S Fischler

T Rivoal

On the denominators of the Taylor coefficients of G-functions

] be a G-function, and, for any n ≥ 0, let δ n ≥ 1 denote the least integer such that δ n a 0 , δ n a 1 , . . . , δ n a n are all algebraic integers. By definition of a G-function, there exists some constant c ≥ 1 such that δ n ≤ c n+1 for all n ≥ 0. In practice, it is observed that δ n always divides D s bn C n+1 where D n = lcm{1, 2, . . . , n}, b, C are positive integers and s ≥ 0 is an integer. We prove that this observation holds for any G-function provided the following conjecture is assumed: Let K be a number field, and L ∈ K[z, d dz ] be a G-operator; then the generic radius of solvability R v (L) is equal to 1, for all finite places v of K except a finite number. The proof makes use of very precise estimates in the theory of p-adic differential equations, in particular the Christol-Dwork Theorem. Our result becomes unconditional when L is a geometric differential operator, a special type of G-operators for which the conjecture is known to be true. The famous Bombieri-Dwork Conjecture asserts that any G-operator is of geometric type, hence it implies the above conjecture.

Introduction

Siegel [START_REF] Siegel | Über einige Anwendungen diophantischer Approximationen[END_REF] introduced the class of G-functions as a vast generalization of the geometric series ∞ n=0 z n = 1 1-z and its primitive -

log(1 -z) = ∞ n=1 z n n . A G-function is a formal power series g(z) = ∞ n=0 a n z n ∈ Q[[z]
] such that there exists C > 0 such that (upon fixing an embedding of Q into C):

(i) the maximum of the moduli of the conjugates of a n is ≤ C n+1 . (ii) there exists a sequence of integers δ n such that 0 < δ n ≤ C n+1 and δ n a m is an algebraic integer for all m ≤ n.

(iii) f (z) is holonomic, i.e. it satisfies an homogeneous linear differential equation with coefficients in Q(z). These properties ensure that g(z) is holomorphic at z = 0. Prototypical examples of Gfunctions are the algebraic functions over Q(z) holomorphic at z = 0, or the hypergeometric series p+1 F p (z) with rational parameters. The latter class includes the polylogarithmic functions Li k (z) = ∞ n=1 z n n k , where k is any given integer. Any holonomic power series with integer coefficients and positive radius of convergence is a G-function.

In this paper, we are interested in the sequence of denominators δ n in (ii). In practice, it has been observed for long that the least denominator δ n always divides D s bn C n+1 where D n = lcm{1, 2, . . . , n}, b, C are positive integers and s ≥ 0 is an integer. We recall that D n = e n(1+o (1)) by the prime number theorem, so that D s bn C n+1 has geometric growth. By Eisenstein's theorem, we have s = 0 for any algebraic function over Q(z). But this is not the general situation, for we have s = k in the case of the polylogarithms Li k (z) for k ≥ 1, because lcm{1 k , 2 k , . . . , n k } = lcm{1, 2, . . . , n} k .

Our main result is a proof of this observation, in very precise terms, conditionally on a well-known conjecture on the generic v-adic radii of convergence of solutions of Goperators, but also unconditionally for an important class of G-operators. We recall that a differential operator L ∈ Q[z, d dz ] is said to be a G-operator if the differential equation Ly(z) = 0 has a G-function solution for which it is minimal for the degree (or order) in d dz . There is no loss of generality in assuming from the beginning that L ∈ K[z, d dz ] for some number field K. Moreover, we can trivially rewrite the differential equation Ly(z) = 0, of order µ say, as a differential system X ′ (z) = G(z)X(z) of order 1, where X(z) = t y(z), . . . , y (µ-1) (z) and G(z) is a square matrix in M µ (K(z)). For any finite place v of K, the generic radius of solvability R v (L) of L at v is the greatest real number R ≤ 1 such that the system X ′ (z) = G(z)X(z) has a solution matrix X (z) at the generic point t relative to v consisting in functions analytic in D(t, R -). See [2, pp. 67 and 70] or [START_REF] Dwork | An introduction to G-functions[END_REF] for these classical definitions. The following conjecture is often attributed to Bombieri.

Conjecture 1. Let K be a number field, and L ∈ K[z, d dz ] be a G-operator. Then the generic radius of solvability R v (L) is equal to 1, for all finite places v of K except a finite number.

This conjecture is a refinement of the following result, which is a consequence of two independent theorems, of André and Bombieri [16, p. 228, Theorem 2.1] on the one hand, and of the Chudnovsky's [12, p. 17, Theorem III] on the other hand: if

L ∈ K[z, d dz ] is a G-operator, then v log(1/R v (L)) < ∞
where the summation is over the finite places v of K; see [START_REF] Dwork | An introduction to G-functions[END_REF]p. 228,Theorem 2.1]. The conjecture is in fact known to be true when L is a Goperator of a special type, namely when it is a geometric differential operator, i.e. a product of factors of Picard-Fuchs differential operators; see [2, pp. 39 and 111]. Roughly speaking, a Picard-Fuchs equation is the differential equation satified by the periods of a family of algebraic varieties (defined over Q in our situation) depending on a complex parameter. It follows that Conjecture 1 is a consequence of the Bombieri-Dwork conjecture, as formulated by André [2,p. 111], which asserts the converse: every G-operator is a geometric differential operator, a conjecture usually phrased in looser terms as "every G-operator comes from geometry".

Our main result is the following. In the case where 0 is not a singularity of L, it is implicit in Bombieri's paper [7, §6]; see also [2, p. 74, Theorem 5.2] and §2 below.

Theorem 1. Let g(z) = ∞ n=0 a n z n be a G-function annihilated by a G-operator L of order µ for which Conjecture 1 holds. Then there exist some positive integers b, b 0 , C such that, for any n ≥ 0, D µ-1 bn+b 0 C n+1 a n is an algebraic integer. Moreover b can be chosen as the least common multiple of the denominators of the exponents of L at 0.

We do not need to assume that L is the minimal G-operator that annihilates g(z), though this assumption obviously provides the smallest value of µ we can achieve by our method. The exponent µ -1 can not be improved in general as the example of polylogarithms shows, but algebraic functions also show it is not always best possible. In the final section, we provide another example where the exponent µ -1 is optimal. If 0 is not a singularity of L, then we can take b = 1.

From the above comments, we also deduce:

Theorem 2. The conclusions of Theorem 1 hold unconditionally if L is a geometric differential operator.

Since any G-function is annihilated by a G-operator, we also deduce the following corollary to Theorem 1 (because

D s bn+b 0 C n+1 divides D s (b+b 0 )n (D s b 0 C) n+1 for any n ≥ 0). Again, it is unconditional if the G-operator is a geometric one.
Corollary 1. Assume that Conjecture 1 holds. Then for any G-function ∞ n=0 a n z n there exist some positive integers b, C and an integer s ≥ 0 such that, for any n ≥ 0, D s bn C n+1 a n is an algebraic integer.

As an application of this result, let us define a filtration on the set G of all G-functions as follows. Given s ≥ 0, we denote by G s the set of g(z) = ∞ n=0 a n z n ∈ G with the following property: there exist b, C ≥ 1 such that, for any n ≥ 0, D s bn C n+1 a n is an algebraic integer. Then G s is a Q-vector space stable under derivation, and we have

G s ⊂ G s+1 and G s G t ⊂ G s+t for any s, t ≥ 0.
Elements of G 0 are exactly globally bounded G-functions (see [START_REF] Christol | Fonctions hypergéométriques bornées[END_REF]); for any s ≥ 1 the s-th polylogarithm Li s (z) belongs to G s and not to G s-1 . Given a hypergeometric function g(z) = p+1 F p (z) with rational coefficients, Christol's criterion [START_REF] Christol | Fonctions hypergéométriques bornées[END_REF] (see also [START_REF] Delaygue | On Dwork's p-adic formal congruences theorem and hypergeometric mirror maps[END_REF]) enables one to compute the least integer

s such that g ∈ G s . Letting G ∞ = ∪ ∞ s=0 G s , Corollary 1 shows that Conjecture 1 implies G ∞ = G (i.e., (G s ) s≥0 is a total filtration of G).
As the proof of Theorem 1 shows (see §3.4), G s is the set of G-functions g(z) = ∞ n=0 a n z n with the following property (where K is a number field containing all coefficients a n ): there exists b ≥ 0 such that for any sufficiently large prime number p, for any place v of K above p, and for any n ≥ 0, we have v(D s bn a n ) ≥ 0. It would be very interesting to find a graduation G ∞ = ⊕ ∞ t=0 F t such that G s = ⊕ s t=0 F t for any s ≥ 0 and F t F t ′ ⊂ F t+t ′ for any t, t ′ ≥ 0.

The structure of this text is as follows. In §2 we comment on some applications of our results, and relate them to linear recurrences with polynomial coefficients. Then we prove Theorem 1 in §3, and apply it in §4 in a situation related to Apéry's proof that ζ(3) is irrational.

Acknowledgment. We warmly thank Yves André and Gilles Christol for many discussions on the subject of this paper and for patiently answering our questions.

Some comments

When a G-function is a solution of a G-operator regular at z = 0, our proof of Theorem 1 can be simplified and appears already implicitly in [7, §6] (see also [2, p. 74, Theorem 5.2]). In this case Theorem 1 follows from the very general theorem of Dwork-Robba [16, p. 120]; in contrary to the proof below, all finite places can be dealt with in the same way, and no shearing transformation is necessary.

However, not every G-function can be solution of a G-operator regular at z = 0. Indeed, assume that an operator L ∈ Q[z, d dz ] is minimal for some G-function g(z) (L is then a Goperator) and that another solution h(z) at z = 0 of Ly(z) = 0 has a non-trivial monodromy at z = 0. (For instance, we can consider the hypergeometric function g

(z) = 2 F 1 [ 1 2 , 1 2 ; 1; z] with a second solution h(z) = f (z) + log(z)g(z) for some f (z) ∈ Q[[z]].) Since L is a right factor of any differential operator M ∈ Q[z, d dz ] such that Mg(z) = 0, we have Mh(z) = 0 as well, so that z = 0 is a singularity of M.
Therefore we need a generalization of the Dwork-Robba Theorem. A first step in this direction has been made by Adolphson, Dwork and Sperber [START_REF] Adolphson | Growth of solutions of linear differential equations at a logarithmic singularity[END_REF]: their result covers the case of a single Jordan block, and can be used to derive Theorem 1 in this case. The general theorem we shall use is due to Christol-Dwork [START_REF] Christol | Effective p-adic bounds at regular singular points[END_REF]; we will make a certain number of transformations to be in position to apply it. So far, there is no known intrinsic way to check if a differential operator L is a Goperator, except if we can find a G-function it annihilates and for which it is minimal, or if we know that it comes from geometry. ( 1 ) Moreover, there is no known algorithm to find such a G-function solution, and in practice the situation is reversed: we are given an explicit G-function g(z) and then we compute its associated G-operator L. Thus, one may think that Theorems 1 and 2 are of limited practical interest because we can directly see that the denominators of the Taylor coefficients of g(z) are of the desired form.

