A Zouaghi

M Bellet

Y Bienvenu

G Perrin

D Cedat

M Bernacki

Modelling of the compaction phase during Hot Isostatic Pressing process at the mesoscopic scale

Keywords: HIP, powder densification, level-set function, remeshing technique, mesoscopic scale

During Hot isostatic pressing (HIP) of metal powder, power-law creep is the dominant mechanism during the densification process. However, the understanding of the global impact of the thermo-mechanical boundary conditions and of the powder granulometry on the microstructure obtained after this first mechanism is not straightforward. A finite element methodology based on the use of a level set framework coupled with a remeshing technique is proposed in order to model the viscoplastic deformation of powder particles during HIP at the mesoscopic scale thanks to a Representative Elementary Volume. The methodology consists in generating, in a finite element mesh, a sphere packing of particles by representing implicitly all particles by means of a limited set of level-set functions. Mesh adaptation is also performed at particle boundaries to describe properly the particles and to manage the discontinuity of the physical properties. Such 2D scale mesoscopic densification simulations are presented and discussed.

Introduction

Because of opportunities that Hot Isostatic Pressing (HIP) offers in both technical and economic points of view, this process is increasingly used in industries for the manufacture of diverse engineering components. This technology allows the production of near-net-shape densified metal or ceramic parts. HIPing process consists of the application of an isostatic pressure through a gaseous or liquid medium to an out-gassed and canned powder mass at an elevated temperature. The modeling of HIP is an open and complex research problem as it is composed by several mechanisms that contribute simultaneously or successively to densification [Atkinson and Davis,00]. To achieve a better understanding of the local heterogeneities, which can appear during HIP process, it is essential to develop computational models at both the macroscopic and the particles scales. During conventional HIPping, the first dominant mechanism taking place in the process is the plastic deformation of powder particles, which at such high temperature, can be reasonably addressed by power-law creep models. Then, when the temperature and pressure are ramped up in phase, several mechanisms like surfacic or volumic diffusion at the particles interfaces appear and become preponderant for the last percent of densification [Ransing and al.,00]. The influence of these mechanisms can be neglected in most of the time of compaction process. That's why, they will not be discussed in this article.

Today, thanks to considerable progress in computational and experimental domains, we can have access to microscopic information and model many processes at the mesoscopic and microscopic scale. Modeling at the REV scale can provide some clarification and explain about the prediction of the behavior of the manufactured parts by studying the impact of process parameters (temperature, pressure, duration, granulometry, …) on final characteristics. The overall main objectives of the work reported in this article are to model the preponderant mechanism at the beginning of HIP, which is the viscoplastic deformation of the particles, and to follow the kinetics of the densification at the REV scale. Powder compaction modeling on the particle scale has been treated by a numbers of research groups. These investigations are based on a finite element method or discrete element method or a combination between the two methods [Ransing and al.,00][ Gething and al.,06]. The finite element model is based on a force balance and ignores any effect of powder acceleration in a Lagrangean domain. Moreover, the inter-particle space is not meshed and a friction between powder and tool set is introduced [Ransing and al.,00]. In addition, the computational cost of these approaches is still a disadvantage especially in 3D. In this paper, a finite element strategy based on a level set framework, coupled with an adaptive automatic remeshing technique, to model the power compaction is presented. Moreover, a Eulerian framework is adopted and a multiphase flow problem is described in a single mesh domain which includes all sub-domains of the REV.

Hence, in this present paper, we shall first recall the governing equations of the mechanical problem, the constitutive laws adopted and boundary conditions applied. Then, the finite element model and the remeshing techniques are described. Afterwards, results are shown and discussed. Finally, we close with concluding remarks and possible future work.

Mechanical modeling

To simulate the consolidation of powder during HIP process at the mesoscopic scale, a REV, composed of 2D circular powder particles and inter-particle space (IPS), is considered. Modelling the particles deformation is one of the main goals of our work. Thus, in order to not follow the evolution of the IPS through boundaries of the domain, a Eulerian framework description is adopted.

Governing equations

The equations governing the mechanical behavior in the REV domain are the momentum and the mass conservation equations which can be written as:

g dt dv ρ σ ρ + ⋅ ∇ = , (1) 0 )
( = ⋅ ∇ + ∂ ∂ v t ρ ρ , ( 2
)
where ν is the velocity, σ the Cauchy stress tensor, ρ the density and g the gravity field.

Material behavior

A gas atomized 316L stainless steel powder has been selected for this study. It is generally assumed that the high temperature deformation mechanism of particles involves a power-law creep behavior [Choi and Gethin,09], [Helle and al.,85], [Wilkinson and Ashby,75] which can be described thanks to the classical following equation:

ε ε & & 1 ) 3 ( 2 - = m K S , (3)
where S, and denote, respectively, the stress deviator tensor, the strain rate tensor and the equivalent strain rate. K is the metal consistency and m the strain rate sensitivity coefficient, both being temperature dependent. The temperature dependent material data for 316L powder are taken from references [Arzt and al.,83] and [Bouaziz and al.,97]. We take as hypothesis that the IPS behaves like a compressible Newtonian fluid (K = η and m = 1 in equation ( 3)):

) ( 2 ε η & dev S = , ( 4
)
where η is the dynamic viscosity and ) (ε& dev denotes the deviatoric part of ε& .

Solution strategy

For a better consideration of the behavior of the IPS (vaccum in HIP), the method used is to introduce a compressibility term (C k ) in the mass conservation equation. So the following equations are solved:

, ) ) 3 ( 2 ( ) ( 1 g p K v v t v m ρ ε ε ρ + ∇ - ⋅ ∇ = ⋅ ∇ + ∂ ∂ -& & (5) 0 = + ⋅ ∇ p C v K & . ( 6
)
where ρ, C K , K and m are introduced by a mixture law. To introduce the incompressibility of the powder particles, the value of the compressibility coefficient C k =ε' is considered very close to zero. Concerning the IPS, values of C k are high (C k >>ε') to take into account the compressibility of the medium.

