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1 RANKIN-COHEN BRACKETS ON QUASIMODULAR FORMS FRANÇOIS MARTIN AND EMMANUEL ROYER Abstract. We give the agebra of quasimoduar forms a coection of Ranin-Cohen operators. These operators extend those defined by Cohen on moduar forms and, as for moduar forms, the first of them provides a Lie structure on quasimoduar forms. They aso satisfy a Leibniz rue for the usua derivation. Ranin-Cohen operators are usefu for proving arithmetica identities. In particuar, we expain why Chazy equation has the exact shape it has. ha , version 2-12 Apr 2008 Introduction The purpose of this paper is to present a generaisation for quasimoduar forms of the Ranin-Cohen bracets for moduar forms: for each n 0,,,s,t positive integers, we define biinear differentia operators [, ] n sending +t to ++2n. We have denoted the vector space of quasimoduar forms of weight and depth ess or equa than s on SL(2, Z (see section 1.1 for the definitions. We give a quite precise description of the image of this biinear form in terms of moduar and paraboic forms. This aows us to obtain efficienty cassica differentia equations and arithmetica identities. Then we prove that the Ranin-Cohen bracets satisfy the Leibniz rue for the normaized usua derivation (D := d 2πidz : D[f,g] n = [D f,g] n + [f,dg] n. The first section is a presentation of the definitions and cassica resuts concerning quasimoduar forms and Ranin-Cohen bracets on moduar forms. In the second section, we prove the foowing theorem. Theorem 1. Let, in Z >0, s {0,..., /2 }, t {0,..., /2 } and n Z 0. Define n ( ( s + n 1 t + n 1 Φ n;,s;,t (f,g := ( 1 r D r f D n r g. n r r Date: Apri 12, Mathematics Subject Cassification. 11F11,11F22,16W25. Key words and phrases. Ranin-Cohen operators, quasimoduar forms, Leibniz rue, Chazy, Ramanujan, diffentia equation. Both authors are partiay supported by the ANR project Modunombres and the BQR project Nomex. The fina version of this paper has been written at the Instituto Naciona de Matemática Pura et Apicada in Rio de Janeiro. Both authors than Hossein Movasati for the invitation and the institution for exceent woring conditions. 1

2 2 FRANÇOIS MARTIN AND EMMANUEL ROYER Then Φ n;,s;,t (, +t ++2n. In some case we get a more precise description in terms of the spaces of moduar forms M and the spaces of paraboic forms S. Proposition 2. Under the hypothesis of theorem 1, if n > 0 then Φ n;,s;,t ( If moreover n > s + t, then Φ n;,s;,t ( s+t, S ++2n D j M ++2n 2j. s+t 1, S ++2n j=1 j=1 D j M ++2n 2j D s+t S ++2n 2s 2t. The same concusion hods if n = s + t and f or g vanishes at infinity. Remar 1. This notion is consistent with the one for moduar forms, the standard Ranin-Cohen bracet of f M and g M is Φ n;,0;,0 (f,g (see paragraph 1.2. Remar 2. For n 0, a biinear differentia operator Ψ sending to v v ++2n is necessariy (for weight compatibiity reasons a inear combination of (f,g D r f D n r f, r {0,...,n}. Such a differentia +t+n operator sends in principe to ++2n (see emma 7. So the operator Φ introduced before has the advantage of reducing the depth of the quasimoduar form obtained, and it was not obvious that such an operator was existing. Remar 3. Theorem 1 is vaid for quasimoduar forms on any subgroup of finite index in SL(2, Z. In the third section, we show that the behaviour of this operator under derivation is natura. Theorem 3. Under the hypothesis of theorem 1, for a f g, D Φ n;,s;,t (f,g = Φ n;,s;+2,t+1 (f,dg + Φ n;+2,s+1;,t (D f,g. Remar 4. For f of weight and exact depth s and g of weight and exact depth t, we write [f,g] n instead of Φ n;,s;,t (f,g. Reca (see proposition 6 that if h has weight w > 0 and depth d then Dh has weight w+2 and depth d + 1. The foowing theorem may then be rewritten as D[f,g] n = [D f,g] n + [f,dg] n. For moduar forms, Zagier, Cohen and Manin showed [CMZ97] that the sum of Ranin-Cohen bracets defines an associative product on the agebra M = 0 M. In a recent paper, Bieiavsi, Tang and Yao [BTY07] showed that this sum is isomorphic to the standard Moya product. Do the Ranin-Cohen bracets for quasimoduar forms introduced here have such a geometric interpretation? and