However, they do have a practical interest, especially if L is a geometric operator because the results are then unconditional. Indeed, let us assume we are given an explicit 1 Another conjecture predicts that a differential operator in Q[z, d dz ] is a G-operator if and only if its p-curvature is nilpotent for all but finitely many primes p. (There is a similar conjecture when Q is replaced by any number field.) The "only if" part is a consequence of a theorem of the Chudnovsky's [START_REF] Chudnovsky | Applications of Padé approximations to Diophantine inequalities in values of G-functions, Number theory[END_REF]. The p-curvature can be computed (see [START_REF] Bostan | A fast algorithm for computing the characteristic polynomial of the p-curvature[END_REF] for an efficient algorithm) from the associated differential system for any p without computing an exact solution of the differential equation. But the drawback of the conjecture is that it has to be computed for infinitely many p, which makes it unpractical. However, if the p-curvature is observed to be nilpotent for a very large number of primes p, the conjecture can be used as an oracle to guess if an operator is a G-operator (and then try to prove this by finding an explicit G-function solution). This procedure was used with success in [START_REF] Bostan | The complete generating function for Gessel walks is algebraic[END_REF].

G-function, or a period function, and its associated G-operator L. By a well-known theorem of André, Chudnovsky and Katz (see [START_REF] André | Séries Gevrey de type arithmétique I. Théorèmes de pureté et de dualité[END_REF] or [START_REF] Fischler | On the values of G-functions[END_REF] for a precise statement), we know that at any α ∈ Q ∪ {∞}, a basis of local solutions at z = α of Ly(z) = 0 can be made of (1) G-functions in the variable zα, if α is a regular point of L;

(2) linear combinations of G-functions in the variable zα, with coefficients some polynomials (over Q) in certain rational powers of zα and integral powers of log(zα), if α is a singularity of L.

(Above and below, we replace zα by 1/z if α = ∞.) In general, we know nothing about the G-functions appearing in these basis, in the sense that there is no general procedure to find closed form formulas for their Taylor coefficients. But Corollary 1 applies to these G-functions, which thus provides non-trivial informations on the denominators of their coefficients. In case ( 1), the G-functions are solutions of Ly(z) = 0, hence we can take s = µ -1. In case ( 2), the G-functions are no longer necessarily solutions of Ly(z) = 0 but they are solutions of G-operators of order at most µ 2 (see [3, p. 110]) so that we can take s = µ 2 -1. This corollary may even be applied unconditionally because if L is a geometric operator, this is also the case of any of the shifted operators L α by any algebraic number α, where by definition L α y(z) = 0 is equivalent to Ly(zα) = 0.

The above remarks can be translated in the language of linear recurrences with polynomial coefficients. We consider a sequence (a n ) n≥0 of algebraic numbers satisfying a recurrence of the form k j=0 p j (n)a n+j = 0, where

p j (n) ∈ Q[n]
. Such recurrences naturally occur (sometimes in many different ways) when trying to prove the irrationality of constants like π, log(2), ζ(3), and it is important to control the growth of the sequence of denominators δ n of a n . The general situation is that δ n grows like (bn + b 0 )! s , and not that δ n divides D s bn+b 0 C n+1 . However, if it is of the latter type, then the series g(z) = ∞ n=0 a n z n is not far from being a G-function because the recurrence relation gives rise to a linear differential equation Ly(z) = 0 with coefficients in Q(z) of which g(z) is solution, and indeed

g(z) is a G-function if a n ∈ Q for instance.
Consider a G-operator L and assume that there exists a local solution ∞ n=0 v n (zα) n of Ly(z) = 0 at some given α ∈ Q ∪ {∞}. To L, we can associate bijectively a linear recurrence

k j=0 q j (n)V n+j = 0 (2.1) with q j (n) ∈ Q[n] of which (v n ) n≥0 is a solution.
Corollary 2. If Conjecture 1 holds, then for any solution (V n ) n≥0 of the recurrence (2.1) with V n ∈ Q for all n ≥ 0, we can find some positive integers b, C and an integer s ≥ 0 such that D s bn C n+1 V n is an algebraic integer for all n ≥ 0. This is unconditional if L is a geometric operator.

Indeed, any solution

(V n ) n≥0 of (2.1) is such that ∞ n=0 V n (z -α) n is a solution of Ly(z) = P (z) for some polynomial P (z) ∈ Q[z] of degree k say (P arises because of the initial conditions). Then ( d dz ) k+1 • L is a G-operator that annihilates ∞ n=0 V n (z -α) n . Hence, ∞ n=0 V n z n is
a G-function by the André-Chudnovsky-Katz theorem. We conclude by an application of Corollary 1. The problem of bounding the growth of denominators of rational solutions of Apéry like linear recurrences in absence of explicit formulas is an important one in Diophantine approximation. It was considered in [START_REF] Beukers | On Dwork's accessory parameter problem[END_REF] and [START_REF] Zagier | Groups and Symmetries: From the Neolithic Scots to John McKay[END_REF] for instance.

In practice, we start with an explicit G-function ∞ n=0 a n z n and compute its minimal Goperator. The above property enables us to construct new linear recurrences with solutions with controlled denominators without having to construct explicitely these solutions. In fact, we don't how to do that in general, except when α = 0 and V n = a n is the sequence we start with. If the G-operator comes from geometry, this is unconditional. We develop this procedure in Section 4 on an explicit example.

Proof of Theorem 1

We split the proof of Theorem 1 in several parts. The strategy is to apply the Christol-Dwork Theorem at all (but finitely many) finite places. To meet the nilpotence assumption (ii) in Theorem 3 below, we use two preliminary steps ( § §3.1 and 3.2). Then we recall and apply the Christol-Dwork Theorem in §3.3, and conclude the proof in §3.4 by considering the (finitely many) remaining finite places. The constant b (resp. b 0 and C) is constructed in §3.1 (resp. after Eq. (3.1) and in §3.4).

Unless otherwise stated, all the tools we use appear in [START_REF] Dwork | An introduction to G-functions[END_REF].

Reduction to the case of integer exponents at 0

We first recall that from the André-Chudnovsky-Katz Theorem, we know that a G-operator is Fuchsian with rational exponents at every algebraic point (and at infinity). We shall prove Theorem 1 in a special case below ( § §3.2 -3.4), namely when all exponents of L at 0 are integers (so that b = 1). To deduce the general case, we recall that all exponents of L at 0 are rational numbers, and denote by b ≥ 1 the least common multiple of their denominators. Then we perform the change of variable x = z 1/b , by letting g

(x) = ∞ n=0 a n x bn . Let L ∈ Q[x, d
dx ] be obtained from L in this way, so that L g(x) = 0. The exponents of L at 0 are those of L multiplied by b: they are integers. Moreover Conjecture 1 holds for L, and L has order µ: the special case of Theorem 1 we shall prove below yields b 0 and C such that D µ-1 bn+b 0 C bn+b a n is an algebraic integer for any n ≥ 0. This concludes the proof of the general case.

Shearing transformation

From now on, we shall prove Theorem 1 under the additional assumption that all exponents of L at 0 are integers (so that b = 1). We write L = B µ (z)( d dz ) µ + . . . + B 0 (z) with B 0 (z), . . . , B µ (z) ∈ Q[z] and B µ (z) = 0. To L, we associate the differential system zY ′ (z) = A(z)Y (z), with

A(z) =            0 1 0 0 . . . 0 0 0 1 1 0 . . . 0 0 0 0 2 1 . . . 0 0 0 0 0 3 . . . 0 0 . . . . . . . . . . . . . . . . . . 0 0 0 0 . . . µ -2 1 -a 0 -a 1 -a 2 -a 3 . . . -a µ-2 µ -1 -a µ-1            ∈ M µ (Q(z))
where a i (z) = z µ-i B i (z)/B µ (z) is holomorphic at z = 0 for i ∈ {0, . . . , µ -1} since 0 is a regular singularity of L. We have Ly(z) = 0 if, and only if, zY ′ (z) = A(z)Y (z) where Y (z) = t y(z), zy(z) ′ , z 2 y ′′ (z), . . . , z µ-1 y (µ-1) (z) ; see for instance [START_REF] Coddington | Linear ordinary differential equations[END_REF]Chapter 6]. This is not exactly the usual way to write the differential system X ′ (z) = G(z)X(z) associated to L for the vector X(z) = t y(z), y(z) ′ , y ′′ (z), . . . , y (µ-1) (z) , but this is a standard variation when dealing with regular singularities. In particular, the generic radii of solvability of L can be computed with the system X ′ (z) = G(z)X(z) or with the system zX ′ (z) = A(z)X(z), giving the same results.

