R De La

Bret Èche

K Destagnol
email: kevin.destagnol@ist.ac.at

Jianya Liu
email: jyliu@sdu.edu.cn

J Wu

Yongqiang Zhao
email: yzhao@wias.org.cn

ON A CERTAIN NON-SPLIT CUBIC SURFACE

In this note, we establish an asymptotic formula with a power-saving error term for the number of rational points of bounded height on the singular cubic surface of P 3 Q

Introduction and results

Let V ⊂ P 3 Q be the cubic surface defined by x 0 (x 2 1 + x 2 2 ) -x 3 3 = 0. The surface V has three singular points ξ 1 = [1 : 0 : 0 : 0], ξ 2 = [0 : 1 : i : 0] and ξ 3 = [0 : 1 : -i : 0]. It is easy to see that the only three lines contained in V Q = V × Spec(Q) Spec(Q) are ℓ 1 := {x 3 = x 1 -ix 2 = 0}, ℓ 2 := {x 3 = x 1 + ix 2 = 0}, and ℓ 3 := {x 3 = x 0 = 0}. Clearly both ℓ 1 and ℓ 2 pass through ξ 1 , which is actually the only rational point lying on these two lines.

Let U = V {ℓ 1 ∪ ℓ 2 ∪ ℓ 3 }, and B a parameter that can approach infinity. In this note we are concerned with the behavior of the counting function N U (B) = #{x ∈ U(Q) : H(x) B}, where H is the anticanonical height function on V defined by

H(x) := max |x 0 |, x 2 1 + x 2 2 , |x 3 | (1.1)
where each x j ∈ Z and gcd(x 0 , x 1 , x 2 , x 3 ) = 1. The main result of this note is the following. Remark. If follows from the arguments in [START_REF] De La Bretèche | Sur le nombre de points de hauteur bornée d'une certaine surface cubique singulière[END_REF] or [START_REF] Liu | Manin's conjecture for a class of singular cubic hypersurfaces[END_REF] that, at least, any ϑ < 1 9 is acceptable in Theorem 1.1, and further improvements are possible.

The Manin-Peyre conjectures for smooth toric varieties were established by Batyrev and Tschinkel in their seminal work [START_REF] Batyrev | Manin's conjecture for toric varieties[END_REF]. Since our cubic surface V is a (non-split) toric surface, the main term of the asymptotic formula (1.2) can be derived from [START_REF] Batyrev | Manin's conjecture for toric varieties[END_REF]. In addition to providing a different proof of the Manin-Peyre's conjectures for V and to getting a power-saving error term of the counting function N U (B), this note also serves to complement the results in [START_REF] Liu | Manin's conjecture for a class of singular cubic hypersurfaces[END_REF], in which Manin's conjecture for the cubic hypersurfaces S n ⊂ P n+1 defined by the equation

x 3 0 = (x 2 1 + . . . + x 2 n )
x n+1 with n = 4k was established. The cubic surface V is the case for n = 2.

We conclude the introduction by a brief discussion of the split toric surface of P 3 Q given by V ′ : x 0 x 1 x 2 = x 3 3 . The variety V ′ is isomorphic to V over Q(i) and was well studied by a number of authors. Manin's conjecture for V ′ is a consequence of Batyrev and Tschinkel [START_REF] Batyrev | Manin's conjecture for toric varieties[END_REF]. Others include the first author [START_REF] De La Bretèche | Sur le nombre de points de hauteur bornée d'une certaine surface cubique singulière[END_REF], the first author and Swinnerton-Dyer [START_REF] De La Bretèche | Fonction zêta des hauteurs associée à une certaine surface cubique[END_REF], Fouvry [START_REF] Fouvry | Sur la hauteur des points d'une certaine surface cubique singulière[END_REF], Heath-Brown and Moroz [START_REF] Heath-Brown | The density of rational points on the cubic surface X 3 = X 1 X 2 X 3[END_REF] and Salberger [START_REF] Salberger | Tamagawa measures on universal torsors and points of bounded height on Fano varieties[END_REF]. Derenthal and Janda [START_REF] Derenthal | Gaussian rational points on a singular cubic surface[END_REF] established Manin's conjecture for V ′ over imaginary quadratic fields of class number one and Frei [START_REF] Frei | Counting rational points over number fields on a singular cubic surface[END_REF] further generalized their work to arbitrary number fields. Of the unconditional asymptotic formulae obtained, the strongest is the one in [START_REF] De La Bretèche | Sur le nombre de points de hauteur bornée d'une certaine surface cubique singulière[END_REF], which yields the estimate

N U (B) = BP (log B) + O B 7/8 exp(-c(log B) 3/5 (log log B) -1/5 ) ,
where U is a Zariski open subset of V ′ , and P is a polynomial of degree 6 and c is a positive constant. In [START_REF] De La Bretèche | Fonction zêta des hauteurs associée à une certaine surface cubique[END_REF], even the second term of the counting function N U (B) is established under the Riemann Hypothesis as well as the assumption that all the zeros of the Riemann ζ-function are simple.

Geometry and Peyre's constant

In [START_REF] Peyre | Hauteurs et nombres de Tamagawa sur les variétés de Fano[END_REF], Peyre proposed a general conjecture about the shape of the leading constant arising in the asymptotic formula for the number of points of bounded height but only for smooth Fano varieties.

The surface V that we study in this note is singular so we can not apply directly this conjecture and [28, Définition 2.1]. To get around this, we construct explicitly in this section a minimal resolution π : V → V of V and show that for U = V {ℓ 1 ∪ ℓ 2 ∪ ℓ 3 } and U = π -1 (U), we have π | U : U ∼ = U. This implies that our counting problem on V can be seen as a counting problem on the smooth variety V since

N U (B) = #{x ∈ U (Q) : H • π(x) B}
where H • π is an anticanonical height function on V . Indeed, by [START_REF] Coray | Arithmetic on singular del Pezzo surfaces[END_REF]Lemma 1.1] the surface V has only du Val singularities which are canonical singularities (see [START_REF] Kollár | Birational geometry of algebraic varieties[END_REF]Theorem 4.20]) and alluding to [25, 2.26, 4.3, 4.4 and 4.5], we can conclude that π * K V = K V where K V and K V denote the anticanonical divisors of V and V respectively. However, V is not a Fano variety and therefore we still can not apply [START_REF] Peyre | Hauteurs et nombres de Tamagawa sur les variétés de Fano[END_REF]Définition 2.1].

We nevertheless establish in this section that V is "almost Fano" in the sense of [START_REF] Peyre | Points de hauteur bornée, topologie adélique et mesures de Tamagawa, Les XXII-èmes Journées Arithmétiques[END_REF]Definition 3.1]. Alluding to the fact that the original conjecture of Peyre has been refined by Batyrev and Tscinkel [START_REF] Batyrev | Tamagawa numbers of polarized algebraic varieties[END_REF] and Peyre [START_REF] Peyre | Points de hauteur bornée, topologie adélique et mesures de Tamagawa, Les XXII-èmes Journées Arithmétiques[END_REF] to this setting, we may refer to [29, Formule empirique 5.1] to interpret the constant C arising in our Theorem 1.1. According to [29, Formule empirique 5.1], the leading constant C in our Theorem 1.1 takes the form

C = α( V )β( V )τ ( V ) (2.1)
where α( V ) is a rational number defined in terms of the cone of effective divisors, β( V ) a cohomological invariant and τ ( V ) a Tamagawa number. For more details, see définition 4.8 of [START_REF] Peyre | Points de hauteur bornée, topologie adélique et mesures de Tamagawa, Les XXII-èmes Journées Arithmétiques[END_REF].

Our main strategy to check that the constant C in Theorem 1.1 agrees with the prediction [29, Formule empirique 5.1] relies in a crucial way on the (non-split) toric structure of the surface V and on results from [START_REF] Batyrev | Tamagawa numbers of polarized algebraic varieties[END_REF].

2.1. Minimal resolution of V and interpretation of the power of log B. We refer the reader to the following references for details about toric varieties over arbitrary fields [START_REF] Oda | Convex bodies and algebraic geometry[END_REF][START_REF] Fulton | Introduction to toric varieties[END_REF][START_REF] Demazure | Sous-groupes algébriques de rang maximum du groupe de Cremona[END_REF][START_REF] Danilov | The geometry of toric varieties[END_REF] and especially [START_REF] Batyrev | Rational points of bounded height on compactifications of anisotropic tori[END_REF][START_REF] Batyrev | Manin's conjecture for toric varieties[END_REF] and [START_REF] Salberger | Tamagawa measures on universal torsors and points of bounded height on Fano varieties[END_REF]End of §8].

The toric surface V is easily seen to be an equivariant compactification of the non-split torus T given by the equation

x 0 (x 2 1 + x 2 2 ) = 1. The torus T is isomorphic to R Q(i)/Q (G m ) where R Q(i)/Q (•)

denotes the Weil restriction functor and is split by the quadratic extension

k = Q(i). We now introduce M = T k := Hom(T, k × ) the group of regular k-rational characters of T and N = Hom(M, Z). Alluding to [34, Lemma 1.3.1], we see that M ∼ = N ∼ = Z × Z with the Galois group G = Gal(k/Q) ∼ = Z/2Z
interchanging the two factors. Let (e 1 , e 2 ) be a Z-basis of N. In a similar manner as in [START_REF] Salberger | Tamagawa measures on universal torsors and points of bounded height on Fano varieties[END_REF]Example 11.50], we denote by ∆ the fan of N R = N ⊗ R given by the rays ρ 1 , ρ 2 , ρ ′ 2 generated by -e 1 -e 2 , -e 1 + 2e 2 and 2e 1 -e 2 .

ρ 1 ρ 2 ρ ′ 2 Figure 1. The fan ∆.
The fan ∆ is G-invariant in the sense of [START_REF] De La Bretèche | Sur le nombre de points de hauteur bornée d'une certaine surface cubique singulière[END_REF]Definition 1.11] and hence defines a non-split toric surface P ∆ over Q. Using the same arguments as in [START_REF] Salberger | Tamagawa measures on universal torsors and points of bounded height on Fano varieties[END_REF]Example 11.50], one easily sees that the k-variety P ∆,k = P ∆ ⊗ Spec(Q) Spec(k) is given by the equation

x 3 3 = x 0 z 1 z 2 with G exchanging z 1 and z 2 . The change of variables x 1 = (z 1 + z 2 )/2 and x 2 = (z 1 -z 2 )/(2i) yields that P ∆,k is isomorphic to the variety of equa- tion x 3 3 = x 0 (x 2 1 + x 2 2
), all the variables being G-invariant. Hence, the surface V is a complete algebraic variety such that V ⊗ Spec(Q) Spec(k) is isomorphic to P ∆,k , the isomorphism being compatible with the Gactions. Then, theorem 1.12 of [START_REF] Batyrev | Manin's conjecture for toric varieties[END_REF] allows us to conclude that V is given by the G-invariant fan ∆ after noting that the assumption that the fan is regular is not necessary.

The fan ∆ is not complete and regular in the sense of [2, Definition 1.9] which accounts for the fact that V is singular. As in [START_REF] Salberger | Tamagawa measures on universal torsors and points of bounded height on Fano varieties[END_REF] The toric surface V defined over Q by the G-invariant fan ∆ is then smooth by [1, Theorems 1.10 and 1.12] and, thanks to [12, 5.5.1] and [20, §2.6], comes with a proper equivariant birational morphism π : V → V which is an isomorphism on the torus T . Here T corresponds to the open subset U = V {ℓ 1 ∪ℓ 2 ∪ℓ 3 }. Now the proof of the proposition 11.2.8 of [START_REF] Cox | Toric varieties[END_REF] yields that π is a crepant resolution and hence that it is minimal since we are in dimension 2.