3 RANKIN-COHEN BRACKETS ON QUASIMODULAR FORMS 3 The existence of Ranin-Cohen bracets (thans to proposition 2 provides a new too to obtain arithmetica identities. For exampe, we recover the Ramanujan differentia equations, Chazy differentia equation (and expain why such a differentia equation has to exist, van der Po equaity and Niebur equaity. As usua, define for h 2 the Eisenstein series: (1 E h (z := 1 2h B h + n=1 where B h is the Bernoui number and σ h (n := d n One of the three Ramanujan equations is It is a direct consequence of σ h (nexp(2πinz d h. DE 2 = 1 12 (E 4 E 2 2. [E 2, ] 1 = E 4 where is the unique primitive form of weight 12 on SL(2, Z. If we write τ(n for the nth coefficient of, Niebur [Nie75] equaity is n 1 τ(n = n 4 σ 1 (n 24 (35a 4 52a 3 n + 18a 2 n 2 σ 1 (aσ 1 (n a a=1 and it foows from [E 2,E 2 ] 4 = 48. Van der Po [vdp51] equaity is n 1 τ(n = n 2 σ 3 (n + 60 a(9a 5nσ 3 (aσ 3 (n a. a=1 It foows from [E 4,DE 4 ] 1 = 960. Many exampes of the two previous type are given in [RS07]. Finay, a quite astonishing equaity is Chazy differentia equation. Its usua form is and it foows from D 3 E 2 = E 2 D 2 E (D E 2 2 (2 [[K, ] 1, ] 1 = 24 K 2 where K = [E 2, ] 1. The most outer bracet is on moduar forms since it may be shown that [K, ] 1 has depth 0. That such a differentia equation has to exist is a consequence of the foowing proposition that we prove using Ranin-Cohen bracets. Proposition 4. Let n 0 and r {0,...,n}. Then D r E 2 D n r E 2 n 4 j=0 j n (mod 2 D j S 2n+4 2j C D n E 4 C D n+1 E 2.

4 4 FRANÇOIS MARTIN AND EMMANUEL ROYER In particuar, [E 2,E 2 ] 0 CE 4 +C DE 2, [E 2,E 2 ] 2 C D 2 E 4, [E 2,E 2 ] 4 C and [E 2,E 2 ] 2n S 4(n+1 D 2 S 4n if n 3. Indeed for n = 2, this proposition impies that both quasimoduar forms E 2 D 2 E 2 and (D E 2 2 are in C D 2 E 4 C D 3 E 2. Hence Vect(E 2 D 2 E 2,(D E 2 2 = Vect(D 2 E 4,D 3 E 2 and D 3 E 2 is a inear combination of E 2 D 2 E 2 and (D E 2 2 : this is the shape of Chazy equation. 1. Overview 1.1. Quasimoduar forms. In this section, we introduce usua definitions and notations and reca some usefu properties of quasimoduar forms. For a more detaied introduction, see [MR05, 17]. ( a b We introduce the foowing notations: et γ = SL(2, Z and c d z H, we define and X(γ,z := c cz + d X(γ: z X(γ,z. As usua, the ( compex upper haf-pane is denoted by H. For 0, f : H a b C and γ = SL(2, Z the function (f γ is defined by (f γ(z = c d (cz + d f(γz. Definition 5. Let Z 0 and s Z 0. An hoomorphic function f : H C is a quasimoduar form of weight, depth s (over SL(2, Z if there exist hoomorphic functions Q 0 (f, Q 1 (f,..., Q s (f on H such that (3 (f γ = s Q i (fx(γ i i=0 for a ( a b c d SL(2, Z and such that Qs (f is not identicay vanishing and f has no negative terms in its Fourier expansion. By convention, the 0 function is a quasimoduar form of depth and any weight. and γ = ( in (3 impies that f is periodic Remar 5. Taing γ = ( of period 1 hence has a Fourier expansion. The definition requires this Fourier expansion to be of the shape f(z = + n=0 f(ne 2πinz. The set of quasimoduar forms of weight and depth s is denoted by s. It is often more convenient to use the C-vectoria space of quasimoduar forms of weight and depth ess or equa than s, which is denoted by. It can be shown that there are no quasimoduar forms (except 0 of negative weight or of depth s > /2 [MR05, emme 120]. Hence we extend our notation by defining M = {0} if < 0 and M = M /2 if s > /2.

