Michel Waldschmidt

Transcendence of Periods: The State of the Art

Keywords: Periods, transcendental numbers, irrationality, integrals, series, Diophantine approximation, irrationality measures, transcendence measures, measures of algebraic independence, Gamma function, Beta function, zeta function, multiple zeta values (MZV). AMS subject classifications 11J81 11J86 11J89

The set of real numbers and the set of complex numbers have the power of continuum. Among these numbers, those which are "interesting", which appear "naturally", which deserve our attention, form a countable set. Starting from this point of view we are interested in the periods as defined by M. Kontsevich and D. Zagier. We give the state of the art on the question of the arithmetic nature of these numbers: to decide whether a period is a rational number, an irrational algebraic number or else a transcendental number is the object of a few theorems and of many conjectures. We also consider the approximation of such numbers by rational or algebraic numbers.

Introduction

In their paper [START_REF] Kontsevich | Mathematics unlimited-2001 and beyond[END_REF] whose title is "Periods", M. Kontsevich and D. Zagier introduce the notion of periods; they give two definitions and claim that they are equivalent. They propose one conjecture, two principles and five problems. The first principle reads as follows: "Whenever you meet a new number, and have decided (or convinced yourself ) that it is transcendental, try to figure out whether it is a period".

If a number is not a period, then it is a transcendental number; indeed, the set of periods is a sub-algebra of C over the field Q of algebraic numbers, hence any algebraic number is a period. We note in passing that there is no explicit known example of a complex number which is not a period (this is Problem 3 in [START_REF] Kontsevich | Mathematics unlimited-2001 and beyond[END_REF]).

The goal of this paper is to study the situation when the answer is positive: what is known on the transcendence of periods?

We also consider the quantitative related aspect of this question, in connection with the following question in [START_REF] Kontsevich | Mathematics unlimited-2001 and beyond[END_REF], § 1.2 before their conjecture 1: we can check easily the equality between two algebraic numbers by computing both of them numerically to sufficiently high precision and using the fact that two different solutions of algebraic equations with integer coefficients of given degree and height cannot be too close to each other (this is the so-called Liouville inequality). In the example they give, due to D. Shanks [START_REF]Shanks -Incredible identities[END_REF]:

( The number N = γ is a rational integer. It suffices to compute its numerical value with an accuracy of one digit to check that it satisfies -1 < N < 1, hence it is zero (this is the easiest case of Liouville inequality [START_REF] Waldschmidt | Diophantine approximation on linear algebraic groups[END_REF] § 3.5). Hence one at least of the 64 factors γ in the product is zero; the equality (1) easily follows.

The question raised by Kontsevich and Zagier in [START_REF] Kontsevich | Mathematics unlimited-2001 and beyond[END_REF] § 1.2 is whether something similar can be done for periods. One should define a notion of complexity for a period which would be analogous to the notion of height for an algebraic number, and then to prove a lower bound of this complexity for a nonzero period, as a substitute for Liouville inequality. One of their suggestions is to count the amount of ink or the number of T E X keystrokes required to write down the considered number.

It would be interesting to know whether there exist periods which are at the same time Liouville number. A positive answer to the following question would mean that there are no such Liouville periods. Question 2. Let θ be real period. Does there exists a constant c(θ) > 0 such that, for any rational number p/q distinct from θ with q ≥ 2, the lower bound θ -p q > 1 q c(θ) holds?

A more ambitious goal would be to ask whether real or complex periods behave, from the point of view of Diophantine approximation by algebraic numbers, like almost all real or complex numbers [START_REF] Bugeaud | Approximation by algebraic numbers[END_REF][START_REF] Waldschmidt | Diophantine approximation on linear algebraic groups[END_REF] Question 3. Given a transcendental period θ ∈ C, does there exists a constant κ(θ) such that, for any nonzero polynomial P ∈ Z[X], we have

|P (θ)| ≥ H -κ(θ)d ,
where H ≥ 2 is an upper bound for the (usual) height of P (maximum absolute value of the coefficients) and d its degree?

Abelian integrals

The arithmetic nature of the value of the integral of an algebraic function of one variable between algebraic (or infinite) limits is now well known, both from the qualitative and quantitative point of view, as we now explain.

2.1. Genus 0: logarithms of algebraic numbers. The main tool is Baker's Theorem on linear independence, over the field Q of algebraic numbers, of logarithms of algebraic numbers. Here we shall use only the following special case. Theorem 4. Let α 1 , . . . , α n be nonzero algebraic numbers, β 1 , . . . , β n algebraic numbers, and, for 1 ≤ i ≤ n, let log α i be a complex logarithm of α i . Then the number

β 1 log α 1 + • • • + β n log α n is either zero or transcendental.
Here is a consequence of Baker's Theorem 4:

Corollary 5. Let P and Q be polynomials with algebraic coefficients which satisfy deg P < deg Q and let γ be either a closed path, or a path whose limits are either infinite or algebraic. If the integral

(6) γ P (z) Q(z) dz
exists, then its value is either zero or transcendental.

A well known example [START_REF] Siegel | Transcendental Numbers[END_REF] p. 97 is

1 0 dt 1 + t 3 = 1 3 log 2 + π √ 3 •
Corollary 5 follows from Theorem 4 by solving the rational fraction P (z)/Q(z) into partial fractions (see for instance [START_REF]Van der Poorten -On the arithmetic nature of definite integrals of rational functions[END_REF]). As a matter of fact, Corollary 5 is equivalent to Theorem 4: it suffices to write down the logarithm of an algebraic number as a period; for the principle value of the logarithm, when α is not a negative real number, we have for instance

log α = ∞ 0 (α -1)dt (t + 1)(αt + 1) , while iπ = 2i ∞ 0 dt 1 + t 2 •
The known measures of linear independence of logarithms of algebraic numbers (lower bounds for linear combinations, with algebraic coefficients, of logarithms of algebraic numbers -see for instance [START_REF] Waldschmidt | Diophantine approximation on linear algebraic groups[END_REF]) imply that the absolute value of a nonzero integral of the form ( 6) is explicitly bounded from below in terms of the heights and degrees of P and Q as well as the heights and degrees of the algebraic numbers which are the limits of the path γ.

2.2. Genus 1: elliptic integrals. The arithmetic nature of elliptic integrals of the first or the second kind has been studied as early as 1934 [START_REF] Schneider | Transzendenzuntersuchungen periodischer Funktionen[END_REF] and then 1937 [START_REF]Arithmetische Untersuchungen elliptischer Integrale[END_REF] by Th. Schneider. Here is Theorem 15 version III in [START_REF] Schneider | Einführung in die transzendenten Zahlen[END_REF].

Theorem 7. The value of any elliptic integral of the first or the second kind with algebraic coefficients between algebraic limits is either zero or else transcendental.

In particular any nonzero period of an elliptic integral of the first or the second kind with algebraic coefficients is transcendental.

Theorem 16 in [START_REF] Schneider | Einführung in die transzendenten Zahlen[END_REF] deals with the transcendence of the quotient of two elliptic integrals of the first kind.

A consequence, which is quoted by Schneider of his Theorem 17 in [START_REF] Schneider | Einführung in die transzendenten Zahlen[END_REF], can be stated: The value of an elliptic integral of the first or the second kind between algebraic limits is the quotient of a period by a factor which is either rational or transcendental.

From Theorem 7 one deduces the result quoted in [START_REF] Kontsevich | Mathematics unlimited-2001 and beyond[END_REF] § 1.1: if a and b are two positive algebraic numbers, the perimeter of the ellipse with radii a and b

(8) 2 b -b 1 + a 2 x 2 b 4 -b 2 x 2 dx
is a transcendental number. More generally the length of any arc whose limits are points with algebraic coordinates is either zero or else transcendental.

Similar results hold for a lemniscate

(x 2 + y 2 ) 2 = 2a 2 (x 2 -y 2 )
when a is algebraic.

These results are proved by Schneider as consequences of his results on elliptic functions. Here is for instance version I of theorem 15 in [START_REF] Schneider | Einführung in die transzendenten Zahlen[END_REF]. Let ℘ be a Weierstrass elliptic function with algebraic invariants g 2 and g 3 :

℘ 2 = 4℘ 3 -g 2 ℘ -g 3 .
Let ζ be the associated Weierstrass zeta function, a and b two algebraic numbers and u a complex number which is not a pole of ℘. Assume (a, b) = (0, 0). Then one at least of the two numbers ℘(u), au + bζ(u) is transcendental.

Also, if one considers two elliptic curves

y 2 = x 3 -x and y 2 = x 3 -x
one deduces that each of the two numbers (9)

1 0 dx √ x -x 3 = 1 2 B(1/4, 1/2) = Γ(1/4) 2 2 3/2 π 1/2 and (10) 1 0 dx √ 1 -x 3 = 1 3 B(1/3, 1/2) = Γ(1/3) 3 2 4/3 3 1/2 π is transcendental.
These two formulae (compare with [START_REF] Masser | Interpolation on group varieties[END_REF] p.21) are special cases of the Chowla-Selberg formula (cf [START_REF] Gross | On the periods of abelian integrals and a formula of Chowla and Selberg[END_REF] and [START_REF] Kontsevich | Mathematics unlimited-2001 and beyond[END_REF] § 2.3) which expresses the periods of CM elliptic curves as products of values of the Euler Gamma function [START_REF] Bloch | Esnault -Gauss-Manin determinants for rank 1 irregular connections on curves[END_REF] Γ

(z) = e -γz z -1 ∞ n=1 1 + z n -1 e z/n .
The extension of Chowla-Selberg formula by G. Shimura to Abelian varieties of CM type gives rise to the relations of Deligne-Koblitz-Ogus on Gamma function (see [START_REF] Brownawell | Papanikolas -Linear independence of Gamma values in positive characteristic[END_REF]).

2.3.