Two types of boundary conditions (BC) are considered. As a first approach, an isotropic pressure is applied (Fig. 2). In a second approach, normal stress is applied on two encompassing domains and symmetry conditions are imposed on the two other boundaries of the REV (Fig. 1).

Numerical modeling

All numerical calculations mentioned in this paper were performed with the Cimlib finite element C++ library developed in Cemef.

Level set framework

In our approach, a level set method is used, in a finite element context, for describing the two media and the virtual tools used to impose the boundary conditions. The level set approach is a numerical method that enables to locate and to track surfaces or interfaces. Powder particles, the surrounding domain (SD) (which transmits all BCs to IPS and powder particles) and IPS are implicitly described using a signed distance function α, to the interface of the considered object c [Coupez,06]. Therefore, the interface c ∂ is given implicitly by the zero isovalue of the level set function α.

         - Ω ∈ ∂ - ∂ ∈ ∈ ∂ = . ) , ( , 0 , ) , ( ) ( c x if c x d c x if c x if c x d x α , (7)
For each node of the FE mesh, the level set function is evaluated. Then, it is possible to determine the function anywhere in the domain. In the first instance, taking into account that all particles have the same behavior and to avoid an important cost of computation, we worked with one level set function representing all powder particles. Assuming that each particle has a distance function

) (x i α
defining its interface, the function

)) ( ( max ) ( x x i i α α =
represents the global distance function that defines all powder particles. In fact, its zero value corresponds to the envelope of the union of all particles.

If a velocity v is applied, the motion of the interfaces (surfaces) is modeled thanks to the following convection-reinitialization equation:

sB U v t λ α λ α = ∇ ⋅ + + ∂ ∂ ) ( , ) ( ) , 0 ( 0 x x t α α = = , ( 8
)
where λ is a parameter having the dimension of a velocity and depending on mesh size and time step, U a vector defined by

α α ∇ - + ∇ = ) 1 ( min min L L s U , 2 2 1       - = E B
πα which represents a calculated parameter in the solver, L min and E two parameters defined as (h mesh < L min < 2*h mesh ) and (10*h mesh < E < 20*h mesh ) where h mesh represents local mesh size at the interfaces in the orthogonal direction and s is the sign of α.

Mixture law

It should be noted that, in this paper, a multiphase flow problem (gas and solid phases) is described. Parameters of the mechanical problem are so defined by a mixture law. To illustrate, let us consider two level-set functions: the first characterizing the interface of particles powder and the second the interface of the SD, the mixture law for a parameter M can be expressed as:

, ) ( f )) ( f - (1 ] )) ( f - (1 ) ( f [ 2 2 1 1 SD IPS particle M M M M α α α α + + = (9)
The choice of the two characteristics functions ) ( f 1 α and ) ( f 2 α is very important because it controls the transition of the physical subdomains characteristics.

Remeshing algorithm

The accuracy of the finite element method depends on the size of the elements. The smaller these elements, the more accurate will be the results. For evident reasons of computational cost, we cannot refine uniformly all the domain. But, it is possible to optimize the mesh by refining only in useful area and coarsening the mesh outside of this area. So a good adaptive remeshing strategy is needed to perform a good description and accuracy of particles interfaces. The computational cost of our model is reduced thanks to an appropriate mesh refinement around the interfaces. In this paper, a predetermined mesh generation is used. In this technique, the size of the mesh refinement which takes place in a narrow zone surrounding the solid-IPS interface is chosen by the user [Bernacki and al.,09]. Furthermore, in HIP, the material undergoes large deformation so remeshing is required in order to avoid a loss of accuracy. Thus, as the interfaces move due to particles deformation, periodic remeshing is performed such that the refinement zone always coincides with the interface position.

Numerical results

The first test case modeled corresponds to the compaction of 2D circular powder particles in a square REV taking into account an isothermal condition (m and K material parameters are considered constant) (Fig. 1). This work has permitted to develop a numerical strategy to model the compaction of a set of 2D powder particles at a constant temperature subjected to velocity or isostatic pressure or stress boundary condition. Moreover, it is important to emphasize that the proposed framework was also applied to other microstructures involving a great numbers of monodisperse or polydisperse particles. An initial microstructure generator, introduced in CimLib, permits to generate complex 2D and 3D statistical virtual microstructures in a finite element (FE) context [Hitti and al.,12]. An optimization technique has been used to suppress unphysical voids encountered when using an advancing front method [Bernacki and al.,09]. As results, in these cases, particles deform significantly over time, leading to increasing necks between particles and relative density. The influence of numerical parameters on mass conservation is studied in this work. These parameters are especially: compressibility coefficient, remeshing period, mixture law and the number of mesh nodes. Results show that the effect of compressibility coefficient is important. A compressibility coefficient C k equal to zero for the powder particles would be ideal to minimize the loss of volume. However, this value would be synonymous of a too stiff mixture law. So 10 -10 Pa -1 and 10 -2 Pa -1 compressibility coefficient values are applied, respectively, to the powder particles and IPS domain. Moreover, when the remeshing period is extended, the conservation is better. Therefore, it is advisable that we perform remeshing only if interfaces of particles are in the situation to leave the refined part of the mesh. This reduces the computational cost and also limits the numerical diffusion due to the transport of variables during remeshing.

Using this initial formalism, several numerical difficulties appeared: unrefinement of the mesh at necks. In fact, considering a global distance function leads to the loss of the description of necks between particles because it just describes the envelope of the union of the particles and the mesh adaptation is performed at the isovalue zero of this function. The method used here, to solve this problem, is to use the classical technique of graph coloring. The idea is to color the particles of a graph (assigning each particle to a level-set function family) such that adjacent particles never share the same color (level-set function) with a minimal number of colors. This method allows keeping the description of the necks between particles and enabled us to limit the number of necessary level-set functions used to describe the particles.