5 RANKIN-COHEN BRACKETS ON QUASIMODULAR FORMS 5 Remar 6. With this definition, the space M of moduar forms of weight 0 for SL(2, Z is exacty the space. Remar 7. A basic exampe of quasimoduar form which is not a moduar form is E 2 defined in (1. It satisfies for a γ SL(2, Z the transformation property (E 2 γ = E πi X(γ, proving that E (see e.g., [MR05, emme 19]. The space =,s is equipped with a natura fitered-graded agebra structure (the graduation according to the weight, the fitration according to the depth. The canonica mutipication (f, g f g defines a +t morphism +. If f, the sequence (Q i(f i Z is defined by the quasimoduarity condition (3, if i {0,...,s}, and Q i (f = 0 for i / {0,...,s}. One can i show that Q 0 (f = f and Q i (f 2i [MR05, Lemme 119]. Quasimoduar forms are the natura extension of moduar forms into a stabe by derivation space, because of the foowing proposition. Proposition 6. If > 0, the normaized derivation D := to s d 2πidz maps s For r Z 0, write f (r := D r (f and f = f (1. The foowing emma connects the transformation equation of f and f (r. Lemma 7. Let f (4 s+r r (D r f γ = +2r i=0. Then, j=0 for a r Z 0 and γ Γ. 1 (2πi j j! ( ( r + r i + j 1 D r j Q i j (f X(γ i j j Proof. The resut is obtained inductivey on r: it is obvious for r = 0, and for the induction suppose that for r 0, formua (4 hods. Let g = f (r. For i Z we have (5 Q i (g = r j=0 ( 1 r (2πi j j! j ( + r i + j 1 Then using proposition 6 (which impies that f (r of [MR05] we find (f (r+1 γ = +2r+2 s+r+1 i=0 ( j Q i j (f (r j +r i +2r 2i. r+s+1 +2r+2 and emma Q i (g + + 2r i + 1 Q i 1 (g X(γ i. 2πi

6 6 FRANÇOIS MARTIN AND EMMANUEL ROYER From (5 we compute Q i (g + + 2r i + 1 Q i 1 (g = 2πi Q i (f (r r i + 1 ( + 2r i Q i r 1 (f+ ( r! (r j! (2πi r+1 r! ( + r i + j 1 j + r ( + 2r i + 1r! (r + 1 j! r 1 (2πi j Q i j(f (r+1 j j=1 ( + r i + j 1. j 1 Formua (4 for r + 1 instead of r foows by expanding the binomia coefficients. Finay, we sha need the foowing structure resut. For competness, we provide a short proof that shoud convince that the theory requires E 2. Proposition 8. Quasimoduar forms can be expressed as inear combinations of derivatives of moduar forms and E 2 : /2 1 /2 = D i M 2i C D /2 1 E 2. i=0 Proof. We proceed by descent on the depth. If f has weight and depth s, we woud ie to have a moduar form g such that f D s g has depth stricty ess than s. For any g M 2s, mutipe use of differentiation theorem [MR05, Lemme 118] ead to ( s 1 (6 Q s (D s g = ( 1 2πi s s! s g. If ( s 1 s 0, which happens if s < /2, we can choose For s = /2, we use and choose to obtain s ( 2s 1! g = (2πi ( s 1! Q s(f M 2s. Q /2 (D /2 1 E 2 = ( 1 /2 1 ( 2πi 2 1! 6 πi α = πi 6 (2πi/2 1 ( 2 1! Q /2(f M 0 = C f αd /2 E 2 / Usua Ranin-Cohen bracets for moduar forms. The Ranin- Cohen bracets have been introduced by Cohen after a wor of Ranin. These are biinear differentia operators, whose main property is to preserve moduar forms. More precisey, et Γ be a finite index subgroup of SL(2, Z. We write M (Γ for the space of moduar forms of weight over Γ. For each