The eigenvalues of A(0) are exactly the exponents of L at 0: by assumption they are integers. Therefore they can be made all equal to 0 by applying a suitable shearing transformation. In precise terms, for any

H(z) ∈ GL µ (Q[z, z -1 ]) (i.e., such that both H(z) and H(z) -1 have entries in Q[z, z -1 ]), let A [H] (z) = zH ′ (z)H(z) -1 + H(z)A(z)H(z) -1 .
Then upon letting Z(z) = H(z)Y (z) we have zY ′ (z) = A(z)Y (z) if, and only if, zZ ′ (z) = A [H] (z)Z(z). Using the matrix H(z) provided by [START_REF] Dwork | An introduction to G-functions[END_REF]p. 106,Corollary 8.3] we have the additional properties that

A [H] (z) ∈ M µ (Q[[z]]
) and 0 is the only eigenvalue of A [H] (0), so that A [H] (0) is nilpotent. Since the differential system zZ ′ (z) = A [H] (z)Z(z) has a regular singularity at 0, it has a solution matrix of the form U(z)z

A [H] (0) with U(z) ∈ GL µ (Q[[z]]) such that U(0) = I µ . Then Y(z) = H(z) -1 U(z)z A [H] (0) is a solution matrix of the differential system zY ′ (z) = A(z)Y (z). Since Y 0 (z) = t (g(z), zg ′ (z), . . . , z µ-1 g (µ-1) (z)) ∈ M q,1 (Q[[z]]
) is a solution of this system, there exists

L 0 = t (ℓ 1 , . . . , ℓ µ ) ∈ M q,1 (Q) such that Y 0 (z) = Y(z)L 0 , so that g(z) = ℓ 1 y 1,1 (z) + . . . + ℓ µ y 1,µ (z) (3.1)
where y i,j (z) is the coefficient of index (i, j) of Y(z). We denote by

b 0 t=-b 0 h i,j,t z t the coefficient of index (i, j) of H(z) -1 and by ∞ n=0 u i,j,n z n that of U(z), where b 0 ≥ 0 is fixed and h i,j,t , u i,j,n ∈ Q. Since A [H] (0) is nilpotent, the coefficient of index (i, j) of z A [H] (0) is a polynomial P i,j (X) ∈ Q[X] taken at X = log(z). Therefore the equality Y(z) = H(z) -1 U(z)z A [H] (0) yields y 1,i (z) = µ j,k=1 b 0 t=-b 0 h 1,j,t z t ∞ n=0 u j,k,n z n P k,i log(z)
so that using Eq. (3.1) we have

g(z) = µ i,j,k=1 ℓ i P k,i log(z) ∞ m=-b 0 min(b 0 ,m) t=-b 0 h 1,j,t u j,k,m-t z m . Recall that g(z) = ∞ n=0 a n z n ∈ Q[[z]
]; then we have the following expression of a n , in which the polynomials P k,i appear only through their constant terms:

a n = µ i,j,k=1 min(b 0 ,n) t=-b 0 ℓ i P k,i (0)h 1,j,t u j,k,n-t (3.2)
This expression allows us to focus now on the coefficients u j,k,n-t .

Application of the Christol-Dwork Theorem

In this section, we apply the Christol-Dwork Theorem [START_REF] Christol | Effective p-adic bounds at regular singular points[END_REF] in the setting of §3.2; we shall deduce Theorem 1 in §3.4 below. We use the following particular case of the version stated in [16, p. 159, Theorem 2.1].

Theorem 3 (Christol-Dwork). Let K be a number field and v a finite place of K over a prime p. Let G(z) ∈ M µ (K(z)) and consider the differential system zZ ′ (z) = G(z)Z(z).

Assume that:

(i) The coefficients of G(z) have no pole in the disk D(0, 1 -).

(ii) The matrix G(0) is nilpotent.

(iii) We have |G(z)| Gauss,v ≤ 1.
(iv) There exists a matrix solution U t (z) at the generic point t (normalized by U t (t) = I µ ) which converges in D(t, 1 -).

Let U(z)z G(0) be a matrix solution of zZ

′ (z) = G(z)Z(z) with U(z) = ∞ n=0 U n z n where U 0 = I µ and U n ∈ M µ (K).
Then for all n ≥ 0, we have

|U n | v ≤ T (n) µ-1+vp((µ-1)!)+βµ , (3.3)
where

T (n) = max 1≤k≤n |k| -1 v if n ≥ 1, T (0) = 1, β µ = min µ -1, v p ( µ j=1 µ j ) , v p is the usual p-adic valuation on Q ⋆ , and |M| v = max i,j |m i,j | v for M = (m i,j ) ∈ M µ (K).
To apply this result, we let K be a number field such that H(z), H(z) -1 , A(z), A [H] (z) belong to M µ (K(z)) and the numbers ℓ i of Eq. (3.1) belong to K. Let P K denote the set of all finite places v of K, i.e. ultrametric absolute values of K normalized by |p| v = 1/p if v is above the prime number p. We shall check the assumptions of Theorem 3 with G(z) = A [H] (z), provided p is large enough. For any v ∈ P K above p, the completion K v of K with respect to v is a finite extension of Q p and for any α ∈ K we have

|α| v = |N Kv/Qp (α)| 1/dv p where d v = [K v : Q p ]
; here |x| p is the usual p-adic absolute value on Q, with |p| p = 1/p. Let v ∈ P K , and t be a generic point with respect to v. Then the local solutions at t of Ly(z) = 0, zY ′ (z) = A(z)Y (z) and zZ ′ (z) = A [H] (z)Z(z) are related to one another as above. Since by assumption the differential operator L has generic radius of solvability equal to 1 (except for finitely many places v over some number field containing its coefficients), the same property holds over K and for the differential system zZ ′ (z) = A [H] (z)Z(z). Let p 1 be such that this radius is equal to 1 (so that assumption (iv) of Theorem 3 holds) for any v above any prime p ≥ p 1 .

For i, j ∈ {1, . . . , µ}, we denote by

R i,j (z) S i,j (z) the coefficient of index (i, j) of the matrix A [H] (z), with R i,j (z), S i,j (z) ∈ O K [z] and S i,j (0) = 0; here O K is the ring of integers of K. Let ω i,j,ℓ , 1 ≤ ℓ ≤ deg S i,j
, denote the roots of S i,j (z). Enlarging K if necessary, we may assume that ω i,j,ℓ ∈ K for any i, j, ℓ; moreover ω i,j,ℓ = 0 since S i,j (0) = 0. At last, let r i,j,t and s i,j,t denote the non-zero coefficients of R i,j (z) and S i,j (z). Let N 2 ∈ K ⋆ denote the product of all numbers ω i,j,ℓ , r i,j,t and s i,j,t as i, j, ℓ, t vary. Then there exists p 2 such that for any prime number p ≥ p 2 , the p-adic valuation of

N K/Q (N 2 ) ∈ Q ⋆ is 0.
At last, let N 3 ∈ K ⋆ denote the product of all numbers ℓ i , P k,i (0) and h 1,j,t of Eq. (3.2) which are non-zero, as i, j, k, t vary; then there exists p 3 such that for any prime p ≥ p 3 , the p-adic valuation of

N K/Q (N 3 ) ∈ Q ⋆ is 0.
Let p 0 = max(µ + 1, p 1 , p 2 , p 3 ), p be a prime number such that p ≥ p 0 , and v ∈ P K be a place above p. Since p ≥ p 2 we have |ω i,j,ℓ | v = |r i,j,t | v = |s i,j,t | v = 1 for any i, j, ℓ, t, so that |R i,j (z)| Gauss,v = max t |r i,j,t | v = 1, |S i,j (z)| Gauss,v = 1, and |A [H] (z)| Gauss,v = 1. Moreover, the coefficients R i,j (z) S i,j (z) of A [H] (z) have no pole in the disk D(0, 1 -). Therefore we have checked all assumptions of Theorem 3 with G = A [H] , since A [H] (0) is nilpotent and p ≥ p 1 . Recall from §3.2 that U(z)z A [H] (0) is a solution matrix of the differential system zZ ′ (z) = A [H] (z)Z(z), with U(z) ∈ GL µ (K[[z]]), U(0) = I µ , and U i,j (z) = ∞ n=0 u i,j,n z n with u i,j,n ∈ K. By (3.3), we have

|u i,j,n | v ≤ T (n) µ-1 for any n ≥ 0.
Here we have used that p > µ so that v p ((µ -

1)!) = v p ( µ j=1 µ j ) = 0. Since |D n | v = p -vp(Dn) = p -⌊ log n log p ⌋ = T (n) -1 we obtain: |D µ-1 n u i,j,n | v ≤ 1 for any i, j, n.
Now recall that g(z) = ∞ n=0 a n z n , where a n is given by Eq. (3.2). Since p ≥ p 3 we have |ℓ i P k,i (0)h 1,j,t | v ≤ 1 for any i, j, k, t, so that this equation yields

|D µ-1 n+b 0 a n | v ≤ 1 for any n ≥ 0.
This upper bound holds for any finite place v ∈ P K above any prime number p ≥ p 0 . We shall now deal with primes p < p 0 .

Conclusion of the proof

Since g is a G-function, there exists a sequence (δ n ) n≥0 of positive rational integers such that δ n a n ∈ O K and δ n ≤ ∆ n+1 for any n, where ∆ > 1 is a positive real number that depends only on g. Then we let

C = p<p 0 p ⌈ log ∆ log p ⌉
where the product ranges over prime numbers p such that p < p 0 . Now let p < p 0 , v ∈ P K be a place above p, and n ≥ 0. Then we have x) for any x ∈ Z \ {0}. Combining this upper bound with the one obtained for p ≥ p 0 , namely |D µ-1 n+b 0 a n | v ≤ 1, we conclude that |D µ-1 n+b 0 C n+1 a n | v ≤ 1 for any finite place v ∈ P K : for any n ≥ 0, D µ-1 n+b 0 C n+1 a n is an algebraic integer of K.

|C n+1 a n | v = |a n | v p -(n+1)⌈ log ∆ log p ⌉ ≤ |a n | v p -log δn log p ≤ |δ n a n | v ≤ 1 since δ n a n ∈ O K and |x| v = p -vp(

Working out an example

In this section, we illustrate our results on the example of the generating function of Apéry's numbers a n = n

L = z 2 (1 -34z + z 2 ) d dz 3 + z(3 -153z + 6z 2 ) d dz 2 + (1 -112z + 7z 2 ) d dz + z -5.
The singularities of L are 0, ( √ 2 -1) 4 and ( √ 2 + 1) 4 . The exponents at 0 are all equal to 0, while those at z = ( √ 2 ± 1) 4 are 0, 1, 1 2 . Moreover, L is minimal for A(z), hence it is a G-operator. Beukers [START_REF] Beukers | Irrationality of π 2 , periods of an elliptic curve and Γ 1 (5), Approximations diophantiennes et nombres transcendants[END_REF] and Dwork [START_REF] Dwork | On Apéry's differential operator[END_REF] independently observed that L is even a geometric operator. A detailed study was made in [START_REF] Beukers | A family of K3 surfaces and ζ(3)[END_REF]; in particular the following representation of A(z) as a period was given, valid for |z| < ( √ 2 -1) 4 :

A(z) = 1 (2iπ) 3 C dudvdw 1 -(1 -uv)w -zuvw(1 -u)(1 -v)(1 -w) where the algebraic cycle C = {u, v, w ∈ C 3 : |u| = |v| = |w| = 1 2 }; this period is a solution of the Picard-Fuchs equation of the families of K3 surfaces (1 -(1 -xy)t - zxyt(1 -x)(1 -y)(1 -t) = 0) z∈C .
See also [START_REF] Lairez | Computing periods of rational integrals[END_REF] for further examples of this nature. Hence Theorem 1 applies unconditionally: for any solution (U n ) n≥0 of (4.1) with U n ∈ Q such that L( ∞ n=0 U n z n ) = 0, we have that D 2 n+b 0 C n+1 U n are algebraic integers for some positive integers b 0 and C. Since this applies to the sequence of integers (a n ) n≥0 , we see that the exponent µ -1 of D bn+b 0 in Theorem 1 is not always sharp.