The consideration of a power law creep behavior for modeling the behavior of particles makes the resolution of the mechanical problem more difficult.

Boundary condition problems in the top-left corner of the REV due to the introduction of two SD to apply BC.

The control of the material conservation is one of the important points. The influence of various parameters on this observable has been studied. A Material loss around 2% was noted.

As a second approach, to avoid the problem of BC, a circular 2D REV is considered. An isostatic pressure is imposed on the external border of the SD. The numerical and physical parameters used for these simulations are presented in table 1. Given this configuration, the mean effective pressure particles is probably significantly different from the pressure applied on the boundary of the REV. For example, in the case of figure 2, a 3.5 ratio between the pressure applied and sustained by the particles is measured. The objective of this second approach is to study the deformation of particles in an isothermal case and also within the case of a cycle modulated from 800 to 1150 °C in one hour (close to the last hour of a rise in pressure and temperature in a HIP cycle). To avoid borders effects, the computing of the average relative density of the REV is performed by calculating the volume (surface in 2D) of powder particles contained in a square situated far from the boundaries of the REV. Some important points from this numerical modelling of the HIP process at the mesoscopic scale can be highlighted: a good profile of relative density evolution (Fig. 3), a remarkable evolution of powder particles, especially in the center of the REV (Fig. 4) and an optimized material loss of around 2%. Fig. 4 illustrates the particles configuration after 1 hour cycle. A good densification is observed. It should be noted that velocity norm field is not homogeneous but depends essentially on the particles arrangement. Current work addresses the modeling of a REV with a real granulometry and the improvement of the proposed technique to optimize numerical results and process parameters.

Acknowledgement

This work was supported by AREVA group. It was carried out as a part of « Nuclear Materials » Chair proposed by AREVA and MINES ParisTech. This support is gratefully acknowledged.

FIG 1 .

 1 FIG 1. Evolution of a square 2D REV (a) initial state (b) after 20% of deformation (relative reduction of the surface area of REV) (c) after 40% of deformation and (d) velocity norm after 40% of deformation

FIG 2 .

 2 FIG 2. Evolution of a circular 2D REV: (a) initial state (b) after 22% of deformation and (c) pressure field after 22% of densification

Table 1 . Numerical and physical parameters used

 1

	K	SD	1075 MPa
		Particles From 371 MPa to 580
		MPa
		IPS	10 -3 MPa
	m	SD	0.2
		Particles From 0.198 to 0.495
		IPS	1
	C K	SD	10 -10 (Pa -1 )
		Particles	10 -10 (Pa -1 )
		IPS	10 -2 (Pa -1 )
	Time step		1.5 s
	h min (mesh size)	0,0012
	remeshing period	30s
	Pressure	From 80 MPa to 120 MPa
	Number of elements	302159
	Number of processors	32

(b) (c)


Laurent Gaubert
email: gaubert@enib.fr

Pascal Redou

Jacques Tisseau

Tissular coupling and frequency locking II infinite population

Introduction

In an accompanying paper [START_REF] Gaubert | Tissular coupling and frequency locking: I finite population[END_REF], we addressed the issue of synchronization under the framework of tissular coupling and in the case of a finite population. This issue is one of the main dynamical emergent property of complex systems. Indeed, this phenomenon actually occurs in natural systems at many scales (from cell to whole ecological systems), as it has been shown in [START_REF] Prigogine | Introduction to Thermodynamics of Irreversible Processes[END_REF][START_REF] Prigogine | Theoretical Physics and Biology[END_REF]. Many examples can be found in various fields of experimentation and theoretical research ([4, 6, 5, 11]). It is in particular a key concept in the field of nonlinear systems' dynamics, especially through the notion of chaotic systems' synchronization [START_REF] Pecora | Fundamentals of synchronization in chaotic systems, concepts, and applications[END_REF][START_REF] Goedgebuer | Laser cryptography by optical chaos[END_REF]. This wide source of examples explain why this field of research is highly interdisciplinary, from pure theory to concrete applications and experimentations. The classical concept of synchronization is related to the locking of the basic frequencies and instantaneous phases of regular oscillations. Those issues are usually addressed by studying specific coupled systems, using classical tools of the field (see for instance [START_REF] Pikovsky | Synchronization: A Universal Concept in Nonlinear Sciences[END_REF]). Convinced that synchronization phenomenon is completely natural in a large variety of coupled dynamical systems, we proposed in a preceding paper a new approach to the subject: we built a general model of coupled systems, called "tissular coupling". Within natural assumptions on interactions of those coupled systems, we proved that, for a wide class of autonomous differential systems, as soon as a finite population exhibits oscillating behaviors, their frequencies are mutually locked to a single value [START_REF] Gaubert | Tissular coupling and frequency locking: I finite population[END_REF]. This paper presents a new piece of answer to those synchronization issues as it addresses the issue of an infinite compact connected population. Even if the general questions addressed in this paper are similar to those addressed in the later one, this study is done with completely different techniques than in the finite situation and the results achieved require assumptions that are specific to the continuous situation. This paper stresses the fact that synchronization is a very natural phenomenon that can be find in many kind of coupled dynamical systems.

In the first section, we recall some parts of the framework already introduced and some mathematical tools specific to our problem. Then we address a sub problem which deals with diffuse coupling, concluding with a general result on synchronization. Finally, we go back to the general case of a uniformly linear tissular coupling and show how those dynamical systems, and based on which assumptions, exhibits the frequencies locking property.

Basic material and notation

The main background of this work can be found in the accompanying paper [START_REF] Gaubert | Tissular coupling and frequency locking: I finite population[END_REF] and in [START_REF] Gaubert | Auto-organisation et émergence dans les systèmes couplés, individuation de données issues de systèmes biologiques couplés[END_REF], so we just recall here some notations and concentrate on the new materials specific to the case of an infinite population.