7 RANKIN-COHEN BRACKETS ON QUASIMODULAR FORMS 7 n 0, (f,g M (Γ M (Γ, define the n-ranin-cohen bracet of f and g by n ( ( + n 1 + n 1 (7 [f,g] n = ( 1 r D r f D n r g. n r r Then [f,g] n M ++2n (Γ. Moreover, if Φ is a biinear differentia operator sending M (Γ M (Γ to M ++2n (Γ for a Γ SL(2, Z a finite index subgroup, then (up to constant Φ(f,g = [f,g] n. For an overview of Ranin-Cohen bracets incuding a proof of these resuts 1, see for instance [Zag94], [Zag92] or [MR05]. Ranin-Cohen bracets appear to be usefu in various mathematica domains as for instance invariant theory ([UU96] and [CMS01] or non-commutative geometry [Yao07]. 2. Ranin-Cohen bracets We prove our main resut (theorem 1. For n 0 and any sequence a = (a r 0 r n, the biinear forms we study tae the form Φ a (f,g = n a r D r f D n r g. We first estabish a sufficient condition on a (emma 9. For s, t and n nonnegative integers, we introduce the set E(s,t,n = { (u,v,α,β Z 4 0 : u s, v t, α + β u + v + n s t 1}. Lemma 9. Let, in Z >0, s {0,..., /2 }, t {0,..., /2 } and n Z >0. For a = (a r 0 r n satisfying n ( ( r n r a r ( + r u 1!( + n r v 1! = 0 α β for a (u,v,α,β E(s,t,n, one has Φ a (, Proof. Let f and g (Φ a (f,g γ = ++2n = +t ++2n.. From emma 7 we deduce n a r (f (r γ(g (n r γ +2r +2(n r s+t+n i=0 C(a;i(f,gX(γ i 1 The uniqueness resut needs expanations: it is proved by using ony agebraic arguments, the demonstration does not depend on the group Γ or on growth conditions. Of course, it is possibe that for some fixed group Γ the uniqueness resut does not hod (for instance if M (Γ = {0}!.

8 8 FRANÇOIS MARTIN AND EMMANUEL ROYER with (8 C(a;i(f,g = n r j 2 =0 ( 1 j2 j 2! 2πi n (i 1,i 2 Z 2 0 i 1 +i 2 =i r a r j 1 =0 ( 1 j1 j 1! 2πi ( ( n r + n r i2 + j 2 1 j 2 j 2 ( ( r + r i1 + j 1 1 j 1 j 1 Q i1 j 1 (f (r j 1 Q i2 j 2 (g (n r j 2. It foows that Φ a (f,g +t ++2n if and ony if C(a;s + t + i = 0 for a i {1,...,n}. This is easiy seen to be equivaent to ( 1 n α β a r (r α!(n r β! 2πi r u v (α,β Z 2 0 α+β=n+u+v s t i ( ( ( ( r n r + r u 1 + n r v 1 α β r α n r β Q u (f (α Q v (g (β = 0 for a i {1,...,n}, the sets of summation being determined by the binomia coefficients. Hence, Φ a +t (, ++2n is impied by ( ( r n r (9 a r ( + r u 1!( + n r v 1! = 0 α β r for a (u,v,α,β E(s,t,n. Remar 8. The statement of the previous emma is in fact an equivaence, +t if we as Φ a to satisfy Φ a ( (Γ, (Γ ++2n (Γ for each finite index subgroup Γ of SL(2, Z: indeed for {a(u,v,α,β} a non identicay zero famiy of compex numbers, if Ψ: (f,g (u,v,α,β E(s,t,n a(u,v,α,βq u(f (α Q v (g (β satisfy Ψ( (Γ, (Γ = 0, then exists M > 0 such that the minimum of dim( (Γ and dim( (Γ is stricty smaer than M. However, as for moduar forms, for each A > 0 exists Γ a finite index subgroup of SL(2, Z such that dim (Γ > A and dim (Γ > A (reca that, Z >0. We sha now give a necessary condition for a satisfying the condition of emma 9. Lemma 10. Let, in Z >0, s {0,..., /2 }, t {0,..., /2 } and n Z >0. If a = (a r 0 r n satisfies n ( ( r n r a r ( + r u 1!( + n r v 1! = 0 α β for a (u,v,α,β E(s,t,n, then there exists λ C such that ( ( + n s 1 + n t 1 a r = λ( 1 r n r r for a r {0,...,n}.