We note that thanks to [13, Corollaire 3] and [1, Proposition 1.15], the minimal resolution V is "almost Fano" in the sense of [START_REF] Peyre | Points de hauteur bornée, topologie adélique et mesures de Tamagawa, Les XXII-èmes Journées Arithmétiques[END_REF]Definition 3.1].

Let us now turn to the computation of the Picard group of V . To this end, we will exploit the exact sequence given by [ ) is a Z-basis of M. Moreover, a function ϕ ∈ PL( ∆) G being completely determined by its integer values on ρ 1 , ρ 2 and ρ1 , ρ2 and ρ3 , we have PL( ∆) G ∼ = Z 5 . Finally, the Z-module M is a permutation module and therefore H 1 (G, M) is trivial. Bringing all of that together yields that rk(Pic( V )) = 5 -1 = 4, which agrees with the prediction coming from Manin's conjecture regarding the power of log B in Theorem 1.1.

2.2.

The factor α. We will use the same method as in [4, Lemma 5] to compute the nef cone volume α( V ) and we refer the reader to [4, Lemma 5] for more details and definitions.

Let T i , T ′ i , T i and T ′ i be the Zariski closures of the one dimensional tori corresponding respectively to the cones R 0 ρ i , R 0 ρ ′ i , R 0 ρi and R 0 ρ′ i . We also introduce the G-invariant divisors

D 1 = T 1 , D 2 = T 2 + T ′ 2 , D 3 = T 1 + T ′ 1 , D 4 = T 2 + T ′ 2 , D 5 = T 3 + T ′ 3 .
Using

D 1 + D 2 + D 3 + D 4 + D 5 ∼ 3D 1 + 2D 2 + D 3
is an anticanonical divisor for V . Following the strategy of [4, Lemma 5] and using the same notations than in [4, Lemma 5], it now follows that C ∨ eff is the subset of R 4 0 given by 2z 1 + z 2 -z 4 0 and that H V is given by the equation 3z 1 + 2z 2 + z 3 = 1. Therefore, a straightforward computation finally yields

α( V ) = 1 0 (1 -z 3 ) 2 dz 3 × 1 2 Vol (z 1 , z 4 ) ∈ R 2 0 : 3z 1 1, 2z 4 -z 1 1 = 1 6 1 3 z 1 =0 1+z 1 2 z 4 =0 dz 4 dz 1 = 7 216
.

2.3. The factor β. Let us now briefly justify that β( V ) = 1. We know that V is birational to the torus

R Q(i)/Q (G m ). But the open immersion G m,Q(i) ֒→ A 2 Q(i) gives rise to an open immersion R Q(i)/Q (G m ) ֒→ A 2
Q by taking the functor R Q(i)/Q (•) and by alluding to [33, Proposition 4.9]. Hence, R Q(i)/Q (G m ) is rational and so is V . Finally this implies that β( V ) = 1 (see [4, section 5] for details).

2.4. The Tamagawa number.

Conjectural expression.

Let us choose S = {∞, 2} and note that our definition will be independent of that choice. We have from [3, Theorem 1.3.2] that Pic( V Q ) is the free abelian group generated by the divisors

T 1 , T 2 , T ′ 2 , T 1 , T ′ 1 , T 2 , T ′ 2 defined in §2.2 with the following G-action σ(T 2 ) = T ′ 2 , σ( T i ) = T ′ i (i ∈ {1, 2})
if σ denotes the complex conjugation. Alluding to [24, Defintion 7.1], we have the following conjectural expression

τ ( V ) := lim s→1 + (s -1) 4 L S (s, χ Pic ( V Q ))ω ∞ p λ -1 p ω p where L S (s, χ Pic ( V Q )) = p ∈S det Id -p -s Frob p | Pic( V Q ) Ip -1
with I p the inertia group and Frob p a representative of the Frobenius automorphism and where

λ p = L p (1, χ Pic ( V Q )), ω ∞ = ω ∞, V ( V (R)), ω p = ω p, V ( V (Q p ))
for measures ω v, V on V (Q v ) whose proper definitions are postponed to the next section (they are the measures ω K ,v defined in [1, §2]) and where λ p is taken to be 1 for p ∈ S. Let ℜe(s) > 1. First we notice that for all p ∈ S, we have that I p is trivial since p is not ramified in Q(i). Then the Frobenius Frob p being trivial for all p ≡ 1 mod 4, it is easy to see that in that case

det Id -p -s Frob p | Pic( V Q ) Ip = 1 - 1 p s 7 .
When p ≡ 3 mod 4, Frob p is of order 2 with the same action than σ on Pic( V Q ) and hence one sees immediately that

det Id -p -s Frob p | Pic( V Q ) Ip = 1 - 1 p s 1 - 1 p 2s 3 .
Bringing all of this together yields that

L S (s, χ Pic ( V Q )) = p>2 1 - 1 p s -4 1 - χ(p) p s -3 and lim s→1 (s -1) 4 L S (s, χ Pic ( V Q )) = L(1, χ) 2 2 4 = 1 2 4 × π 4 3 and therefore τ ( V ) = π 4 3 ω ∞ × ω 2 2 4 × p>2 1 - 1 p 4 1 - χ(p) p 3 ω p .
2.4.2. Construction of the Tamagawa measure. Let us write here V (i) for the affine subset of V where x i = 0 with coordinates x (i) j = x j /x i for j = i. Note that V (i) is defined by the equation

f (i) (x (i) 0 , . . . , x (i) i , . . . , x (i) 3 ) = x 3 x i 3 - x 0 x i x 1 x i 2 + x 2 x i 2
where (x

(i) 0 , . . . , x (i) i , . . . , x (i)
3 ) denotes (x

(i) 0 , x (i) 2 , x (i)
3 ) after removing the i-th component. The same arguments as in [22, §13] go through to yield that ω V ∼ = O V (-1) and that such an isomorphism is given on V (i) by

x -1 i -→ (-1) i+t ∂f (i) /∂x (i) j dx (i) k ∧ dx (i) ℓ for k < ℓ ∈ {0, 1, 2, 3} {i}, {i, j, k, ℓ} = {0, 1, 2, 3} and t = k + ℓ if k < i < ℓ and t = k + ℓ -1 otherwise. Moreover, we have already seen that ω V ∼ = π * ω V . The dual sections τ i of s i in ω -1 V ∼ = O V (1)
define the embedding V ֒→ P 3 under consideration in this note and the morphism V → V ֒→ P 3 is given by the sections

π * τ i of H 0 ( V , ω -1 V ). Consider now the subsequent Arakelov heights (ω -1 V , (||.|| v ) v∈Val(Q) ) and (ω -1 V , (||.|| ′ v ) v∈Val(Q) )
defined by these global sections where for all

v ∈ Val(Q), x ∈ V (Q v ), y ∈ V (Q v ), τ ∈ ω -1
V and σ ∈ ω -1 V we use respectively the v-adic metrics defined by

||τ || v = min 0 i 3 π * τ i =0 τ π * τ i (x) v if v is finite and ||τ || ∞ = min    min i∈{0,3} π * τ i =0 τ π * τ i (x) ∞ , π * τ 1 (x) τ 2 ∞ + π * τ 2 (x) τ 2 ∞ -1 2    if v
is the archimedean place and τ = 0 and

||σ|| ′ v = min 0 i 3 τ i =0 σ τ i (y) v if v is finite and ||σ|| ′ ∞ = min    min i∈{0,3} τ i =0 σ τ i (y) ∞ , τ 1 (y) σ 2 ∞ + τ 2 (y) σ 2 ∞ -1 2    if v
is the archimedean place and σ = 0. These heights correspond to the heights H on V and H • π on V that we used in our counting problem. Applying the definition 2.2.1 of [START_REF] Peyre | Points de hauteur bornée, topologie adélique et mesures de Tamagawa, Les XXII-èmes Journées Arithmétiques[END_REF], we get a measure ω v, V on V (Q v ) associated to the v-adic metric ||.|| v which is the measure defined in [1, §2] and used in §2.4.1 and a measure ω v,V on V (Q v ) associated to the v-adic metric ||.|| ′ v . 2.4.3. Computation of the archimedean density. We follow once again the strategy adopted in [22, §13]. One sees easily that U = V (3) and the same argument as in [22, §13] shows that

ω ∞ = ω ∞, V ( V (R)) = ω ∞, V (π -1 (U)(R)).

Now the local coordinates x

(3) 1 -1 and x

(3) 2 at the rational point (1, 1, 0) of U give an isomorphism

U(R) ∼ = W = {(z 1 , z 2 ) ∈ R 2 : (z 1 + 1) 2 + z 2
2 = 0} and a similar computation as in [22, §13] yields

ω ∞, V (π -1 (U)(R)) = R 2 dz 1 dz 2 max {1, z 2 1 + z 2 2 , (z 2 1 + z 2 2 ) 3/2 } = z 2 1 +z 2 2 1 dz 1 dz 2 + z 2 1 +z 2 2 >1 dz 1 dz 2 (z 2 1 + z 2 2 ) 3/2 = 3π
after a polar change of coordinates. We can therefore conclude that ω ∞ = 3π. 2.4.4. Computation of ω p for odd p. Thanks to the remarks of [32, Page 187], one can construct a model V over Spec(Z) satisfying the conditions of [29, Notation 4.5] with S = {∞, 2}. Hence, one can consider the reduction V p modulo p of V for every prime number p.

The torus T has good reduction T p for every prime p > 2 since p is not ramified in Q(i) and T p is a split torus of rank 2 if p ≡ 1 mod 4 and a non-split torus of rank 2 split by F p 2 if p ≡ 3 mod 4. Hence, the reduction V p modulo p can be realized as the toric variety over F p under the torus T p given by the fan ∆ ′ which is invariant under Frob p . Since the fan stays regular and complete, we can conclude that V p is smooth and hence that V has good reduction modulo p > 2 (see [START_REF] Danilov | The geometry of toric varieties[END_REF]).

We can therefore apply [START_REF] Jahnel | Tamagawa and rational points on algebraic varieties[END_REF]Corollary 6.7] to obtain for all odd p the following expression

ω p = # V (F p ) p 2
. Now alluding to Weil's formula, we obtain

ω p = 1 + Tr(Frob p |Pic( V Q )) p + 1 p 2 = 1 + 4 + 3χ(p) p + 1 p 2
by using the description of the action of Frob p on Pic( V Q ) given in §2.4.1.