• P is a measured space with finite measure λ. Typical populations are finite populations with counting measure or measurable subset of R m with Lebesgue measure, so that P has a topology. Every p ∈ P is called a cell; • A state of P is an element of S = M b (P, R n ), the space of measurable bounded applications from P to R n . A state of P is denoted s, s(p) standing for the state of the cell p; • let I be a real interval and r ∈ N. A trajectory of P is an element of F(I, S). For concision purpose, any trajectory will be written s again, and s(t, p) stands for the state of cell p at time t. Then, the space of trajectories is:

T r = C r (I, S), ∀ compact J ⊂ I, sup J×P s(t, p) < ∞
A trajectory s is regular only along the time t (excepted in terms of measurability), thus we use the following unambiguous notation :

s ′ (t, p) = ∂ 1 s(t, p)
• All norms are denoted in the same manner :

s ∈ S : s = sup P s(p) s ∈ T : s = sup I×P s(p, t) s(t, .) = sup P s(t, p) s(., p) = sup I s(t, p)
• a period on P is a map τ from P to R * + . A trajectory s ∈ T r is said to be τ -periodic if for any p ∈ P, s(., p) is τ (p)-periodic. τ (p) is then called the period of p. The space of such trajectories is written T r τ . For any s ∈ T 0 τ we define the τ -mean I τ (s) of s as:

I τ (s)(p) = 1 τ (p) τ (p) 0 s(t, p)dt
• If C is a coupler on P and τ is a period on P, the set A(τ, C) is defined as:

A(τ, C) = T ∞ ∪ C -1 (T ∞ )
In this paper, P stands for an infinite compact connected population, with finite measure λ, mostly a measurable subset of R m with Lebesgue measure. Any period τ on P is assumed to be measurable and bounded. On the contrary of the finite population case, to define a tissular coupler one needs to specify both its diffuse part c d and its atomic part c a . Such a coupler will be called diffuse if c a = 0.

Partition associated to a periodic motion

Let suppose that s is a τ -periodic trajectory of P, it divides P into a partition according to the period of each cell. Let p ∈ P and a ∈ R * , we define the following application:

τ a (p) = aτ (p)
and for any subset A ⊂ R :

P A p = a∈A τ -1 a (τ (p))
In other words:

P A p = {q ∈ P : ∃ a ∈ A, aτ (q) = τ (p)}
As soon as 1 ∈ A, the set of all distinct P A p realizes a partition of P, we write P A this partition (with non-empty sets). The main examples are:

• if all the periods constitute a dependent set, then P Q = {P}; • if all the periods are identical (P is synchronized), then P 1 = {P}.

Measures associated to an uniformly linear coupling

The next tool we need is a method to measure the latter sets. But this cannot be done independently from the coupler itself. This is why we introduce the following family of measures on P, one for each cell p:

λ p (B) = B c d (p, q) dq
We recall that the support of a measure is defined to be the largest closed subset of P for which every open neighborhood of every cell of the set has positive measure. Let S p stands for the support of c d (p, .), this support indicates where the measure λ p lives and then which cells influence p in its evolution. In virtue of c d 's continuity, we know that for any measurable subset B of P, the following equivalence holds:

λ p (B) > 0 ⇔ λ(B ∩ S p ) > 0
Moreover, in the case of a diffuse coupler, this family of measures is uniformly absolutely continuous with respect to λ: Lemma 1. If C is an uniformly linear and diffuse coupler, then there exists a constant m C > 0 such that for any measurable subset B of P we have:

λ p (B) ≤ m C λ(B) , ∀p ∈ P Useful subsets of A(τ, C)
The kind of systems we want to handle comes from differential equations studied near limit cycles, so, like in the finite population case, we need to avoid some degenerated trajectories. In order to do this, if s ∈ T τ , we define the set of all cells on which the variations of s are not negligible. Precisely, if ε > 0, we define the following set:

V ε (s) = p ∈ P inf c∈R n s(., p) -c > ε
We call ε-kernel of C the set:

Ker ε (C) = {s ∈ T τ , V ε (C(s)) = P}
An element in the ε-kernel of C contains, for at least one cell p, an application s(., p) which image under C is uniformly near zero. Those solutions are out of interest here because they can be seen as solution of a system on a sub-population of C. Indeed for such a p s ′ (., p) is negligible and s(., p) nearly constant. In other words, we only consider trajectories which dynamic in each cell is "truly" periodic.

Given a period τ on P, any element s of A (ω, C) and δ > 0, the following set gathers cells which period are "not isolated", it will be naturally linked to

V ε (s) : R δ C (τ ) = p ∈ P λ p P Q p > δ If p ∈ R δ C (τ )
, there is a set, which measure is at least δ, consisting of cells with periods commensurable with τ (p). Moreover, those cells are situated in the neighborhood of p defined by S p .

Ergodic aletrnative

Like in the finite case, we will use this alternative within the following form [START_REF] Gaubert | Auto-organisation et émergence dans les systèmes couplés, individuation de données issues de systèmes biologiques couplés[END_REF]: Proposition 1. Let τ be a period on P and C a tissular diffuse coupler, then for all s in A (τ, C) we have:

C(s)(t, p) = P Q p c d (
s(t, p), p, q)s(t, q)dq+ P-P Q p c d (s(t, p), p, q)τ (q)I τ (q) (s)(q)dq This results clearly shows that the ergodic theorem acts like a filter on periodic solutions, separating commensurable periods from the others. It is a key element of our proofs.