9 RANKIN-COHEN BRACKETS ON QUASIMODULAR FORMS 9 Proof. Define b = (b r 0 r n by b r = a r ( + r s 1!( + n r t 1! for a r. Then n ( ( ( ( r n r + r u 1 + n r v 1 b r = 0 α β s u t v for a (u,v,α,β E(s,t,n. Choosing u = s, t = v and β = 0 eads to F (α (1 = 0 for a α {0,...,n 1} where F is the generating (poynomia function of b defined by n F(x = b r x r. This impies the existence of µ C such that F(x = µ(x 1 n and thus b r = µ( 1 r( n r. The resut foows by defining n! λ = µ ( s + n 1!( t + n 1!. We obtain the existence of the Ranin-Cohen operator for quasimoduar forms in showing that the vector a we found in emma 10 is admissibe. Lemma 11. Let, in Z >0, s {0,..., /2 }, t {0,..., /2 } and n Z >0. Let a = (a r 1 r n be defined by ( ( s + n 1 t + n 1 a r = ( 1 r. n r r Then Φ a ( +t, ++2n. Proof. By emma 9 it suffices to chec that (10 ( 1 r ( ( ( ( 1 r1 r2 u 1 + r1 v 1 + r2 = 0 r 1!r 2! α β s u t v (r 1,r 2 Z 0 Z 0 r 1 +r 2 =n for a (u,v,α,β E(s,t,n. Fix (u,v,α,β E(s,t,n, then (10 is the coefficient of order n in the product P 1 (XP 2 (X where + ( 1 r ( ( 1 r1 u 1 + r1 P 1 (X = X r 1 r 1! α s u We have with P 2 (X = r 1 =0 + r 2 =0 Q 1 (X = 1 r 2! ( r2 β ( v 1 + r2 t v P 1 (X = Xα α! Q(α 1 (X + r 1 =0 X r 2. ( 1 r ( 1 u 1 + r1 X r 1 r 1! s u

10 10 FRANÇOIS MARTIN AND EMMANUEL ROYER and with Q 1 (X = X +s+1 (s u! R(s u 1 (X R 1 (X = + r 1 =0 ( 1 r 1 r 1! = X u 1 e X. X r 1+ u 1 We therefore may write P 1 (X = Π 1 (Xe X where Π 1 is a poynomia of degree α + s u. Simiary, P 2 (X = Π 2 (Xe X where Π 2 is a poynomia of degree β + t v. It foows that P 1 P 2 is a poynomia of degree α + β + s + t u v. Finay, since, by definition, α + β u v < n s t we get (10. Remar 9. With the hep of the hypergeometric methods [PWZ96, Chapter 3], we obtain that and s u+α Π 1 (X = ( 1 α r=α t v+β Π 2 (X = ( 1 β r=β ( + α u 1 + r s 1 ( 1 r ( + β v 1 + r t 1 ( r α X r r! ( r X r β r! Previous emmas prove theorem Ranin-Cohen bracets and derivation In this section, we prove theorem 3. First, we remar that [ n 1 ( ( s + n 1 t + n 1 (11 Φ n;,s;,t (f,g = ( 1 r n r r ( ( ] s + n 1 t + n 1 f (r+1 g (n r n r 1 r + 1 ( ( s + n 1 t + n 1 + fg (n+1 + ( 1 n f (n+1 g. n n Next, ( s + n 1 Φ n;,s;+2,t+1 (f,g = fg (n+1 n n 1 ( ( s + n 1 t + n ( 1 r f (r+1 g (n r n r 1 r + 1

11 RANKIN-COHEN BRACKETS ON QUASIMODULAR FORMS 11 so that (12 Φ n;+2,s+1;,t (f,g + Φ n;,s;+2,t+1 (f,g = ( ( s + n 1 t + n 1 fg (n+1 + ( 1 n f (n+1 g n n [ n 1 ( ( s + n t + n 1 + ( 1 r n r r ( ( ] s + n 1 t + n f (r+1 g (n r n r 1 r + 1 and equaity from (11 and (12 foows by expanding the binomia coefficients. 4. A more precise structure resut In this section, we prove proposition 2. Let n > 0. If f s and g t then Φ n;,s;,t (f,g has weight + + 2n and depth ess than s + t. Since n > 0 this depth is not maxima since s + t < + + 2n. 2 Then it foows from proposition 8 that s+t Φ n;,s;,t (f,g M ++2n D j M ++2n 2j. However, the definition of Φ n;,s;,t (f,g impies that its Fourier coefficient at 0 is 0 and since this is aso true for derivatives of moduar forms we get j=1 s+t Φ n;,s;,t (f,g S ++2n D j M ++2n 2j. j=1 The contribution to Φ n;,s;,t (f,g coming from s+t 1 S ++2n D j M ++2n 2j j=1 has depth stricty ess than s + t. Hence where g M ++2n 2s 2t. Since Q s+t (Φ n;,s;,t (f,g = Q s+t (D s+t g Q s+t (D s+t g = (2πi s t ( + + 2n s t 1! ( + + 2n 2s 2t 1! g