Genus ≥ 1: Abelian integrals. In [START_REF]Zur Theorie des Abelschen Funktionen und Integrale[END_REF], Th. Schneider proves similar results for Abelian integrals. His proof is an extension in several variables of his previous arguments for the one dimensional case. In the analytic part of the proof, the main tool is Schwarz Lemma. He proves a several variables version of Schwarz' Lemma by means of an interpolation formula for Cartesian products. This method enables him to prove results on Abelian functions. The most important example of results he obtains is the following:

Theorem 12. Let a and b be rational numbers which are not integers and such that a + b is not an integer. Then the number

(13) B(a, b) = Γ(a)Γ(b) Γ(a + b) = 1 0 x a-1 (1 -x) b-1 dx is transcendental.
Many papers deal with the arithmetic nature of Abelian integrals: the works by Th. Schneider in 1940 have been pursued first by S. Lang in the 1960's, then by D.W. Masser using Baker's method in the 1980's, until an essentially complete solution of the question was reached in 1989 by G. Wüstholz [START_REF]Wüstholz -Algebraische Punkte auf analytischen Untergruppen algebraischer Gruppen[END_REF]. He proved a generalization of Baker's Theorem 4 to commutative algebraic groups. Essentially we know the answer to the questions of transcendence and linear independence over the field of algebraic numbers of periods of Abelian integrals of the first, the second or the third kind. For instance J. Wolfart and G. Wüstholz [START_REF] Wolfart | Wüstholz -Der Überlagerungsradius gewisser algebraischer Kurven und die Werte der Betafunktion an rationalen Stellen[END_REF] have shown that the only linear dependence relations with algebraic coefficients between the values B(a, b) of Euler Beta function at points (a, b) ∈ Q 2 are those which follow from the Deligne-Koblitz-Ogus relations. Moreover we know quantitative estimates which give lower bounds for an Abelian integral which is not zero. The most recent and sharp estimates in this context for the general situation of commutative algebraic groups are due to É. Gaudron [START_REF] Gaudron | Mesure d'indépendance linéaire de logarithmes dans un groupe algébrique commutatif[END_REF][START_REF]Mesures d'indépendance linéaire de logarithmes dans un groupe algébrique commutatif[END_REF].

Linear relations with algebraic coefficients between Abelian integrals are well understood now; , we cannot say the same so far for algebraic relations. In a footnote [START_REF]Grothendieck -On the de Rham cohomology of algebraic varieties[END_REF], A. Grothendieck suggests a conjectural statement on the transcendence of periods of Abelian varieties over the field of algebraic numbers. The first precise formulation of this conjecture in the projective smooth case is given by S. Lang in his book [START_REF] Lang | Introduction to transcendental numbers[END_REF] on transcendental numbers (see also [START_REF] Deligne | Cycles de Hodge absolus et périodes des intégrales des variétés abéliennes[END_REF]); in the same book is given for the first time the statement of Schanuel's Conjecture on the algebraic independence of values of the exponential function (which includes the conjecture 43 on algebraic independence of logarithms of algebraic numbers). This topic has been revisited by Y. André (see [START_REF]Quelques conjectures de transcendance issues de la géométrie algébrique[END_REF][START_REF]Une introduction aux motifs (motifs purs, motifs mixtes, périodes)[END_REF]) who proposes his generalized periods Conjecture ([7] 24.3.1) which contains both the Conjectures of Grothendieck and Schanuel. The recent booklet [START_REF]Une introduction aux motifs (motifs purs, motifs mixtes, périodes)[END_REF] contains a detailed exposition of Grothendieck's Conjecture, and its various versions (including the one in [START_REF] Kontsevich | Mathematics unlimited-2001 and beyond[END_REF]) and extensions.

For 1-motives attached to products of elliptic curves, C. Bertolin [START_REF]Bertolin -Périodes de 1-motifs et transcendance[END_REF] rephrased André's conjecture more explicitly by stating her elliptico-toric conjecture, which contains the expected transcendence and algebraic independence results concerning the values of the exponential function, of the Weierstrass ℘ and ζ functions, of elliptic integrals and of the modular invariant j (see also [START_REF]Open diophantine problems[END_REF][START_REF]Variations on the six exponentials theorem[END_REF]).

Values of the Euler Gamma function

From the definition (13) of Euler Beta function as an integral, it follows that its values at points in Q 2 where it is defined are periods. From the relation [START_REF] Brownawell | Transcendence in positive characteristic[END_REF] between Gamma and Beta functions, one deduces

Γ(a 1 ) • • • Γ(a n ) = Γ(a + • • • + a n ) n-1 i=1 B(a 1 + • • • + a i-1 , a i ).
Hence for any p/q ∈ Q with p > 0 and q > 0, the number Γ(p/q) q is a period. For instance

π = Γ(1/2) 2 = 1 0 x -1/2 (1 -x) -1/2 dx.
From ( 9) and ( 10) one derives also expressions for Γ(1/3) 3 and Γ(1/4) 4 as periods.

The arithmetic nature of the values of Euler Beta function at rational points a much better understood (thanks to Schneider's Theorem 12) than the values of the Gamma function. The fact that the number Γ(1/2) = √ π is transcendental follows from Lindemann's Theorem on the transcendence of π. The transcendence of the numbers Γ(1/4) and Γ(1/3) has been proved in 1976 by G.V. Čudnovs kiȋ [START_REF]Čudnovs kiȋ -Algebraic independence of constants connected with the exponential and the elliptic functions[END_REF].

Theorem 14. The two numbers Γ(1/4) and π are algebraically independent, and the same is true for the two numbers Γ(1/3) and π.

As noticed by D.W. Masser, one can formulate this statement by saying that the two numbers Γ(1/4) and Γ(1/2) are algebraically independent, and the same for Γ(1/3) and Γ(2/3).

From the standard relations (see [START_REF]Čudnovs kiȋ -Algebraic independence of constants connected with the exponential and the elliptic functions[END_REF] below) satisfied by the Gamma function one deduces other transcendence results; for instance Γ(1/6) is also a transcendental numbers. Apart from these values, no proof of the transcendence of another value of the Gamma function at a rational point is known. Transcendence results are bound to denominators which divide 6: for other values the genus of the corresponding Fermat curve is larger than 1, and so far the transcendence results on Abelian varieties are not as strong as their elliptic counterparts.

The proof by G.V. Čudnovs kiȋ of his Theorem 14 rests on the following result [START_REF]Čudnovs kiȋ -Algebraic independence of constants connected with the exponential and the elliptic functions[END_REF], dealing with periods and quasi-periods of Weierstrass functions, which he applies to the elliptic curves y 2 = x 3 -x and y 2 = x 3 -1 using ( 9) and [START_REF] Beukers | Wolfart -Algebraic values of hypergeometric functions[END_REF] Let ℘ be a Weierstrass elliptic function with invariants g 2 and g 3 . Let ω be a nonzero period of ℘ and let η be the corresponding quasi-period of the ζ function:

℘ 2 = 4℘ 3 -g 2 ℘ -g 3 , ζ = -℘, ζ(z + ω) = ζ(z) + η.
Then two at least of the numbers g 2 , g 3 , ω/π, η/π are algebraically independent.

From this statement one deduces also that the number ( 8) is transcendental, not only over Q, but also over the field Q(π) (cf. [START_REF] Kontsevich | Mathematics unlimited-2001 and beyond[END_REF] § 1.1).

So far the only known proof of the transcendence (over Q) of Γ(1/4) as well as of Γ(1/3) is Čudnovs kiȋ's one which establishes the stronger result that these numbers are transcendental over Q(π).

From the quantitative point of view, sharp transcendence measures for these numbers have been obtained by P. Philippon and then by S. Bruiltet [START_REF] Bruiltet | D'une mesure d'approximation simultanée à une mesure d'irrationalité: le cas de Γ(1/4) et Γ(1/3)[END_REF]: This shows that neither Γ(1/4) nor Γ(1/3) is a Liouville number.

Theorem 15. Let P ∈ Z[X, Y ] be a non-constant polynomial of degree d and height H. Then log |P (π, Γ(1/4)| > -10 326 (log H + d log(d + 1) d 2 log(d + 1)
One might expect that the next step could be the transcendence of the number Γ(1/5) (cf. [START_REF] Masser | Interpolation on group varieties[END_REF], p. 2 and p. 35). From Schneider's Theorem 12 one deduces the transcendence of the number B(1/5, 1/5), hence one at least of the two numbers Γ(1/5), Γ(2/5) is transcendental. A stronger statement follows from the work of P. Grinspan [START_REF]Grinspan -Measures of simultaneous approximation for quasi-periods of abelian varieties[END_REF] (see also [START_REF] Vasil Ev | On the algebraic independence of the periods of Abelian integrals[END_REF]): Theorem 16. One at least of the two numbers Γ(1/5), Γ(2/5) is transcendental over the field Q(π).

In other terms, two at least of the three numbers Γ(1/5), Γ(2/5) and π are algebraically independent. The proof of [START_REF]Grinspan -Measures of simultaneous approximation for quasi-periods of abelian varieties[END_REF] also yields a quantitative result.

Since the Fermat curve x 5 + y 5 = z 5 with exponent 5 has genus 2, its Jacobian is an Abelian surface; to prove Theorem 16, one needs to replace, in Čudnovs kiȋ's proof, the elliptic functions by Abelian functions; this explains the difficulty to separate the two numbers Γ(1/5) and Γ(2/5) when one is interested to prove the transcendence of one of them.

Before continuing with the denominator 5, let us come back briefly to the denominators 3 and 4. Theorem 14 has been extended by Yu.V. Nesterenko [START_REF] Nesterenko | Modular functions and transcendence questions[END_REF][START_REF]Introduction to algebraic independence theory[END_REF] in 1996, who succeeded to obtain the algebraic independence of three numbers: Theorem 17. The three numbers Γ(1/4), π and e π are algebraically independent, and the same is true with the three numbers Γ(1/3), π and e π √ 3 .

To prevent confusion, let us mention that the numbers e π and e π √ 3 are not expected to be periods.

The proof by Yu.V. Nesterenko of his Theorem 17 involves Eisenstein series E 2 , E 4 and E 6 -here we shall use the notations P , Q, R introduced by S. Ramanujan:

(18)                    P (q) = E 2 (q) = 1 -24 ∞ n=1 nq n 1 -q n , Q(q) = E 4 (q) = 1 + 240 ∞ n=1 n 3 q n 1 -q n , R(q) = E 6 (q) = 1 -504 ∞ n=1 n 5 q n 1 -q n •
The first transcendence results on the values of these functions are due to D. Bertrand in the 1970's. A remarkable progress was achieved in 1996 by K. Barré-Sirieix, G. Diaz, F. Gramain and G. Philibert [START_REF] Barré-Sirieix | Philibert -Une preuve de la conjecture de Mahler-Manin[END_REF], who solved the following problem raised by K. Mahler (and by Yu.V. Manin in the p-adic case) on the modular function J = Q 3 /∆, where ∆ = 12 -3 (Q 3 -R 2 ): for any q ∈ C with 0 < |q| < 1, one at least of the two numbers q, J(q) is transcendental.

This breakthrough enabled Yu.V. Nesterenko [START_REF] Nesterenko | Modular functions and transcendence questions[END_REF] to prove the next result, which is quoted in § 2.4 of [START_REF] Kontsevich | Mathematics unlimited-2001 and beyond[END_REF]:

Let q ∈ C be a complex number satisfying 0 < |q| < 1. Then three at least of the four numbers q, P (q), Q(q), R(q) are algebraically independent.