Synchronization with uniformly linear diffuse tissular coupler

As it is suggested in the title, in this section the kind of coupler we handle has the following shape:

C(s) = P c d (., q)s(q)dq = L C .s
Here is the first result explaining the link between the sets V and R. It will be generalized in the next result: Proposition 2. Let τ be a period on P, for any s in A (τ, C) the following inclusion holds:

V 0 (C(s)) ⊂ R 0 C (τ ) Noticing that the elements of A (τ, C) are uniformly bounded on I, we can then consider s = sup I×P s(t, p) and use it to acquire the generalization: Proposition 3. Let τ be a period on P, for any s in A (τ, C) we have:

V ε (C(s)) ⊂ R ε s C (τ )
In particular, as P is a compact set, we can state: Corollary 1. Let τ be a period on P and suppose there exists ε > 0 and e ∈ A (τ, C) -Ker ε (C), then there exists a finite number of cells p 1 , . . . , p j such that:

P = j i=1 P Q pi
The latter result leads us to this somewhat different result: Corollary 2. Let τ be a period on P and suppose there exists ε > 0 and e ∈ A (τ, C) -Ker ε (C), then there exists a sequence {p i } i∈N such that :

P Q pi = i P 1 pi
These results indicate a great reduction of the problem as they show that the periods cannot be arbitrary scattered. We are now able to prove the main result of this section. In order to handle only true cases of synchronization, we add an assumption on τ so that we won't have to deal with sub-period stuff:

Definition 1. τ is said to be simple if 2 Conv(τ (P)) ∩ Conv(τ (P)) = ∅.
Under this hypothesis, we have the following global synchronization result:

Theorem 1. Suppose τ is a simple period on P, if there exists ε > 0 and s in A (τ, C) -Ker ε (C), then τ is a constant map on P.

Remark 1. To prove this result, we used the continuity of τ , deduced from the one of c d . Nevertheless, leaving this assumption aside, we can get to an interesting result which could lead to generalization. We only need corollary 2 (we point that its validity does not directly depends on c d 's continuity but on the uniform absolute continuity of the family λ p and on the fact that c d (., .) is uniformly bounded on P 2 ).

Definition 2. The set of point with isolated period, written P 0 , is defined as:

P 0 = p ∈ P, λ p P 1 p = 0
The following proposition shows that under minimal assumptions, there are almost no points with isolated period:

Theorem 2. Suppose there exists a partition as in the corollary 2 and that P ⊂ p∈P S p then λ(P 0 ) = 0.

The latter result gives a global information that is verified in a general case: almost no cell has an isolated period. Those two results are really different, the first one is less theoretical but refers to stronger hypotheses than the second one, which comes under the field of measure theory.

Synchronization with uniformly linear tissular coupler

Now, we consider the general case, leaving aside the assumption that C is diffuse. Thus, in this last and short section, we consider a uniformly linear tissular coupler on P (c a needs no more to be zero). Like in the finite population case, the first step consists in detecting and eliminating degenerated solutions:

Definition 3. Let s ∈ T (I, E b ) and ε ≥ 0, s is said to be an ε-eigenvector

(for C) if: C(s) -c a ∈ Ker ε (C)
An ε-eigenvector is therefore a s which contains a map e(., p) uniformly near a hypothetic eigenvector of C for the eigenvalue c a (p).

Finally, here is the generalization of theorem 1:

Theorem 3. Let τ be a simple period on P and ε > 0. If there exists s ∈ A (τ, C) which is not an ε-eigenvector, then τ is a constant map.

Conclusion

In this work, we have given another example of how the notion of tissular coupling can help to address synchronization issues within a wide variety of coupled systems. Our approach is completing the classical methods showing that within suitable conditions, synchronization must occur along time. Those results are often qualitatively dependent to the studied systems. After the first positive result of our accompanying paper [START_REF] Gaubert | Tissular coupling and frequency locking: I finite population[END_REF], we hold to our approach and didn't prove that synchronization ultimately happens but demonstrated that if coupled systems oscillate, then they must be synchronized. As in the finite population situation, the loss in time evolution informations is compensated by very general results, almost independent from the individual differential systems to be coupled. Furthermore, conclusions are less restrictive than in [START_REF] Gaubert | Tissular coupling and frequency locking: I finite population[END_REF] (there is no threshold like "half of the population is synchronized"). Moreover, we hope that this complementary approach to synchronization issues will lead to future developments as it brings about, also in this paper, many mathematical tools unfamiliar to this field of research. Furthermore, other examples of the use of tissular coupling can be found in [START_REF] Gaubert | Auto-organisation et émergence dans les systèmes couplés, individuation de données issues de systèmes biologiques couplés[END_REF] concerning emergence of spatial patterns. All those works and results are inspiring and seem to prove that tissular coupling will be a prolific framework, completing what already exists.


François Dumas
email: francois.dumas@math.univ-bpclermont.fr

Emmanuel Royer
email: emmanuel.royer@math.univ-bpclermont.fr

POISSON STRUCTURES AND STAR PRODUCTS ON QUASIMODULAR FORMS

Keywords: 2010 Mathematics Subject Classification. 17B63, 11F25, 11F11, 16W25, Poisson brackets, Rankin-Cohen brackets, formal deformation, Eholzer product, star product

We construct and classify all Poisson structures on quasimodular forms that extend the one coming from the first Rankin-Cohen bracket on the modular forms. We use them to build formal deformations on the algebra of quasimodular forms.

Introduction

In [START_REF] Cohen | Sums involving the values at negative integers of L-functions of quadratic characters[END_REF]Theorem 7.1], Henri Cohen defined a collection of bi-differential operators on modular forms. Let n be a positive integer, f a modular form of weight k and g a modular form of weight ℓ. The n-th Rankin-Cohen bracket of f and g is the modular form of weight k + ℓ + 2n defined by

RC n (f , g) = n r=0 (-1) r k + n -1 n -r ℓ + n -1 r D r f D n-r g, D = 1 2πi d dz .
The algebraic structure of these brackets has been studied in Zagier's seminal work [START_REF] Zagier | Modular forms and differential operators[END_REF]. That Rankin-Cohen brackets define a formal deformation of the algebra of modular forms has been widely studied. Important contributions are [UU96, CMZ97, Yao07, BTY07, Pev12, KP13].