2.4.5. Computation of ω 2 . For p = 2, the model Ṽ having bad reduction, we appeal to the lemma 6.6 of [START_REF] Jahnel | Tamagawa and rational points on algebraic varieties[END_REF] to compute ω 2 . By [1, Proposition 2.10] and noting that the smooth assumption is not necessary, one gets

ω 2, V ( V (Q 2 )) = ω 2, V (π -1 (U)), ω 2,V (V (Q 2 )) = ω 2,V (U).
Now an analogous computation as the one in §2.4.3 yields that both quantities ω 2, V (π -1 (U)) and ω 2,V (U) are equal to the expression

W dz 1 dz 2 max {1, |z 2 1 + z 2 2 | 2 , |z 1 (z 2 1 + z 2 2 )| 2 , |z 2 (z 2 1 + z 2 2 )| 2 } with W = {(z 1 , z 2 ) ∈ Q 2 2 : (z 1 + 1) 2 + z 2 2 = 0}. Therefore, ω 2 is equal to ω 2,V (V (Q 2 )
) and [START_REF] Jahnel | Tamagawa and rational points on algebraic varieties[END_REF]Remark 6.8] implies that

ω 2 = lim n→+∞ N(2 n ) 2 3n
, where

N(2 n ) := # x (mod 2 n ) : x 0 (x 2 1 + x 2 2 ) ≡ x 3 3 (mod 2 n ) . Let v 2 (x 2 1 + x 2 2 ) = k and v 2 (x 0 ) = k 0 . If k = 1 + 2k ′ is odd and 1 + 2k ′ < n, then the number of (x 1 , x 2 ) satisfying v 2 (x 2 1 + x 2 2 ) = k is 2 2n-2k ′ -2 .
There are 2 n-(1+2k ′ +k 0 )/3-1 ways to choose x 3 and then 2 2k ′ +1 choices for x 0 . Then, in the case where v 2 (x 2 1 + x 2 2 ) is odd, the number of solutions is asymptotic to

2 3n 3|1+2k ′ +k ′ 0 k ′ 0 2 -2-(1+2k ′ +k ′ 0 )/3 ∼ 5 6 2 3n .
The number of (x 1 , x 2 ) satisfying v 2 (x 2 1 + x 2 2 ) = 2k ′ is, at least for 2k ′ < n, equal to 2 2n-2k ′ -1 . There are 2 n-(2k ′ +k 0 )/3-1 ways to choose x 3 and then 2 2k ′ choices for x 0 . Summing over 3 | 2k ′ + k 0 and k ′ 0 we get the contribution of the case v 2 (x 2 1 + x 2 2 ) even in N(2 n ), which is asymptotic to 7 6 • 2 3n . It follows that

ω 2 = 2 = 1 + 2 + 3χ(2) + 2χ 2 (2) 2 + χ 2 (2) 2 2 .
2.4.6. Conclusion. Bringing everything together yields the following expression for the Peyre constant

α( V )β( V )τ ( V ) = 7 216 (3π) π 4 3 τ.
This is in agreement with the constant C in (1.3).

Proof of Theorem 1.1

By symmetry, we have

N U (B) = # x ∈ E : x 0 (x 2 1 + x 2 2 ) = x 3 3 , max x 0 , x 2 1 + x 2 2 B ,
where

E := {x ∈ N × Z 2 × N : gcd(x 0 , x 1 , x 2 , x 3 ) = 1} and N = Z 1 .
As in [START_REF] De La Bretèche | Sur le nombre de points de hauteur bornée d'une certaine surface cubique singulière[END_REF], we parametrize x 2 1 + x 2 2 , x 0 and x 3 by

x 2 1 + x 2 2 = n 1 n 2 2 n 3 3 , x 0 = n 2 1 n 2 n 3 4 , x 3 = n 1 n 2 n 3 n 4 ,
where n 1 and n 2 are squarefree and gcd(n

1 , n 2 ) = 1 which is equivalent to µ 2 (n 1 n 2 ) = 1. It follows that N U (B) = 4 n∈N 4 µ 2 (n 1 n 2 )=1 n 2 1 n 2 n 3 4 B n 1 n 2 2 n 3 3 B 2 r(n 1 n 2 2 n 3 3 , n 1 n 2 n 4 )
where

r(n, m) := 1 4 # (x 1 , x 2 ) ∈ Z 2 : x 2 1 + x 2 2 = n, ((x 1 , x 2 ), m) = 1 .
Here, we remark that our choice of height function is particularly well suited to handle the expression r(n, m).

Let χ be the non-principal character modulo 4 and r 0 := 1 * χ. The quantity r(n, m) is a multiplicative arithmetic function in n, and we have r(n, m) := p r p vp(n) , p vp(m) .

We use the fact that, when ν 1,

r(p ν , p) =          2 if p ≡ 1 (mod 4), 0 if p ≡ 3 (mod 4), 1 if ν = 1, p = 2, 0 if ν 2, p = 2.
Proof. The proof is very similar to the one in [START_REF] Pieropan | Torsors and generalized Cox rings for Manin's conjecture[END_REF]Proposition 2.71] and that is why we will not repeat all the details here. Since ṼQ is a split toric variety, we know by [START_REF] Salberger | Tamagawa measures on universal torsors and points of bounded height on Fano varieties[END_REF] that a Cox ring of identity type for ṼQ is given by R = Q[t 1 , t 2 , t ′ 2 , t1 , t′ 1 , t2 , t′ 2 , t3 , t′ 3 ] where t i = div(T i ), t ′ i = div(T ′ i ), ti = div( Ti ) and t′ i = div( T ′ i ). We then have by [START_REF] Pieropan | Torsors and generalized Cox rings for Manin's conjecture[END_REF]Remark 2.51] that every Cox ring of injective type λ is isomorphic to the ring of invariant of

Theorem 1 . 1 . 1 p 2 and χ the non-principal character modulo 4 .

 11124 There exists a constant ϑ > 0 and a polynomial Q ∈ R[X] of degree 3 such thatN U (B) = BQ(log B) + O(B 1-ϑ ).(1.2)The leading coefficient C of Q satisfies The constant C agrees with Peyre's prediction [29, Formule 5.1].

2 Figure 2 .

 22 Figure 2. The fan ∆.

[ 1 ,

 1 Proposition 1.15], one immediately sees that Pic( V ) is generated by D 1 , D 2 , D 3 , D 4 , D 5 with the relation D 5 = 2D 1 + D 2 -D 4 and that the divisor

e 1 T 1 + e 2 T 2 + e ′ 2 3 = [a 1 D 1 + a 3 D 3 + 1 ẽ′ 2 + 3 ,η 3 4 . 1 = η 5 + η 5 2 , x 2 = η 5

 112223113312341225 m∈T R m where R m is the vector space generated by the degree m elements of R. For m ∈ T given by m = [a 1 D 1 + a 3 D 3 + a 4 D 4 ] , we have to solve the following linear system with e i , e ′ i , ẽi , ẽ′i 0 to determine R m a 4 D 4 ] .Alluding to the fan ∆ ′ and [1, Proposition 1.15], we get that this linear system is equivalent to  ẽ3 -ẽ′ 3 = ẽ2 + ẽ′ 3 -ẽ3 e 2 + ẽ′ 3 -2ẽ 3 = e ′ 2 + ẽ3 -2ẽ ′ 3 = 0. This easily yields that R is generated byη 1 = t 1 , η 2 = t1 t′ 1 , η 3 = t2 t′ 2 , η 4 = t 2 t ′and η 5 the conjugate of η 5 with the relationη 5 η 5 = η 2 η 2 3Using the Galois invariant variablesx -η 5 2i one finally ensures that every Cox ring of injective type λ is isomorphic to R.Foundation of China under Grant 11531008, the Ministry of Education of China under Grant IRT16R43, and the Taishan Scholar Project of Shandong Province. The hospitality and financial support of these institutions are gratefully acknowledged.

  , Example 11.50], there exists a complete and regular refinement ∆ of ∆ given by the extra rays ρ1 , ρ2 , ρ3 , ρ′ 1 , ρ′ 2 , ρ′ 3 generated by -e 1 , -e 1 + e 2 , e 2 , -e 2 , e 1 -e 2 and e 1 .

  1, Proposition 1.15]. With the notations of [1, Proposition 1.15], we have M G ∼ = Z generated by e

* 1 + e * 2 if (e * 1 , e * 2

Acknowledgements. The authors are grateful to thank Ulrich Derenthal and Marta Pieropan for useful discussions. This article was initiated during a workshop on number theory held at the Weihai Campus of Shandong University in July 2017. The first, third and fourth authors are partially supported by the program PRC 1457 -AuForDiP (CNRS-NSFC). The third author is supported by the National Science R. DE LA BRET ÈCHE, K. DESTAGNOL, J. LIU, J. WU & Y. ZHAO Jie Wu, CNRS, UMR 8050, Laboratoire d'Analyse et de Mathématiques Appliquées, Université Paris-Est Créteil, 61 Avenue du Général de Gaulle, 94010 Créteil cedex, France E-mail address: jie.wu@math.cnrs.fr

Then, when ν 1 + ν 2 1, the value of r(p ν 1 +2ν 2 +3ν 3 , p ν 1 +ν 2 +ν 4 ) is given by

, ν 2 , ν 3 , ν 4 ) = (0, 0, 0, ν 4 ), r 0 (p 3ν 3 ) if (ν 1 , ν 2 , ν 3 , ν 4 ) = (0, 0, ν 3 , 0), 0 if (ν 1 , ν 2 ) = (0, 0), min{ν 3 , ν 4 } 1, p ≡ 2, 3 (mod 4), 2 if (ν 1 , ν 2 ) = (0, 0), min{ν 3 , ν 4 } 1, p ≡ 1 (mod 4),

The Dirichlet series associated to this counting problem is

It can be written as an Euler product of F p (s 1 , s 2 ), where

and

Let s stand for the pair (s 1 , s 2 ). Then there exists G such that

The above quantity G(s) can be written as an Euler product of G p (s) where G 2 (1/3, 1/3) = 2 -3 while, for p ≡ 3 (mod 4)

and for p ≡ 1 (mod 4),

The series F is absolutely convergent when ℜe(s 1 ) > 1 3 and ℜe(s 2 ) > 1 3

and the function G can be analytically continued to ℜe(s 1 ) > 1 6 and ℜe(s 2 ) > 1 6 . Moreover, we have

Thus F satisfies the assumptions of Theorem 1 of [START_REF] De La Bretèche | Estimation de sommes multiples de fonctions arithmétiques[END_REF] with (

It follows that there exists a constant ϑ > 0 and a polynomial Q ∈ R[X] of degree 3 such that

Now alluding to Theorem 2 of [START_REF] De La Bretèche | Estimation de sommes multiples de fonctions arithmétiques[END_REF] to get the leading coefficient C of Q, we obtain

and therefore the leading coefficient C of Q is given by

.

3 ) = τ , from which (1.3) follows. This completes the proof.

The descent argument

Our main argument in order to derive Theorem 1.1 in section 3 consists of a descent from our original variety Ṽ onto the variety of equation

Although this is not required to verify Peyre's conjecture since Ṽ is a rational variety, it is particularly interesting to find out which torsor were used during this descent argument because Ṽ is a non-split variety. Indeed, as mentioned in [START_REF] Derenthal | Cox rings over nonclosed fields[END_REF], versal torsors parametrizations (see [START_REF] Colliot-Thélène | La descente sur mes variétés rationnelles II[END_REF] for precise definitions) are mostly used in the case of split varieties and the question of the right approach in the case of non-split varieties is quite natural. Using the Cox ring machinery over nonclosed fields developed in [START_REF] Derenthal | Cox rings over nonclosed fields[END_REF], all known examples of Manin's conjecture in the case of non-split varieties derived by means of a descent rely on a descent on quasi-versal torsors in the sense of [START_REF] Colliot-Thélène | La descente sur mes variétés rationnelles II[END_REF]. For example, the descent in [START_REF] Destagnol | La conjecture de Manin pour certaines surfaces de Châtelet[END_REF] is a descent on torsors of injective type Pic(V Q(i) ) ֒→ Pic(V Q ) whereas it is shown in [START_REF] Derenthal | Cox rings over nonclosed fields[END_REF] that the ad hoc descent used in [START_REF] De La Bretèche | On Manin's conjecture for singular del Pezzo surfaces of degree four II[END_REF] is a descent on the torsor of injective type Pic(V ) ֒→ Pic(V Q ). Here, we now show in the following lemma that the descent corresponds to a torsor of a different type, which is not quasi-versal.