12 12 FRANÇOIS MARTIN AND EMMANUEL ROYER (see (6, to prove that g is paraboic we sha prove that the Fourier coefficient at 0 of Q s+t (Φ n;,s;,t (f,g is 0. From (8 we get (13 Q s+t (Φ n;,s;,t (f,g = u v (α,β Z 2 0 α+β=n+u+v s t ( ( ( ( r n r + r u 1 + n r v 1 α β r α n r β ( 1 n α β a r (r α!(n r β! 2πi r Q u (f (α Q v (g (β. Since derivatives of quasimoduar forms have Fourier coefficients vanishing at 0, the ony contribution to the Fourier coefficient of Q s+t (Φ n;,s;,t (f,g at 0 is given by (α,β = (0,0 in (13. However, the summation is on (α,β such that α + β = n + u + v s t and we have n + u + v s t > 0 if n > s+t. Thans to (13 we aso see that if f s + t > 0 and ĝ(0 = 0 then and g satisfies s+t 1 Φ s+t;,s;,t (f,g S ++2s+2t D j M ++2s+2t 2j D s+t S +. j=1 5. Appications An easy but usefu consequence of the fact that D = E 2 is the foowing emma. Lemma 12. Let n 0. Let f and g +t ++2n such that Φ n;,s;,t (f, g = h.. There exists h For exampe, we have Φ 1;+12,s;12,0 ( f, = Φ 1;,s;12,0 (f, Homogoneous products of derivatives of E 2. In this section we prove proposition 4 by recursion on n. For n = 0 we have E 2 2 = E D E 2 CE 4 C DE 2. Assume that: D r E 2 D n r E 2 n 4 j=0 j n (mod 2 D j S 2n+4 2j C D n E 4 C D n+1 E 2 (0 r n. Dea first with the case where n = 2m is even. By recursion hypothesis, we have D ( D r E 2 D n r E 2 = D r E 2 D n+1 r E 2 + D r+1 E 2 D n r E 2 n 4 j=0 j n (mod 2 D j+1 S 2n+4 2j C D n+1 E 4 C D n+2 E 2.

13 RANKIN-COHEN BRACKETS ON QUASIMODULAR FORMS 13 The set {D r E 2 D n r E 2, 0 r n} has m+1 distinct terms (corresponding to 0 r m. The set {D r E 2 D n+1 r E 2, 0 r n + 1} has aso m + 1 distinct terms (corresponding to 0 r m. It foows that { D r E 2 D n+1 r E 2 + D r+1 E 2 D n r E 2, r {0,...,m} } and { D r E 2 D n+1 r E 2, r {0,...,m} } are basis of the same space with change of basis matrix given by It foows that for any r {0,...,m} (hence any r {0,...,n} we have D r E 2 D n+1 r E 2 n 3 j=0 j n+1 (mod 2 D j S 2n+6 2j C D n+1 E 4 C D n+2 E 2. We now dea with the case where n = 2m 1 is odd. Again, by recursion hypothesis, we have D ( D r E 2 D n r E 2 = D r E 2 D n+1 r E 2 + D r+1 E 2 D n r E 2 n 4 j=0 j n (mod 2 D j+1 S 2n+4 2j C D n+1 E 4 C D n+2 E 2. The subspace generated by a the quasimoduar forms D r E 2 D n+1 r E 2 + D r+1 E 2 D n r E 2 when r runs over {0,...,2m 1} is the hyperpane { 2m } 2m α r D r E 2 D 2m r E 2 ( 1 r α r = 0 hence it is sufficient for the proof of our recursion step to find a inear combination with 2m α r D r E 2 D 2m r E 2 2m 4 j=0 j even 2m D j S 4m+4 2j C D 2m E 4 C D 2m+1 E 2 ( 1 r α r 0. This is the step where we use Ranin-Cohen bracets. Since [E 2,E 2 ] 2m+2 2 4m+8 we have Q 2 ([E 2,E 2 ] 2m+2 S 4m+4 (see (13 for the cuspidaity.