In this paper, we construct formal deformations of the algebra M ≤∞ * of quasimodular forms. This algebra is generated over C by the three Eisenstein series E 2 , E 4 and E 6 . The algebra M * of modular forms is the subalgebra generated by E 4 and E 6 . As a first step, we classify the admissible Poisson structures of M ≤∞ * . A Poisson bracket { , } on M ≤∞ * is admissible if i) the restriction of { , } to the algebra M * of modular forms is the first Rankin-Cohen bracket RC 1 ii) it satisfies {M ≤s k , M ≤t ℓ } ⊂ M ≤s+t k+ℓ+2 for any even integers k, ℓ and any integers s and t where M ≤s k is the vector space of quasimodular forms of weight k and depth less than s. The vector space of parabolic modular forms of weight 12 is one dimensional. We choose ∆ = E 3 4 -E 2 6 a generator.

Proposition A (First family of Poisson brackets)-For any λ ∈ C * , there exists an admissible Poisson bracket { , } λ on the algebra of quasimodular forms defined by the following values on the generators:

ii) More generally, for any a ∈ C, the brackets defined for any integer n ≥ 0 by Remark-A generator is π defined by linear extension of π(f ) = kf E 2 for f any quasimodular form of weight k. For this choice, the derivation d a is defined on the generators by

[f , g] d a ,n = n r=0 (-1) r k + n -1 n -r ℓ + n -1 r d r a (f ) d n-
d a E 2 = 2a E 2 2 - 1 12 E 4 , d a E 4 = 4a E 4 E 2 - 1 3 E 6 , d a E 6 = 6a E 6 E 2 - 1 2 E 2 4 .
We construct a second family of Poisson brackets.

Proposition B (Second family of Poisson brackets)-For any α ∈ C, there exists an admissible Poisson bracket ( , ) α on the algebra of quasimodular forms defined by the following values on the generators:

(E 4 , E 6 ) α = -2∆, (E 2 , E 4 ) α = α E 6 E 2 , (E 2 , E 6 ) α = 3 2 α E 2 4 E 2 .
Moreover, i) for any α ∈ C \ {4}, the Poisson bracket ( , ) α is not unimodular. For α = 4, the Poisson bracket ( , ) 4 is Jacobian (hence unimodular) of potential k 0 = -2∆ E 2 . ii) The Poisson algebras M ≤∞ * , ( , ) α and M ≤∞ * , ( , ) α ′ are Poisson modular isomorphic if and only if

α = α ′ . iii) For any α ∈ C, a) if α Q, the Poisson center of M ≤∞ * , ( , ) α is C b) if α = 0, the Poisson center of M ≤∞ * , ( , ) α is C[E 2 ] c) if α = p/
q with p ∈ Z * and q ∈ N * , p and q coprimes, the Poisson center of

M ≤∞ * , ( , ) α is                      C if p < 0 C ∆ p E 4q 2 if p ≥ 1 is odd C ∆ u E 2q 2 if p = 2u for odd u ≥ 1 C ∆ v E q 2 if p = 4v with v ≥ 1.
Remark-The bracket ( , ) 0 is the trivial bracket.

This second family provides a new set of formal deformations of the algebra of quasimodular forms. Following [Zag94, Eq. ( 38)], we consider the derivation v defined

on M ≤∞ * by v(f ) = (∆, f ) α 12∆ .
Let us define K α : M ≤∞ * → C by setting K α (f ) = k -(3α + 2)s if f has weight k and depth s. The set of complex-like derivations δ such that K α (f )f δ(g) -K α (g)gδ(f ) = 0 for any f ∈ M ≤s k and g ∈ M ≤t ℓ , for any k, ℓ, s, t, is a one dimensional vector space over C. Let π α be a generator. We note

M s k = M k E s 2 . Theorem B-Let α ∈ C. For any b ∈ C, let δ α,b be the derivation on M ≤∞ * defined by δ α,b = bπ α + v. i) For all f ∈ M s k and g ∈ M t ℓ , we have (f , g) α = (k -(3α + 2)s)f δ α,b (g) -(ℓ -(3α + 2)t)gδ α,b (f )
for any f ∈ M s k and g ∈ M t ℓ . ii) Moreover, the brackets defined for any integer n ≥ 0 by

[f , g] K α δ α,b ,n = n r=0 (-1) r k -(3α + 2)s + n -1 n -r ℓ -(3α + 2)t + n -1 r δ r α,b (f )δ n-r α,b (g), for any f ∈ M s k and g ∈ M t ℓ define a formal deformation of M ≤∞ * satisfying M ≤s k , M ≤t ℓ K α δ α,b ,n ⊂ M ≤s+t k+ℓ+2n for all k, ℓ in 2N and s, t in N if and only if b = 0.
Remark-A generator π α is defined by linear extension of:

π α (f ) = [k -(3α + 2)s]f E 2 (f ∈ M s k ).
For this choice, the derivation δ α,b is defined on the generators by

δ α,b E 2 = -3bα E 2 2 , δ α,b E 4 = 4b E 4 E 2 - 1 3 E 6 , δ α,b E 6 = 6b E 6 E 2 - 1 2 E 2 4 .
To complete the classification of Poisson structures, we introduce a third family of Poisson brackets. We note however that, when µ 0, this third family does not lead to a formal deformation of M ≤∞ * with the shape of Rankin-Cohen brackets (see §4.3).