Theorem 17 follows by specializing q = e -2π and q = -e -π √ 3 , since J(e -2π ) = 1728,

P (e -2π ) = 3 π , Q(e -2π ) = 3 ω π 4 , R(e -2π ) = 0 with (cf. ( 9
)
)

ω = Γ(1/4) 2 √ 8π = 2.6220575542 . . . while J(-e -π √ 3 ) = 0, P (-e -π √ 3 ) = 2 √ 3 π , Q(-e -π √ 3 ) = 0, R(-e -π √ 3 ) = 27 2 ω π 6 with (cf. ( 10
)
)

ω = Γ(1/3) 3 2 4/3 π = 2.428650648 . . . In [38] § 2.
3 there are comments on relations between periods and Eisenstein series (as well as theta functions, which also occur in the work [START_REF] Nesterenko | Modular functions and transcendence questions[END_REF] of Nesterenko -see [START_REF]Introduction to algebraic independence theory[END_REF]).

Nesterenko's Theorem 17 and Grinspan's Theorem 16 suggest the following statements, which is likely to be a special case of André's extension of Grothen -dieck's Conjecture (see § 2 and [START_REF]Une introduction aux motifs (motifs purs, motifs mixtes, périodes)[END_REF] This problem has been considered by F. Pellarin (see [START_REF] Pellarin | Idéaux stables dans certains anneaux différentiels de formes quasimodulaires de Hilbert[END_REF]).

A more ambitious goal is to give a complete list of all algebraic relations among the values of the Gamma function at rational points. D. Rohrlich considered the question for multiplicative relations. Let us first recall what are the standard relations. For any a ∈ C which is not a pole of Γ(x), Γ(x + 1), Γ

(1 -x) nor of Γ(nx), ( 20
) (Translation) Γ(a + 1) = aΓ(a), (Reflection) Γ(a)Γ(1 -a) = π sin(πa) (Multiplication)      n-1 k=0 Γ a + k n = (2π) (n-1)/2 n -na+(1/2) Γ(na)
for any positive integer n. [START_REF]Čudnovs kiȋ -Algebraic independence of constants connected with the exponential and the elliptic functions[END_REF].

A formalization of this Conjecture in terms of "universal distribution" has been proposed by S. Lang in [START_REF]Séminaire Delange-Pisot-Poitou, 19e année: 1977/78[END_REF].

A stronger conjecture than 21 is the Rohrlich-Lang Conjecture which deals not only with monomial relations, but more generally with polynomial relations. The precise relation between conjectures 21, 22 and the Grothendieck's Conjecture (for certain Abelian varieties) is explained in detail in [START_REF]Une introduction aux motifs (motifs purs, motifs mixtes, périodes)[END_REF] 24.6.

Values of the Riemann zeta function

We start with an elementary result on the values of the Riemann zeta function at the positive integers.

Lemma 23. For s ≥ 2, the number

ζ(s) = n≥1 1 n s is a period. Proof: The formula (24) ζ(s) = 1>t 1 >•••>ts>0 dt 1 t 1 • • • dt s-1 t s-1 • dt s 1 -t s is easily checked.
It will be convenient to use the notation of iterated Chen integrals ([17] § 2.6) and to write [START_REF] Diaz | Utilisation de la conjugaison complexe dans l'étude de la transcendance de valeurs de la fonction exponentielle[END_REF] as [START_REF] Edixhoven | Yafaev -Subvarieties of Shimura varieties[END_REF] ζ(s)

= 1 0 ω s-1 0 ω 1 with ω 0 = dt t and ω 1 = dt 1 -t •
The arithmetic nature of the values of the Riemann zeta function at positive even integers is known since Euler:

(26) π -2k ζ(2k) ∈ Q for k ≥ 1.
These rational numbers involve Bernoulli numbers (see for instance formula [START_REF] Kontsevich | Mathematics unlimited-2001 and beyond[END_REF] in [START_REF]Cartier -Fonctions polylogarithmes, nombres polyzêtas et groupes pro-unipotents[END_REF]).

Lindemann's Theorem asserts that the number π is transcendental, hence so is ζ(2k) for any integer k ≥ 1.

To prove that given numbers are rational or algebraic is most often more fruitful than to prove that they are transcendental. For instance, in the case of Euler's formula [START_REF] Fel | Transcendental numbers[END_REF], the Bernoulli numbers carry a lot of interesting arithmetic information. However, our purpose here is to give the state of the art on transcendence results: the goal of transcendence statements is to ensure that all the algebraic relations are known, hence that the source of all arithmetic information is already available.

The main Diophantine question raised by Euler numbers is to decide which are the algebraic relations among the numbers andC. Viola [START_REF] Rhin | Viola -The group structure for ζ(3)[END_REF]:

ζ(3) - p q > q -µ
for sufficiently large q, with µ = 5, 513 . . .

The recent works initiated by T. Rivoal and followed by T. Rivoal, K. Ball and W. Zudilin among others, provide the first information on the arithmetic nature of the values of the Riemann zeta function at odd positive integers. In particular the dimension of the Q-vector space spanned by the number ζ(2k + 1), k ≥ 1 is infinite (cf. [START_REF] Shafarevich | Fischler -Irrationalité de valeurs de zêta (d'après Apéry, Rivoal[END_REF]).

A preliminary step towards a proof of Conjecture 27 is to linearize the situation. Indeed, available Diophantine methods for proving linear independence are much more efficient than methods for proving algebraic independence. An example of this claim is Baker's Theorem 4 which solves the questions of linear independence over the field of algebraic numbers of logarithms of algebraic numbers, while it is not known that there exists two algebraically independent logarithms of algebraic numbers. An earlier example, showing the power of linear independence methods, is the Lindemann-Weierstraß' Theorem, which can be stated either as a result of algebraic independence, ([26] Th. 2.3'), or as a result of linear independence ([26] Th. 2.3) -the proof by F. Lindemann et K. Weierstraß is a proof of linear independence.

Euler already noticed that the product of two values of the Riemann zeta function (as it is called now) is again the sum of a series. Indeed, from the relation

n 1 ≥1 n -s 1 1 n 2 ≥1 n 2 -s 2 = n 1 >n 2 ≥1 n -s 1 1 n 2 -s 2 + n 2 >n 1 ≥1 n -s 2 2 n 1 -s 1 + n≥1 n -s 1 -s 2 ,
one deduces, for s 1 ≥ 2 and s 2 ≥ 2,

ζ(s 1 )ζ(s 2 ) = ζ(s 1 , s 2 ) + ζ(s 2 , s 1 ) + ζ(s 1 + s 2 ) with ζ(s 1 , s 2 ) = n 1 >n 2 ≥1 n -s 1 1 n 2 -s 2 .
For k, s 1 , . . . , s k positive integers with s 1 ≥ 2, we set s = (s 1 , . . . , s k ) and ( 28)

ζ(s) = n 1 >n 2 >•••>n k ≥1 1 n s 1 1 • • • n s k k •
These numbers are called "Multiple Zeta Values" or "MZV". For k = 1 we recover of course Euler's numbers ζ(s).

Remark 29. Each of the numbers ζ(s) is a period: indeed, using the iterated Chen integrals notation [START_REF] Edixhoven | Yafaev -Subvarieties of Shimura varieties[END_REF] we have (cf. [START_REF]Cartier -Fonctions polylogarithmes, nombres polyzêtas et groupes pro-unipotents[END_REF] § 2.6):

(30) ζ(s) = 1 0 ω s 1 -1 0 ω 1 • • • ω s k -1 0 ω 1 .
The product of series ( 28) is a linear combination, with integer coefficients, of such series. Therefore the vector space (over Q or over Q) spanned by the numbers ζ(s) is also an algebra over the same field. Moreover, the product of two integrals [START_REF] Goncharov | Multiple ζ-values, Galois groups, and geometry of modular varieties[END_REF] is also a linear combination of such integrals. These linear combinations are not the same as the relations occurring from the product of two series: therefore, taking the difference yields nontrivial linear relations with integer coefficients between MZV. Further linear relations can be derived -for instance Euler's relation ζ(2, 1) = ζ(3) -by taking into account the relations which are obtained using the same argument for divergent series and using a regularization argument [START_REF]Cartier -Fonctions polylogarithmes, nombres polyzêtas et groupes pro-unipotents[END_REF].

A complete description of all linear relations among MZV should in principle yield a description of all algebraic relations among these numbers, hence should provide a solution to Conjecture 27 concerning the algebraic independence, over the field Q(π), of the values at odd positive integers of the Riemann zeta function. The contrast is interesting between the situation suggested on the one hand by Conjecture 27, where no algebraic relation is expected, and where the underlying algebraic structure is uninteresting, and on the other hand by the rich structure provided by the known linear relations between MZV.

The goal now is to describe all linear relations among MZV. Let Z p be the Q-subspace of R spanned by the numbers ζ(s) for s of "weight"

s 1 + • • • + s k = p, with Z 0 = Q and Z 1 = {0}.
Here is Zagier's Conjecture (Conjecture 108 of [START_REF]Cartier -Fonctions polylogarithmes, nombres polyzêtas et groupes pro-unipotents[END_REF]) on the dimension d p of Z p . The precise relation between conjectures 21, 22 and the Grothendieck's Conjecture (for certain mixed Tate motives) is explained in detail in [START_REF]Une introduction aux motifs (motifs purs, motifs mixtes, périodes)[END_REF] 25.4 and 25.7. See also the paper by P. Deligne and A.B. Goncharov [START_REF] Deligne | Goncharov -Groupes fondamentaux motiviques de Tate mixte[END_REF].

Conjecture 31. The numbers d p satisfy, for p ≥ 3,

d p = d p-2 + d p-3 .
Hence the sequence (d 0 , d 1 , d 2 , . . .) starts with (1, 0, 1, 1, 1, 2, 2, . . .). This conjecture can be written

p≥0 d p X p = 1 1 -X 2 -X 3 • A candidate
ζ {2} 5 , ζ {2} 2 , {3} 2 , ζ {2, 3} 2 , ζ (2, {3} 2 , 2 , ζ 3, {2} 2 , 3 , ζ {3, 2} 2 , ζ {3} 2 , {2} 2
provide a basis of the vector space Z 10 .

Exemple 33.