For any fixed α ∈ Q, α = 0, we now consider the local solutions of Ly(z) = 0 at z = α. It amounts to the same to solve locally at z = 0 the (shifted) differential equation L α y(z) = 0 where 4 , there is at least one holomorphic solution ∞ n=0 c n z n of L α y(z) = 0 because at least one exponent at z = 0 is an integer (they are in fact equal to 0, 1, 1 2 ). No closed form expression is known for the sequence (c n ) n≥0 but it satisfies the linear recurrence (trivially associated to L α for any α ∈ C)

L α = (z + α) 2 1 -34(z + α) + (z + α) 2 d dz 3 + (z + α) 3 -153(z + α) + 6(z + α) 2 d dz 2 + 1 -112(z + α) + 7(z + α) 2 d dz + z + α -5. If α = ( √ 2 + 1) 4 or α = ( √ 2 -1)
(n + 1) 3 V n + (2α -17)n 2 + (6α -51)n + 5α -39 (2n + 3)V n+1 + (6α 2 -102α + 1)n 2 + (24α 2 -408α + 4)n + 25α 2 -418α + 4 (n + 2)V n+2 + α(2α 2 -51α + 1)(2n + 5)(n + 3)(n + 2)V n+3 + α 2 (α 2 -34α + 1)(n + 4)(n + 3)(n + 2)V n+4 = 0. (4.2)
Note that for α = ( √ 2 ± 1) 4 , the recurrence is of order 3 only because α 2 -34α + 1 = 0. Since L α is a geometric operator, we can apply Theorem 1 unconditionally to any solution (V n ) n≥0 of (4.2) with V n ∈ Q such that L α ( ∞ n=0 V n z n ) = 0, which includes (c n ) n≥0 . It follows that there exist some positive integers b 0 and C such that D 2 2n+b 0 C n+1 V n is an algebraic integer for all n ≥ 0.

We now deal with the case α / ∈ {0, ( √ 2 + 1) 4 , ( √ 2 -1) 4 }. Then z = 0 is a regular point of L α and thus its exponents are integers. In fact they are equal to 0, 1 and 2 for any such α. Let us denote by ∞ n=0 c j,n z n ∈ Q[[z]], j = 1, 2, 3, a basis of local holomorphic solutions at z = 0 of L α y(z) = 0. They are G-functions but closed forms for the coefficients c j,n are not known. Since ∞ n=0 c j,n z n are independent solutions of L α y(z) = 0, the three sequences (c j,n ) n≥0 are independent solutions of the recurrence (4.2). The latter is of order 4 so that the (c j,n ) n≥0 do not generate all the solutions (V n ) n≥0 of (4.2). However, they generate all the sequences (V n ) n≥0 with V n ∈ Q such that L α ( ∞ n=0 V n z n ) = 0. By Theorem 1, we are ensured unconditionally that for any such (V n ) n≥0 , we have D 2 n+b 0 C n+1 V n ∈ Z for all n ≥ 0 and for some positive integers b 0 , C. .

If (U n ) n≥0
Indeed ( a n ) n≥0 is a solution of the recurrence (4.1) but L( ∞ n=0 a n z n ) = 5 (see [START_REF] Beukers | A family of K3 surfaces and ζ(3)[END_REF]). We thus have D 3 n+b 0 C n+1 a n ∈ Z for some integer b 0 ≥ 0. In this case, this is sharp as it is known that the exponent 3 can not be improved; moreover we can take b 0 = 0 and C = 1. Similar observations can be formulated for the solutions (V n ) n≥0 of the recurrence (4.2) such that

V n ∈ Q but L α ( ∞ n=0 V n z n ) = 0.

2 ,

 2 which appeared in Apéry's famous proof of the irrationality of ζ(3). The series A(z) = ∞ n=0 a n z n is a G-function because the sequence of integers (a n ) n≥0 satisfies the linear recurrence(n + 2) 3 U n+2 -(34n 3 + 153n 2 + 231n + 117)U n+1 + (n + 1) 3 U n = 0,(4.1)which, together with the initial values of a n , implies that A(z) is a solution of the differential equation Ly(z) = 0 with

  is a solution of the recurrence (4.1) withU n ∈ Q but L( ∞ n=0 U n z n ) = 0, then in fact L( ∞ n=0 U n z n ) = P (z) for some polynomial P (z) ∈ Q[z] of degree k say. Hence, ∞ n=0 U n z n is a G-function because ( d dz ) k+1 • L is still a G-operator.It is of order k + 4, of geometric type, with integer exponents at z = 0. Hence, we can apply Theorem 1 unconditionally: there exist some positive integers b 0 , C such that D k+3 n+b 0 C n+1 U n are algebraic integers for all n ≥ 0. This situation applies to the second sequence of rational numbers used by Apéry to prove the irrationality of ζ(3), namely


"From collective action to a pressure group" The birth of the National Syndicate of wine brokers in France and its action (1920s-1949)

Le Bras Stéphane

The goal of this communication is to present the birth and the role of the National Syndicate of Wine brokers in France form the 1920s to 1949 and to show how a social conflict and social tensions can lead an unorganized corporation to a strong and effective Union.

The mid-1920s in France are marked by a series of unrest and collective actions from many pressure groups, from the left wing (socialists, communists) to the right wing (leagues and fascists paramilitary groups). The French society experienced at this time many economic difficulties which resulted in a troubled and disturbed social situation.

This unrest is as well tangible in the wine society which is characterized since the 1900s by a profound restructuration of its field and an irregular economy (like shown by professors Gavignaud-Fontaine or Lachiver, crises are frequent in the wine economy and these crises are often followed by period of high but swift prosperity).

In the heart of this restructuring and of its hazards, the wine brokers, the vital wheels in the smooth running of the market, the essential go-betweens between the world of the property and that of the trade, have to face the fluctuations -ceaseless and excessive -of the prices, new commercial practices that leave them puzzled and the legislative interventionism of the State, which is often seen as disturbing in a profession marked by a liberal state of mind.

In front of these multiple pressures, the answers, in particular in Languedoc, are at first epidermic and anti-authority, to disrupt and perturb the market of wines, for secondly, on the initiative of the Southern brokers, make more constructive and more ambitious efforts to give birth to a real pressure group with widened skills.

These are the reasons why we'll see here the different causes of the problem, the long way to the Union and the birth of the National syndicate and its first actions. These are just the outlines of my research

To the sources of the dissatisfaction: the question of the brokerage in the South

The role of the broker

The broker in wine is a major character of the wine-producing society. In four Southern departments (Aude, Gard, Hérault, Pyrénées-Orientales), which are in the Interwar period, the most producing four departments of the national territory, every village counts at least one broker. Called " Broker of goods ", its main activity -even exclusive -lies in the brokerage in wines. Officially recognized by the State with whom they acquit a registration fee, they assure the flow and display of the production. By his pivotal position, he is so able to know at best the state of the market, the capacities of some and wills of the others. If he knows which quantity this one wants to sell, he also knows at which moment the other one is be going to want to buy.

As a liaison officer, he visits some and the others to permit a bigger fluidity in the flow of stocks and in the establishment of contracts. The various testimonies often evoke the subtlety and the psychology of the brokers who have the difficult tries to make match an ambitious offer and a careful demand. In addition to this every day labor, the broker adds a weekly activity when he goes(surrenders) on markets of wines which are everyday of the week in the South of France. The go-between, allowing this time the physical meeting of the parts accompanies owners and traders there to ensure and secure the transactions or to present new buyers and new sellers.

The brokers brake into two categories, the rural brokers and the urban brokers. The first ones are originated from the villages and work on their own when the second are working in more or less big companies (sometimes up to 50 employees).

Their activity as well as their probity and experience are an essential key to the market.

Protesting and refusing the situation

Nevertheless, while open the 1920s, the sector is in crisis. The number of brokers did not stop progressing since the end of the First world war " stressing the disorder within the profession ", disorder owed according to the testimonies to a lack of gratitude and to a proliferation of the " brown brokers " who try to benefit from periods of strong profitabilities on the wine markets, the whole without settling registration fees, realizing there of subsequent and fraudulent profits.

In front of this situation, the labor unions of local brokers propose the solutions which seem to them the most right and the most fundamental within the framework of a rational structuralization of the market: legislative gratitude of the status of the wine broker (the last definition of which dates 1866 and which joins in the non-specialized frame(executive) of the " broker of goods ") and the purification of the profession by the institution of a professional ID card or at least of an union card recognized by the Property and the Business as well as the rise of the rate of brokerage. But these measures are not heard by the various actors of the market of wines: state, traders, owners. It explains largely by two reasons. The first one comes from the relative youth of labor unions. Rare are the ones which have more than about fifteen years of existence. The second lies in the atomization of the local and regional syndicates: for the department of the Hérault in 1925, there are not less than six brokers' local labor unions.

To reduce this deficit of attention, the local labor unions(syndicates) in the South, in a context of fast and steady increase of the high cost of living and further to a new failure of the negotiations at the beginning of 1926, try then to disrupt(perturb) the market to make listen again their demands. In April, 1926, they dash into a long-term unitarian strike, an unique(only) fact due to its reach(impact) and its identical character. For the first time, the brokers claim only, and by the strength, rights for their profession.

Facing this movement, the traders, little inclined to see a part of their profits flying away, propose an appreciation of the brokerage and in May, the " Syndicate of the Wholesale trade of Montpellier " proposes the first one, followed by the other syndicates, an increase of the right of brokerage, then carried to 0,75 Fr, proposition which is accepted at once by the president of the Labor union of the brokers of Montpellier.

The long way to the Union

A profession under domination

Hardly of this first victory, the necessity of uniting in a federalist frame becomes more marked in the bend of the years 1920-1930.

Because, first of all, the problem is that the brokers stay profoundly dominated by the powerful Business. So, in the South, the rules of brokerage are always imposed by the Trade. Indeed, the brokers are dominated by and subject to the regional Trade under the shape of a professional ID card and in the commitment written not to sell to the traders foreign to the region while the traders from their part make only a commitment to trade with the brokers owners of this card. Moreover, any deal will be officialized by a contract on which will be registered all the information relative to the deal.