With the notations of §2.


B O Berndtsson

Dario Cordero-Erausquin

A Z Bo'

Yanir A Klartag
email: klartagb@post.tau.ac.il

Rubinstein

COMPLEX LEGENDRE DUALITY

We introduce complex generalizations of the classical Legendre transform, operating on Kähler metrics on a compact complex manifold. These Legendre transforms give explicit local isometric symmetries for the Mabuchi metric on the space of Kähler metrics around any real analytic Kähler metric, answering a question originating in Semmes' work.

INTRODUCTION

For a function ψ : R N → R its classical Legendre transform is defined as [START_REF] Fenchel | On conjugate convex functions[END_REF][START_REF] Mandelbrojt | Sur les fonctiones convexes[END_REF] (1.1)

ψ * (y) = sup x [x • y -ψ(x)].
This transform plays an important role in several parts of mathematics, notably in classical mechanics and convex geometry. Being the supremum of affine functions (of y), ψ * is always convex and in case ψ is convex it equals the Legendre transform of its Legendre transform. One easily verifies (see Section 2) that ψ * = ψ if and only if ψ(x) = x 2 /2, so the Legendre transform is a symmetry on the space of convex functions around its fixed point x 2 /2.

In particular this applies when R N = C n . In this case we change the definition slightly and put

(1.2) ψ(w) = sup z [2Re (z • w) -ψ(z)],
where a • b := n i=1 a i b i . The reason for this change is that while the supremum in (1.1) is attained if y = ∂ψ(x)/∂x, the supremum in (1.2) is attained if w = ∂ψ(z)/∂ z, and one verifies that the unique fixed point is now ψ(z) = |z| 2 .

In this connection a very interesting observation was made by Lempert, [START_REF] Lempert | Symmetries and other transformations of the complex Monge-Ampère equation[END_REF]: Put ω ψ := i∂ ∂ψ. Assuming that ψ is smooth and strictly convex, g * ω ψ = ω ψ , if g(z) = ∂ψ(z)/∂ z. It follows from this that (1.3) g * ω n ψ = ω n ψ . The measure ω n ψ /n! =: MA C (ψ) is the complex Monge-Ampère measure associated to ψ, and Lempert's theorem thus implies that MA C (ψ) and MA C ( ψ) are related under the gradient map g. This may be compared and perhaps contrasted to the way real Monge-Ampère measures transform under gradient maps, see Section 2.

At this point we recall the definition of the Mabuchi metric in the somewhat nonstandard setting of smooth, strictly plurisubharmonic functions on C n . The idea is to view this space as an infinite dimensional manifold; an open subset of the space of all smooth functions. Its tangent space at a point φ should consist of smooth functions χ such that φ + tχ remains strictly 1 plurisubharmonic for t close enough to zero. This set of course depends on the particular φ we have chosen, but at any rate the tangent space will always contain smooth functions of compact support, so we take by definition the space of such functions as our tangent space. The Mabuchi norm of a tangent vector χ at a point φ is now defined by (1.4)

χ 2 φ := C n |χ| 2 ω n φ /n!.
We will interpret (1.3) as saying that the Legendre transform is an isometry for the Mabuchi metric on the space of convex functions (see section 4). It follows, at least formally, that the Legendre transform maps geodesics for the Mabuchi metric to geodesics, which reflects the so called duality principle for the complex method of interpolation [START_REF] Berndtsson | Complex interpolation of real norms, duality, and foliations[END_REF]. This is only a special case of Lempert's result, which implies that a much more general class of gradient-like maps are isometries for the Mabuchi metric. In this note we will develop this scheme and define 'Legendre tranforms' for Kähler potentials over a manifold M, usually compact.

For this we note first that the usual Legendre transform is not involutive on all plurisubharmonic functions, but just on convex functions, hence in particular on functions that are close to its fixed point |z| 2 . Imitating this, we start with a (local) Kähler potential φ on M, and define a 'Legendre tranform' depending on φ that is defined for potentials close to φ, fixes φ and is an isometry for the Mabuchi metric. For this to work, we need to assume that φ is real analytic. The definition of the φ-Legendre transform involves a polarization of our real analytic potential, which is locally a function φ C (z, w) defined near the diagonal in M × M. φ C is holomorphic in z, antiholomorphic in w and coincides with φ on the diagonal (these properties determine φ C uniquely). Roughly speaking, the idea is then to replace z • w by φ C and define our transform as

(1.5) (L φ ψ)(w) := sup z [2Re φ C (z, w) -ψ(z)].
When φ(z) = |z| 2 this gives us back the Legendre transform of (1.2). Let us first examine this transform in the case of a linear space, the cradle of the classical Legendre transform. Write ∆ C n = {(z, z) ; z ∈ C n } for the diagonal. We say that a smooth function φ on C n is strongly plurisubharmonic if its complex Hessian is bounded from below by a positive constant, uniformly at all points of C n .

Theorem 1.1. Let φ : C n → R be a real-analytic, strongly plurisubharmonic function. Then there are an open set

V φ ⊆ C n × C n containing ∆ C n and a neighborhood U of φ in the C 2 -norm on C n with the following properties: 0. For u ∈ U the function L φ (u) : C n → R ∪ {+∞} is well-defined,
where the supremum in (1.5) runs over all z with (z, w)

∈ V φ . 1. L φ (u) = u if and only if u = φ. 2. For u in a smaller neighbourhood U ′ ⊂ U, the function L φ (u) lies in U and L 2 φ (u) = u. 3. L φ is an isometry for the Mabuchi metric restricted to U.
The transform in (1.5) works fine if φ, ψ and φ C are defined on all of C n or C 2n , but for functions that are only locally well defined we need to find a variant of the definition that has a global meaning on a manifold. For this it turns out to be very convenient to use a remarkable idea of Calabi, [START_REF] Calabi | Isometric imbeddings of complex submanifolds[END_REF]. The Calabi diastasis function is defined as

D φ (z, w) = φ(z) + φ(w) -2Re φ C (z, w).
We then change the above definition by applying it to ψ + φ instead of ψ, and then subtract φ afterwards. This way we arrive at the equivalent transform

L φ (ψ)(w) := L φ (ψ + φ)(w) -φ(w) = sup z (-D φ (z, w) -ψ(z)).
Notice that in the classical case when φ(z) = |z| 2 , D φ (z, w) = |z -w| 2 and the transform becomes the familiar variant of the Legendre transform

sup z -(|z -w| 2 + ψ(z)).
The point of this is that, as is well known from the work of Calabi, D φ only depends on ω φ = i∂ ∂φ, i e it does not change if we add a pluriharmonic function to φ. As we shall see this implies that our construction of L φ = L ω φ globalizes and becomes well defined on functions ψ on a manifold M that are close to 0 in the C 2 -norm. Following the ideas, but not the precise proof, of Lempert, we can then verify that L ω φ is an isometry for the Mabuchi metric on U ω φ . Our main result is as follows: Theorem 1.2. Let M be a compact Kähler manifold, and let ω be a real analytic Kähler form on M. Let H ω := {u ∈ C ∞ (M); i∂ ∂u + ω > 0}. Then the generalized Legendre transform, L ω (defined in section 4) is defined on a neigbourhood U of 0 in H ω in the C 2 -toplogy and 1. L ω (u) = u if and only if u = 0. 2. For u in a smaller neighbourhood U ′ ⊂ U, L ω (u) lies in U and L 2 ω (u) = u. 3. L ω is an isometry for the Mabuchi metric on H ω restricted to U.

THE CLASSICAL LEGENDRE TRANSFORM

As a warm up and for comparison we first briefly look at the classical Legendre transform, (1.1). If ψ is differentiable, and if the supremum in the right hand side is attained in a point x, then y = ∂ψ/∂x =: g ψ (x). Hence we have that

(2.1) x • y ≤ ψ(x) + ψ * (y)
with equality if and only if y = g ψ (x), or in other words x • g ψ (x) -ψ(x) = ψ * (g ψ (x)). If moreover ψ is assumed smooth and strictly convex, g ψ is invertible. It follows that ψ * is also smooth, and by the symmetry of (2.1) that the inverse of g ψ is g ψ * . Recall that the (real) Monge-Ampère measure of a (smooth) convex function is MA R (ψ) := det(ψ j,k (x))dx. It follows from the above that g * ψ (dy) = MA R (ψ), and g * ψ * (dx) = MA R (ψ * ). We next turn to the Legendre transform of functions on C n and its relation to complex Monge-Ampère measures. We then redefine the Legendre transform by (1.2). Equality now occurs when w = ∂ψ(z)/∂ z =: g ψ , where we have also redefined the gradient map g to fit better with complex notation. We now give the first case of Lempert's theorem; it should be compared to how real Monge-Ampère measures transform.

(2.2) g * ψ (ω ψ) = ω ψ ,
so the complex Monge-Ampère measures of ψ and ψ are related by

g * ψ (MA C ( ψ)) = MA C (ψ). Proof. Let Λ = {(z, w); w = g ψ (z)} be the graph of the gradient map g ψ considered as a submanifold of C 2n . On Λ d(z • w) = ∂ψ(z) + ∂ ψ(w).
(This is simply because when (z, w) lie on Λ, then ∂ψ(z) = wj dz j and ∂ ψ(w) = z j d wj .) Since the left hand side is a closed form, it follows that

∂∂ψ(z) = d∂ψ(z) = -d ∂ ψ(w) = -∂ ∂ ψ(w).
If we pull back this equation under the map z → (z, g ψ (z)) we get

∂ ∂ψ = g * ψ (∂ ∂ ψ), which proves the theorem.
We remark that the apparent discrepancy between how the gradient map transforms the real versus the complex Monge-Ampère measures can be rectified as follows. First, since

[ψ ij ] := [ψ ij ] -1 = [ψ * ij ]
under appropriate regularity assumptions, the Riemannian metric ψ ij dx i ⊗ dx j is the pull-back of ψ * ij dy i ⊗ dy j via the gradient map ∇ψ : R n → R n . Therefore the gradient map pulls back the measure det

[ψ * ij ]dy 1 ∧• • •∧dy n to the measure det[ψ ij ]dx 1 ∧• • •∧dx n .
When M has toric symmetry, Theorem 2.1 precisely produces this observation via a careful translation between the real notation and the complex notation (cf. the proof of [6, Proposition 2.1]).