Here are the first values of d p :

• d 0 = 1 because we agree that ζ(s 1 , . . . , s k ) = 1 for k = 0. • d 1 = 0 since {(s 1 , . . . , s k ) ; s 1 + • • • + s k = 1, s 1 ≥ 2} = ∅. • d 2 = 1 since ζ(2) = 0 • d 3 = 1 since ζ(2, 1) = ζ(3) = 0 • d 4 = 1 since ζ(4) = 0 and ζ(3, 1) = 1 4 ζ(4), ζ(2, 2) = 3 4 ζ(4), ζ(2, 1, 1) = ζ(4) = 2 5 ζ(2) 2 .
The first value of d p which is not known is d 5 . Conjecture 31 predicts d 5 = 2, and we know

d 5 ∈ {1, 2} because ζ(2, 1, 1, 1) = ζ(5), ζ(3, 1, 1) = ζ(4, 1) = 2ζ(5) -ζ(2)ζ(3), ζ(2, 1, 2) = ζ(2, 3) = 9 2 ζ(5) -2ζ(2)ζ(3), ζ(2, 2, 1) = ζ(3, 2) = 3ζ(2)ζ(3) - 11 2 ζ(5).
Hence d 5 = 2 is equivalent to the irrationality of the number ζ( 2

)ζ(3)/ζ(5).

Conjecture 31 predicts an exact value for the dimension d p of Z p . The Diophantine open problem is to prove the lower bound. The upper bound has been established recently thanks to the work of A.B. Goncharov [START_REF] Goncharov | Multiple ζ-values, Galois groups, and geometry of modular varieties[END_REF] and T. Terasoma [START_REF]Terasoma -Mixed Tate motives and multiple zeta values[END_REF] (see also Theorem 6.4 of [START_REF]Hoffman -The algebra of multiple harmonic series[END_REF] and Cor. 5.25 of [START_REF] Deligne | Goncharov -Groupes fondamentaux motiviques de Tate mixte[END_REF]. ):

The integers δ p which are defined by the linear recurrence relation of Zagier's Conjecture 31 δ p = δ p-2 + δ p-3 with initial conditions δ 0 = 1, δ 1 = 0 give an upper bound for the dimension d p of Z p . Surprisingly enough, this bound is not obtained through a combinatorial study of the known relations between MZV, but by a fairly general motivic argument.

Hypergeometric functions

For a, b, c and z complex numbers with c ∈ Z ≤0 and |z| < 1, Gauss hypergeometric function (see for instance [START_REF] Fel | Transcendental numbers[END_REF], Chap. 1 § 3.6, Chap. 2 § 3.2) is defined by

2 F 1 a, b ; c z = ∞ n=0 (a) n (b) n (c) n • z n n! where (a) n = a(a + 1) • • • (a + n -1).
Exemple 34. Let K, P n and T n denote respectively Jacobi's elliptic integral of the first kind, the n-th Legendre polynomial and the n-th Chebyshev polynomial:

K(z) = 1 0 dx (1 -x 2 )(1 -z 2 x 2 ) , P n (z) = 1 n! d dz n (1 -z 2 ) n ,
orbit of a special (CM) point on a curve in a Shimura variety which is defined over Q. The set E is infinite when the monodromy group is arithmetic, it is finite when the monodromy group is not arithmetic.

To Gauss hypergeometric function is associated Gauss continued fraction

G(z) = G(a, b, c; z) = 2 F 1 a, b + 1 ; c + 1 z / 2 F 1 a, b ; c z = 1/ 1 -g 1 z/ 1 -g 2 z/(• • • ) with coefficients g 2n-1 = (a + n -1)(c -b + n -1)/ (c + 2n -2)(c + 2n -1) , g 2n = (b + n)(c -a + n)/ (c + 2n -1)(c + 2n) .
J. Wolfart [START_REF]Values of Gauss' continued fractions[END_REF] showed that if the parameters a, b, c are rational numbers, c = 0, -1, -2, • • • , and if G(z) is not an algebraic function, then, for almost all algebraic values of the argument z, the value G(z) is a transcendental number. He uses Wüstholz's Theorem [START_REF]Wüstholz -Algebraische Punkte auf analytischen Untergruppen algebraischer Gruppen[END_REF] together with the results of G. Shimura and Y. Taniyama on Abelian varieties.

For the number we just considered, which are related to hypergeometric functions, when a transcendence proof is known, an effective measure of Diophantine approximation by algebraic numbers is also available (see in particular [START_REF] Fel | Transcendental numbers[END_REF] and [START_REF] Gaudron | Mesure d'indépendance linéaire de logarithmes dans un groupe algébrique commutatif[END_REF][START_REF]Mesures d'indépendance linéaire de logarithmes dans un groupe algébrique commutatif[END_REF]).

Gauss hypergeometric functions are particular members of a much more general family, which is constituted by the generalized hypergeometric functions (see for instance [START_REF] Fel | Transcendental numbers[END_REF] Chap. 2, § 6).

Let p ≥ 2 be an integer, a 1 , . . . , a p , b 1 , . . . , b p-1 and z complex numbers with b i ∈ Z ≤0 and |z| < 1. Set

p F p-1 a 1 , . . . , a p b 1 , . . . , b p-1 z = ∞ n=0 (a 1 ) n • • • (a p ) n (b 1 ) n • • • (b p-1 ) n • z n n! • Exemple 39. The functions 1 F 0 1/n z n = n √ 1 -z n and 3 F 2 1/4, 1/2, 3/4 1/3, 2/3 2 8 z 3 3 = ∞ k=0 4k k z k are algebraic.
In connection with these examples, let us quote the main result of a recent Oberwolfach report on integral ratios of factorials and algebraic hypergeometric functions by F. Rodriguez-Villegas. Let γ = (γ ν ) ν≥1 be a sequence of integers which are zero except for finitely many. Define F. Rodriguez-Villegas proves that the generating series u is algebraic if and only if u n ∈ Z for every n ≥ 0 and d = 1.

By induction on p starting from (36) one can check:

Proposition 40. For a 1 , . . . , a p , b 1 , . . . , b p-1 rational numbers with p ≥ 2, b i ∈ Z ≤0 and for z ∈ Q with |z| < 1, we have p F p-1 a 1 , . . . , a p b 1 , . . . , b p-1 z ∈ 1 π p-1 P.
The reference [START_REF] Fel | Transcendental numbers[END_REF] gives the state of the art on the arithmetic nature of values of generalized hypergeometric functions, both from the qualitative point of view (irrationality, transcendence, linear independence, algebraic independence) and from the quantitative one (approximation measures, linear independence measures, measures of algebraic independence).

Mahler's measure of polynomials in several variables

Let P ∈ C[z 1 , . . . , z n , z -1 1 , . . . , z -1
n ] be a nonzero Laurent polynomial in n variables. Mahler's measure M(P ) and Mahler's logarithmic measure µ(P ) are defined by

µ(P ) = log M(P ) = 1 0 • • • 1 0 log P (e 2iπt 1 , . . . , e 2iπtn ) dt 1 • • • dt n .
In the easiest case n = 1 we write

P (z) = d i=0 a d-i z i = a 0 d i=1 (z -α i )
and we have

M(P ) = |a 0 | d i=1 max{1, |α i |}.
One deduces µ(P ) ∈ P.

More generally, for P ∈ Q[z 1 , . . . , z n , z -1 1 , . . . , z -1 n ] we have µ(P ) ∈ 1 π n P. D. Boyd and C.J. Smyth (see the references in [START_REF] Boyd | Mahler's measure and special values of L-functions-some conjectures[END_REF]) computed several examples of the value of µ which they expressed in terms of special values of the L-function attached to Dirichlet's characters. Then D. Boyd, F. Rodriguez Villegas, V. Maillot and S. Vandervelde wrote some logarithmic Mahler's measures as combinations of the dilogarithm function. Apart from the well known example 3), due to C.J. Smyth, known results deal mainly with polynomials in two variables. However, thanks to the works of C. Deninger, there is some hope for a better understanding of more general cases.

µ(1 + z 1 + z 2 + z 3 ) = 7 2π 2 ζ(
7. Exponential of periods and exponential periods 7.1. Exponential of periods. One of the suggestions of Kontsevich and Zagier in [START_REF] Kontsevich | Mathematics unlimited-2001 and beyond[END_REF] § 1.2 is that the numbers 1/π and e may not be periods. Among the candidates for being nonperiods one may include e π and e π 2 .

The transcendence of the number e π is known thanks to A.O. Gel'fond in 1929 -it is now also a consequence of Gel'fond-Schneider's solution of Hilbert's seventh problem (hence of Baker's Theorem 4). The transcendence of e π 2 is not yet proved.

Conjecture 41. Let α 1 , α 2 , α 3 be nonzero algebraic numbers. For j = 1, 2, 3 let log α j ∈ C \ {0} be a nonzero logarithms of α j , which means a nonzero complex number such that e log α j = α j . Then

(log α 1 )(log α 2 ) = log α 3 .
Exemple 42. With log α 1 = log α 2 = iπ, the transcendence of the number e π 2 follows from Conjecture 41. Another example is the transcendence of the number

2 log 2 .
Other conjectures are proposed in [START_REF]Variations on the six exponentials theorem[END_REF], both for the exponential function (three, four, five exponentials conjectures) and for elliptic functions. Very few special cases of these conjectures have been proved. The appendix of [START_REF]Variations on the six exponentials theorem[END_REF] by H. Shiga provides a link with periods of Kummer surfaces.

The proofs of the few partial results which are known in the direction of Conjecture 41 and of the four exponentials Conjecture [START_REF] Lang | Introduction to transcendental numbers[END_REF][START_REF] Waldschmidt | Diophantine approximation on linear algebraic groups[END_REF][START_REF]Variations on the six exponentials theorem[END_REF] rest on the method which enabled Th. Schneider in 1934 to solve Hilbert's seventh problem. In this method the addition Theorem of the exponential function, namely e x+y = e x e y , plays the main role, while the differential equation is not used, in contrast with what is suggested at the end of § 2.4 of [START_REF] Kontsevich | Mathematics unlimited-2001 and beyond[END_REF]. A second example of a transcendence proof which does not use derivatives is by G. Diaz [START_REF] Diaz | Utilisation de la conjugaison complexe dans l'étude de la transcendance de valeurs de la fonction exponentielle[END_REF] for the special case of Hermite-Lindemann's Theorem on the transcendence of e β when β is algebraic and β ∈ R∪iR. A third transcendence method where derivatives are not involved is Mahler's one, which is the main topic of K. Nishioka's Lecture Notes [START_REF] Nishioka | Mahler functions and transcendence[END_REF]. By the way, Conjecture 5.4 of [START_REF]Open diophantine problems[END_REF] on the transcendence of numbers whose development in a basis is given by an automatic sequence, which had been investigated by J.H. Loxton and A.J. van der Poorten by means of this method of Mahler, has been settled by B. Adamczewski, Y. Bugeaud and F. Lucas [START_REF] Adamczewski | Lucas -Sur la complexité des nombres algébriques[END_REF][START_REF] Adamczewski | On the complexity of algebraic numbers i. expansions in integer bases[END_REF], thanks to completely different arguments based on the very efficient Subspace Theorem of W.M. Schmidt.