This first stage towards the raising of moral standards of the profession is a half-victory for the brokers who see there recognizing the importance of their role but who have to accept an undeniable appointment of a guardian. As a matter of fact, the conditions and the standards of the brokerage are "From collective action to a pressure group". The birth of the National Syndicate of wine brokers in France andits action (1920s-1949). International Association Strikes and Social Conflict Conference, Lisbon, march 2011 3 defined thus always by the Business then accepted by the brokers by means of asymmetric local collective agreements. Furthermore, The strong subordination of the brokers in the wine-producing world is also perceptible during the coming in the South Mediterranean of the Commission of inquiry led by E. Barthe in 1930-31 on the situation of the vine growing of France and Algeria in the name of the Committee(Commission) of Drinks. During the hearings the opinion of the brokers leaders are always linking or subordinated to the Trade leaders.

The first attempts

In front of the degradation of a nevertheless profitable situation till the end of 1920s, takes place a reversal of tendency in the profession. Indeed, in the financial difficulties due to the economic crisis which strikes France quite hard, depreciation caused by the brown broker in increase and a market of wines which "overflows" comes to be added the drastic increase of the registration fees.

While up to there, liberalism was lauded, from now on, following the various headways having punctuated the last years, the corporatist movement, in a wide movement of defense of its interests, decide to unite. So, pushed by the acknowledgement in favour of the concentration and of the federalism, the major national federations (South, Bordeaux region, Algeria in particular) try to make contacts, to the heart of the 1930s and of the wine crisis which continues. These are diffuse contacts at first then solid secondly. This action with as main objective the recognition of a particular status, is interrupted by the degradation of the international situation in 1938 while the first signs of a national impact of the message are indisputable (correspondence with E. Barthe, private bill for the status of the Rural Brokers in June, 1938). It is the fundamental step there which has been crossed and which only the war will repel.

Indeed, the failures but also the success of the interwar period made become aware to the leaders of the big regional corporatist movements of the necessity of an union. The advantages of a concerted action is not to be demonstrated anymore and it's very clearly established around that a national union allowing them to press more effectively on the fates of their field, the whole in a triple optics clearly established: the rationalization, the purification and the control of the profession.

The National Syndicate

The birth of the Syndicate If the idea was born even before 1938, international events but also, after the Liberation, disagreements, difficulties of understanding, regional issues (opposition of some local unions refractory to any idea of national unity) delayed the completion of negotiations.

The disagreements are numerous : the impossibility of finding an agreement on the agenda of a meeting, acrimony related to travel, endless negotiations about the venue and date of the meeting about the development of statutes and the election of a federal office.

Yet the urgent need for concerted action could make disappear the past reluctance and early 1946, long-term contacts are established between Bordeaux and southern representatives. These contacts help to settle a meeting March 20, 1946 in Bordeaux. Finally in September 1946, representatives of Montpellier invited their counterparts from Bordeaux and from Algeria to a meeting in the hotel of the Chamber of Commerce in Montpellier where discussions began around four central themes: the commission rates, the status of wine brokers, the creation of a National Federation, and "one of the finer points to consider" the financing of such a body.

September 5 at night, the bases of the "National Federation of Wines and Spirits Brokers from France and Algeria" are discarded and those articles are submitted by 1 January 1947. The new body, resulting of months of negotiations but also of several years of fighting has two basic goals:

-Be a permanent and lasting relationship with the unions of dealers, the National Union of wholesale trade in wines and spirits, bodies of qualified intermediaries, governments;

"From collective action to a pressure group". The birth of the National Syndicate of wine brokers in France andits action (1920s-1949). International Association Strikes and Social Conflict Conference, Lisbon, march 2011 -Allow adoption of a status brokers.

The first secretary is based in Montpellier, where is the secretary general administrative, permanent authority and effective interface between different parts of the Federation. He is the indispensable cog in the joint action. Decisions are taken by the majority of board members, elected annually, engaging the entire. Corporatist solidarity has indeed taken over the manic individualism that prevailed in the profession a few years ago.

Its first actions

It is a tool independent, territorially and economically powerful who sees the day. Designed and served as a real strength of peer group pressure, it will be able to show all its power, especially as it is joined in November 1947 by the Federation of Dealers in Burgundy and in January 1948 by two unions of Maine and lower Loire and Loire. With more than 3,000 members and twenty-eight unions in 1949, it focuses on its primary purpose, the voting official status of wine broker. For this, he resumed the basis for discussions which had almost resulted in 1938 and certain laws of the Vichy era that were to its advantage.

Thus, after an effective lobbying action towards the world of the Property and the State, on December 31, 1949, the law establishing the "status of rural broker" is adopted by Parliament. It's a victory of the "National Federation of Wines and Spirits Brokers" which here becomes a source of proposals and a pulse while a few years earlier the unions that formed the Union were just auxiliary to the property and especially to the Commerce. From secondary actors, brokers, high weight from federalism and corporatist national awareness of their responsibilities, becoming major players in the organization of wine market.

The statute ensures the monopoly of the sector to the profession. Indeed, the law establishes a professional identity card issued by the prefecture on the proposal of a commission of six members where brokers generally are the majority. This commission even has a right to regulate the profession by a "role of screening, to limit the constant number of brokers in a given region". Finally, this monopoly is reinforced by the requirement to have the card to link buyers and sellers. This is one more step to the consolidation of the profession and the elimination of "brown brokers" by heavy fines and forfeitures. Finally, it is forbidden to practice a profession that is incompatible with the job of broker, including that of the wholesaler. It's the end of an unacceptable situation for most brokers where some southern traders exercised, legally and in defiance of conflicts of interest reported by the profession, trading and brokering.

Here is a clear marker of a successful work accomplished and thoughtful regulation. It aims to clearly separate -and officially -the various intermediaries in the wine world and marks the independence -and ultimately recognized by the state -of the profession without which the wine market can no longer function.

Conclusion :

Thus, after a maturation that has lasted nearly twenty-five years, an organized and national union was born at a price of local struggles and a major effort to get out of a position of submission. It achieved its main goal: the control of its industry, its practices and its standards in a time of restructuration of the wine market. The brokers became a leader group in the wine market, ensuring through a political and unitarian offensive, embodied by the birth of the National Federation, the control, the monopoly and the regulation of the profession and above all its own destiny.


R Phillips
email: stephane.lebras@univ-pau.fr

"Corporatism in the Warehouses" Wine Wholesalers in

The first half of the XX th century marks for wine-producing Languedoc a period of trouble and disorders. While the effects of the phylloxera 1 crisis are still felt in the turn of the century, the wine-producing world tries to find the maximal extension which was his in the years 1860-70, both in term of production and surface.

While the effects of the phylloxera crisis are fading away and while the vineyard from the Languedoc is reconstituted in the bend of the XX th century, the Midi 2 plains cover themselves with new vines which turn the region into an industrious and hyper-productive vine growing area. Carignan and Aramon -both key vines of the reconstruction -or hybrid vines commit the local producers in an unlimited race for productivity. It is the victory of a "vine growing of mass" 3 that sets the region into an excessive production period. Besides, the market is blocked by so-called "artificial" wines which allowed to answer a steady demand the previous decades while the yields were at the lowest. So, at the beginning of the century, the market is disrupted by the presence of defrauded wines and artificial wines and by the raid of wines stemming from these new vineyards with high profitability. Languedoc enters then in a cycle of retailing crises of which it will go out really only at the end of 1970s when is set up a qualitative policy on the initiative of the EEC (in particular through a policy of bonuses with qualitative vines).

Sensitive to the chances of the macroeconomic situation, to the climatic variations, to the evolutions of the legislation, to the competitive balance of power which are at stake there, the market of regular wines knows, between 1900 and 1940, a particularly irregular dynamics, alternating the short phases of fast prosperity to which are linked longer periods of recessions 4 . In these conditions, the distribution and sale is a central problem for the economic survival of a region which is economically quasi-exclusively bound to the vine growing 5 .

In this context of crisis, the wine wholesalers from the Languedoc -Or the "Trade" as we call it in the region -strive to limit the disastrous effects of a situation which they can only undergo. A central and major character of the wine-producing society, the wholesaler is the intermediary between the world of the production and the public retailing. Necessary and essential cog in the flow of stocks, it assumes, by its pivotal position, a crucial role on markets. In this period of generalized crisis, its effort of domination of the market of wines in Languedoc is even more marked because it tries to limit at most the sudden irregularities, to assert itself in front of producers better and better organized and to assure at the same time the structuring but also the fluidity of the market.

And so, during these few years, while the wine economy is affected by numerous jolts, the Trade defines its hegemonic temptation by the power of its corporation, the imposition of normative practices and the rigorous control of the market of wines. From 1900, the date of the creation of the regional Syndicate to the year 1940 which upsets the economic conditions of the whole country, in a corporatist logic of protection of its interests, the Trade sets up different logics of defense with uneven but certain effects. It is these strategies and their impacts on the market of wines that will analyze this presentation.

I. Strategies of conquest: the control of the wine market

In a context tensed by the irregularity of markets but also by the presence of new and better equipped actors6 , the regional trade sets up various corporatist strategies of defense of its interests. But beyond this defensive tactics, it also gets organized around ambitious objectives of conquest.

A. The appropriation of prestigious and high-duty positions

The first objective of the regional trade is to master the instruments of control of the economic competences. In a hegemonic logic, the wholesalers dash into the conquest of prestigious but also fundamental positions for the economic orientation of the wine market.

First of all, it is the Chambers of commerce which are the target of the appetite of the local trade. In all France, Employers Unions dominate the chambers of commerce, with in particular a power which increased after the law of 18987 and the Midi does not make an exception. Ph. Lacombrade showed that, for period 1870-1939, about 18 % of the members of Chambers of commerce from the Languedoc are wholesalers8 . In Montpellier and in Sète, figures exceed the 25 %, the proof of a seizure of the decision-making power by the trade which is appropriating the power and the politico-economic territory.

In Béziers, wine heart of the region, the Chair of the Chamber of Commerce is held by an industrialist close to commercial circles, A. Gaillard, between 1912 and 1934 Through these personal trajectories we can see the will of the regional trade to padlock the access to the posts which can influence the future of the wine-producing network and as a consequence the regional economy, profoundly dependent on the wine production and on its trade.

The market of wines is also an economic frame in which numerous conflicts occur. So, the trials are legion between traders, wine growers, brokers, service providers (as the representatives or the railroads). To ensure the defense (literally as figuratively) of their interests, the traders also try to obtain responsible jobs in the court authorities. It is the case in the 1920s with G. Malet in Montpellier which is first judge of the Commercial court of Montpellier from 1921 till 1931. If we take as an example the year of 1923 in Montpellier, the president of the commercial court is a trader (E. Leenhardt) and five judges on ten also. In Agde, the figure amounts to two out of three whereas in Béziers, the Capital of the wine, the president is a former trader and four judges on seven are traders. In Sète, the president and five judges on eight are traders.

