly due to the difference of Schemes proposed to farmers.

V) CONCLUSION

Estimation results support the presence of fixed transaction costs among fixed compliance costs and their significant negative effect on participation. The policy implications of this case study are straight forward. In order to increase farmers' adoption, the State seems to have two possibilities, whether to compensate farmers' transaction costs with a higher subsidy, or to reduce farmers' transaction costs. The first option requires an evaluation of the magnitude of transaction costs borne by the farmer, which is not an easy operation. The second option is to act on transaction cost significant determinants, namely trust and bounded rationality. For instance, knowing that farmers' trust in the implementation process has a major role, the State may work on the clarity of contract requirements so as to narrow its implementation interpretation spectrum. Trust may also be restored by balancing the State and the farmers' rights when a case is brought to private negotiation or to court. Finally, it is important to remind that this study focuses on transaction costs borne by the farmer only, whereas the agro-environmental transaction generates transaction costs borne by the State too. Consequently, reducing farmers' transaction costs may increase the State's one. This relation remains unknown and provides interesting future researches.

ANNEX : Measure of the WTA determinants

Determinants of transaction costs, which we identified and defined in the previous section, are notions that do not allow any direct measurement. When available, previous studies characterizing conditions in which these determinants may increase or decrease helped us to build a measurement framework.

Explanatory variables measuring "Trust"

Trust is an expectation held by an agent that its trading partner will behave in a mutually beneficial manner [START_REF] Sako | Determinants of trust in supplier relations: evidence from the automotive industry in Japan and the United States[END_REF].

Conditions favouring trust [START_REF] Sako | Determinants of trust in supplier relations: evidence from the automotive industry in Japan and the United States[END_REF] :

-"Long-term trading and future expectations of customer commitment": (i) the longer the informal commitment made by the customer to continue trading with the supplier, the higher is the supplier's trust for its customer; (ii) the longer the duration of past trading, the higher is the supplier's trust of its customer. -"Reciprocity in information exchange": the more a supplier's disclosure of information to its customer is matched by the customer's provision of information to the supplier, the higher the supplier's trust for its customer. -"Technical assistance": the more technical assistance is provided by the customer, the higher the supplier's trust in the customer

TRUST IMPL

This variable encompasses farmers' opinions (strongly disagree; somewhat disagree; somewhat agree; strongly agree; do not know) on the following statements:

-"The eligibility rules are fair" -"The financial compensation is sufficient to cover the extra costs incurred by the farmer" -"Compensation payments are always made on time" -"The rules and requirements are easy to understand" -"The measures can easily be implemented on my farm" -"The intended environmental benefits are clear and easy to understand" -"The current policy rules and regulations will remain constant over a longer period" -"There is a lot of control when implementing measures" -"The sanctions for not carrying out the contract are reasonable" These statements were chosen because they tend to describe farmers' expectation that the State will behave in a mutually beneficial manner. The global variable TRUST IMPL was created from these primary data using a Multiple Correspondence Analysis. A positive TRUST IMPL is assumed to indicate the farmer trusts in the State. The higher it gets, the more the farmer trusts in the State.

TRUST INST

In addition of TRUST IMPL's statements, this variable encompasses:

-"The local government can be trusted" -"Our government can be trusted for their commitments" -"The agricultural administration can be trusted" -"The European Union can be trusted" -"Generally speaking, other farmers' can be trusted" -"Generally speaking, most people can be trusted" As for TRUST IMPL, a positive TRUST INST is assumed to indicate the farmer trusts in the State. The higher it gets, the more the farmer trusts in the State.

A positive ORGA indicates the farmer's perception about future conditions is rather clear and thus shows low uncertainty. The higher it gets, the less uncertain the farmer perceives future conditions.

ADVICES

This variable gathers information on organizations or persons farmers get technical, financial or administrative advices from. It was created using a Multiple Correspondence Analysis. ADVICES follows the same logic as ORGA, namely a positive ADVICES indicates the farmer beneficiate from conditions lowering uncertainty about future conditions. The higher it gets, the less uncertain the farmer perceives future conditions.

Explanatory variables measuring the "Similarity of Transactions" Except from the production system type which is a qualitative variable, every variables describing the similarity of transactions are continuous. As [START_REF] Masten | The Costs of Organization[END_REF], the similarity of transactions is measured by comparing the initial technology with the technology required by enrolled environmental friendly practices. Therefore, from what we know about farm technology dedicated to the production of environmental services, the similarity of transactions is characterized as follows:

the higher the grassland share, the more similar the transactions; -the higher the farms land area, the more similar the transactions; -the lower the arable land share, the more similar the transactions; -the more labour force, the more similar the transactions; -the bigger the animal population, the less similar the transactions; -the bigger the milk quota, the less similar the transactions; -An organic production system indicates similar transactions.

Explanatory variables measuring " Utility" Conditions favouring utility in the transaction:

We argue any form of interest for environmental or nature issues is an indicator of farmers' utility in environmental services he may produce through an AEC.