14 14 FRANÇOIS MARTIN AND EMMANUEL ROYER Equation (8 combined with the fact that Q 1 (E 2 is constant impies that [ (14 Q 2 ([E 2,E 2 ] 2m+2 = 24 (2πi 2 (2m + 2D2m+1 E (2πi 2 2m+2 ( 2m ( ( r r + 1 ( 1 r D r 2 E 2 D 2m+2 r E 2 r 2 2 r=2 2m+1 ( 2m ( ( ] r + 1 2m + 3 r + ( 1 D r r 1 E 2 D 2m+1 r E 2. r 2 2 r=1 Let ( ( ( r N N α r (N = 2( 1 r (N + 1 2r. 2 r r 1 Equation (14 gives 2m+2 r=2 α r (2m + 2D r 2 E 2 D 2m+2 r E 2 = (2πi 2 Q 2 ([E 2,E 2 ] 2m+2 24(2m + 2D 2m+1 E 2 Let β r (N = ( 1 r α r (N. We prove that N A(N = r=2( 1 r α r (N = β r (N r Z S 4m+4 C D 2m+1 E 2. is stricty negative (hence differs from 0. Zeiberger s agorithm (e.g., on the open-source computer agebra system Maxima [PWZ96, Chapter 6] provides a function K(N,r such that 2 2(N + 1(2N 1β r (N N(N 1β r (N + 1 = More precisey (15 K(N, r = K(N,r + 1β r+1 (N K(N,rβ r (N. (r 2(r 1(N + 1[3N 3 + 8N 2 (1 r + N(4r 2 6r + 3 2r 2 + 4r 2]. (N 2r + 1(N r + 1(N r + 2(N 1 We deduce the recursive formua A(N + 1 = A(N 2(N + 1(2N 1 N(N 1 which, since A(2 = 4, impies ( 2N 2 A(N = N(N 1 < 0. N 1 Finay, we have found a function which beongs to the hyperpane. This competes the proof. 2 Note that no agorithm is needed to chec that K(N, r as defined in (15 wors.

15 RANKIN-COHEN BRACKETS ON QUASIMODULAR FORMS Niebur formua. From proposition 4 we obtain Φ 4;2,1;2,1 (E 2,E 2 S 12 = C. The computation of the first coefficients gives Φ 4;2,1;2,1 (E 2,E 2 = 48. This is the differentia equation proved by Niebur in [Nie75] : = 18(D 2 E E 2 D 4 E 2 16D E 2 D 3 E 2 and comparing the Fourier expansions gives Niebur formua van der Po formua. From proposition 2 we obtain Φ 1;4,0;6,1 (E 4,DE 4 S 12. The computation of the first coefficient gives Φ 1;4,0;6,1 (E 4,DE 4 = 960. This is the differentia equation proved by van der Po: It eads to τ(n = n 2 σ 3 (n E 4 D 2 E 4 5(D E 4 2 = 960. a+b=n (4b 5abσ 3 (aσ 3 (b n 1 = n 2 σ 3 (n + 60 (9a 2 13an + 4n 2 σ 3 (aσ 3 (n a a=1 n 1 = n 2 σ 3 (n + 60 (9b 2 5bnσ 3 (aσ 3 (n a b=1 and the summation of the two ast equaities impies the van der Po formua in its origina form [vdp51, eq. (53]: n 1 τ(n = n 2 σ 3 (n + 60 (2n 3a(n 3aσ 3 (aσ 3 (n a. a= Chazy equation. Reca that we proved at the end of the introduction that an equation of the shape αe 2 D 2 E 2 + β(d E 2 2 = D 3 E 2 has to exist. Coefficients α and β can be computed by identifications of the first Fourier coefficients. Our aim in this section is to give an interpretation of this equation in terms of Ranin-Cohen bracets. We have hence and hence so that Φ 1;2,1;12,0 (E 2, 1 4 = C E 4 Φ 1;2,1;12,0 (E 2, = E 4 Φ 1;4,0;12,0 (E 4, M 6 = C E 6 Φ 1;4,0;12,0 (E 4, = 4 E 6 Φ 1;16,0;12,0 (Φ 1;2,1;12,0 (E 2,, = Φ 1;4,0;12,0 (E 4, = 4 2 E 6.