COMPLEX LEGENDRE TRANSFORMS

Let Ω denote a domain in C n . Denote by φ a real-analytic psh function on Ω. Denote by φ C the analytic extension of φ to a holomorphic function on a neighborhood W φ of the diagonal in C n × C n (whenever (M, J) is a complex manifold we denote by M the complex manifold (M, -J)). Such an extension exists since the diagonal

∆ C := {(p, p) ∈ C n × C n : p ∈ C n } is totally real in C n × C n .
We immediately switch point of view and work from now on in C n × C n , where φ C can be considered as a function on

W φ ⊂ C n × C n
that is holomorphic in the first factor and anti-holomorphic in the second. Explicitly, if in local coordinates

φ(z) = c α,β z α zβ , then φ C (z, w) = c α,β z α wβ .
The Calabi diastasis function associated to a real-analytic strongly psh function φ on Ω ⊂ C n is the function

(3.1) D φ (p, q) := φ(p) + φ(q) -φ C (p, q) -φ C (q, p) = φ(p) + φ(q) -2Re φ C (p, q). defined on W φ ⊂ Ω × Ω. Clearly D φ (p, q) = D φ (q, p) with D φ (p, p) = 0. In the local coordi- nates, D φ (p, q) = c α,β (p α -q α )(p β -q β ).
Note that the first non-zero term in the Taylor series is non-negative as φ is psh. Moreover, denote by π i :

C n ×C n → C n , i = 1, 2, the natural projections, i.e., π 1 (z, w) = z, π 2 (z, w) = w.
Calabi proves the following [START_REF] Calabi | Isometric imbeddings of complex submanifolds[END_REF].

Lemma 3.1.

There exists an open neighborhood

V φ of ∆ Ω contained in W φ on which D φ ( • , q)
is strongly convex with

(3.2) D φ (z, q) ≥ C|z -q| 2 on π 1 (V φ ∩ Ω × {q}).
We can now define a Legendre type transform associated to φ. For simplicity, whenever we refer to a function in our discussions below, we do not allow the constant function +∞. We denote by uscf the upper semi-continuous (usc) regularization of a function f :

X → R, uscf (x) := lim δ→0 sup y∈X |y-x|<δ f (y).
It is the smallest usc function majorizing f . Definition 3.2. The complex Legendre transform L φ is a mapping taking a function

ψ : Ω → R ∪ {∞} to L φ (ψ)(q) := usc sup p∈Ω (p,q)∈V φ [2Re φ C (p, q) -ψ(p)].
Since Re φ C (p, q) is pluriharmonic in q, L φ is psh. The definition depends on V φ , and we discuss that dependence later.

When

φ(z) = |z| 2 , then φ C (z, w) = z • w, while D φ (z, w) = |z -w| 2 , C = 1 and W φ = V φ = C n × C n ;
we recover, up to a factor of 2, the Legendre transform on R 2n . w). Setting z = w gives ψ ≥ φ. Since the complex Legendre transform is order-reversing then also ψ ≤ φ.

Lemma 3.3. Let ψ : Ω → R ∪ {∞}. Then L φ (ψ) = ψ if and only if ψ = φ. Proof. According to Lemma 3.1, whenever (z, w) ∈ V φ , 2Re φ C (z, w) ≤ φ(z) + φ(w), with equality iff z = w. Thus L φ φ = φ. Conversely, suppose L φ ψ = ψ. Then, whenever (z, w) ∈ V φ , ψ(z) + L φ ψ(w) = ψ(z) + ψ(w) ≥ 2Re φ C (z,

Definition 3.4. Say that a function

ψ : Ω → R ∪ {∞} is φ-convex if ψ = L φ η for some usc function η : Ω → R ∪ {∞}. Lemma 3.5. Let η : Ω → R ∪ {∞} be usc. Then L 2 φ η ≤ η with equality iff η is φ-convex. Proof. Whenever (z, w) ∈ V φ , η(z) + L φ η(w) ≥ 2Re φ C (z, w).
Thus,

(3.3) L 2 φ η(z) = usc sup w [2Re φ C (w, z) -L φ η(w)] ≤ usc η(z) = η(z).
Next, if L 2 φ η = η, then by definition η is φ-convex. It remains therefore to show the converse, and for this it suffices to show that

L 3 φ ν = L φ ν for any usc function ν : Ω → R ∪ {∞}. By (3.3), L 3 φ ν ≤ L φ ν. However, L 2 φ ν ≤ ν by (3.3), thus L 3 φ ν ≥ L φ ν.

LEGENDRE DUALITY ON COMPACT MANIFOLDS

Remarkably, a variant of these transforms can be defined on any Kähler manifold M. Let ω be a closed strictly positive real-analytic (1, 1)-form on M. Locally then ω equals √ -1∂ ∂u for some strongly psh real-analytic function u, and we define u C and subsequently

D ω := D u ,
locally. To check that these definitions are actually consistent globally and give rise to a diastasis function on a non-empty neighborhood of ∆ M it suffices to observe [START_REF] Calabi | Isometric imbeddings of complex submanifolds[END_REF] that whenever h is a realvalued function on a ball in C n that is pluriharmonic, i.e., √ -1∂ ∂h = 0, then h = h 1 (p) + h 1 (p) with h 1 holomorphic; thus, h C (p, q) = h 1 (p) + h 1 (q), so D h ≡ 0. Once again, by a variant of Lemma 3.1 [5, Proposition 5] we obtain an open neighborhood V ω of the diagonal on which D ω is nonnegative and strongly convex in each variable and on which ω admits local real analytic Kähler potential u for which u C exists. This neighborhood contains a δ-tubular neighborhood (with respect to some Riemannian metric) of the diagonal, at least whenever M is compact. Now, fix a real-analytic Kähler form ω on M.

We can now define a Legendre transform with respect to ω.

Definition 4.1. The complex Legendre transform L ω maps a function ψ :

M → R ∪ {∞} to L ω (ψ)(q) := usc sup p∈M (p,q)∈Vω ϕ [-D ω (p, q) -ψ(p)].
As in the setting of Ω ⊂ C n , the transform also depends on V ωϕ .

Definition 4.2. Say that

ψ is ω-convex if ψ = L ω η for some usc function η : M → R ∪ {∞}.
The following lemmas follow in the same manner as Lemmas 3.3 and 3.5. In fact, intuitively, L ω is locally given by L ω (ψ)(q) = L u (u + ψ) -u, where u is a local Kähler potential for ω. ω η ≤ η with equality iff η is ω-convex.

A GENERALIZED GRADIENT MAP

The fact that Calabi's diastasis D ω is locally uniformly convex in each variable on a neighborhood of the diagonal should, intuitively, ensure the supremum in the definition of L ω is attained in a unique point. In this section we make this intuition rigorous by giving a condition that ensures the supremum is attained. We will discuss only the case of compact manifolds. The case of non compact manifolds, leading up to Theorem 1.1, is proved similarily. Theorem 5.1. Let (M, ω) be a compact closed real-analytic Kähler manifold. There exists ǫ = ǫ(ω) > 0 such that for every function η satisfying ||η|| C 2 (M ) < ǫ, the supremum in Definition 4.1 is for any q in M attained at a unique point, z = G(η)(q).

1. If η is of class C k then G(η) is a diffeomorphism of M of class C . 2. L ω (η) is of class C k and the map η → L ω (η) is continuous for the C k -topology. 3. L 2 ω (η) = η. 4. G(L ω (η)) = G(η) -1 .
Proof. Because M is compact, the neighborhood V ω contains a ball of fixed size, call it δ > 0 (with respect to the the distance function d of the reference metric ω, say), around every point on the diagonal. Fix q ∈ M. Let (5.1)

f q (z) = -D ω (z, q) -η(z), z ∈ π 1 (V ω ∩ (M × {q})).
We claim that f q attains a unique maximum in π 1 (V ω ∩ (M × {q})). First, Lemma 3.1 implies that f q (z) ≤ -Cd(z, q) 2 -η(z), z ∈ π 1 (V ω ∩ (M × {q})),

and if ||η|| C 2 (M ) is sufficiently small, f q is uniformly concave on π 1 (V ω ∩ (M × {q})). If ||η|| C 0 (M ) < ǫ, f q (q) ≥ -ǫ, while, f q (z) ≤ -Cd(z, q) 2 + ǫ, z ∈ π 1 (V ϕ ∩ (M × {q})), So, if ǫ is small enough, f q (z) ≤ -2ǫ, z ∈ π 1 (V ϕ ∩ (M × {q})) \ B δ/2 (q),
Thus we see that the maximum of f q over π 1 (V ω ∩ (M × {q})) must be attained at a point in B δ/2 (q), which moreover is unique by the strict concavity of f q . This maximum point is the unique solution z of (5.2)

F q (z) := ∇ z f q (z) = 0 in B δ/2 (q) ⊂ π 1 (V φ ∩ (M × {q})).
We denote this unique solution by z = G(η)(q). Thus,

(5.3) L ω (η)(q) = f q (G(η)(q))
Since f q is uniformly concave in B δ/2 (q), the Implicit Function Theorem (IFT) implies that G(η)(q) is of class C k-1 in q whenever η ∈ C k . Thus, by (5.3) it follows that L ω ψ ∈ C k-1 .

Next, we claim that G(η) is invertible. To see that, let (5.4) F t,q (z) := ∇f t,q (z), f t,q (z) := -D ω (z, q) -tη(z).

The IFT, applied to F t,q , implies that (5.5) ∇G(tη) = -(∇ z F t,q ) -1 ∇ q F t,q = (∇ 2 z D ω (z, q) -t∇ 2 z η) -1 ∇ q F q . When t = 0, Lemma 4.3 implies that G(0)(q) = q, so (5.6)

I = ∇G ϕ (0) = -(∇ 2 z F q ) -1 ∇ q F q = (∇ 2 z D(z, q
)) -1 ∇ q F q . Combining (5.5) and (5.6) we see that whenever ||η|| C 2 is sufficiently small, the Jacobian of G(tη) is positive definite for all t ∈ [0, 1], hence the Jacobian of G(η) is invertible. This means that G(η) is locally injective, i e that if q = q ′ and d(q, q ′ ) is sufficiently small, then G(η)(q) = G(η)(q ′ ). Since we moreover know that G(η) is uniformly close to the identity, this gives that G(η) is globally injective. Since it is also open, it is a diffeomorphism onto its image.

That the supremum in Definition 4.1 is obtained for z = G(η)(q) means that

(5.7) L ω (η)(q) = -D ω (G(η)(q), q) -η(G((η)(q)).
Since, for (z, q) ∈ V ω we always have

(5.8) η(z) ≥ -D ω (z, q) -L ω (η)(q)
it follows that (5.9)

∇L ω (η)(q) = -∇ q D ω (z, q).
when G(η)(q) = z, so (5.10)

∇L ω (η)(q) = -(∇ q D ω )(G(η)(q), q).
But D ω is smooth, in fact real analytic, so we get, since

G(η) is of class C k-1 that ∇L ω (η) is of class C k-1 too. In other words L ω (η) is of class C k .
On the other hand, if η is close to zero, we know that G(η) is close to the identity, which is G(0). Hence, by (5.9), ∇L ω (η) is close to ∇L ω (0) = 0. Since it follows directly from the definition that the C 0 -norm of L ω (η) is small if the C 0 -norm of η is small, it follows that L ω (η) is close to zero in the C k -norm if η is close to zero in the C k -norm. In particular, with k = 2, this implies that we can apply the arguments in the beginning of this proof to L ω (η). Then (5.6) implies that L 2 ω (η) = η and that G(L ω (η)) = G(η) -1 . This completes the proof.

THE INVERSE GRADIENT MAP AND THE COMPLEX MONGE-AMPÈRE OPERATOR

As we shall comment later on, our next result can be viewed as a variant of a result of Lempert. G(ψ) ⋆ ω ψ = ω Lωψ . Therefore, the complex Monge-Ampère measure of ψ, ω n ψ , is pulled-back under G(ψ) to the complex Monge-Ampère measure of L ω ψ.