ENV AW This variable contains information on farmers' hobbies, readings and involvement in environmental associations. It was created using a Multiple Correspondence Analysis. We assumed a positive SENSI indicates farmers' environmental awareness and thus derive utility from environmental services. The higher SENSI gets, the more the farmer derives utility from environmental services.

We follow the same logic for the variable "free time dedicated in nature related hobbies". As regard to children, which gives the number and age of farmers's children, we assumed a farmer with young children will be more aware of environmental issues than farmers with no child.

  ranking variables according to their estimated effect on participation. The most influential factors are the type of farming, past experience in AEC and farmer's trust in the implementation process.

Table 1 : Overview of the origin population Sub-samples Farmer population Participation rate Farm production system

 1

	France	34000	7%	Predominance of dairy farms
	Netherlands	6000	11%	Predominance of dairy farms
	Belgium	20667	15%	Mixed farming
	England	5690	33%	Mainly cattle and sheep farms
	Germany	1054	10%	?
	Italy Em. Rom.	34919	40%	?
	Italy Ven.	?	2.5%	Arable farms and dairy farms
	Ireland	13233	26%	Mainly cattle farming
	Finland	4600	15%	Mainly grain crops
	Czech Republic 46400	26.5%	Combined production

Table 2 : Synthesis of the WTA determinant expected effects

 2

	Determinants of WTA	Y Δ or y Δ	T Δ or t Δ	WTP
	Trust		-
	Uncertainty		+
	Bounded rationality		+
	Utility		-	+
	Technology/ Similarity -	-
	-means a reduced variation
	+ means an increased variation

Table 4 : Average annual payment per hectare of contracted area

 4

		France	Netherlands	Belgium	England	Germany	Ireland	Finland	Czech Republic	Italy_ Em. R.	Italy_ Veneto
	Mean (€/ha)	119.7	168.4	455.6	389.1	163.4	164.7	124.1	90.8	317.4	334.4
	Median (€/ha)	109.4	137.1	180.9	76.9	129.9	169.2	105.8	90.9	277.8	183.6

Table 5 . Model of participation (pooled sample with country dummies)

 5

	PARAMETERS (Mac Fadden R² = 30.01%) Estimation	Standard error	Pr > Khi 2
	Intercept		-0.7666	0.1268	<.0001
	TRUST
	TRUST IMPL		0.6756	0.0393	<.0001
	TRUST INST		0.1802	0.0376	<.0001
	BOUNDED RATIO
	GEN EDUC	Non response	-0.1714	0.1508	0.2557
		None	0.3177	0.1351	0.0187
		Low	0.0872	0.0830	0.2935
		High	0.1683	0.1133	0.1375
	UTILITY
	NAT HOBBIES		-0.1735	0.0432	<.0001
	ENV AW		0.2423	0.0471	<.0001
	CHILDREN		0.3659	0.0892	<.0001
	SIMILARITY
	PROD	Non response	1.1070	0.5030	0.0278
		Organic	0.8475	0.1389	<.0001
	GRASSLAND		0.8191	0.1123	<.0001
	CONTROL VARIABLES
	EXPERIENCE		0.9820	0.0754	<.0001
	COUNTRY	Netherlands	-0.2252	0.1766	0.2023
		Belgium	0.4276	0.1297	0.0010
		United Kingdom	0.2569	0.1332	0.0538
		Germany	-0.0672	0.1508	0.6560
		Ireland	-0.6353	0.1442	<.0001
		Finland	-0.9524	0.1734	<.0001
		Czech Republic	0.0769	0.1404	0.5837
		Italy E. Romagna	-0.5395	0.1674	0.0013
		Italy Veneto	0.2729	0.1604	0.0888

Table 6 : Marginal effects (pooled sample with country dummies)

 6

	Variable	Marginal effect
	TRUST
	TRUST IMPL	0.25
	TRUST INST	0.07
	BOUNDED RATIONALITY
	GEN EDUC	0.12
	UTILITY
	CHILDREN	0.14
	NAT HOBBIES	-0.06
	ENV AW	0.09
	SIMILARITY
	PROD	0.33
	GRASSLAND	0.31
	CONTROL VARIABLES
	EXPERIENCE	0.37
	BE	0.17

Table 7 : Estimation of the proportion of Farmland which is enrolled (Heckman model)

 7

	PARAMETERS		Estimate	Std Error	Pr > |t|
	Intercept		-.8727	0.1674	<.0001
	λ		-.0439	0.1041	0.6736
	UTILITY
	ENV AW	Environmental	0.0807	0.0344	0.0192
		awareness
	SIMILARITY
	PROD	Non response	0.2143	0.4045	0.5964
		Organic	0.2723	0.1192	0.0225
	GRASSLAND	Share of grassland	0.8977	0.1316	<.0001
	DAIRY	No	0.2394	0.0940	0.0110
	UAA	Surface	-.00164	0.0004	<.0001
	UAA²	Square-surface	0.0001	0.0001	<.0001
	CONTROL VARIABLES
	CHANGE	Past change	0.0856	0.0457	0.0616
	COUNTRY	Netherlands	-.9754	0.1404	<.0001
		Belgium	-.9450	0.1274	<.0001
		United Kingdom	-.3397	0.1659	0.0408
		Germany	0.0818	0.1913	0.6690
		Ireland	0.3155	0.3239	0.3302
		Finland	0.5412	0.2114	0.0106
		Czech Republic	-.1289	0.1401	0.3576
		Italy E. Romagna	0.2997	0.1833	0.1024
		Italy Veneto	-.3639	0.1706	0.0332
	R² = 23.17%. Number of observations= 1996. Number of participants = 981

A contract is said to be flexible when some terms are left open to ex post renegotiation.