Finally, the traders also appropriate a last instrument of control that are the Commissions of price listing. These commissions which meet in the region at the end markets are composed only with traders. They refuse to see sitting in the Commission the Producers when these last ones make the request. In 1925, the Chamber of Commerce of Béziers authorizes moreover, by virtue of the law of 1866 which establishes the Commission, the presence of the agricultural organizations that "in an advisory role" 10 . It is also an evident will there to control the economic territory from the Languedoc through a central institution in the establishment of the prices and thus an essential instrument in the market of wines.

B. The development of syndicalism

In the meantime is set up the frame of syndicalism. Until 1900, the local syndicates are uncoordinated ones, sometimes even opposed on certain conflicting points. There is at the beginning of the century about a dozen syndicates, essentially located in the area of production-expedition: Carcassonne, Narbonne, Perpignan, Montpellier, Béziers, Sète, Nimes for the most important, with more limited ones in number and in influence in Lunel, Marseillan, Pézenas.

Syndicates are create with the aim of "representing the general interests of the Trade, everywhere where need will be and to act on its behalf for the care of its collective interests "11 . Created for the greater part in the 1880s, they get organized around a Syndical Chamber in the head of which sits a president. In Sète, where the Syndicate is create in 1888, for example, the Syndical Chamber is elected for three years and consists of 12 members, eligible for reelection by two-third every three years. This Syndical Chamber is the government of the Syndicate and it makes the decisions on its behalf. In the head of the Syndicate, we find an elected president, two vice-presidents, a secretary and a treasurer. It gathers every month for ordinary meetings and every year for the General assembly (in March in Sète). On the initiative of the Chamber or twelve members of the Syndicate, extraordinary assemblies can be held. The role of the Syndicate is to defend its members but also the interests of the Trade.

It is invested with "any powers to defend by all the licit means the interests of the Trade in front of or against any competent authority, administrative such as customs, the Régie12 , railroad, etc."13 . So, if a legal action is brought to court in the name of a member, it is made in the interest and in the name of the Syndicate which contributes at the expense of representation but also to the sentences, if any. In case of attribution of damages, the Syndicate keeps 20 % of the sum. To defend the Syndicate and its members, the Syndicate has a counselor. All the regional Syndicates work, more or less, under this shape and in a corporatist logic try to defend the interests of their members (approximately 80 to 90 Bret in its strater speech, the objective of the Syndicate is clearly expressed: " Fight together the difficulties and the encroachments of the administrations ". For Bret, it is evident that if every Syndicate has to keep its autonomy and its freedom of initiative, it is fundamental to group together to defend common values on the southern and the national markets. The ambition is "to join forces against all which stops the progress and the ease of our transactions". Thus, the purpose is to build a solid and powerful organization, capable of carrying the voice of the Trade to defend its interests and ensure the tranquility of its company. It is thus in the logic of fluidification of the commercial activities that this new organization has been given birth.

So, Bret continues by stigmatizing the laws "inequitable and inequitable which paralyze us". The purpose is clear, the enemy is identified: the State and its by-products: the Régie and the Customs. Moreover, from this first speech, Bret advances the abuses of the State control and the harassments which it imposes to the wholesalers. And to make a success, it is not only necessary, according to Bret to protest or to speak out but it is, above all, necessary to act, where from the necessity of a union and the implementation of an action of lobbying with the local and national public authorities. To insist indeed on the necessity of grouping together, Bret ends his speech with a letter which he sent to Millerand, then Minister of Trade, and stayed unanswered. And he ends on his conviction of a bigger efficiency and so, a bigger respect for the interests of the Trade, if the grouping is voted. Following some exchanges, the statuses are unanimously adopted. Thus, in the middle of 1920s, while the

II. An efficient and targeted propaganda

In the optics of a strengthening of its weight and its presence on the market of wines and in the sector in general, the Trade elaborates a speech which aims above all at defending its interests and at legitimizing its action. The purpose is double: to offer a bigger visibility to a slandered profession and make sure the control of the wine market.

A. A multidimensional message

The message aims, first of all, at the defense of the commercial interests of the Trade: limitation of the administrative harassments, the reduction of taxes and expenses which burden the transactions. Finally, the traders keep throughout period this priority objective which is the control of the market. As soon as 1900, it is P. Bret who considers necessary to "increase the opportunities of our Trade". This is confirmed by the numerous demands uttered this day by the present Syndicates, in particular those aiming at the obstacles made for the circulation of wines. That's why the companies of railroads are a privileged target from 1900. Indeed, the wine from the Languedoc is quasi-exclusively sent by railway, except for rare exceptions (Canal du Midi but in sharp drop or automobile transports which develop but remain still minority especially for the distances exceeding 200 km). To facilitate the expeditions, the demands are numerous and varied: Abolition of the rights of Octroi, obligation for the rail companies to inform the addressees, the revision of the rates of the carriers, the preferential prices for complete trains as done in other sectors, etc. But the control issue is also dealt through the question of quality. In 1900, P. Bret evokes the question to make " appreciate the real wine of France " 23 , while at the end of period, the Trade subscribes to the comment of deputy Barthe 24 which underlines that "the production should rather have by its side a prosperous Trade directed to the defense of quality wines " 25 . In this particular case, the Syndicate authorities always defended the necessary quality of wines, that it is the natural quality of the wine through the control and fights of the frauds (sugaring, watering, alcoholisation) or the gustative quality of wines, in particular after 1920.

B. Some effective tools

To make its message go through, the Trade has effective tools allowing it to connect with a public of specialists but also, at the same time, from 1920s, in a desire of an extended control, the whole population from the Languedoc.

The main propaganda instruments are the various publications stemming from the world of commerce. So, in a profoundly disturbed context and a wine landscape profoundly From the first number, the Trade justifies its action through an entitled circular "The question of wines" which legitimizes the activities of the wholesalers who are "provided with beautiful reserves of diverse natural wines, chosen among the best (…) to whom it is up to correct the defects by looking for the mixture of natural wines, giving a perfect body and a modest price" 29 . Beyond this propagandist and very justifiable effort in a corporatist publication, the Bulletin aims at propagating the official speech of the syndical institution through well targeted axes of defense such as the promotion of the natural wine, the regulation of transport (routing and expeditions), the role of the other economic actors of the wine network (brokers, landlords). The Bulletin also aims at a better knowledge of the wineproducing techniques and happens the echo of the oenological conferences or the technical improvements 30 . It spreads also widely the orders of justice, what is important for a period (1900-1920) when the jurisprudence shapes the market of wines and while the legislation hardens. Finally, he proposes to its members classified ads, in particular concerning phytosanitary products, wine making process, services of sales representatives or different business affairs. because the creation of a powerful corporate organ was imposed upon the Southern Federation ". Besides, this new format will allow to suit "in the importance of our grouping [and] in the extension for which it is justifiable to hope" 31 . Now a public and official organ, l'Action Méridionale becomes then voice of the Federation and the tone is there different.

Appearing semimonthly, on four pages (eight in rare exceptions -Congresses or extraordinary laws), it is naturally sent to all the members but is also available in cafés and kiosks of the region. It is also sent to all the members of Parliament of the wine departments. Naturally, we find articles connected to the news ( The "wine-producing Chronicle" section) and to the commercial interests but also the reports of the markets of the region, the prices of wines, reports of union meetings ("The federal life") as well as more non-specialized articles linked to the wine civilization. The tone is there voluntary polemical, so much in the titles ("For our good wines", "Controversy", "An adjustment", "Economic at first!", " Illicit Speculation") that in the themes treated. These last ones often catch up with the usual themes of corporatist defense: legitimization of the activity, the free-enterprise, the ease of the Trade, the tax relief, the control of the market. We also find there articles dealing with major fiscal reforms or legislative ones as in 1928 about the reform on "the tax on the sales" the title of which blocks the headlines on 5 columns 32 or in 1930 when the "Parliamentary Commission on the causes of the crisis" visits the region 33 . Nonetheless, a new axis is defined, that of the propaganda for the promotion of wines from the Languedoc through the question of the exports and the participations in the Exhibitions and the others competitions. Finally, we find also numerous advertisements there and announces, proof of the wide distribution of the newspaper.

III. The impacts of a well-planned process

Thanks to strategies which show to be particularly effective, the traders are capable, in the bend of the 1920s, to position as an indispensable actor of the market of wines and especially as a dominant actor. It becomes a reality of two manners: by the audience which it experiences and by the imposition of normative practices on the market.

A. A respected and privileged interlocutor

The Trade through his leading position in the consular authorities or thanks to the action and to the activity of its Syndicates becomes little by little a privileged interlocutor. (1900-1940) situation seems to improve, both as for the question of the saturation and congestion or the thefts. Nevertheless, in September, a new controversy arises, regarding large delays, between the station of Montpellier and the Syndicate which decides then to appeal to the Chamber of Commerce to solve the problem. At the end of September, the engineer of the company who manages the station (PLM 36 ) commits himself to the fact that works of enlargement are soon undertaken 37 . In 1914, it is through the members of Parliament as Aldy, a deputy of the Aude, that the Trade pressed the government so that this one in his turn intervenes with rail companies, in particular in the optics of a bigger fluidity in the expeditions 38 . Time as we can see is very important issue for the wholesalers and always in the midst of negotiations between authorities, Syndicates and train companies. After the war, the contacts are also many, mainly reports on the rates of transportation, like the ones presented by A. Gaillard in the 1920s for the Chamber of Commerce of Beziers 39 or in some articles of l'Action

Méridionale.

In (1900-1940) what is already a victory in itself when we know the balance of power in the 1900s or in the 1910s. Besides, certain points promoted by the trade are adopted by the commission and integrated into the future law, in particular the question of the wine lockers, the quality control or the analysis of wines42 .

Lastly, the Trade is also in constant dialogue with the Administration (in particular

B. The normalization and the standardization of the market

Beyond the visibility and the respectability which they underlie, all the strategies settled by the Trade aim the imposition of transactional practices on the market of wines.

These practices concern, first of all, the shapes of the deals themselves. In this context, the question of the brokerage is, in this trend of normalization and standardization, a central question. The broker is the one who puts in relation the trader and the wine-grower. It is an important intermediary for wine companies because it is the reliable person who proposes quality wines and who knows well the production -in all its varieties -of a given land. Thus, (1900-1940) the other local syndicates, an increase of the rate of brokerage, now up to 75 centimes an hectoliter. But very quickly, the union representatives of the traders get the situation in hand and near the end of 1926, a project of uniform rules is proposed by the Southern Federation.