Proposition C (Third family of Poisson brackets)-For any µ ∈ C, there exists an admissible Poisson brackets , µ on the algebra of quasimodular forms defined by the following values on the generators:

E 4 , E 6 µ = -2∆, E 2 , E 4 µ = 4 E 6 E 2 +µ E 2 4 , E 2 , E 6 µ = 6 E 2 4 E 2 -2µ E 4 E 6 .

Moreover, i) this Poisson bracket is Jacobian with potential

k µ = -2∆ E 2 +µ E 2 4 E 6 .
ii) Remark-We note that , 0 = ( , ) 4 .

Finally, the following result implies that our classification is complete. Remark-From an algebraic point of view, classifications of Poisson structures and associated (co)homology for polynomial algebras in two variables appear in [START_REF] Monnier | Poisson cohomology in dimension two[END_REF], [START_REF] Pichereau | homologie de poisson et singularités isolées en petites dimensions, avec une application en théorie des déformations[END_REF] or [START_REF] Roger | Poisson cohomology of the affine plane[END_REF] for a Poisson bracket on C[x, y] defined by {x, y} = ϕ(x, y) with ϕ an homogeneous or a square free weight homogeneous polynomial in C[x, y]. The algebra of modular forms M * = C[E 4 , E 6 ] with the Poisson bracket defined by RC 1 is the case A 2 in the classification theorem 3.8 in [START_REF] Monnier | Poisson cohomology in dimension two[END_REF]. Applying propositions 4.10 and 4.11 of [START_REF] Pichereau | homologie de poisson et singularités isolées en petites dimensions, avec une application en théorie des déformations[END_REF], or theorems 4.6 and 4.11 of [START_REF] Monnier | Poisson cohomology in dimension two[END_REF], we can deduce that the Poisson cohomology spaces HP 1 (M * ) and HP 2 (M * ) are of respective dimensions 1 and 2. In three variables, the Poisson structures on the algebra M ≤∞ * = C[E 2 , E 4 , E 6 ] of quasimodular forms arising from theorem C above do not fall under the classification of [START_REF] Dufour | Rotationnels et structures de Poisson quadratiques[END_REF] since there are not quadratic. The (co)homological study of [START_REF] Pichereau | homologie de poisson et singularités isolées en petites dimensions, avec une application en théorie des déformations[END_REF] and [START_REF] Pichereau | Poisson (co)homology and isolated singularities[END_REF] does not apply to the brackets { , } 1 and ( , ) α since they are not Jacobian, or to the Jacobian bracket , 1 because its potential k 1 does not admit an isolated singularity at the origin.

Number theoretic and algebraic background

2.1. Quasimodular forms. The aim of this section is to provide the necessary background on quasimodular forms. For details, the reader is advised to refer to [START_REF] Zagier | Elliptic modular forms and their applications[END_REF] or [START_REF] Martin | Formes modulaires et périodes[END_REF]. On SL(2, Z), a modular form of weight k ∈ 2N, k 2, is a holomorphic function on the Poincaré upper half plane H = {z ∈ C : ℑz > 0} satisfying

(cz + d) -k f az + b cz + d = f (z)
for any a b c d ∈ SL(2, Z) and having Fourier expansion

f (z) = n≥0 f (n)e 2πinz .
We denote by M k the finite dimensional space of modular forms of weight k. The algebra of modular forms is defined as the graded algebra

M * = k∈2N k 2 M k .
Let k ≥ 2 be even. We define the Eisenstein series of weight k by

E k (z) = 1 - 2k B k +∞ n=1 σ k-1 (n)e 2πinz .
Here the rational numbers B k are defined by their exponential generating series

+∞ n=0 B n t n n! = t e t -1
and σ k-1 is the divisor function defined by

σ k-1 (n) = d|n d>0 d k-1 (n ∈ N * ).
If k ≥ 4, the Eisenstein series E k is a modular form of weight k and M * is the polynomial algebra in the two algebraically independent Eisentsein series E 4 and E 6 . In other words,

M * = C[E 4 , E 6 ], M k = (i,j)∈N 2 4i+6j=k C E i 4 E j 6 .
However, the Eisenstein series E 2 is not a modular form. It satisfies

(cz + d) -2 E 2 az + b cz + d = E 2 (z) + 6 πi c cz + d (z ∈ H)
for any a b c d ∈ SL(2, Z). Moreover, the algebra of modular forms is not stable by the normalised complex derivation

D = 1 2πi d dz .
For example, we have the following Ramanujan differential equations

D E 2 = 1 12 E 2 2 -E 4 D E 4 = 1 3 (E 4 E 2 -E 6 ) D E 6 = 1 2 E 6 E 2 -E 2 4 .
To account for these observations, and using the fact that E 2 , E 4 and E 6 are algebraically independent, we introduce the algebra M ≤∞ * of quasimodular forms defined as the polynomial algebra

M ≤∞ More intrinsically, if for γ = a b c d ∈ SL(2, Z) we define X(γ) = z → c cz + d and f | k γ = z → (cz + d) -k f az + b cz + d
then a quasimodular form of weight k ∈ 2N and depth s ∈ N is a holomorphic function f on H such that there exist holomorphic functions f 0 , . . . , f s (f s 0) satisfying

f | k γ = s j=0 f j X(γ) j
for any γ ∈ SL(2, Z). Moreover, it is required that any f j has a Fourier expansion

f j (z) = n≥0 f j (n)e 2πinz (z ∈ H).
The zero function is supposed to have arbitrary weight and depth 0. We write M ≤∞ k for the space of quasimodular forms of weight k and M ≤s k for the space of quasimodular forms of weight k and depth less than or equal to s. We have

M ≤0 k = M k and M ≤s k = s j=0 M k-2j E j 2 , M ≤∞ * = k∈2N M ≤k/2 k . Moreover D M ≤s k ⊂ M ≤s+1 k+2 .
Since the depth of a quasimodular form is nothing but its degree as a polynomial in E 2 with modular coefficients we note that

