Coproducts and cobrackets on multiple zeta values
Samuel Baumard
Abstract. The aim of this talk is to describe a coproduct on a multiple zeta values introduced by A. Goncharov, and to discuss its connections with the Lie bracket on the double shu e algebra. In a rst part, we de ne the coproduct following Goncharov, and detail some of its properties in a particular space of formal zeta values. Then, the de nition of the double shu e Lie algebra is recalled, whose Lie bracket is in some way dual to a cobracket deduced from Goncharov's coproduct. The last part is devoted to introduce a recent version of a conjecture concerning Ihara's congruences in the double shu e algebra.
Contents 1. Goncharov's coproduct on multiple zeta values 1 2. Dual structures: the double shu e Lie algebra 5 3. Ihara's congruences 7 References 11
1. Goncharov's coproduct on multiple zeta values 1.1. De nition of the coproduct. In the article [Gon05], Goncharov de- nes a coproduct on several spaces of iterated integrals, which gives in particular a coproduct on real multiple zeta values. Let us rst describe the iterated integrals he considers, and then the coproduct he builds on his version of formal multiple zeta values.
Let (a0, . . . , an+1) be a (n + 2)-uple of complex points, and let γ be a path connecting a0 to an+1. One can then de ne the iterated integral Z def 1 dt1 dtn (1) Iγ (a0; a1, . . . , an; an+1) = n ··· (2πi) ∆n,γ t1 − a1 tn − an where ∆n,γ is the set of all ordered n-uples (t1, . . . , tn) of points on the path γ, provided that this integral converges (if it is not the case, it is however possible to give it a signi cation by regularizing it). If the ai's are in a number eld F , and having xed an embedding σ : F,→ C, it is possible to map the integrals above into a graded algebra σ , which we shall not discuss here. The image P• (F ) of an integral depends no more on the choice of the path γ, and the elements thus de ned obey several rules such as a shu e product formula corresponding to the shu e product of two iterated integrals.
Note that if the ai's equal 0 or 1, one recovers the integral de nition of classical multiple zeta values. By analogy with this algebra of iterated integrals, Goncharov de nes for any set S a graded commutative algebra I•(S) as follows. a) The algebra has formal generators for any inte- I•(S) I(s0; s1, . . . , sm; sm+1) ger and -uple of elements of . m > 0 (m + 2) si S b) For any pair of elements a, b ∈ S, the generators I(a; ∅; b) are equal and yield the unit of I•(S). 1 2 S. BAUMARD c) These elements satisfy the shu e product formula, that is, if one xes two elements a and b of S and two integers m and n, then the identity X (2) I(a; s1, . . . , sm; b) · I(a; sm+1, . . . , sm+n; b) = I(a; sσ(1), . . . , sσ(m+n); b) σ∈Sm,n
holds for any choice of elements si ∈ S, where Sm,n denotes the set of all (m, n)- shu es1. d) Every generator splits in the following way, called path I(s0; s1, . . . , sm; sm+1) composition formula: for any x ∈ S, one has m X . (3) I(s0; s1, . . . , sm; sm+1) = I(s0; s1, . . . , sk; x) · I(x; sk+1, . . . , sm; sm+1) k=0 e) The element equals zero if (and otherwise). I(a; s1, . . . , sm; a) m > 0 1 Each of the four latter conditions is the analogue of a similar property of iterated integrals. Moreover, the algebra has a natural grading L , I•(S) I•(S) = m 0 Im(S) obtained by giving to the generator the weight .> I(s0; s1, . . . , sm; sm+1) m In the case of multiple zeta values, the set S is simply chosen to be {0, 1}, and it is then convenient to be able to express an element I(1; ... ; 0) in terms of the I(0; ... ; 1)'s. This is done by the following proposition, proved by Goncharov by induction on m: Proposition 1. If is an integer and are elements of , m > 0 a0, . . . , am+1 S then m . (4) I(a0; a1, . . . , am; am+1) = (−1) I(am+1; am, . . . , a1; a0)
Goncharov then de nes a coproduct ∆ on the algebra I•(S) by giving the image of the generators (but the fact that this de nition is coherent with the shu e product formula is not obvious). A generator is sent by I(a0; a1, . . . , am; am+1) ∆ on the element k X Y (5) I(ai0 ; ai1 , . . . , aik ; aik+1 ) ⊗ I(aip ; aip+1, . . . , aip+1−1; aip+1 ) p=0 where the sum is over all (k + 2)-uples (i0, . . . , ik+1) with k ∈ [[0, m]] satisfying the conditions 0 = i0 < i1 < ··· ik < ik+1 = m + 1. 1.2. Interpretation in the Homann setting. The Homann notations are a useful way of encoding multiple zeta values. Let us recall them.