As pointed out in the introduction and section 2, this should be compared with the contrasting fact that the real Monge-Ampère operator is pulled-back under the inverse gradient map to the Euclidean measure.

Proof. Since the statement is local we look at a neighborhood W ⊂ M where we have a realanalytic Kähler potential φ of ω. By definition, for (z, w)

∈ V ω -D ω (z, w) ≤ ψ(z) + L ω ψ(w)
with equality precisely when z = G(ψ)(w). Let Λ be the set where this holds. Then, when (z, w) ∈ Λ,

-∂ z D ω (z, w) = ∂ z ψ(z), or, equivalently, -∂w D ω (z, w) = ∂w L ω ψ(w).
In other words (since

D ω (z, w) = φ(z) + φ(w) -φ C (z, w) -φC (z, w)), (6.1)
∂ z φ C (z, w) -∂ z φ(z) = ∂ z ψ(z), and (6.2)
∂w φ C (z, w) -∂w φ(w) = ∂w L ω ψ(w).
This means that the identity holds when both sides are considered as forms on C 2n and (z, w) lies on Λ. Since Λ is the graph of G(ψ), Λ is a manifold of real dimension 2n. Let p 1 and p 2 be the projections of W × W to the first and second factors, and let π 1 and π 2 be their restrictions to Λ. By Theorem 5.1, π 1 and π 2 are invertible maps and

(6.3) G(ψ) = π 1 • π -1 2 .
By (6.1) and (6.2),

dφ C (z, w) = π ⋆ 1 ∂(φ + ψ)(z) + π ⋆ 2 ∂(φ + L ω ψ)(w), when (z, w) ∈ Λ.
Hence the same identity holds when we restrict both sides to Λ as differential forms. Since the left hand side is a closed form, it follows that

π * 1 d∂(φ + ψ)(z) + π * 2 d ∂(φ + L ω ψ)(w) = 0, on Λ.
If we apply (π -1 2 ) * to this equation we get

(π -1 2 ) * π * 1 ∂∂(φ + ψ) + ∂ ∂(φ + L ω ψ) = 0.
By (6.3) it follows that

G(ψ) ⋆ ω ψ = G(ψ) ⋆ ( √ -1∂ ∂(φ + ψ)) = √ -1∂ ∂(φ + L ω ψ) = ω Lωψ ,
so we are done.

THE MABUCHI METRIC

Let M be a closed compact Kähler manifold. Recall that if ω is a Kähler form on M, the space of ω-plurisubharmonic functions, H ω is the space of smooth functions on M such that

ω ψ := √ -1∂ ∂ψ + ω > 0.
This is an open subset of the space of smooth functions and inherits a structure as a differentiable manifold from the one on C ∞ (M). The tangent space to H ω is the space of smooth functions on M and one defines a weak Riemannian metric on H ω by

g M (ν, χ) ψ = M νχω n ψ , for every ν, χ ∈ T ψ H ω ∼ = C ∞ (M).
Proposition 7.1. Let ω be real-analytic. There exists a neighborhood U ω of 0 in H ω in the C 2 topology such that L ω defines a Fréchet differentiable map from U ω to U ω . Its differential is

dL ω (η).χ = -χ • G(η), ∀η ∈ U ω .
Proof. Let q ∈ M. Define (cf. (5.1)),

f q (z, η) := -D ω (z, q) -η(z), z ∈ π 1 (V ω ∩ (M × {q})),
and let

F q (z, η) := ∇ z f q (z, η). Then F q is of class C k-1 if η is of class C k .
By the implicit function theorem the equation F q (z, η) = 0 defines z as a function of η, z = z(η), and since z(η) is the point maximazing f q (z, η) for given η we have that z(η) = G(η)(q) (which we now regard as a function of η, while q is fixed). Hence we see that z(η) = G(η)(q) is of class C k-1 . Moreover

L ω (η)(q) = f q (z(η), η).

Hence, by the chain rule

d/dt| t=0 L ω (η + tχ) = d/dt| t=0 f q (z(η), η + tχ), since ∇ z f q (z, η) = 0 for z = z(η). Since d/dt| t=0 f q (z(η), η + tχ) = -χ(z(η)) = -χ(G(η)(q))
we are done.

Theorem 7.2. Let M be a closed compact Kähler manifold and let ω be a real analytic Kähler form. There exists a C 2 neighborhood U ω of 0 in H ω such that L ω defines a Fréchet differentiable map from U ω to itself with the following properties:

(i) L ω is an isometry for the Mabuchi metric on U ω .

(ii) L 2 ω ψ = ψ for ψ ∈ U ω . (iii) L ω ψ = ψ if and only if ψ = 0.

Proof. Properties (ii) and (iii) are the content of Lemma 4.3 and Theorem 5.1. We turn to proving (i). Indeed, by Theorem 5.1 and Proposition 7.1

g M (dL ω ψ(χ), dL ω ψ(ν))| Lωψ = M χν • G(ψ)ω n Lωψ = M χν • G(ψ)G(ψ) ⋆ (ω n ψ ) = M χνω n ψ = g M (χ, ν)| ψ , proving (i).
Finally, if U ω is a neighborhood satisfying properties (i)-(iii), then replacing U ω by U ω ∩L(U ω ), we may assume L maps U ω to itself. This theorem should be seen in the light of the picture of H ω as a symmetric space, put forward by Mabuchi, Semmes and Donaldson, [START_REF] Mabuchi | Some symplectic geometry on compact Kähler manifolds I[END_REF][START_REF]Complex Monge-Ampère and symplectic manifolds[END_REF][START_REF] Donaldson | Symmetric spaces, Kähler geometry and Hamiltonian dynamics[END_REF]. In these works H ω is first studied as a Riemannian manifold, its curvature tensor is computed and is found to be covariantly constant. In the finite dimensional case, this implies the existence of symmetries around any point in the space. As described in §8.1, Semmes has also found symmetries for the Mabuchi metric, but to our knowledge the ω ϕ -Legendre transforms are the first examples of explicit symmetries for H ω . It would be interesting to generalize the theorems of Artstein-Avidan-Milman [START_REF] Artstein-Avidan | Hidden structures in the class of convex functions and a new duality transform[END_REF] and Böröczky-Schneider [START_REF] Böröczky | A characterization of the duality mapping for convex bodies[END_REF] to this setting and investigate whether these are all the symmetries of H ω under some reasonable regularity assumptions. From Theorem 7.2 it follows in particular that the ω-Legendre transform maps geodesics in H ω to geodesics. By the work of Semmes [START_REF]Complex Monge-Ampère and symplectic manifolds[END_REF], geodesics in H ω are precisely given by solutions of the homogenous complex Monge-Ampère equation, so that a curve t → ψ t (z) = ψ(t, z) , where t lies in a strip 0 < Re t < 1 and ψ depends only on the real part of t, is a geodesic in H ω if and only if ( √ -1∂ ∂t,z ψ + ω) n+1 = 0.

One main motivation for Lempert's work was to find symmetries of the inhomogenous complex Monge-Ampère equation. Here we find a somewhat different kind of symmetries for the homogenous complex Monge-Ampère equation (HCMA). The applicability of this may be somewhat limited by the absence of positive existence results for geodesics, but if we change the set up slightly and consider functions ψ defined for t in a disk instead of a strip, there is at least one setting in which our theorem applies. Considering boundary data s → ψ s on the unit circle that happen to extend to a smooth solution of the HCMA, then the same thing holds for sufficiently small perturbations of the data [START_REF] Donaldson | Holomorphic discs and the complex Monge-Ampère equation[END_REF], see also [START_REF] Moriyon | Regularity of the Dirichlet problem for the complex Monge-Ampère equation[END_REF]. Taking the given boundary data to be identically equal to 0, for which trivially an extension exists, we see that any boundary data that are sufficiently small can be extended to a solution of the HCMA, ψ t with t in the disk ∆. Theorem 7.2 shows that then L ω (ψ t ) also solves the HCMA. Indeed, solutions of the HCMA are critical points of the energy functional induced by the Mabuchi metric

E(ψ) = ∆×M ∂ t ψ ∧ ∂t ψ ∧ ω n ψt ,
and as shown above the energy functional is preserved under the ϕ-Legendre transform.

RELATIONS WITH LEMPERT'S AND SEMMES' WORK

8.1. Comparison to Lempert's theorem. Lempert starts with a complex manifold M and its holomorphic cotangent bundle T * (M). If z are local coordinates on M it induces local coordinates (z, ξ) on T * (M), so that a one-form, i e a point in T * (M) can be written ξ j dz j . There is a standard holomorphic symplectic form Ω on T * (M) that in such coordinates is written Ω = dξ j ∧ dz j . A (local) holomorphic map from T * (M) to itself, F , is symplectic if F * (Ω) = Ω, and Lempert's construction depends on the choice of such a symplectic map. Another ingredient is a differentiable real valued function ψ on M. From ψ we get a gradient map (8.1) z → (z, ∂ψ) =: ∇ψ which is a section of T * (M). Lempert's generalized gradient map is the map from M to itself

G ψ = π • F • ∇ψ,
where π is the projection from T * (M) to M. He then defines a generalized Legendre transform by

L F (ψ)(G ψ (z)) = ψ(z) + 2Re Σ(∇ψ),
where Σ is a generating function of the symplectic tranformation F . This means that Σ is holomorphic on T * (M) and satisfies

dΣ = ξ • dz -F * (ξ • dz).
Such a generating function exists at least locally since the right hand side is a closed form if F is symplectic.

We indicate briefly how this translates to our set up. First, there is a minor difference that we work with a symplectic form and generating function that is holomorphic in z and antiholomorphic in ξ, but the major difference is that we chose a different kind of generating function. The symplectic transformation F gives a map from T * (M) to M by w = π(F (z)). For special symplectic maps (sometimes called free canonical transformations) one can choose (z, w) as coordinates on T * (M) and express the generating function in terms of these coordinates instead. Locally, our construction amounts to choosing φ C (z, w) as such a generating function. If we define a symplectic transformation using φ C as a generating function one can check that our Legendre transform coincides with Lempert's. 8.2. Semmes' work. Another major motivation for our work is Semmes' work [START_REF]Complex Monge-Ampère and symplectic manifolds[END_REF] and we now relate the previous theorem to his work. Semmes starts by endowing the holomorphic cotangent bundle (T * ) 1,0 M with the complex structure Ĵ induced by pulling back the standard complex structure (induced by the complex strucutre J on M) under the (locally defined) maps [21, p. 530] (z, λ) → (z, λ + ∂ z φ).

This is well-defined and independent of the choice of local potential φ for ω since ∂ z φ -∂ z φ ′ is holomorphic whenever φ ′ is another such choice. To any smooth Kähler potential ψ, Semmes then associates the submanifold Λ ψ , the graph of ∂ψ in (T * ) 1,0 M. Under the biholomorphism between ((T * ) 1,0 M, Ĵ) and ((T * ) 1,0 M, J) the standard tautological 1-form α = λ i dz i and holomorphic symplectic form Ω = dz i ∧ dλ i on the latter are pulled back to forms that we denote by α and Ω. Then √ -1 Ω| Λ ψ = √ -1∂ ∂(φ + ψ) = ω ψ . Semmes goes on to observe that whenever ϕ is real-analytic, there exists an involutive anti-biholomorphism of a neighborhood of Λ ϕ in ((T * ) 1,0 M, Ĵ) whose fixed-point set equals Λ ϕ . Thus, if ψ is sufficiently close to ϕ in C 2 then Λ ψ is mapped to another submanifold that must be of the form Λ η for some η. Theorem 6.1 precisely establishes that this involution is given by our generalized gradient map G ϕ (ψ), so G ϕ (ψ)(Λ Lωψ ) = Λ ψ .