Fatally, this rule subjects the brokers to the regional Trade through a professional ID card designed by the Southern Federation and at the written commitment not to sell to the traders foreign to the region. From their part, the traders make a commitment to deal only with the brokers owners of this card. Above all, proof of the institutionalization of the commercial circuits, any deal is officialised from now on by a slip on which will be registered all the information relative to the contract, in particular the names of the seller, the buyer and the broker. Finally, in case of dispute, a Chamber of equal arbitration will be gathered to judge 51 .

Less than twenty years after their first tries, the wholesalers fulfilled their thirst of domination over the brokers.

Lastly, the syndicate effort for a standardization of the qualitative criteria is also

Conclusion

In conclusion, it is evident that the corporatist effort paid off. Patiently but resolutely, the regional Trade, through the institutions which it dominated or through direct contacts with public authorities applied its program of domination on the wine market. This power which leans on positions of choice in the economic sector of the region, effective propaganda and targeted message is elaborated around recurring themes such as the purge, the normalization and the standardization of the market. Beyond these simple demands, it is the security and tranquility of the transactions that look for the traders, in a paradoxical context where they are legitimized (as profession but also as intermediary bodies) but also questioned by more and more aggressive competitors.

This syndical power becomes a reality in the quantitative weight of the representative authorities (local or regional) but also during public demonstrations as the meeting of protests which regroups thousands of persons in Montpelier on November 30th, 1909 57 or the corporatist reunion of Lézignan-Corbières in May, 1930 58 in honor of deputy Léon Castel 59 .

Another significant example is when P. Cassan is buried in February, 1935 in Narbonne, Castel, mayor-deputy of the city, Mr Sarrault, the former senator, the whole City Council, the prefect, the president of the Commercial court, the administrator of the Bank of France among others, form a procession which is followed by hundreds of persons and the quasi-totality of the commercial elite of the region 60 . Nevertheless, the period which opens is the one questionings. The wine growers organize Wine-growers' Cooperative Warehouse better and better equipped commercially and which are going to dash, massively, into the trade war. The brokers set up syndical structures which give birth after the war to a powerful national Federation and which brings the brokers out of the submission of the wholesalers. Above all, the Second war discredits the traders who took advantage of the conflict to grow rich and who pay their image of major interlocutor of public authorities. Thus, while is established the golden age of the syndical unionism in the Region, it closes almost so early, plunging the traders into rejection and into the oblivion.

  . The next Chair who succeeds him, is P. Granaud, a wholesaler, between 1934 and 1939. In Sète, it is Jean Prats, of the powerful house "Cazalis and Prats", who carries out the function during more than 25 years, between 1908 and 1933. Prats, who becomes the president of the National Syndicate of the wine in 1925, is named in 1931 president of the quite recent Xth Economic region, post which succeeds him in 1935, Gustave Malet, himself preside of the Chamber of Commerce of Montpellier from 1935 until 1941. In Carcassonne, Emile Génie, president of the Syndicate of wines of the city is at the same time a judge then a president of the Commercial court and the member of the Interministerial Committee of the Vine growing. In the department of Aude, A. Pradal succeeds in the 1920s to P. Cassan in the head of the powerful Syndicate of Lézignan-Corbières then to the Chamber of Commerce of Narbonne 9 .

  Malet, then president of the Syndicate, complains about a stand offered by these newspapers to Mr Ravel, the President of the Federation of the Wine-growers' Cooperative Warehouse, who attacked the wholesalers 21 . Ravel considers the traders as "parasites living to the detriment of the producers and consumers ". Here, the corporatist defense aims at protecting and redeeming the image and the legitimacy of the traders, often questioned by the producers, in a context of crisis and hyper-competition. Malet also blame the Wine-growers' Cooperative Warehouse for benefiting from grants and from fiscal exemptions which discriminate the Trade and because owing to the Wine-growers' Cooperative Warehouse actions, the wholesalers are now excluded of distribution networks 22 . Proof that this theme is a recurring theme, the terms -from a side as the other one -are exactly the same in 1939 in an article referring to the comments of E. Orsetti, then president of the regional Syndicate. In December, 1931, always in the same logic, it is question of the sales reprersentatives (which quickly develop in the bend of the 1930s) and of their welfare. The traders hope that the vagueness of the law voted in 1928 and modified in 1930 will not be made in their disfavor and will not add additional expenditure.

First

  of all, it is in constant connection with the State either through the members of Parliament, or through the Ministers. The contacts between the Trade and the representatives of the State are then consistent and through numerous channels. It is the case in particular regarding railroads. So, ten years after the letter presented by P. Bret whom Millerand has never answered, the same Millerand is coming to Montpellier. Become Minister of the Public works, he is contacted by the Syndicate of Montpellier and he agrees to receive them in January, 1910 in order to hear their complaints. Eleven months later, he answers, in a letter to these complaints concerning the dysfunctions of the train station of Montpellier. But, following an investigation of his services, he does not see the necessity of giving a favorable recommendation to the requests of the trade (loading of Tanker-Wagons, excessive rates, thefts of wine). In front of this refusal, the Syndicate sends a new letter which, eventually, conducts to certain changes a few months later like the improvement of the platforms or the limitation of price rates. It's a first victory for the Syndicate but it's not enough since the Syndicate from Béziers engaged the same procedure and was proved right in the enlargement of the station 34 . At the end of 1911, the Syndicate from Montpellier pressures the station of Montpellier for the delays which continue to accumulate in spite of the observations of the Minister 35 . Same thing with the thefts which raise in 1913. Eventually, in June, 1913, after a visit of E. Leenhardt, president of the Syndicate from Montpellier, with the Stationmaster, the 34 BM, dec 1910 9 th European Social Science History Conference, Glasgow -11-14 April 2012 Session : Corporatism in Rural Europe, XIX th -XX th Centuries Le Bras S. -Corporatism in the wine warehouses

  the 1930s, a new turn is taken and the message is now oriented over the purification of the market, disrupted essentially by Algerian wines and bad wines. So, in a report presented in front of the Chamber of Commerce of Béziers in April, 1932 and sent to the members of Parliament of the region as well as to Ministers of Trade and the Industry, to the State Minister of the Education and to the other Chambers of commerce of the Midi, Bastide d'Izard, which is an influential wholesaler from Béziers, presents its solutions to cleanse the market, in particular the compulsory distillation and the quota system 40 . This approach joins in logic of a demand which peak is the coming in the region of the Parliamentary committee charged with investigating into the wine crisis in 1931 41 . During the hearingswhich are held in Montpellier, to Béziers, in Perpignan, in Carcassonne in February, 1930 and in Nimes in June, 1931, the representatives of the Trade are heard every time, sometimes even several times for H. Campanaud then President of the Southern Federation and G. Malet, honorary president of the Federation. The voice of the trade is then listened to, as much as that of the Production, 36 Paris-Lyon-Mediterranée, the most powerful company of the area connecting Paris to the South of France

Fraud

  control and The Régie), what allows to improve the conditions of exercise of the wine companies. So in July, 1912, the President of the Syndicate is notified by the Director of Indirect taxes about certain irregularities committed by the carters who discharge the barrels of wines in front of the station when the station is closed. After a dialogue between the President and the Director, this last one accepts, temporary, the situation. This way, the Syndicate has the time to let know to its members that this practice is forbidden and passable of pursuits. In the 1920s, the various reports of the chambers of commerce where dominate tend themselves to alert the Administration of the necessity of watching the frauds which seem to continue massively in the region. So in a report presented by Gaillard, Oustric and Araou in May, 1929, on the causes of the crisis, wheter natural (regional overproduction and wines of Algeria) or artificial (sugaring and abnormal wines), the Trade asks for the stricter control of the practices of sugaring, for the distillation of abnormal wines and more active surveillance of the movements of wines 43 .

  important. From 1907, the traders by the voice of their Syndicates or of the Chambers of commerce insist on the qualitative orientation that the market of wines has to set. Thus, the session of June 3rd, 1907, chaired by F. Guy who is of one of the biggest trading families of the city, enumerates the solutions of the wine crisis which affects the region. All are associated to the purification of the sector by pursuing and by fighting more effectively watering, sugaring, the wines of synthesis 52 . Moreover in 1911, during the annual congress, Roussel-Lacombe, then General Secretary of the regional Syndicate rejoices with the creation of the department of the repression of the frauds in 1907 53 .In the 1920s and the 1930s, the qualitative argument is in the center of the debate. So, the quality notion is ceaselessly, from the end of 1920s, in the heart of the demands of the Trade to the State. It is a crucial step and a major awareness towards the conceptualization of a quality market of which the traders would be the champions54 . Above all, all these measures are largely spoken out in the hearings during the visit of the Committee Barthe and they appear mainly in the laws which define from 1931 the wine Status 55 . The traders become Ibid. ADH 8 M179, PV Chambre de commerce de Béziers sur la crise de 1907 BM, May 1911 AM, 15/08/25, « Nos bons vins », Bagnol J.-M., Op. cit.moreover the first defenders as proven by the numerous reports of the sessions of the Committee of wines of the Chamber of Commerce of Béziers which demands a better control in the execution of the quota systems as well as a fiercer repression of sprains in the status of 1931 56 . It is the proof there, if it was necessary need to underline it, of the undeniable dimension that has taken the Trade of wines, at the same time upstream and downstream to the legislative or administrative decision-making.