First, the letter h denotes the algebra Qhx, yi of noncommutative polynomials in the two formal variables x and y with rational coe cients. It contains the subalgebra 1 , which is generated (as an algebra) by the monomials h = Q ⊕ h y yi = xi−1y, and which in turn contains the subalgebra h0 = Q ⊕ x h y generated (as a vector space) by the monomials with . yi1 ··· yid i1 > 2 The multiple zeta values are encoded in the following manner. One de nes a linear map ˆ 0 which maps the monomial to the multiple zeta ζ : h → R ys1 ··· ysd value X 1 (6) ζ(s1, . . . , sd) = s1 sr n1 ··· nr n1>···>nr and which respects the units. The conditions de ning h0 are precisely the ones needed to ensure the convergence of the above sum. Moreover, as mentioned before,
1 Recall that a (m, n)-shu e is a permutation of the set [[1, m + n]] which reciprocal is in- creasing on the subsets [[1, m]] and [[m + 1, m + n]]. COPRODUCTS AND COBRACKETS ON MULTIPLE ZETA VALUES 3 there is a well-known representation of those multiple zeta values in terms of iterated integrals. Both representations of multiple zeta values give rise to an expression of the product of two multizetas as a sum of multizetas, which translates to an algebra structure on h0. More precisely, the shu e product ø on h corresponds to the product of iterated integrals, and is given recursively by 0 0 0 (xiw) ø (xjw ) = xi (w ø (xjw )) + xj ((xiw) ø w ) (7) where xi ∈ {x, y}. There is also an equivalent de nition in terms of the sets Sm,n previously introduced. As for the stu e product ∗ on h1, it corresponds to the product of series and is given by the recursive de nition 0 0 0 0 (yiw) ∗ (yjw ) = yi (w ∗ (yjw )) + yj ((yiw) ∗ w ) + yi+j(w ∗ w ). (8) Both multiplication laws restrict to the subalgebra h0, and the linear map ζˆ becomes an algebra morphism for each of them.
It is possible to extend the morphism ζˆ to a morphism from h1 (with the shu e product) to R, and even from h to R, by means of a regularization formula (due to Furusho) involving shu es and projection onto words that end with a y. One can also de ne a formal version of these zeta values, considering the com- mutative algebra FZ generated by formal symbols Z(f) for any polynomial f in the x's and y's, subject to the following conditions: the symbols Z(f) must be li- near in f, and satisfy identities corresponding to the regularization and the shu e and stu e relations. For instance, one asks that for any divergent word w, the element Z(w) is equal to the element Z(w0) given after shu e regularization of w, and that one has the identities Z(w) Z(w0) = Z(w ø w0) for any two words w and w0. Goncharov's coproduct should give a coproduct on the formal zeta values thus de ned, or equivalently a coproduct on Qhx, yi compatible with the shu e product and the relations of regularization and stu e. If one follows directly Goncharov's de nition (5), it yields to the following expression: let w = w1 ··· wn be a word in x and y; then k def X (9) ∆G(w) = wi1 ··· wik ⊗ Ø εi,p(wip+1 ··· wip+1−1) p=0 16i1<··· Df (x) = 0 and Df (y) = [y, f]. (10) The dual of the coproduct ∆G has a simple expression in terms of these deriva- tions, provided that the right hand side of the tensor product is a Lie element: Proposition 2. Let w be any word in x and y and let f and g be elements of Qhx, yi. If g is a Lie polynomial, then h∆G(w), f ⊗ gi = hw, fg − Dgfi. (11) One can give a variant of this result by transforming a little the coproduct ∆G. Let denote the antiautomorphism of that changes the sign of and , SX Qhx, yi x y and τ the automorphism of Qhx, yi⊗2 that swaps the sides of any pure tensor. Then one can prove the following Proposition 3. With SX and τ as above, def i) The linear map 0 is still a coproduct in relation ∆G = τ ◦ (SX ⊗ SX ) ◦ ∆G ◦ SX to the shu e product; ii) Let w be any word in x and y and let f and g be elements of Qhx, yi. If f is a Lie polynomial, then 0 . (12) h∆G(w), f ⊗ gi = hw, fg + Df gi 1.4. From coproducts to cobrackets. We brie y describe here how a copro- duct in a graded Hopf algebra A gives a Lie coalgebra structure on a vector space deduced from A. First, one says that a vector space E over a eld k is a Lie coalgebra if its dual E∗ has a structure of Lie algebra. If E is nite dimensional, or if it is graded and all its graded parts are nite dimensional, it amounts to saying that there is a linear map δ : E → Λ2E (dual to the bracket of a Lie algebra g, which factors through a linear map Λ2g → g), satisfying the identity (δ ∧ 1 − 1 ∧ δ) ◦ δ = 0 (dual to the Jacobi identity). Suppose one is given a Hopf algebra A over a eld k, with coproduct ∆ and counit ε. Let I denote its augmentation ideal, which is by de nition the kernel of ε, and ∆¯ the restricted coproduct, de ned by ∆(¯ x) = ∆(x) − x ⊗ 1 − 1 ⊗ x for x ∈ A. Using the direct sum decomposition A = k 1 ⊕ I, one can easily check that this restricted coproduct maps I into I ⊗ I. Moreover, it maps I2 into I2 ⊗ I + I ⊗ I2, and thus gives a map from I/I2 to its tensor square (I/I2)⊗2; as a consequence, one gets a linear map δ : I/I2 → Λ2(I/I2). Moreover, one can check that the identity (∆ ⊗ 1 + 1 ⊗ ∆) ◦ ∆ = 0 (which is the de nition of the coassociativity of A) remains true if one replaces the coproduct by the restricted one; the identity (δ ∧ 1 − 1 ∧ δ) ◦ δ = 0 therefore holds. To sum up, one has the following Proposition 4. Let A be a Hopf algebra with augmentation ideal I. Then the restricted coproduct provides the quotient I/I2 with a Lie coalgebra structure. Returning to the multiple zeta values, a variant of Goncharov's coproduct could give a Lie cobracket on the space of so-called new formal zeta values, that is to say the quotient of formal zeta values by the subspace containing ζ(2) and the nontrivial products of zeta values. We will see below another point of view on this cobracket, given by considering directly the dual of the new formal zeta values space. COPRODUCTS AND COBRACKETS ON MULTIPLE ZETA VALUES 5 2. Dual structures: the double shu e Lie algebra 2.1. General facts. Let us go back to the algebra FZ of section 1.2 and introduce some more notations. The space nfz (for new formal zeta) will be the quotient of the subspace of FZ spanned by the Z(w) for all words w having more than three letters by the subspace spanned by the products Z(w) Z(w0) for any two nontrivial words w and w0. This space inherits a natural grading from the weight grading on , the subspace being generated by the images of FZ nfzn Z(w) for the words w with exactly n letters. Moreover, it comes with an increasing ltration by depth, the subspace d being composed of elements where nfzn Z(f) f is a homogeneous polynomial of degree n in x and y, in which the number of y's in all monomials is at most d. On the dual side, the Q-vector space ds (for double shu e) is the set of non- commutative polynomials f in x and y, of minimal degree at least 3, such that, for any nonempty words w and w0 for which it makes sense, hf, w ø w0i = hf ∗, w ∗ w0i = 0 (13) where f ∗ is the corrected polynomial X (−1)n−1 f ∗ = π f + hf, xn−1yi yn (14) y n n>1 coming from the regularization of multizetas (here πy stands for the orthogonal projection onto the words ending on y). The space ds is again graded by weight, and has a ltration, decreasing this time: the subspace d is the set of polynomials dsn f ∈ ds of homogeneous weight n, and in which the number of y's in every monomial is at least d. For instance, here are the rst two nontrivial graded parts of ds, as well as the dimension of the rst twelve parts: ds3 = Q · ([x, [x, y]] − [y, [x, y]]) ds5 = Q · [x, [x, [x, [x, y]]]] − 2 [y, [x, [x, [x, y]]]] + 2 [y, [y, [x, [x, y]]]] − 1 3 − [y, [y, [y, [x, y]]]] − 2 [[x, y], [x, [x, y]]] + 2 [[x, y], [y, [x, y]]] n 1 2 3 4 5 6 7 8 9 10 11 12 dim dsn 001010111 1 2 2 Note that every element in ds is