16 16 FRANÇOIS MARTIN AND EMMANUEL ROYER Next we compute hence and Φ 1;30,0;12,0 ( 2 E 6, = 2 Φ 1;6,0;12,0 (E 6, 3 M 8 = C 3 E 2 4 Φ 1;30,0;12,0 ( 2 E 6, = 6 3 E 2 4 = 6 Φ 1;2,1;12,0 (E 2, 2 Φ 1;30,0;12,0 (Φ 1;16,0;12,0 (Φ 1;2,1;12,0 (E 2,,, = 24 Φ 1;2,1;12,0 (E 2, 2. This is (2. We deduce the usua form of the Chazy equation in the foowing way. From we get K := Φ 1;2,1;12,0 (E 2, = E 2 D 12D E 2 = (E D E 2 L := Φ 1;16,0;12,0 (K, = 16K 12D K = 4 2 (E E 2 DE 2 +36D 2 E 2 and since Φ 1;30,0;12,0 (L, = 30LD 12D L = 24 3 ( E E2 2 DE E 2 D 2 E (D E D 3 E 2 the equaity Φ 1;30,0;12,0 (L, = 24 K 2 gives the Chazy equation 2D 3 E 2 2E 2 D 2 E 2 + 3(D E 2 2 = 0. References [BTY07] Pierre Bieiavsy, Xiang Tang, and Yijun Yao, Ranin-Cohen bracets and forma quantization, Adv. Math. 212 (2007, no. 1, MR MR [CMS01] Y. Choie, B. Mourrain, and P. Soé, Ranin-Cohen bracets and invariant theory, J. Agebraic Combin. 13 (2001, no. 1, MR MR (2002a:11039 [CMZ97] Paua Beazey Cohen, Yuri Manin, and Don Zagier, Automorphic pseudodifferentia operators, Agebraic aspects of integrabe systems, Progr. Noninear Differentia Equations App., vo. 26, Birhäuser Boston, Boston, MA, 1997, pp MR MR (98e:11054 [MR05] François Martin and Emmanue Royer, Formes moduaires et périodes, Formes moduaires et transcendance, Sémin. Congr., vo. 12, Soc. Math. France, Paris, 2005, pp MR MR (2007a:11065 [Nie75] Dougas Niebur, A formua for Ramanujan s τ-function, Iinois J. Math. 19 (1975, MR MR (52 #3059 [PWZ96] Maro Petovše, Herbert S. Wif, and Doron Zeiberger, A = B, A K Peters Ltd., Weesey, MA, 1996, With a foreword by Donad E. Knuth, With a separatey avaiabe computer dis. MR MR (97j:05001 [RS07] [UU96] [vdp51] [Yao07] B. Ramarishnan and Brundaban Sahu, Ranin-Cohen Bracets and van der Po-Type Identities for the Ramanujan s Tau Function, arxiv: v1 [math.nt] (2007, 14 pages. André Unterberger and Juianne Unterberger, Agebras of symbos and moduar forms, J. Ana. Math. 68 (1996, MR MR (97i:11044 Bath van der Po, On a non-inear partia differentia equation satisfied by the ogarithm of the Jacobian theta-functions, with arithmetica appications. I, II, Neder. Aad. Wetensch. Proc. Ser. A. 54 = Indagationes Math. 13 (1951, , MR MR (13,135a Yi-Jun Yao, Autour des dformations de ranin-cohen, thse, Écoe poytechnique, january 2007, Avaiabe at Theses/Fies/Yao.pdf.

17 RANKIN-COHEN BRACKETS ON QUASIMODULAR FORMS 17 [Zag92] [Zag94] Don Zagier, Introduction to moduar forms, From number theory to physics (Les Houches, 1989, Springer, Berin, 1992, pp MR MR (94e:11039, Moduar forms and differentia operators, Proc. Indian Acad. Sci. Math. Sci. 104 (1994, no. 1, 57 75, K. G. Ramanathan memoria issue. MR MR (95d:11048 Université Baise Pasca Cermont-Ferrand, Laboratoire de Mathématiques Pures, Les Cézeaux, F Aubière cedex, France E-mai address: Francois.Martin@math.univ-bpcermont.fr Université Baise Pasca Cermont-Ferrand, Laboratoire de Mathématiques Pures, Les Cézeaux, F Aubière cedex, France E-mai address: Emmanue.Royer@math.univ-bpcermont.fr