9 th European Social Science History Conference, Glasgow -11-14 April 2012 Session : Corporatism in Rural Europe, XIX th -XX th Centuries Le Bras S. -Corporatism in the wine warehouses (1900-1940)

9 th European Social Science History Conference, Glasgow -11-14 April 2012 Session : Corporatism in Rural Europe, XIX th -XX th Centuries Le Bras S. -Corporatism in the wine warehouses (1900-1940)

9 th European Social Science History Conference, Glasgow -11-14 April 2012 Session : Corporatism in Rural Europe, XIX th -XX th Centuries Le Bras S. -Corporatism in the wine warehouses (1900-1940) members

  for Sète over the period, there is 100 -120 in Montpellier and between 50 in 80 in Béziers).But in front of the disorganization of the market, the local Syndicates decide to unite. So, October 23rd, 1900, P. Bret, the president of the Syndicate of Montpellier receives in the prestigious Hotel of the Metropolis the delegates of the Syndicates of Nîmes, Perpignan, Béziers and Narbonne as well as the numerous members of the Syndicate of Montpellier, while Caffarel of the Syndicate of Sète is present but non-officially 14 . J. Génie of Carcassonne is not present but reassures Bret, by telegram, of his support. All the largest syndicates of the region are here, directly or indirectly, represented. In a context of crisis recalled by president

  regional Syndicate became the Southern Federation of the Wholesale trade of Wines and Spirit, the Syndicate includes the Syndicates of Béziers, Carcassonne, Sète, Lézignan, Lunel, Montpellier, Nimes and Perpignan, the eight largest Syndicates of the region. All in all, in the middle of 1920s, about 650 members give to it a considerable strength and weight 15 .

th European Social Science History Conference, Glasgow -11-14 April 2012 Session : Corporatism in Rural Europe, XIX th -XX th Centuries Le Bras S. -Corporatism in the wine warehouses (1900-1940) expeditions

  Octroi 17 in the cities which are provided with it, professional taxes, general taxes. Moreover, is added the action of the "Administration of the indirect taxes" which verifies in the quality and the quantity of the goods held, received and sent by the Trade. Besides the regular and sometimes binding control (it is necessary to declare, for every transaction, the and the receptions), the State control can intervene by surprise, in case of suspicion or of denunciation, what is, generally very badly experienced by the traders. So, in 1912, the Syndicate is becoming concerned by the condemnation of a trader from Montpellier for "Watering" 18 . While very clearly the investigations showed that it was the producer who was responsible for the fraud, the Magistrates' court convicted the trader for the reason that he was the only signatory of the receipt. It is for the Trade a judgment which risks "to deteriorate terribly our situation " 19 . Criticism against the civil or penal jurisdictions are recurring as in 1913 when E. Leenhardt criticizes the decisions "which surprise and amaze by the ignorance of the particular conditions of certain deals". Leenhardt proposes that the Syndicate of Montpellier is named as expert for the information to be supplied20 . In 1924, in

The question of the limitation of the expenditure is a recurring question in the debate which is established within labor unions. In May, 1904, during the meeting of the Syndical chamber of Sète, it is stated that the wholesalers are facing "the fatal tendency of the regime to require more and more heavy sacrifices (…) which strike heavily the Trade and specially the Trade of drinks " 16 . It is necessary to notice here that expenditure pressure on the traders are very heavy: franchise tax to set up a business, License for the detention of drinks, taxes on circulation, 15 ADH, Annuaire départemental, 1925 16 AMS, 7F727, Op. cit. 17 Commercial boundaries in the cities. You had to pay a tax to get through them in. They are progressively suppressed from late XIX th Century to early XX th 9 a letter sent to the regional newspapers (L'Eclair, Le Petit Méridional and la Dépêche), G.

th European Social Science History Conference, Glasgow -11-14 April 2012 Session : Corporatism in Rural Europe, XIX th -XX th Centuries Le Bras S. -Corporatism in the wine warehouses (1900-1940)

Adding illegally water to the wines in order to get more quantity ADH PAR 2043, Bulletin Mensuel (BM), n°60-61, dec-janv 1913 BM, feb-mar 1913 BM,sept 1924 Which is really abusive and excessive for the period 9

th European Social Science History Conference, Glasgow -11-14 April 2012 Session : Corporatism in Rural Europe, XIX th -XX th Centuries Le Bras S. -Corporatism in the wine warehouses (1900-1940) remodeled

  in 1907 26 , the regional Syndicate decides to publish a bulletin of connection, the Monthly Bulletin. Its real name is the Monthly Bulletin of the Syndicate of the wholesale Trade of Wines and Spirit of Montpellier and the Syndicates affiliated to the Regional Syndicate of the Midi. It shows here the initiative and effort of the powerful Syndicate from Montpellier which gives birth to this instrument of connection and propaganda in October, 1907. The objective of the Bulletin is simple: make the voice of the Trade carried out, legitimize its action and spread publically its decisions. It is important all the more as the Trade was one of the targets of the movements of the spring, 1907 and of the parliamentary debates which accompanied them 27 . Besides, the wine growers are meeting in a powerful organization, the CGV 28 and it is necessary, even if, at first, the traders are associated to it, not to leave too much space to this new body.

23 AMM, NC, Op. Cit 24 Bagnol J.-M., 2011. Le midi viticole au parlement : Edouard Barthe et les députés du vin de l'Hérault

(années 1920-1930)

, Montpellier, PULM 25 L'Action Méridionale (AM), August 1939 9

  In a logic of development and broader influence, the Southern Federation decides to change the format of its publication. It explains its initiative in the last number of the Bulletin in September, 1924. It is then decided to transform the Bulletin into " a real newspaper (…)

	26 In 1907, a huge crisis stroke the region, leading hundreds of thousands of wine-growers to protest in the
	biggest cities of the region, eventually giving ways to new reforms

27

Wholesalers are often referred to scientists and fraud makers-which is largely incorrect 28 Gavignaud-Fontaine G., « Aux origines de la CGV dans le Midi : combats pour la défense du vin naturel et la maîtrise des marchés des vins »,

Etudes héraultaises, 2007-2008, n° 37-38, p. 141-154

29 BM n°1, oct 1907 30 BM, jan 1910, « Résumé des conférences pratiques d'oenologie pratique de M. Guiraud »

th European Social Science History Conference, Glasgow -11-14 April 2012 Session : Corporatism in Rural Europe, XIX th -XX th Centuries Le Bras S. -Corporatism in the wine warehouses (1900-1940)

	9
	BM, sept 1924
	AM, 31/12/1928
	AM, 01/03/1930

9 th European Social Science History Conference, Glasgow -11-14 April 2012 Session : Corporatism in Rural Europe, XIX th -XX th Centuries Le Bras S. -Corporatism in the wine warehouses

  Chambre de commercede Béziers, Transports des vins, 1919-1921 40 ADH 8 M 231, Chambre de Commerce de Béziers, Commissions des Vins 41 Archives de l'Assemblée Nationale, Rapport fait au nom de la Commission des boissons chargée de procéder à une enquête sur la situation de la viticulture deFrance et d'Algérie, Paris, 1930-1931

	37 BM, june and october 1913
	38 BM, sept 1914
	39 ADH, 8 M 308,

9 th European Social Science History Conference, Glasgow -11-14 April 2012 Session : Corporatism in Rural Europe, XIX th -XX th Centuries Le Bras S. -Corporatism in the wine warehouses

  56 ADH, 8 M 231, Chambre de Commerce de Béziers, Commissions des Vins, Délibérations sur la crise viticole, 1932-1933 57 BM, Nov 1909 58 AM 19 May 1930 59 The MP who presented the law reforming the professional tax in 1928 60 AM, mars 1935

Lachiver M., Vins, vignes et vignerons, Paris, Fayard, 1988

Dugrand R., Villes et campagnes enBas-Languedoc, Paris, PUF, 1963

The CGV ("Confederation générale des vignerons" -General confederation of Wine-growers) or the Winegrowers' Cooperative Warehouses

7 Chatriot A.et Lemercier C., « Les corps intermédiaires » in Duclert V. et Prochasson C., Dictionnaire critique de la République, 2002, p. 691-

698. 8 Lacombrade P., « Les Chambres de Commerce du Languedoc face à la question viticole sous la Troisième République » in Vin etRépublique (1907République ( -2007)), Actes du colloque, L'Harmattan, 2010, p.

114. 9 th European Social Science History Conference, Glasgow -11-14 April 2012 Session : Corporatism in Rural Europe, XIX th -XX th Centuries Le Bras S. -Corporatism in the wine warehouses

Archives Municipales de Sète (AMS), 7F737, Statut du Syndicat de Commerce en Gros deCette, 1904

The Régie is the other name of the Administration of indirect taxes. It oversees all the transactions and movements of goods -here wines -in the department.

AMS, 7F737, Op. Cit.

Archives municipales de Montpellier (AMM), NC, Syndicat des vins de Montpellier, Procès-Verbal de la réunion du 23/10/1900 9 th European Social Science History Conference, Glasgow -11-14 April 2012 Session : Corporatism in Rural Europe, XIX th -XX th Centuries

AAN, Rapport Barthe, op. cit., T. I, p. 44 et p. 48.

9 th European Social Science History Conference, Glasgow -11-14 April 2012 Session : Corporatism in Rural Europe, XIX th -XX th Centuries Le Bras S. -Corporatism in the wine warehouses (1900-1940) 9 th European Social Science History Conference, Glasgow -11-14 April 2012 Session : Corporatism in Rural Europe, XIX th -XX th Centuries Le Bras S. -Corporatism in the wine warehouses

the question of the rate of brokerage, which is the remuneration for the brokers, and the one of the respectability of the brokers are recurring subjects in the speech of the traders. For example, in 1909, we learn in the Bulletin that the Syndicate of Montpellier distributed brokers' card by which these last ones make a commitment on several points: to not take out of their role of intermediary, to respect the local uses, to propose the best goods 44 . In exchange, the members of the Syndicate make a commitment to appeal to these very brokers in priority. This safety of the exchanges is essential for the Trade so much that in 1910, an additional step is taken. A draft contract is proposed by which the brokers should sign the slip of sample 45 and thus to be responsible for sold goods. With this contract, the Trade makes a commitment to pay the brokers at the level of 25 centimes an hectoliter, what is not still the case at the beginning of the century. But the attempt does not finally succeed in front of the refusal of the brokers of a modification of the local uses. On the other hand, the Syndicate from Montpellier plans, in 1911, to edit a "black book" 46 in which would be registered the incorrect brokers, in particular those who deal directly with the Outside 47 . In the following years, Syndicates try to limit the importance of the brokers-agents who work at the same time as brokers and as commission agents 48 , there also with a sure success, these last ones disappearing more or less except in some places (Olonzac where they reappear at the end of 1930s). Eventually, the issue of the brokers seems settled because, scanning the Bulletin, in 1908-1909, it is discussed in 9 articles while for 1912 and 1913, it concerns only two articles 49 .

It is necessary to wait the 1920s for the question of the brokerage to return. These last ones distinguish themselves in the 1920s by conducting movements of contestation, especially of the rates of brokerage. During the most sharp cases (on 1921 or 1926), it is every time the same pattern. In April, 1926, the brokers dash into a long-term unitarian strike in the course of which they demand a considerable increase of the rate of brokerage 50 . In front of this movement, the traders, little inclined to see a part of their profits flying away, propose an appreciation and in May, the Syndicate of Montpellier is the first one to propose, followed by 44 BM, jan 1909 45 Before the agreement, the buyer tastes samples which are accompanied with a receipt signed by the seller 46 BM, oct 1911 47 Outside is the name given to the region where the wines were shipped (Paris, Lyon, East of France) 48 The Commission agents are wholesalers with no stock and no warehouses. They are buying in the name of another wholesaler, generally from the Outside. They receive a commission, hence their name. (1900-1940)