a Lie polynomial, which results from the shu e relations. The de nitions of nfz and of ds are quite similar, and in fact these two spaces happen to be dual to each other: Proposition 5. There exist a canonical isomorphism of graded vector spaces between ds and the graded dual of nfz. This is easily seen by noting that the set of linear forms on is exactly hf, · i nfzn the set of forms on FZn which are identically zero on the subspace of the de ning relations, which amounts to saying that f satis es both shu e and stu e relations. The dimension of the rst graded parts of ds are relatively easy to compute, and it is conjectured that one has X n (15) dim dsn = µ d Ud d|n 6 S. BAUMARD where µ is the Möbius function and U the Perrin sequence de ned by the gene- rating series P n 2X2+3X3 . By duality, this is closely related to the n 0 UnX = 1−X2−X3 Broadhurst-Kreimer> conjectures on dimensions of spaces of multiple zeta values. 2.2. Racinet's result on the Poisson bracket. We previously saw that the space nfz could be provided a structure of Lie coalgebra, and so its dual ds becomes a Lie algebra. In his PhD thesis [Rac00], Racinet gives another approach of this Lie structure, constructing explicitly a Lie bracket on ds. Let us shortly review his result. The Lie bracket considered here is the Poisson bracket, de ned by {f, g} = [f, g] + Df (g) − Dg(f); (16) it satis es the identity [Df ,Dg] = Df ◦ Dg − Dg ◦ Df = D{f,g}. (17) The fact that the Poisson bracket preserves the Lie elements and thus the shu e elements is almost obvious on the de nition; the fact that it preserves ds is far less easy. Let us sketch the main ideas of Racinet's proof. i−1 Let Y stand for the alphabet formed by all formal symbols yi = x y with i ∈ . If one denotes by the coproduct on de ned by N ∆∗ QhY i ⊂ Qhx, yi n X ∆∗(yn) = yi ⊗ yn−i (18) i=0 for , with the convention , then the stu e elements are exactly the ones n > 1 y0 = 1 whose corrected polynomial is primitive for the coproduct ∆∗, or in other terms the elements f that satisfy ∆∗(f∗) = f∗ ⊗ 1 + 1 ⊗ f∗. Racinet's proof then consists in showing that, if f and g are two elements of ds, then their Poisson bracket {f, g} is still primitive for ∆∗. The strategy is to build a linear endomorphism Y of such that s{f,g} QhY i 2 a) this morphism is the Lie bracket of two coderivations for the coproduct ∆∗; b) one has Y . s{f,g}(1) = πy{f, g} = {f, g}∗ As one readily checks that the coderivations of a Hopf algebra are stable by Lie bracket, and that the image of the unit by a coderivation is itself primitive, this directly implies that the Poisson bracket of f and g is also in ds. The map Y is de ned for any element by Y , and is itself sf f ∈ QhY i sf = πy ◦sf sf de ned by the identity sf (g) = fg + Df (g). The condition (b) above is rather easy to check. As for the condition (a), Racinet shows that sY = [sY , sY ] for {f,g} sec(f∗) sec(g∗) any f and g in ds, where the map sec is de ned on Qhx, yi by X (−1)i sec(f) = ∂i (f)xi. (19) i! x i>0 One then wants to prove that sY is a coderivation for every element f ∈ ds, sec(f∗) and this is in fact equivalent to showing that the map de ned by dsec(f∗) dsec(f∗) = is itself a coderivation. It is facilitated by the fact that adsec(f∗) +Dsec(f∗) dsec(f∗) is, by de nition, a derivation, and one is nally led to show that (20) ∆∗ ◦ dsec(f∗)(yn) = (dsec(f∗) ⊗ 1 + 1 ⊗ dsec(f∗)) ◦ ∆∗(yn) 2 In an algebra A with product m, a derivation is a map D : A → A such that D ◦ m = m◦(D⊗1+1⊗D). Dually, in a coalgebra A with coproduct ∆, a coderivation is a map D : A → A such that ∆ ◦ D = (D ⊗ 1 + 1 ⊗ D) ◦ ∆. COPRODUCTS AND COBRACKETS ON MULTIPLE ZETA VALUES 7 for every integer n > 0. This last goal brings into play some rather involved calcu- lations. 