Theorem 2 . 1 .

 21 (Lempert) With the above notation

Lemma 4 . 3 .Lemma 4 . 4 .

 4344 Let ψ : M → R ∪ {∞}. Then L ω (ψ) = ψ if and only if ψ = 0. Let η : M → R ∪ {∞}. Then L 2

Theorem 6 . 1 .

 61 Fix a real-analytic Kähler form ω. Then, for each smooth function ψ such that √ -1∂ ∂ψ + ω > 0, we let ω ψ := ω + √ -1∂ ∂ψ. Then if ψ satisfies the assumptions of Theorem 5.1

ACKNOWLEDGMENTS

This work is based on the SQuaREs project award "Interactions between convex geometry and complex geometry" from the American Institute of Mathematics (AIM). The authors are grateful to AIM and its staff for the funding, hospitality, and excellent working conditions over the years 2011-2013. YAR is grateful to R.J. Berman and Chalmers Tekniska Högskola for their hospitality and support in Summer 2014 when an important part of this work was carried out. Finally, part of this work took place while BB and YAR visited MSRI (supported by NSF grant DMS-1440140) during the Spring 2016 semester. This research was supported by grants from ANR, BSF (2012236), ERC, NSF (DMS-0802923,1206284,1515703), VR, and a Sloan Research Fellowship.


Motion dynamics of Si (100) positive and negative islands under electromigration and thermomigration

Stefano Curiotto, Ali El-Barraj, Fabien Cheynis, Pierre Müller, Frédéric Leroy Aix Marseille Université, CNRS, CINaM UMR7325, 13288 Marseille, France

Mass transport processes play an important role in nano devices. They determine the morphology and the stability of a working device, and eventually its lifetime. When a direct electric current is applied to a material, it affects the mass transport by adding a bias to the atomic diffusive motion. This effect, called electromigration, can play an important role on the dynamics and the morphological stability of nanostructures at surfaces. In this work we have studied the motion of single-step holes and islands under the effect of an electric field on Si(100), in real time, by Low Energy Electron Microscopy (LEEM). The details of the motion depend on the strong diffusion anisotropy of the Si dimer rows. Studying the velocity and the shape of the nanostructures, we discuss a model to explain and guide the motion of holes and islands. The manipulation of nanostructures is an important challenge for nanoelectronics and nanofabrication technologies. External fields and thermal gradients are natural candidates to move and control the nanostructure position on a surface. Particularly promising is the use of electric fields to induce mass transport in specific directions, phenomenon known as electromigration (EM).

The elementary process of electromigration of an atom on a surface (adatom) is well known: the electric current biases random diffusion inducing mass transport in specific directions. The motion of many adatoms results in the motion of an entire nanostructure.

While it has been neglected in most studies, the crystalline anisotropy of the surface may play an important role on adatom motion and therefore on nanostructure motion.

We have used Low Energy Electron Microscopy (LEEM), to observe Si(100) surfaces in-situ and in real time with high temporal and spatial resolution, and in a large temperature range (1000-1500 K).

In order to support the experimental results and obtain more insights in the motion mechanism, we have developed a Kinetic Monte Carlo model based on both anisotropic and electromigration-biased diffusion at surfaces.

Si(100) surface reconstruction

The Si(100) surface structure has dimer rows running perpendicularly on alternating terraces. Diffusion is faster along dimer rows than perpendicularly to them

Activation energy

Schematics of a 2D void.

The dimer rows inside the hole are oriented horizontally, in the same direction of u. The force exerted by the electric field is oriented along the blue arrow and forms an angle q with u.

The void displaces in a direction forming an angle g with u.

J ext and J int are the atomic fluxes at the exterior and at the interior of the void. D are diffusion coefficients parallel and perpendicular to dimer rows; v are hole and island velocities; a is an atomic length; c is the concentration of diffusion species

FF

  45º from the dimer rows With a force applied from left to right, the voids move towards the top, at 90º from the electric-field direction and 135º from the [011] axis Void

  taking into account the dimer rows and the diffusion anisotropy reproduce all the observed behaviors of void and island motion Motion direction of holes and islands for different force directions Conclusions • The motion of nanostructures on Si(100) is possible thanks to the difference of diffusion coefficient parallel or perpendicular to dimer rows. • The effective activation energy for the motion of 2D islands and voids at Si(100) surfaces is 2.0±0.1 eV, given by the sum of a diffusion energy and a creation energy of the diffusing species 5 nm lateral resolution atomic vertical resolution Displacement of voids under thermomigration 2D voids moving under thermomigration (TM), the left side is colder than the right side Ad-dimers (presumed mobile species) move from the hot to the cold side Motion of voids and islands similar to EM


Aldon Gilles
email: gilles.aldon@ens-lyon.fr

COLLABORATION BETWEEN TEACHERS AND RESEARCHERS: A THEORETICAL FRAMEWORK BASED ON META-DIDACTICAL TRANSPOSITION

The meta-didactical transposition framework is an important reference for studying existing relationships in collaborative research between teachers and researchers. Born in a training context, it is important to extend it to describe and analyze these same relationships in the context of research. It is by finely analyzing the objects on which the protagonists focus that we can identify significant elements of interaction description leading to a better understanding and improvement both from a theoretical and methodological point of view of collaborative research.

Introduction

The framework of meta-didactical transposition, born in Turin in a teacher training context [START_REF] Aldon | The meta-didactical transposition: a model for analysing teachers education programs[END_REF][START_REF] Arzarello | Meta-didactical transposition: a theoretical model for teacher education programmes in[END_REF], has been taken up, expanded and enriched to adapt to the analysis of interactions in so-called collaborative research, involving actors from different institutions, researchers, teachers, computer scientists, etc. The purpose of this paper is to show how the theorizing of relationships between actors can help to better understand and act on the improvement of collaborative research. The fundamental hypotheses that support this work are developed in order both to justify the scientificity and the usefulness of collaborative research in general and particularly in mathematics education. This construction is being tested in a case study coming from the work done with teachers in the European FaSMEd project1 . The work presented in this paper is based on the work done in the EducTice team of the French Institute of Education (IFÉ -ENS de Lyon), in particular in the FoRCE project (Formation et Recherche Collaborative en Éducation), a Franco-Canadian project involving IFÉ, the Lyon's Local Education Authority and the Faculty of Pedagogy of the Université de Sherbrooke [START_REF] Nizet | L'analyse de valuations dans une démarche collaborative de recherche[END_REF][START_REF] Nizet | L'analyse de valuations dans une démarche collaborative de recherche[END_REF].

An attempt to theorize the relationships between researchers and teachers

Fundamental hypotheses

As anthropologist, Marcel Mauss (1923) pointed out, education is a total fact, "a fact that sets in motion the entire society and its institutions" (p. 102). It is therefore a complex phenomenon, which, as the philosopher Edgar Morin points out, can only be understood through complex thinking [START_REF] Morin | Introduction à la pensée complexe[END_REF]:

Aldon

It can be said that what is complex is on the one hand empirical, uncertain, unable to be certain of everything, to formulate a law, to design an absolute order. On the other hand, it is also a matter of logic, that is, the inability to avoid contradictions. (chap. 27/57)2

Educational research thus has the need to take this complexity into account and integrate it into its research paradigms:

We are in an uncertain battle and we do not yet know who will win. But it can already be said, however, that if simplifying thinking is based on the domination of two types of logical operations: separation and reduction, both of which are brutalizing and mutilating, then the principles of complex thinking will necessarily be principles of distinction, conjunction and implication. (Id. chap. 33/57)3

Asking questions about the type of research and the position of actors in research raises at least three types of questions, of an epistemological, methodological and ethical nature. Epistemological questions relate to the positions of the research regarding the actors and the aims of this research. They make it possible, in a way, to position research in the tensions between antagonistic poles: academic research to produce new knowledge or research to train actors. And from another point of view, research related to the construction of teaching materials or research to understand a practice. What are the researcher-practitioner relationships? Is it a question of considering the teacher in the research as a parameter, i. e., as an element of the system to be studied, or as a variable, i. e., as an element of the system to be characterized, or finally as an object of research? [START_REF] Roditi | Recherches sur les pratiques enseignantes et relations chercheurs-praticiens[END_REF]. What are the targeted productions and what are the conditions for their scientificity?

From a methodological point of view, the organization of collaborative work is not self-evident and must be thought of in relation to the theoretical frameworks at the same time as the goals that the actors set for themselves. The construction of tools to provide collaborative working conditions is essential.

Finally, ethical issues are essential in any work with actors from different institutions. Everyone has different issues in the common work that need to be considered and addressed in the research. Issues relating to institutional recognition and the capitalization of work are at the heart of the "contract" that must be concluded between the actors of collaborative research. This contract seems to me to have two very distinct facets: the first concerns the setting in stone of the conditions and objectives expected by the research actors: what is expected of each party and how it will be possible to achieve these objectives, but also the expected results and the methods for disseminating these results (training, text, actual achievements, etc.). The second facet concerns the object of the research, which can only be defined a priori through questions or hypotheses about an object whose boundaries have not yet been precisely constructed. The contract is then similar to the didactic contract of the Theory of Didactic Situations [START_REF] Brousseau | Le contrat didactique et le concept de milieu: Dévolution[END_REF], i.e. it cannot be fully explained. The contract is negotiated through a devolution of research, just as learning is the result of a devolution of the didactic situation negotiated between teachers and students. Ethical issues are at the root of the conceptualization of collaborative research and cannot be neglected under penalty of breach of contract and therefore failure.

These reflections naturally lead to hypotheses on which we base our research and which tend to consider all actors as partners in research. It is therefore a question of working with teachers rather than on teachers, considering the tensions already encountered.

What answer can be given by research to formalize collaborative research between researchers and teachers?

The Meta-Didactical Transposition

The meta-didactical transposition is based on Chevallard's anthropological approach of didactics (1989,1992). It describes a dynamic of relationships between teachers, researchers, trainers and any actor involved in collaborative research. In the research team "EducTice" where I was working, we were seduced by this approach [START_REF] Sanchez | Recherche collaborative orientée par la conception: Un paradigme méthodologique pour prendre en compte la complexité des situations d'enseignement-apprentissage[END_REF][START_REF] Aldon | The meta-didactical transposition: a model for analysing teachers education programs[END_REF][START_REF] Aldon | La genèse de praxéologies partagées entre enseignants et chercheurs dans un cadre d'évaluation formative[END_REF][START_REF] Nizet | L'analyse de valuations dans une démarche collaborative de recherche[END_REF] and we tried to extend this theoretical model to describe and analyze collaborative research. This model describes a dynamic of relations between researchers and teachers based on five pillars:

1. Double dialectic: didactical dialectic between the thee vertices of the didactical triangle knowledge, teaching and learning and meta-didactical dialectic between didactical dialectic and pragmatical or theoretical justification of this dialectic.