Conjecture 41 is a very special case of the Conjecture on algebraic independence of logarithms of algebraic numbers (which is in turn only a special case of Schanuel's Conjecture):

Conjecture 43. (Algebraic independence of logarithms). Logarithms of algebraic numbers which are Q-linearly independent are algebraically independent.

There are several examples of numbers which can be written as the value of a polynomial at a point whose coordinates are logarithms of algebraic numbers. Among the most important ones are the determinants of square matrices whose entries are logarithms of algebraic numbers. Some regulators have this form; to decide whether they vanish or not may be considered as a transcendence problem. When they do not vanish, one expects them to be transcendental, and also not to be Liouville numbers. 7.2. Exponential periods. The next definition is from [START_REF] Kontsevich | Mathematics unlimited-2001 and beyond[END_REF] § 4.3. The remark after the introduction of [START_REF] Kontsevich | Mathematics unlimited-2001 and beyond[END_REF] includes the sentence: "The last chapter, which is at a more advanced level and also more speculative than the rest of the paper, is by the first author only".

Definition An exponential period is an absolutely convergent integral of the product of an algebraic function with the exponent of an algebraic function, over a real semialgebraic set, where all polynomials entering the definition have algebraic coefficients.

Exemple 44. In the algebra of exponential periods one finds of course all periods, as well as the numbers

e β = β -∞ e x dx when β is algebraic, the number √ π = ∞ -∞ e -x 2 dx,
the values of the Euler Gamma function at rational points

Γ(s) = ∞ 0 e -t t s • dt t ,
as well as values of Bessel functions at algebraic points

J n (z) = |u|=1 exp z 2 u - 1 u du u n+1 •
These examples are viewed by S. Bloch and H. Esnault [START_REF] Bloch | Esnault -Gauss-Manin determinants for rank 1 irregular connections on curves[END_REF] as periods coming from a duality between homologic cycles and differential forms for connections with irregular singular points on Riemann surfaces. The referee pointed out to me that there are parallel works due to Masaaki Yoshida, Yoshishige Haraoka and Hironobu Kimura. I thank Cristiana Bertolin for providing me more information on this question.

Euler constant.

No proof is available yet for the irrationality of Euler's constant [START_REF]Oort -Canonical liftings and dense sets of CM-points[END_REF] γ = lim

n→∞ 1 + 1 2 + 1 3 + • • • + 1 n -log n = 0.5772157 . . .
(see [START_REF]Sondow -Criteria for irrationality of Euler's constant[END_REF]), even when a stronger assertion is expected, namely:

Conjecture 46. The number γ is transcendental.

A still stronger statement is suggested in [START_REF] Kontsevich | Mathematics unlimited-2001 and beyond[END_REF]

Finite characteristic

Diophantine questions concerning real or complex numbers have analogs in function fields. In the case of finite characteristic a good reference is [START_REF]The arithmetic of function fields[END_REF]. The first investigations go back to L. Carlitz (1935); the tools he introduced have been used in particular by I.I. Wade (1941) who proved the first transcendence statements in this context. After several works, in particular by J.M. Geijsel and P. Bundschuh in 1978, the theory has been developed in a deep way by Jing Yu after 1980, firstly in the situation of elliptic modules which were introduced by V.G. Drinfel'd in 1974, then, for the t-motives of G. Anderson after 1986. For a while, the results which were obtained in the situation of finite characteristic were analogues of the classical results related to complex numbers, until Jing Yu succeeded to prove new results which go much further than their complex counterparts [START_REF] Yu | Transcendence in finite characteristic[END_REF].

The use, introduced in this context by L. Denis in 1990, of the derivative operator with respect to the variable of the function field, produces statements (like the transcendence of the derivative of the periods) which have no counterpart in the classical situation of complex numbers. Another peculiarity of finite characteristic is the possibility to use tensor products; they allow sometimes to reduce questions of algebraic independence to problems of linear independence -a nice example is given by S. David and L. Denis in [START_REF] David | Denis -Périodes de modules de Drinfel'd -"indépendance quadratique en rang II[END_REF]).

Two surveys on this subject give further information on this topic, the first one in 1992 by Jing Yu in [START_REF] Yu | Transcendence in finite characteristic[END_REF], the other one in 1998 by W.D. Brownawell [START_REF] Brownawell | Transcendence in positive characteristic[END_REF].

8.1. Euler constant in finite characteristic. A remarkable result in finite characteristic is the transcendence of the analog of Euler's constant. The complex number γ (see [START_REF]Oort -Canonical liftings and dense sets of CM-points[END_REF] ) can be defined as

γ = lim s→1 ζ(s) - 1 s -1
where ζ denotes the Riemann zeta function

ζ(s) = p (1 -p -s ) -1 .
In this last product p runs over the set of prime numbers. In finite characteristic the corresponding product is over the set of monic irreducible polynomials over a finite field F q with q elements. Here the product converges at the point s = 1, and the value at this point is an analog of Euler's constant (it belongs to a completion C of an algebraic closure of F q ((1/T ))). This element of C is transcendental over F q (T ): this was proved by G.W. Anderson and D. Thakur in 1990 [START_REF] Anderson | Thakur -Tensor powers of the Carlitz module and zeta values[END_REF]. An interesting remark of them is that the tools which were available to I.I. Wade were already sufficient and the proof could have been given as early as in 1940. 8.2. Thakur Gamma function. By analogy with the definition [START_REF] Bloch | Esnault -Gauss-Manin determinants for rank 1 irregular connections on curves[END_REF] of the Euler Gamma function as an infinite product, D. Thakur defines (cf. [START_REF] Brownawell | Transcendence in positive characteristic[END_REF])

Γ(z) = z -1 ∞ n∈A + 1 + z n -1 ,
where A = F q [T ] is the ring of polynomials over the finite field F q and where A + is the set of monic polynomials. This Gamma function is meromorphic over the field C. It satisfies similar relations as the standard relations [START_REF]Čudnovs kiȋ -Algebraic independence of constants connected with the exponential and the elliptic functions[END_REF] which are satisfied by the Euler Gamma function. Also, in finite characteristic, the counterpart of the relations of Deligne-Koblitz-Ogus has been obtained by Deligne, Anderson and Thakur (see [START_REF] Brownawell | Papanikolas -Linear independence of Gamma values in positive characteristic[END_REF]).

In 1992 G. Anderson introduced the notion of soliton, which, according to [START_REF] Brownawell | Papanikolas -Linear independence of Gamma values in positive characteristic[END_REF], is a higher dimensional version of the shtuka function for rank 1 Drinfeld modules. The meromorphic functions which were used by R. Coleman in his study of Frobenius endomorphisms on the curves of Fermat and Artin-Schreier had been interpreted by D. Thakur in terms of his Gamma function. While he was developing the parallel between these questions and the theory of partial differential equations, G. Anderson was led to introduce his solitons. This theory was used in 1997 by S.K. Sinha, who constructed t-modules whose periods have coordinates which are product of an algebraic number and values of Thakur Gamma function at rational points a/f , with a and f in A + . By means of the earlier transcendence results of Jing Yu, S.K. Sinha could derive the transcendence of certain values of Thakur Gamma function at rational points. These works were pursued by W.D. Brownawell and M.A. Papanikolas who show in [START_REF] Brownawell | Papanikolas -Linear independence of Gamma values in positive characteristic[END_REF] that the only linear relations with algebraic coefficients between the values of Thakur Gamma function are those which arise from Deligne-Anderson-Thakur relations. It is fair to say that this is the analog for function fields over a finite field of the Theorem of Wolfart and Wüstholz [START_REF] Wolfart | Wüstholz -Der Überlagerungsradius gewisser algebraischer Kurven und die Werte der Betafunktion an rationalen Stellen[END_REF] on the linear independence of values of the Beta function.

A most remarkable fact is that it is possible to go further in the function field case. In [START_REF] Anderson | Papanikolas -Determination of the algebraic relations among special Gamma-values in positive characteristic[END_REF] For the values of the Gamma function, the situation in finite characteristic is much more advanced in the case of function fields than in the case of the complex field, where the Conjecture 22 of Rohrlich-Lang seems out of reach.

Very recently, M.A. Papanikolas succeeded to prove for Drinfeld modules the analog of the Conjecture 43 on algebraic independence of logarithms [START_REF] Cortona | Papanikolas -Tannakian duality for Anderson-Drinfel'd motives and algebraic independence of Carlitz logarithms[END_REF]. He develops a theory of Tannakian Galois groups for t-motives and relates this to the theory of Frobenius semilinear difference equations. He shows that the transcendence degree of the period matrix associated to a given t-motive is equal to the dimension of its Galois group. Using this result he proves that Carlitz logarithms of algebraic functions that are linearly independent over the rational function field are algebraically independent.

Another method (related to Mahler's method which was alluded to in § 7.1) has been worked out by L. Denis, who also obtains results of algebraic independence of logarithms of algebraic points on Drinfeld modules.

2 andlog

 2 |P (π, Γ(1/3)| > -10 330 (log H + d log(d + 1) d 2 log(d + 1) 2 .

  23.4.3): Conjecture 19. Three at least of the four numbers Γ(1/5), Γ(2/5), π and e π √ 5 are algebraically independent.

Conjecture 21 .

 21 (D. Rohrlich). Any multiplicative dependence relation of the form π b/2 a∈Q Γ(a) ma ∈ Q with b and m a in Z is a consequence of the standard relations

Conjecture 22 .

 22 (Rohrlich-Lang). The ideal over Q of all algebraic relations among values of the function (1/ √ 2π)Γ(a) for a ∈ Q is generated by the distribution relations, the functional equation and the oddness of the Gamma function.

ζ( 2 )

 2 , ζ(3), ζ(5), ζ(7) . . . ? One expects that there are no relation at all ([17] and [27] Conjecture 0.1). In other terms: Conjecture 27. The numbers ζ(2), ζ(3), ζ(5), ζ(7) . . . are algebraically independent. Very few is known on this conjecture. In 1978 R. Apéry proved that the number ζ(3) is irrational . Apéry's proof shows that the number ζ(3) is not a Liouville number. The sharpest known irrationality measure for ζ(3) is due to G. Rhin

  for a basis of the space Z p is suggested by M. Hoffman ([36], Conjecture C): Conjecture 32. A basis of Z p over Q is given by the numbers ζ(s 1 , . . . , s k ), where s 1 + • • • + s k = p and each s i is either 2 or 3. This conjecture is compatible with what is known for p ≤ 16 (work of Hoang Ngoc Minh and others). For instance, if {a} b denotes the sequence of b occurences of a, it is expected that the 7 following values

  ! γν . Assume γ = 0 and ν≥1 νγ ν = 0, which, by Stirling's formula, is equivalent to the generating series u := ν≥1 u n λ n having finite non-zero radius of convergence. Set d := -ν≥1 γ ν .