3. Ihara's congruences In this last part, we describe the so-called Ihara's congruences, rst observed by Yasutaka Ihara around 1992. These congruences involve linear combinations of Poisson brackets of particular elements of ds. Let us rst discuss the original remark of Ihara, and then several steps towards a more precise conjecture. 3.1. Stable derivation algebra and a theorem of Ihara and Takao. In the articles [Iha92] and later [Iha02], Yasutaka Ihara describes a Lie algebra grt over Z which derives from the spherical braid group on ve strands. It is conjectured that one has a Lie algebra isomorphism between ds and grt⊗Q; the best comparison obtained so far is due to Furusho [Fur10], who showed that grt ⊗ Q is in fact a subalgebra of ds. The Lie algebra grt can be de ned as follows. The group P5 is the spherical braid group on ve strands, to which is attached a graded Lie algebra P5 in a canonical way. This latter Lie algebra is generated by elements xij where i and j span Z/5Z. One then de nes the n-graded part grngrt of grt as being P {f ∈ Lie [x, y]n | f(xi,i+1, xi+1,i+2) = 0} (21) Z i∈Z/5Z where f(xi,i+1, xi+1,i+2) stands for the image of f in P5 by the Lie algebra mor- phism that send x on xi,i+1 and y on xi+1,i+2. The Lie algebra grt is then the direct sum L n . It is called stable derivation algebra or Grothendieck-Teichmüller n 0 gr grt algebra. > The Lie bracket on the space grt thus de ned is not trivial, but there is an equivalent (although slightly dierent) way to build grt as a subquotient of the Lie algebra of derivations of P5 that allows to de ne a straightforward Lie bracket on the stable derivation algebra. As grt is de ned as a subset of Zhx, yi, one has a notion of monomial coe cients, n and this leads to the de nition of particular elements Dn ∈ gr grt. Namely, for any odd integer , one takes an element which is a homogeneous polynomial of n > 3 Dn weight n, such that the coe cient of xn−1y is nonzero and minimal among all such coe cients in grngrt (in other words, it is taken to be a generator of the abelian group hgrngrt, xn−1yi). One can show that such elements always exist, but are not unique in general. One expects that a linear combination P should be of depth i+j=n aij [Di,Dj] 2 in general, as each Lie bracket contains a part of depth 2 which could not cancel with the others. Ihara then asks which are the combinations that are of depth greater than or equal to 3; for instance, he nds that it is the case for the element 2 [D3,D9] − 27 [D5,D7] (22) in weight 12, and for another linear combination in weight 16. The study of such linear combinations shows a rather surprising connection with the space of parabolic modular forms for , as shown by Ihara and SL2(Z) Takao (recall that stands for the space of weight cusp forms): Sm(C) m Theorem 6 (IharaTakao [Iha02]). Let m > 4 be an even integer. Then, m among the elements [Di,Dj] ∈ gr grt for i and j odd integers, greater than or equal to and satisfying , there are exactly independent 3 i + j = m dimC Sm(C) linear relations modulo depth 3. 8 S. BAUMARD 3.2. Link with period polynomials. The above theorem suggests a connec- tion between coe cient of combinations of the form (22) and cusp forms, but this connection is not obvious on the stable derivation side. However, when one considers what happen on the conjecturally equivalent double shu e side, the relation becomes clearer, as observed by Herbert Gangl around 2006. The elements Dn ∈ D n−1 can indeed be related to polynomials fn ∈ dsn such that hfn, x yi = 1. With this new normalization, the combination (22) is rewritten {f3, f9} − 3 {f5, f7}. (23) Gangl's remark is then that the coe cients 1 and −3 in combination (23) happen to be, up to a multipicative constant, coe cients of the even period polynomial of Ramanujan's modular form3. Let us recall the de nition of this polynomial. Let f be a parabolic modular form of weight k. Then its period polynomial is de ned to be k−2 Z ∞ X k − 2 r(f) = f(z)(X − z)k−2 dz = (−1)i r (f) Xk−2−i (24) i i 0 i=0 where the integral is taken on any path connecting 