2. Meta-didactical praxeologies: according to [START_REF] Chevallard | On didactic transposition theory : some introductory notes[END_REF], any human activity can be described as a praxeology modeled by a quadruplet (T, τ, θ, Θ): in f ): in front of a task t, one (or several) technique (s) τ allows the task to be solved in the same way that all tasks of the same type T can be solved. This technique, which depends very closely on the institution in which the type of task is proposed, is justified in the institution by a discourse (θ, technology: discourse on techniques) and a theory Θ): in f . The double dialectic then leads us to consider two types of praxeologies: a didactical praxeology in which the type of task is built with reference to the knowledge to be taught, the technique being recognized in an institution and the technological and theoretical justifications depending both on institutional habits and learned knowledge; and a meta-didactical praxeology in which the type of task is didactical praxeology: for example in the European project FaSMEd which dealt with formative assessment in science and mathematics, the praxeology of a lesson on fractions, where writing a fraction is based on a technique justified by calculation rules and more theoretically by algebra, is the basis for a formative assessment lesson, itself justified by empiricism and theories of formative assessment.

3. Institutional aspects: for each of the actors, the praxeologies are directed by the fact that the actors belong to an institution; in the case of a type of task, the technique used and the justifications for this technique depend functionally on the actors' institution.

Aldon 4. Internalization: this phenomenon describes the exchanges between actors from different institutions and is at the heart of the collaborative work that could not exist if the actors did not take advantage of the interactions to modify their knowledge system. Internalization is linked to the knowledge at stake, whether practical or theoretical.

Internalization can be considered as a modification of a praxeology at a didactical or meta-didactical level.

5. Brokering: for dialogue to take place in a fruitful way, actions are needed to make the ideas at stake explicit (brokering) in order to facilitate dialogue by clarifying each other's positions. The broker is essential in the phenomenon of internalization to make the link between the points of view expressed and to reformulate in the language of each of the institutions involved the concepts and knowledge at stake.

Figure 1 shows an initial schematization of meta-didactical transposition that describes the dynamics that can be created in collaborative research: through joint work, with the help of brokering actions, teachers and researchers build a shared praxeology by internalizing a priori external components to one or the other community (Fig. 1). At the meta-didactical level, i.e. at the level of the discussion of didactical practices, teachers and researchers have developed praxeologies in their own institutions: they have techniques as well as justifications for these techniques. The transposition phenomenon must thus lead to a shared praxeology at the same time as the external components are internalized in the practices.

This schematization shows the dynamics that can be created, but it also raises questions: how can this dynamic be set in motion? How and why is it maintained? What is the meaning of the term "shared" when describing a shared praxeology? To go a little further in the concept is to precisely define this work object. By maintaining the conceptualization of meta-didactical transposition, it is possible to provide some answers to these questions by formalizing the nature of the work object (or object of interactions) that can be seen as a boundary object in the sense given by [START_REF] Star | Institutional ecology, "translations" and boundary objects: Amateurs and professionals in Berkeley's Museum of Vertebrate Zoology, 1907-1939[END_REF] [START_REF] Robutti | Boundary objects in mathematics education and their role across communities of teachers and researchers in interaction[END_REF]. This idea Aldon of a boundary object, resulting from an anthropological approach, is particularly interesting for describing interactions between communities and developing the concept of shared praxeology: Boundary objects are objects which are both plastic enough to adapt to local needs and constraints of the several parties employing them, yet robust enough to maintain a common identity across sites. (Star & Griesemer, 1989, p.393) In addition to this interpretative flexibility, the structure of the boundary object is also of great importance and the "information needs" trigger the actions that can be carried out on the boundary object at stake. The metaphor of the object as an object in the object-oriented programming paradigm can help to better understand this structure and the scale issue highlighted by [START_REF] Star | This is Not a Boundary Object: Reflections on the Origin of a Concept[END_REF]:

The two other aspects of boundary objects, much more rarely cited or used, are (1) the material/organizational structure of different types of boundary objects and (2) the question of scale/ granularity. Boundary objects are a sort of arrangement that allow different groups to work together without consensus. However, the forms this may take are not arbitrary. They are essentially organic infrastructures that have arisen due to what Jim Griesemer and I called "information needs" in 1989 (Star 2010, p. 602) But what is particularly interesting in this quotation is the fact that the boundary object exists if and only if the actors act on it. It is through interactions that this frontier can be widened, enlightened by the collective activity resulting from all the individual actions of the protagonists. Following [START_REF] Carlile | Transferring, Translating, and Transforming: An Integrative Framework for Managing Knowledge Across Boundaries[END_REF], we distinguish three types of activity on the components of a border object:

[…] we scale the relative complexity of the circumstances at a boundary using Shannon and Weaver's (1949) three levels of communication complexity: syntactic, semantic, and pragmatic. (p. 557) At a syntactic level, the "transfer" action proposes a clarification of the knowledge of the object; it corresponds to a situation where a common vocabulary is established or is a priori constituted. These actions keep the frontier in the initial state but allow all actors to explore together by agreeing on the terms used, by highlighting different components of the object but without disrupting the relationships between the objects. At a semantic level, the "translation" action moves the boundaries of the frontier or affects the relationships between the components of the boundary object. The search for common sense concerns the relations between the components, the constituents of the boundary object as well as a common construction of the meaning of one of its components. The translation activity leads the protagonists to build a sufficient compromise to agree on the subject of study within the specific framework of their discussion.

Finally, at a pragmatic level, "transformation" actions lead to a sharing of knowledge with a view to its use in the rest of the work; we affirm the pragmatic nature of the intervention because there is an awareness of a modification of a technique with regard to a type of task. This "negotiation" [START_REF] Nizet | L'analyse de valuations dans une démarche collaborative de recherche[END_REF]) can be accepted as an evolution of the work perspectives with the object in a dimension of training and professional development that we can relate to the phenomenon of internalization of meta-didactical transposition.

Transfer and translation activities highlight the mutual understanding of the manipulated components by acting both at a syntactic level of mutual understanding of the manipulated components and at a semantic level of the meaning that can be given and shared to these Aldon components. Transformation actions modify the relationships to objects by including them in all the tools that can be used for action at a didactical level. To make the link with Meta-Didactic Transposition, transfer leads to a shared understanding of the type of tasks, translation leads to a modification of technologies (in the Chevallard sense) and theories when the transformation leads to a modification of techniques related to the types of tasks related to the components of the boundary object at stake allowing a practical investment of the studied component (Fig. 2).

Fig.2 Boundary objects and MDT. Illustration

The announced methodology of the project was built on the paradigm of Design Based Research [START_REF] Swan | Design Research in Mathematics Education, Encyclopedia of Mathematics Education[END_REF]. In this project, the general discussion on formative assessment at the meta-didactical level, using in particular the results of the research, promotes the construction of classroom activities and their implementation in the classrooms. At the same time, these classroom implementations provide feedback that, through observations and analyses, evolves the model and participates in the professional development of actors, teachers and researchers. During the experimental phases of FaSMEd (task design, a priori analyses, a posteriori reflections, etc.), researchers share research results while teachers mainly offer their professional knowledge and pragmatic justifications of their practices. Thus, during the meta-didactical transposition process, the researchers' praxeologies come up against those of the teachers, and it may happen that components of praxeologies that were external to a certain community gradually become internal, within a praxeology "shared" by the two communities. The concepts of "formative assessment" and "use of technology", for example, are the subject of internalization phenomena that have been highlighted and studied in the conduct of FaSMEd. The awareness of the phenomena of internalization appear clearly in the discussions between researchers and teachers; the dialogues clearly show the two didactical and meta-didactical levels, in particular in lines 8-10 and 13-15 where the comparison of class observation with analyses of the constructed situation calls into question the didactic positioning of teachers in a violent way (line 15): A second meeting illustrates the actions on the boundary object. In the dialogue, the boundary object, as a container, is indeed the formative assessment, which is however never mentioned. On the other hand, interactions focus on Quiz, open questions and technology. First, the question of the relationship between Quiz and technology objects arises and is quickly reduced to discussing the type of question that can be asked, thus integrating the "open question" object as a component of the boundary object. Later on, the protagonists distinguish the role of the use of technology: mediamethod relationships exist and it is these relationships that need to be studied (line 10-11) from both the student and teacher's point of view; the dialogue then leads to a point of widening the object's boundary by linking "evaluation modalities" to the differentiation strategies specific to formative assessment.

In terms of action on boundary objects, the transfer appears to agree on the relationship that may exist between the assessment and the means of assessment; thus, lines 9 and 10, the distinction to be made between means and end is highlighted: 9 C2: yes, it's not due to technology 10 CP1: no, it's something else This is followed by a translation action:

11 E1 : this is the evaluation method

In the "Quiz" component of the boundary object, the questioning mode is questioned and linked to the assessment methods, which is then repeated to specify the relationships that may exist between the way of questioning.

Conclusion

Using the example of design-based research from the European FaSMEd project, I presented theoretical and methodological frameworks for analyzing the interactions between teachers and researchers in collaborative research. The essential elements of the Meta-Didactical Transposition are not all developed but remain as a watermark of this text. In particular, the importance of the institutional dimension of transposition, which is not well developed here, remains a fundamental framework for studies. It is clear that analyses of praxeologies but also of the components of a boundary object only make sense in a given institution and that the dialogue between the actors takes into account the institutional positions of each (Fig. 2). Similarly, the dialogues that have been

Aldon

more finely analysed are only proposed here as illustrations of the theory and should be more broadly detailed. Finally, actions on the components of a boundary object can only lead to internalization to the extent that they are constructed in a sequence, not necessarily linear, of transfer, translation and transformation. The drivers of these actions therefore necessarily involve brokering acts that can be highlighted in the analysis of interactions.

Fig. 1 :

 1 Fig. 1 : Schematization of the Meta-didactical Transposition (from Aldon & al. 2013) Boundary objects

  8 H1 And then after that, that's how we expect from... they're in check because maybe what we wanted to do is not what... it's not that in fact... Aldon 9 H2 yes yes yes... yes and then it's... 10 H1 ...and, and it's interesting to have it... this exchange there and I think it's great yeah it's great yeah it's great interesting there's that... really if we can continue at the school level it's good eh: really because there's a wealth and an exchange like that and it's... not people like that as you see it in hindsight... sometimes we are actually in our thing aah ! fuck and shit (blow on the table) what the fuck they don't understand and in fact it's because yes in fact it's us we didn't propose, put the problem of, of, of, of, and it's really true it's.... But there is also an internalization of the reality of the class that must fit into the model (lines 16-23) and which illustrates this double movement of internalization necessary for a real sharing of praxeologies.

	[…]
	15 H1 because

Formative Assessment for Science and Mathematics Education, The research leading to these results has received funding from the European Community's Seventh Framework Programme fp7/2007-2013 under grant agreement No[612337].

Translated by us from: "On peut dire que ce qui est complexe relève d'une part du monde empirique, de l'incertitude, de l'incapacité d'être certain de tout, de formuler une loi, de concevoir un ordre absolu. Il relève d'autre part de quelque chose de logique, c'est-à-dire de l'incapacité d'éviter des contradictions."

Translated by us from: "Nous sommes dans une bataille incertaine et nous ne savons pas encore qui l'emportera. Mais l'on peut dire, d'ores et déjà, que si la pensée simplifiante se fonde sur la domination de deux types d'opération logiques : disjonction et réduction, qui sont l'une et l'autre brutalisantes et mutilantes, alors les principes de la pensée complexe seront nécessairement des principes de distinction, de conjonction et d'implication."