  , G.W. Anderson, W.D. Brownawell and M.A. Papanikolas show that all algebraic dependence relations between the values of Thakur Gamma function are consequences of the relations of Deligne-Anderson-Thakur.

  .4. A two dimensional analog of Euler constant. For each integer k ≥ 2, denote by A k the minimal area of a closed disk in R 2 containing at least k points of Z 2 . For n ≥ 2, set δ n = -log n + These quantities have been introduced and investigated by D.M. Masser in connection with Gel'fond's proof in 1929 of the transcendence of the number e π , and then by F. Gramain[START_REF] Gramain | Computing an arithmetic constant related to the ring of gaussian integers[END_REF] who conjectures:

	Conjecture 48.		δ = 1 +	4 π	γL (1) + L(1) ,
	where γ is Euler's constant (45) and
		L(s) =	(-1) n (2n + 1) -s .
				n≥0
	is the L-function of the quadratic number field Q(i) (Dirichlet's Beta function).
	Since L(1) = π/4 and
	L (1) =	n≥0	(-1) n+1 •	log(2n + 1) 2n + 1
	=	π 4	3 log π + 2 log 2 + γ -4 log Γ(1/4) ,
	Conjecture 48 can be written
				n k=2	1 A k	and δ = lim n→∞	δ n .

§ 1.1 and § 4.3:

Conjecture 47. The number γ is not a period, it is not even an exponential period.

7δ = 1 + 3 log π + 2 log 2 + 2γ -4 log Γ(1/4) = 1.82282524 . . . The best known estimate for δ is [32] 1.811 • • • < δ < 1.897 . . .

It seems likely that this number δ is not a period (hence that it is transcendental); however, given the lack of information we have on it, this number is probably not the best candidate to solve the question of

[START_REF] Kontsevich | Mathematics unlimited-2001 and beyond[END_REF]

( § 1.2 problem 3): exhibit at least one number which is not a period.

Michel WaldschmidtInstitut de Mathématiques de Jussieu -UMR 7586 du CNRS, Université P. et M. Curie (Paris VI) 175 rue du Chevaleret, F-75013 Paris E-mail: miw@math.jussieu.fr http://www.math.jussieu.fr/∼miw

T n (cos z) = cos(nz). Then:

For c > b > 0 rational numbers, we have (Euler, 1748)

using Gamma's reflexion formula in [START_REF]Čudnovs kiȋ -Algebraic independence of constants connected with the exponential and the elliptic functions[END_REF] together with the connection (13) between the Beta and Gamma function, one deduces

Therefore [START_REF] Kontsevich | Mathematics unlimited-2001 and beyond[END_REF]

3, due to

Fricke and Klein:

The transcendence of the values 2 F 1 a, b; c z of hypergeometric functions when a, b, c and z are rational numbers has been investigated as early as 1929 by C.L. Siegel [START_REF] Siegel | Transcendental Numbers[END_REF]. A.B. Shidlovskii and his school produced a number of important results on this matter (see [START_REF] Fel | Transcendental numbers[END_REF]).

In 1988, J. Wolfart [START_REF] Shiga | Wolfart -Werte hypergeometrischer Funktionen[END_REF] studied the set E of algebraic numbers ξ such that 2 F 1 (ξ) is also an algebraic number. When 2 F 1 is an algebraic function, E = Q is the set of all algebraic numbers. Assume now that 2 F 1 is a transcendental function. J. Wolfart [START_REF] Shiga | Wolfart -Werte hypergeometrischer Funktionen[END_REF] proved that there is a bijective map between the set E and a set of Abelian varieties of CM type -this is an extension to higher dimension of a theorem due to Th. Schneider on the transcendence of the modular invariant j ( [START_REF]Arithmetische Untersuchungen elliptischer Integrale[END_REF], [START_REF] Schneider | Einführung in die transzendenten Zahlen[END_REF] Th. 17).

Wolfart 's proof uses the fact that the numbers 2 F 1 (a, b, c; z) are related with the periods of differential forms on the curve

aN , while N is the smallest common denominator of a, b, c (see [START_REF] Koblitz | Rohrlich -Simple factors in the Jacobian of a Fermat curve[END_REF]). His transcendence tool is the analytic subgroup Theorem of Wüstholz [START_REF]Wüstholz -Algebraische Punkte auf analytischen Untergruppen algebraischer Gruppen[END_REF].

J. Wolfart was able to give a complete list of algebraic numbers among these hypergeometric values. In the course of his study, he found with F. Beukers [START_REF] Beukers | Wolfart -Algebraic values of hypergeometric functions[END_REF] new relations which apparently were not stated before, like When the monodromy group of the hypergeometric differential equation satisfied by 2 F 1 is an arithmetic triangular group, the set E is infinite. Wolfart [START_REF] Shiga | Wolfart -Werte hypergeometrischer Funktionen[END_REF] claimed that conversely, the set E is finite when the monodromy group is not arithmetic but his proof was not complete. The works of P. Cohen and J. Wolfart [START_REF] Cohen | Wolfart -Modular embeddings for some nonarithmetic Fuchsian groups[END_REF], then of P. Cohen and G. Wüstholz [START_REF] Cohen | Wüstholz -Application of the André-Oort conjecture to some questions in transcendence[END_REF], provide a link between this question and the André-Oort Conjecture [START_REF] André | G-functions and geometry[END_REF][START_REF]Oort -Canonical liftings and dense sets of CM-points[END_REF], according to which the special subvarieties of the Shimura varieties are precisely the subvarieties which contain a Zariski dense subset of special points. P. Cohen then noticed that a special case of the André-Oort Conjecture in dimension 1 suffices; the crucial fact has been proved by B. Edixhoven and A. Yafaev [START_REF] Edixhoven | Yafaev -Subvarieties of Shimura varieties[END_REF]: in a Shimura variety, a curve contains infinitely many points on the same Hecke orbit of a special point if and only if it is special. This provides an answer to Siegel's initial question. More precisely, if we combine the results of [START_REF] Cohen | Wolfart -Modular embeddings for some nonarithmetic Fuchsian groups[END_REF][START_REF] Cohen | Wüstholz -Application of the André-Oort conjecture to some questions in transcendence[END_REF][START_REF] Edixhoven | Yafaev -Subvarieties of Shimura varieties[END_REF][START_REF] Shiga | Wolfart -Werte hypergeometrischer Funktionen[END_REF], one deduces: there is a bijective map between the exceptional set E and a set of points in the same Hecke


Thibault Prouteau

Nicolas Dugué

Simon Guillot

Anthony Perez
email: anthony.perez@univ-orleans.fr

Nicolas Thibault

SINr: a python package to train interpretable word and graph embeddings

Keywords: Graph embedding, Word embedding, Interpretability, Frugal machine learning 1 Introducing SINr

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

With neural approaches, tremendous progress was made in natural language processing, notably to represent the vocabulary of the language at hand, those representations are then used as input for machine learning algorithms. These representations are dense numeric vectors named word embeddings. Some examples of approaches to train such vectors using large textual corpora are Word2vec [START_REF] Mikolov | Efficient estimation of word representations in vector space[END_REF], Glove [START_REF] Pennington | Glove: Global vectors for word representation[END_REF], and the Transformer-based approached for contextualized representations, Camembert [START_REF] Martin | Camembert: a tasty french language model[END_REF] or Flaubert [START_REF] Le | Flaubert: Unsupervised language model pre-training for french[END_REF] in French. This progress was transfered to the graph universe, allowing the emergence of graph embedding, a whole field of research with Word2vec inspired approaches such as Node2vec [START_REF] Grover | node2vec: Scalable feature learning for networks[END_REF], matrix factorization methods like HOPE [START_REF] Ou | Asymmetric transitivity preserving graph embedding[END_REF] and auto-encoding paradigms [START_REF] Cao | Deep neural networks for learning graph representations[END_REF]. SINr was introduced to take advantage of this progress, it allows to embed words and nodes just like the aforementioned methods. However, it is based on community detection: for each node, the vector of embedding is calculated as the proportion of its links going to the communities as described Figure 1. This approach allows to avoid some flaws inherent to the usual approaches:

-As far as we know, SINr is the first approach specifically designed to deal with both word and graph embeddings. Textual corpora are represented as graphs, and with the adequate preprocessing provided by the package, word embeddings can easily be extracted with SINr. For graph embedding, no specific pre-processing is required. -Contrary to the neural approaches that require complex GPU calculations, SINr is based on the Louvain [START_REF] Vincent D Blondel | Fast unfolding of communities in large networks[END_REF] algorithm to detect community and thus runs in linear-time, it can be executed on standalone laptops. -Contrary to the usual approaches, because dimensions are based on the communities, the space in which words and graphs are embedded with SINr is interpretable.

The performances of SINr were evaluated on several tasks, including link prediction on graphs, and pair of words similarities for textual data [START_REF] Prouteau | Sinr: Fast computing of sparse interpretable node representations is not a sin! In IDA[END_REF]. While providing good performances, it runs faster than most of the other embedding approaches. Furthermore, the interpretability of the model was also demonstrated to be comparable to the state-of-the-art when considering word embedding [START_REF] Prouteau | Are embedding spaces interpretable? results of an intrusion detection evaluation on a large french corpus[END_REF]. In this paper, we consider the SINr package that we distribute on https://github.com/SINr-Embeddings/sinr and that can also be found on Pypi to be installed with pip. Fig. 2: Because of the sparsity of the embedding space, one can see that related words have non-zero values for the same dimensions (abscissa), mother and father for dimensions 4, 7, 8, 9 and 11 for instance. The non-zero dimensions are distinct when comparing mother and car that are unrelated.

Playing with SINr

One can setup SINr using pip install sinr, and train word and graph embeddings using the package features. While these embedding features are shared among models, SINr brings the option to probe and get an understanding of the resulting embedding space as one can see Figure 2. Using SINr leads to sparse vectors: a node is not connected to all the communities of a graph, and similarly, a word is not related to all the topics of a corpus. As shown by [START_REF] Subramanian | Spine: Sparse interpretable neural embeddings[END_REF], sparsity is one of the features required to enforce interpretability. The other one is to embed data in spaces larger than the classic 128 dimensions ones. The γ resolution parameter of the Louvain [START_REF] Vincent D Blondel | Fast unfolding of communities in large networks[END_REF] algorithm allows to vary the number of communities, thus controlling the number of dimensions.

Conclusion

We presented SINr, a package to train word and graph embeddings based on community detection. It runs in linear-time on standalone laptops and leads to interpretable spaces that can be inspected using features provided in the package. In the future, we plan to deal with temporal graphs and thus temporal embeddings with SINr and to include more features to visualize vectors.