0 to the in nity in Poincaré's upper half plane. The even part of this polynomial is r(f)(X)+r(f)(−X) . r+(f) = 2 For instance, the even period polynomial of Ramanujan's delta function is 36 10 8 2 6 4 + (25) r+(∆) = 691 (X − 1) − (X − X ) + 3 (X − X ) ω∆ with + , and one recovers the coe cients and of the combina- ω∆ ' 0, 114 379 i 1 −3 tion (23). Gangl's remark led to a theorem of Leila Schneps [Sch06], of which Luque, Novelli and Thibon [LNT07] gave an alternative proof one year later: Theorem 7 (L. Schneps). Let n > 8 be an even integer. Let [ n−4 ] X4 S = ai {f2i+1, fn−2i−1} ∈ dsn. (26) i=1 Then the depth of S is greater than or equal to 3 if and only if the polynomial n−4 [ 4 ] X 2i n−2−2i ai (X − X ) (27) i=1 agrees modulo Xn−2 − 1 with the even period polynomial of a cusp form. n−2 As the map r+ is an injection (even after rducing modulo X −1), this result gives a stronger version of theorem 6. 3 In fact, this remark could already have been made from the article [Iha02]; indeed, the com- bination in weight is ∗ ∗ ∗ ∗ ∗ ∗ , where ∗ stands for the quotient 16 2 [D3 ,D13] − 7 [D5 ,D11] + 11 [D7 ,D9 ] Di i−1 of Di by its coe cient in x y, and with an appropriate determination of D11 and D13. The ele- ments ∗ correspond exactly to the double shu e elements , and this time the coe cients , Di fi 2 −7 and 11 are proportional to coe cients of the period polynomial of the cusp form ∆E4. COPRODUCTS AND COBRACKETS ON MULTIPLE ZETA VALUES 9 3.3. Comparison with a result of Gangl, Kaneko and Zagier. In the article [GKZ06], the three authors state a theorem that seems to be very similar to Schneps' one. Let us rst describe the vector space in which they work. For any even natural integer , the formal double zeta space is the -vector k Dk Q space generated by formal symbols Zk, Zrs and Prs for r+s = k and r, s ∈ [[1, k−1]], subject to the relations Prs = Zrs + Zsr + Zk (28) X i − 1 i − 1 and P = + Z (29) rs r − 1 s − 1 ij i+j=k for every (r, s) as above. This space is the exact analogue of the space of formal zeta of weight k and maximal depth 2. The authors consider the subspace ev of generated by the for and Pk Dk Prs r s even, and similarly to Ihara in [Iha02] ask themselves to what condition a linear combination P falls into ev. The answer is the following. r,s odd crs Zrs Pk Theorem 8. Let p be any even polynomial of degree at most k − 4, without constant term. Write Pk−1 k−2 r−1. Then one has p(X + 1) = r=1 r−1 qr X X ev (30) qr Zrs ∈ Pk r+s=k r,s odd if and only if p is the even period polynomial of some cusp form, modulo Xk−2 − 1. To study the relationship between theorems 7 and 8, one could consider the universal enveloping algebra of ds, which can be seen as a subspace of Qhx, yi. The duality relation between this algebra and a coalgebra of multiple zeta values should be explored, in which the desired relations seem to translate into orthogonality relations with respect to particular subspaces of these (co)algebras. This approach reduces the problem to a combinatorial and linear algebraic one. 3.4. Shari 's condition, and a conjecture. There is another striking fact concerning Ihara's linear combinations such as (23): the numerators of their coef- cients seem to be multiples of very particular prime numbers. For instance, one observes the congruences {f3, f9} − 3 {f5, f7} ≡ 0 (mod 691) (31) and 2 {f3, f13} − 7 {f5, f11} + 11 {f7, f9} ≡ 0 (mod 3617). (32) Here again, the second expression involves an appropriate choice of elements f11 and f13, and both congruences are to be interpreted as taking place in the locali- zations and . Z(691) Z(3617) The prime numbers 691 and 3617 that appear above are not completely in- 4 nocent: they also show up in the numerators of Bernoulli numbers B12 and B16. This suggests another connection with modular forms, since the constant terms of Eisenstein series are essentially Bernoulli numbers; namely, one has ∞ Bk X G (z) = − + σ (n) qn (33) k 2k k−1 n=1 4 The rational numbers Bn called Bernoulli numbers are usually de ne by the exponential generating series t = P Bn tn. Although their denominators are well understood (by et−1 n>0 n! means of the ClaudenVon Staudt theorem), their numerators are much more complex and have, among others, interpretations in terms of algebraic number theory (Kummer's criterion). 