These propositions are derived from[START_REF] Ducos | The asset specificity issue in the private provision of environmental services: Evidence from agro-environmental contracts[END_REF].

See the annex for a more detailed description of how explanatory variables measure determinants of the WTA.

This document presents results obtained within the EU project SSPE-CT-2003-502070 on Integrated tools to design and implement Agro Environmental Schemes (http//:merlin.lusignan.inra.fr/ITAES). It does not necessary reflect the view of the European Union and in no way anticipates the commission's future policy in this area. This research also benefited from a grant under the S3E program (Ministère de l'Ecologie, du Développement et de l'Aménagement Durable).

We use the same linear specification, with the same explanatory variables, to model the expected optimal area. If there were no fixed costs both steps would bring more or less the same results, scaled by the estimated parameter of the inverse Mills ratio. With fixed costs, the results will diverge if the determinants of fixed costs are different from the determinants of the marginal WTA. These determinants will more affect the first step than the second, possibly in opposite directions.

III.2) Explanatory variables

Several types of variables were collected so as to capture the notion of the determinants previously described. They describe the farmer (education level, environmental awareness…), his production system (farm legal status, number of Full Time Equivalent workers…), his professional environment (involvement in agricultural organizations, administrative and technical external services,…) and his relationship with the State (trust in administrations, …). From these raw data, we created variables providing a measure of the determinants of the WTA as presented in table 3.

GOODWILL

Similarly, GOODWILL gathers the following statements:

-"The procedures for contract applications are easy" -"The rules and requirements are easy to understand" -"It is easy to find the right person to contact in the administration when there are problems" -"Regarding agro-environmental schemes, administration behaviour is fair and responsible" A positive GOODWILL indicates the farmer trusts in the State. The higher it gets, the more the farmer trusts in the State.

Explanatory variables measuring "Bounded Rationality" Same definition as [START_REF] Williamson | The Economic Institution of Capitalism-Firms, Markets and Relational Contracts[END_REF]. Bounded rationality implies "economic agents do not know all the solutions to the problems they face, are unable to calculate the possible outcomes of these solutions, and cannot perfectly arrange these outcomes in order in their space of preferences. With regard to contracts, this means that they are unable to design the optimal solutions (behavioral rules) taking into account every relevant contingency without high, and sometimes prohibitive, costs and delays" [START_REF] Brousseau | Incomplete Contracts and Governance Structures: Are incomplete Contract Theory and New-Institutional Economics Substitutes or Complements?[END_REF]. Conditions favouring bounded rationality: We argue initial education and any form of improved knowledge reduce each individual rationality.

GEN EDUC + AGRI EDUC

Bounded rationality is characterized with qualitative variables by creating classes of variables. Variables describing farmers' education were assumed to measure their rationality since education is expected to provide solutions to problems and unable farmers to calculate the possible outcomes to these solutions (cf. section II.3.1). The higher the education level, the less bounded the farmer's rationality. Explanatory variables measuring "Uncertainty" Conditions favouring uncertainty [START_REF] Mahul | Les outils de gestion des risques de marché, étude commandité par le Ministère de l[END_REF]: The agricultural production is subject to various risk sources. A classification of those risks is provided in numerous studies (see for instance USDA 1999) and is provided bellow:

-"Production risks" from climatic uncertainties (drought, flooding,…), technological uncertainties, sanitary conditions and epidemics. -"Market or price risks" characterizes price production factor changes.

-"Institutional risk" created by policy or regulation changes.

-"Financial risks" includes interest rate variation uncertainties or those of exchange rate, non-payment risk,… -"Human and personal risks" are common to any individual. They are related to illness, accidents,…

ORGA

This variable reports farmers' membership and participation frequency in agricultural organizations. It was created using a Multiple Correspondence Analysis. We assumed a farmer regularly meeting other farmers and people from the profession or involved in professional organizations better feel what is going on and what are policy orientations. This refers to the volatility aspect of uncertainty.


M A Bchir

M Willinger Imebe

Provision of club goods in the lab

Model

Experimental design

• Baseline :

-Choice between private account and collective -Step level mechanism : the group has to reach a provision point in order to provide the public good. Unless tokens are lost.

• Exclusion :

-

  The individual has to contribute in order to benefit from the collective account.-No minimum level of contribution required Parameters • Endowment = 20 tokens • Three levels of threshold tested :