  Bipartite projection of G into graph G′ = (⊤, ⊥, E) along the communities. Weight on the edges is the proportion of neighbors in that community. Adjacency matrix of G′, each row is a SINr embedding.

Fig. 1 :

 1 Fig. 1: Illustration of SINr, vertices are represented based on the communities they are linked to.

Acknowledgements This work was funded by ANR-21-CE23-0010 DIGING.


A Time for Memory in Lebanon? (Book Review)

by Candice Raymond French Institute of the Near East c.raymond@ifporient.org About Sune HAUGBOLLE, War and Memory in Lebanon, New York, Cambridge University Press, 2010, 260 p., and Franck MERMIER and Christophe VARIN (dir.) Mémoires de guerres au Liban (1975Liban ( -1990)), Paris, Sindbad/Actes Sud, 2010, 618 p.

During the general uprising that arose in Lebanon on the 17 th of October 2019 observers were struck by its cross-sectarian character and capacity to mobilise throughout most of the country's regions and across socio-professional categories. The initial unanimity in the protesting, with the prevalence of national symbols (flag, anthem), made plain in the uncompromising condemnation of the entire political class, without exception. It was soon interpreted as a sign of a national union almost unheard of in contemporary Lebanese history. Some were even tempted to see it as the real end of Lebanon's long civil war that engulfed the country between 1975 and 19901 the fractures of which seemed unresolvable ever since.

One of the hypotheses put forward to explain the scale of such an event, when all previous protests had failed to mobilise the population so widely, is worth examining: the idea that, with the generational renewal, the grip of fear of a return to war had finally been loosened. Nevertheless, it only took a hastily assembled wall to block the Nahr al-Kalb motorway tunnel, north of Beirut, or clashes between the Beirut neighbourhoods of Ayn al-Remmaneh and Chiyah, to rekindle a spate of anxieties. By taking up positions in these different sites of memory of the war, the protesters rekindled the fear of a violent "regression" and of a resurrection of the sectarian demons -all fuelled with the help of the traditional media and social networks. Initiatives were quickly taken to dismantle the wall and women's marches took place across the old demarcation lines to defuse this perceived danger and ward off the curse. But they were nonetheless insufficient in curtailing the mobilisation from partially ebbing. These recent episodes show to what extent the memory of the civil war still lingers thirty years after the official end of the fighting. It continues to be an effective factor in how social and political life in Lebanon functions. The persistence of vivid, often traumatic memories of this long war, transmitted within the family circle and creating just as many "post-memories" [START_REF] Hirsch Marianne | Family Pictures: Maus, Mourning, and Post-Memory[END_REF], infuses the imagination, feeds fears, gives shape to anticipations, and influences behaviour. But these episodes also demonstrate the effectiveness of the motto "tendhakar ma ten'ad" ("Let it [the war] be remembered and not repeated"), produced by three decades of memory work carried out by a civil society that is now at the heart of the protest movement; here, the reminiscence of the dark hours of the war is as much an anxiety-provoking reemergence of repressed or hidden memory as it is a "obligated memory" [Ricoeur, 2000: 105] with its correlated injunction to responsibility.

The official policy of the Lebanese state in the aftermath of the war, which consisted in promoting the non-remembrance (if not the forgetting) of the violence and pronouncing the amnesty of its protagonists2 as the main path towards national reconciliation, places Lebanon in what the literature on post-conflict policies has grouped under the paradigm of the "politics of amnesia" [Makdisi and Silver-Stein, 2006]. However, this policy was more prescriptive than performative, since this oblivious amnesty has not been enough to extinguish the living memories of past violence and the resentment that they feed on, nor to fully restore the pólis as its ancient Greek precedent aimed to do [START_REF] Loraux | L'oubli dans la mémoire d'Athènes[END_REF], nor permanently silence the supporters of a "duty to remember" sacrificed on the altar of security appeasement and post-war political and economic interests. Banning memory certainly deprived Lebanon up to the early 2000s of any large-scale public discussion on the subject of the civil war, but without being able to generate the "collective amnesia" [START_REF] Khalaf Samir | From a Geography of Fear to a Culture of Tolerance. Reflections on Protracted Strife and the Restoration of Civility in Lebanon[END_REF] that was prematurely diagnosed by some social scientists. Since then, socially differentiated analyses have highlighted the multiplicity of memories constituted and transmitted in different social spaces despite the official injunction to forget, while other studies allow for the historicization of one of the main intellectual debates of the Lebanese post-war period, namely the very necessity of memory work and, more broadly, the modalities of managing the legacy of the war.

On this double level of memories and politics of memory, Lebanon has indeed proved to be a particularly fertile ground for researchers from different social science disciplines over the last two decades. The two books selected for this book review, both published in 2010 at the end of several years of research, actually ushered in a series of publications in memory studies in Lebanon that illustrate how dynamic this field of research is; no less than six books on this theme were published in the space of only two years3 , with others to follow, not to mention the numerous scientific articles that have been written on these issues. This temporality of research echoes the political chronology of the country. Insofar as the crisis sequence that began in 2005 with the dramatic assassination of former Prime Minister Rafic Hariri, and the subsequent lifting of Syrian control over Lebanon enabled a series of new contentions, the return of war (the Israeli offensive in the summer of 2006, and the increase in political violence culminating in the interparty fighting of May 2008) acted as a powerful emotional trigger and led to the first nationwide debates about war memory [START_REF] Haugbolle Sune | War and Memory in Lebanon[END_REF][START_REF] Aïda | La Guerre et la Mémoire[END_REF]. This topicality of the memory issue within Lebanese society has since found its counterpart in the production of social sciences, leading to this genuine memory moment in Lebanese studies.

Memory cultures and the politics of remembering

This contextualisation helps to place the contributions of the two books at the heart of this book review. Sune Haugbolle's book, based on his doctoral thesis at Oxford [START_REF] Haugbolle Sune | The Politics of Remembering in Post-War Lebanon: civil war, memory and public culture[END_REF], is centred on field research conducted between 1997 and 2005, a period during which public interest in the memory of the civil war grew. The author starts from the observation that in the absence of any effort on the part of the state to establish what happened during the war and who was responsible for the tragedies that accompanied it, it is mainly through cultural production (understood in a broader sense than only the production of the cultural sector) that "politics of remembering" have been deployed by various non-state actors seeking to define the historical, political and ethical significance of the war [Haugbolle, 2010:4]. By placing observation at the intersection between personal memories and cultural forms that frame their public expression, the book aims to be both "an ethnography of social memory and a history of a central debate in the cultural and intellectual life of postwar Lebanon" [ibid.: 3]. However, the ethnography of living memories and post-memories, of the way in which they are inscribed in various social practices, transmitted or recomposed, only occupies a marginal place 4 . The book's central question is much more about publicizing certain individual or group memories, or in other words, the conditions (in particular the changing norms of the sayable and the unsayable), the strategies and modalities according to which they gain visibility in the public sphere and participate in the progressive elaboration of historical memories. S. Haugbolle mobilises a wide range of sources, from the most common ones such as the press and books (whether testimonies or novels) to other less common ones such as television programmes, films, graffiti, and other forms of urban iconography, while reporting on the public debates, artistic exhibitions, and intellectual encounters that he had the opportunity to observe. This rich and eclectic material is certainly one of the strengths of the book, although the restitution of its social contexts of production and reception proves patchy throughout the chapters. At any rate, it allows us to document the interplay of multiple "memory cultures", a notion that the author prefers to the all-encompassing notion of collective memory so as to underline the fragmentary, processual, and negotiated character of the production of Lebanese historical memories. It also makes it possible to historicise and, to a certain degree, sociologise the emergence of a memory issue and the mobilisations that have taken hold of it, drawing as many sequences in the political trajectory of the memory question: during the war itself, after the Taif peace agreement (1989) and the consecration of Syrian tutelage over Lebanon, and during the "Independence Intifada" in 2005.

The seven chapters of the book follow a thematic and chronological progression. The introductory chapter shapes the book's research problem by mobilising a multidisciplinary theoretical framework that draws on both memory studies and the literature on nationalism. It usefully returns to the master narratives of the war and the debates about the nature of the Lebanese sectarian system, recalling how the discourse that assimilates the Lebanese War to a "war for others" (according to the famous expression of the journalist and diplomat Ghassan Tuéni), i.e. a proxy war between regional powers on the Lebanese terrain, has made it possible to erase the most troublesome internal aspects of conflict and has thus imposed itself as the dominant representation of the conflict. The fascinating second chapter proposes a genealogy of contemporary memory culture by reviewing the transformations that modern institutions of public representation, in its double meaning of political representation and cultural expression, have undergone since the 1950s and 1960s. Seeking to historicize how Lebanese society pictures itself to be and negotiates the meaning of its own history, the author revisits the upsurge in mass culture media and its consequences on the expansion of an increasingly inclusive and decentralized public sphere, where cultural production constituted a space for debate on the political and social developments of pre-war Lebanon. It highlights the rise of the educated middle class, among whom the first dissident voices would be recruited and the first attempts at civil resistance to the war would be made during the 1980s.

The next four chapters take us into the post-war years and explore the different manifestations of these competing memory cultures. In a context that was not very conducive to the opening of a public debate on the meaning of the war (in addition to the adoption of the amnesty law in 1991, the law on audiovisuals voted in 1994 reinforced the tendency of many actors to self-censorship), the discussions that accompanied the start of the huge reconstruction project in downtown Beirut, addressed in Chapter 3, gradually grew to form a much more encompassing debate on the memory of the civil war. The "intellectual party" (as Samir [START_REF] Kassir Samir | Dix ans après, comment ne pas réconcilier une société divisée ?[END_REF] dubbed it -iterating Péguy) played a central role in objecting to a building project that oscillated between two geographic visualizations: Beirut as a forward-looking 'Hong Kong' of the Middle East, or as a middle-eastern Paris trading in nostalgia for a fantasized pre-war era. This theme of nostalgia dealt with in Chapter 3 and especially Chapter 4, gives rise to a wide-ranging discussion of its versatility, stoked up by an examination of its various expressions and uses within different generational groups and social classes. In contrast to these nostalgic graspings of the past, Chapter 5 addresses the much more marginal cultural productions that confront the darker aspects of the civil war and focus on the instigators of violence rather than its victims. Apart from a few literary and cinematographic works that share the transgressive ambition of unabashedly denouncing the militia experience, it is mainly the autobiographies and testimonies published by former political cadres or ordinary combatants that fall into this category. These public speaking engagements often take on the appearance of a cathartic exercise, but they also contain their share of euphemised violence and externalised guilt. According to Haugbolle, they are above all a way for their authors to renegotiate their membership in the moral community of post-war Lebanon, by aligning themselves with the dominant watchwords of reconciliation and condemnation of sectarianism. These productions, however, reveal representations in which the enemy of yesterday, now today's partner in a supposedly reconciled nation, is still the 'Other', outside a sectarian "we" that has been perpetuated as the main category of identification. Not surprisingly, the memory cultures of political parties discussed in Chapter 6 take this logic of sectarian identity to the extreme and inscribe their own "sectarian hagiography" in urban space, centred on the cult of dead leaders and war "martyrs", through a series of visual markers designed to circumscribe as many exclusive territories. Observing the graffiti, banners, flags, political posters, and other memorials, sometimes of a makeshift nature, is probably a clever way to circumvent the difficulty for an 'outsider' to enter certain partisan circles. It allows us to address new issues related to the spatialization of memory cultures and the particular way of making war memories visible that transform the spaces of everyday life in Beirut into out-andout propagandist palimpsests.