10 S. BAUMARD for any even , with the classical notations 2πiz and P k−1. k > 4 q = e σk−1(n) = d|n d Considering the equivalent of congruences (31) and (32), Ihara raised the conjecture that one had this type of congruence in every even weight k and modulo every prime number dividing the numerator of Bk . Their are two problems with 2k this conjecture, making quite di cult both to check it numerically and to progress towards a proof: the elements fi are not uniquely determined, and the coe cients of the Poisson brackets were not completely understood indeed, these coe cients are known to be periods of parabolic forms, but the question of which parabolic form was to be considered remained still open. Although little progress seems to have been made concerning the rst issue, works of McCallum and Shari [MS03,Sha09] have recently led to a better understanding of the latter coe cients. Let us brie y explain their contribution, and the nal conjecture obtained. The aforementioned authors place themselves in a Lie algebra over for g Zp some odd prime number p. This Lie algebra is built from the action of the absolute Galois group Gal(Q, Q) on the pro-p-completion of the group π1(P1(Q)\{0, 1, ∞}). It is conjectured that the Lie algebra is isomorphic to , and g ⊗Zp Qp grt ⊗Z Qp from there to . In this algebra, McCallum and Shari consider linear ds ⊗Q Qp combinations similar to Ihara's ones. Then, Shari conjectures that the coe cients in play are related to very speci c parabolic forms. Recall that the space is endowed with the action of the Hecke algebra , Sk(C) Tk formed of commuting endomorphisms that are self-adjoint with respect to Peters- son's scalar product. One can then consider the simultaneous eigenforms of the algebra with rst Fourier coe cient equal to (one then call them primitive Tk 1 forms), and the latter are known to have Fourier coe cients in the ring of inte- gers O of some number eld, and, up to a multiplicative constant, algebraic Q(Sk) even periods. Furthermore, for any prime ideal p above Bk in O , one can 2k Q(Sk) choose a primitive form which is congruent to the Eisenstein series Gk modulo p (see for instance [DG96]). Those are precisely the parabolic forms considered by Shari , and this leads to the most precise version of the conjecture, which can be formulated as follows: Conjecture 9. Let k > 12 be an even integer and let f a primitive modu- lar form congruent to the Eisenstein series Gk modulo p. Then there exists a choice of elements fi ∈ dsi as in 3.2 and a complex number ω, such that the num- bers k−2 ( odd) are algebraic integers, and that one has a nontrivial ω i−1 ri−1(f) i congruence X k−2 . (34) ω i−1 ri−1(f) {fi, fj} ≡ 0 (mod p) i+j=k One other way to state the latter conjecture is to use the fact that, thanks to the EichlerShimura isomorphism, the space of period polynomials is endowed with an action of the algebra . The polynomials considered above are then polynomials Tk with algebraic integer coe cients, which are eigenvectors for all , and whose Tk eigenvalues are congruent modulo p to those of the Eisenstein series (which in turn are of the form σk−1(n)). Conjecture 9 is quite simple to check numerically, notably thanks to explicit cal- culations of the Hecke action on period polynomials by Fukuhara [Fuk07]. Heuristi- cal experimentation has been performed in Maple up to weight 36, and in Magma up to weight 44 (with the help of Aurel Page). This never invalidated the conjecture. 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