The concluding chapter traces the turning point of the years 2000-2005, providing us with a fascinating reading of the political movement of spring 2005 through the prism of memory discourses. It shows in particular how the memory culture developed in intellectual circles, which had gained momentum in the early 2000s, was co-opted by the anti-Syrian opposition front in the form of a demand for truth (initially relating to the assassination of Rafiq Hariri but extended to all crimes of the past), that was heard in different ways among the actors of the movement, and was ultimately more divisive than expected: «Telling the truth about the war for many people meant displaying their disagreements over national history and identity in antagonistic terms. This was not the truth memory makers had in mind. » [Haugbolle, 2010: 224]. According to Haugbolle, the "independence intifada" and the various memorial events that accompanied it thus appeared to be the moment of consecration of a liberal memory culture (in the Anglo-Saxon sense of the term), while at the same time rapidly showing its limits; its moorings in the particular social imagination of cultural elites largely from the left would lead it to be overly optimistic about the reconciling virtues of the work of remembrance and to underestimate the strength of competing " politics of remembering". If 2005 did indeed mark the ending of the injunction to silence, the subsequent crisis sequence bore witness to the strong politicisation of war memory, to which the actors in the political field had uninhibited recourse to publicly discredit their opponents of the moment.

A mosaic of memory

The richness of the empirical material and of the analyses scattered throughout the book, despite the occasionally convoluted writing style, makes it an essential reference in this field of research, although it is evident that the subject has not yet been depleted. Many of the avenues opened up by S. Haugbolle are refined, developed, or completed by the collective work directed by F. Mermier and C. Varin [2010]. Inspired by a rich francophone literature in sociology, anthropology, and history of memory, ranging from Maurice Halbwachs to Michèle Baussant and Marie-Claire Lavabre, via Paul Ricoeur, Pierre Nora, and Henri Rousso, the work presented in this imposing volume goes beyond this theoretical framework and adopts a wide variety of approaches and investigative devices to deal with a wide variety of issues. Memories (personal and collective, living and transmitted, social and historical) are sometimes the subject of the investigation itself, or the documentary material that allows us to address other subjects. Thus, although this was not the primary objective of the book, as the editors point out, we also learn a great deal about the war itself; some of its episodes (in particular the clashes between the two Shiite organisations Amal and Hezbollah at the end of the 1980s, dealt with by O. Lamloum) or some of the social spaces that were subjected to its effects (the school environment, for example, studied by S. Slim). The eclecticism of the collection extends to the modes of writing adopted by the contributors, with some very beautiful chapters of creative non-fiction such as the "Diary of G.S., displaced by war" reconstructed by W. Charara from interviews conducted with one of his students.

The thirty contributions are grouped into three parts. It is perhaps regrettable that they were not preceded by introductions that would have allowed us to relate their immense diversity to a few theoretical or transversal thematic issues. The first part, entitled "Writings of Memory", brings together a set of texts that examine the complex relationship between memory(ies) and historiography (whether in academic history, history textbooks, or competing testimonies published by former actors in the conflict), and that look at how different memory entrepreneurs have taken over certain sites of memory, commemorative dates, or media (in particular the web). This part, which focuses on the strategies and practices of the actors, is very complementary to the monograph mentioned above, and extends its effort to historicise the Lebanese memory issue by dealing with another historical dispute that is embedded in contemporary social memory, namely the sectarian conflicts of the nineteenth century. The second part deals with the political dimension of war memories, firstly by looking at the memory construction and competition among the former armed protagonists, and then at the issues relating to the fate of the victims (questions of the disappeared, reconciliations organised by the state in the mixed villages of Mount Lebanon, or the psychological shields developed by individuals following traumatic experiences). Finally, three types of "spaces of memory" are explored in the last part: the city, the cultural field, and the religious field. The city, first of all, reduced here to Beirut alone (as in the previous book, one would have appreciated some excursions outside the capital, as the cities and regions of the provinces remain the poor cousins of the historiography of the war), but intensely explored through six chapters forming perhaps the most ethnographic part of the book. The examination of the social and spatial reconfigurations following the demographic reconfigurations resulting from the war gives a central place to the question of the displaced, whose memory stakes are grasped by scrutinizing the cemeteries of the southern suburbs, by collecting the accounts of former Christian inhabitants of a district that has become Shiite, by settling narratively in a squatted building on the famous Hamra Street, or by surveying the streets of a district close to the city centre awaiting renovation. Artistic creation is another space of memory that is approached both from the angle of the post-war avant-gardes that have made the archive their idiom (a subject that has been abundantly dealt with elsewhere, but which is the subject of a particularly enlightening sociological approach here) and from the angle of the partisan commemorative posters produced during the war itself. Finally, the last two chapters examine the politico-religious dimension of the social frameworks of memory, through Shiite rituals and Christian pilgrimages. The book ends with a short but didactic conclusion by Fawaz Traboulsi defending the necessity of remembrance and the need to forget.

It is easy to see from this overview of the themes and objects dealt with that this is a richly informative work that explores the multiple ways in which these fragmented war memories are inscribed and mediated. Incidentally, the memory entrepreneurs from organised civil society find their rightful place: not quite marginal but nonetheless minor, compared to the dynamics at work in sectarian and partisan spaces. The book also shows what work of memory is indeed being done in Lebanese society; it is a type that does not fit into the weave of an untraceable national narrative, but one that diffracts among several "imagined communities", which are themselves worked on by several politics of remembering that are more or less elaborate and adaptable to a shifting political environment. Over the last decade, various studies have continued to explore the manufacture, circulation, and reception of sectarian or partisan memory cultures in the mixed Druze-Christian region of Mount Lebanon [De Clerck, 2015] or within the Shiite community [START_REF] Chaib Kinda | Culture du martyre au Liban Sud : entre fabrication de catégories et enjeux mémoriels[END_REF][START_REF] Fournier | Géographie des lieux de mémoire du Sud du Liban[END_REF][START_REF] Saade Bashir | Hizbullah and the Politics of Remembrance[END_REF]. And in the same manner as Lebanon. Mémoires de guerre they show that a view from below allows for a detailed reading of the cleavages and tensions within communities and even individuals. The dynamics of memory can therefore prove to be a valuable analyser of the power relationships in certain territories or social groups, and all the more so as they call for a reflection on variations in scale. If the political hegemonies sharing the control of Lebanese society retain an undeniable capacity to frame memories and imaginations, it is in any case not certain that they always manage to keep the public space closed to the claims made by the actors of organised civil society. The adoption of Law 105 in November 2018 by the newly-elected Parliament on enforced disappearances, and the appointment in June 2020 of the national commission responsible for shedding light on the fate of the missings, thus appear to be a further victory won by the tenacious mobilisation of the families of the disappeared [Abou Jaoudé, 2020].

More than a decade of intensive production on the issue, now begs the question of whether memory studies have become a saturated field of research in Lebanon. If the fragmentation of war memories implies that multiple fields remain to be explored, it is nonetheless up to each discipline to consider which thematic and theoretical renewals are possible. For the history discipline, it is a given that memories are now among the common objects of research. However, the larger project to be pursued is that of a history of the war itself and its effects on society. Academic research, also subject to the effects of generational renewal, has taken on this essential "history work" by shifting the investigation to less studied phases of the war, geographical areas, or research objects5 . Working with and against the memories that permeate the historiographic operation from one side to the other, this research lays the foundations for a revisiting of the political, social, and cultural history of this crucial period for the understanding of contemporary Lebanon.

Without necessarily pretending to "set the tone" for memory dynamics, the conviction remains that this historical research is playing its part in the effort of the Lebanese society to critically distance itself from a painful past.

Marwan Chahine, Is this finally the end of the civil war? (« La fin de la guerre civile, enfin ? »), L'Orient-Le Jour, 1 Nov.

The Amnesty Law of the 26 th of August 1991 exempts from prosecution the perpetrators of political crimes committed before March 1991, with the exception of crimes committed against political leaders, religious leaders, or diplomats.

[START_REF] Haugbolle Sune | War and Memory in Lebanon[END_REF] MERMIER and VARIN, 2010;[START_REF] Volk | Memorials and Martyrs in Modern Lebanon[END_REF][START_REF] Aïda | La Guerre et la Mémoire[END_REF][START_REF] Sawalha Aseel | Reconstructing Beirut: memory and space in a postwar Arab city[END_REF][START_REF] Larkin | Memory and Conflict in Lebanon: remembering and forgetting the past[END_REF] Let us also mention the more recent publications of S. Hermez[2017], who places the study of living memories of the civil war in a broader anthropological reflection on violence in post-2005 Lebanon, and of C.[START_REF] Calargé | Liban, mémoires fragmentées d'une guerre obsédante : l'anamnèse dans la production culturelle francophone[END_REF], who examines studies on anamnesis of war carried out in post-war Francophone cultural production.

For this type of work, readers can turn to[START_REF] Aïda | La Guerre et la Mémoire[END_REF], whose third part analyses the memory narratives solicited from Christian and Druze victims of the Mountain War(1982)(1983)(1984), and Craig[START_REF] Larkin | Memory and Conflict in Lebanon: remembering and forgetting the past[END_REF], who deals with the post-memories of Lebanese high school and university students. See also, from a more historical perspective, the various articles published by Dima de Clerck on the Christian and Druze memories of southern Mount Lebanon, notablyDE CLERCK, 2014, and her chapter in MERMIER and VARIN, 2010.

Several dissertations and theses recently defended or in progress show this renewed interest in the history of the Lebanese war, which was also reflected in the holding of two conferences in Beirut in 2015 and 2018 (the works presented on the occasion of the latter forming the subject of the recent issue of Confluences Méditerranée, 112, 2020/1: "Liban : la guerre de1975-1990 dans le rétroviseur")