M ≤s k = s j=0 M t k , M ≤∞ * = k∈2N k/2 t=0 M t k where M t k = M k-2t E t 2 = (i,j)∈N 2 4i+6j=k-2t C E i 4 E j 6 E t 2 .
An important element in our study will be the discriminant function

∆ = E 3 4 -E 2 6 . We note that D ∆ = ∆ E 2 .
Let n be a non negative integer, f a modular form of weight k and g a modular form of weight ℓ. The n-th Rankin-Cohen bracket of f and g is

RC n (f , g) = n r=0 (-1) r k + n -1 n -r ℓ + n -1 r D r f D n-r g.
This is a modular form of weight k + ℓ + 2n. If f and g are quasimodular forms of respective weights k and ℓ and respective depths s and t, their n-th Rankin-Cohen bracket is defined in [START_REF] Martin | Rankin-Cohen brackets on quasimodular forms[END_REF] by

RC n (f , g) = n r=0 (-1) r k -s + n -1 n -r ℓ -t + n -1 r D r f D n-r g. (1)
This is a quasimodular form of weight k + ℓ + 2n and minimal depth (that is s + t).

2.2. Poisson algebra. The aim of this section is to give a brief account on what is needed about Poisson algebra. For more details, the reader is advised to refer to [START_REF] Laurent-Gengoux | of Grundlehren der Mathematischen Wissenschaften[END_REF].

A commutative C-algebra A is a Poisson algebra if there exists a bilinear skew-symmetric map b : A × A → A satisfying the two conditions:

• Leibniz rule b(f g, h) = f b(g, h) + b(f , h)g • Jacobi identity b (f , b(g, h)) + b (g, b(h, f )) + b (h, b(f , g))
= 0 for all f , g and h in A. The bilinear map b is given the name of Poisson bracket. If A is a finitely generated algebra with generators x 1 , . . . , x N , a Poisson bracket b is entirely determined by its values b(x i , x j ) for i < j where A is generated by x 1 , . . . , x N . More precisely, we have

b(f , g) = 0≤i<j≤N f x i g x j - g x i f x j b(x i , x j ) (2)
for f and g expressed as polynomials in x 1 , . . . , x N .

If A = C[x, y], any p ∈ A determines a Poisson bracket satisfying b(x, y) = p. However, if A = C[x, y, z],
for any p, q and r in A, there exists a Poisson bracket on A defined by b(x, y) = r, b(y, z) = p, and b(z, x) = q if and only if curl(p, q, r) • (p, q, r) = 0 (3) where

curl(p, q, r) = r y - q z , p z - r x , q x - p y .
If condition (3) is satisfied, then (p, q, r) is called a Poissonian triple. A particular case is obtained if there exists k ∈ C[x, y, z] such that curl(p, q, r) = (p, q, r) ∧ gradk. The bracket b is said then to be unimodular. Among unimodular brackets are the Jacobian brackets.

A bracket b is Jacobian if (p, q, r) = grad k for some polynomial k. The bracket b satisfies then

b(f , g) = jac(f , g, k) (f , g ∈ A) In this case, C[x,
A polynomial h ∈ C[x, y, z] is indecomposable if there is no polynomial p ∈ C[x] with deg p ≥ 2 such that h = p • ℓ for some ℓ ∈ C[x, y, z]. Lemma 1-Let C[x, y, z] be endowed with a Jacobian Poisson structure of non constant potential k. Its Poisson center is C[k] if and only if k is indecomposable. Proof. Assume that k is not indecomposable: k = p • ℓ with p ∈ C[x], deg p = 2. Then jac(ℓ, g, k) = (p ′ • ℓ) jac(ℓ, g, ℓ) hence ℓ is in the Poisson center however not in C[k].
Assume conversely that k is indecomposable. Let f be in the Poisson center, then the rank of the Jacobian matrix of (f , g, k) is at most 2 for any g. If it is 1 for any g then grad f and grad k are zero, it contradicts the fact that k is not constant. Hence, for some g, the rank is 2. It follows (see e.g. [GS06, Theorem 6]) that there exists

q ∈ C[x, y, z], F ∈ C[x] and K ∈ C[x] such that f = F • q and k = K • q.
Since k is indecomposable and non constant, we have deg K = 1 hence q, and f , are polynomials in k.

(f , g) = b A (f , g) b(x, f ) = σ(f )x + δ(f )
for all f and g in A if and only if σ is a Poisson derivation and δ is a Poisson σ-derivation.

In this case, the Poisson algebra

A[x] is said to be a Poisson-Ore extension of A. It is denoted A[x] σ,δ .
We describe also a general process to obtain Poisson brackets from a pair of derivations. A pair (δ, d) of two derivations of A is solvable if there exists some scalar α such that δ • d-d •δ = α d. In particular, a solvable pair (δ, d) is abelian when α = 0. Proposition 3-Let A be a commutative algebra, d and δ two derivations of A.

Let b : A × A → A be defined by b(f , g) = δ(f ) d(g) -d(f )δ(g) (f , g ∈ A).
A ⊗ A ⊗ A ⊗ A → A is defined by B(f , g, h) = b (f , b(g, h)) then B = α(d ⊗ d ⊗δ -d⊗δ ⊗ d) + (δ ⊗ (d •δ) ⊗ d-d ⊗δ ⊗ (d•δ)) + (d ⊗(d •δ) ⊗ δ -δ ⊗ d⊗(d •δ)) + (δ ⊗ δ ⊗ d 2 -δ ⊗ d 2 ⊗δ) + (d ⊗ d ⊗δ 2 -d ⊗δ 2 ⊗ d).
Point iii) is obtained by direct computation.

A direct consequence of this Proposition is the following Corollary. If A = n≥0 A n is a commutative graded algebra, a map κ : A → C is graded-additive if, for any k and ℓ, for any f ∈ A k and any g ∈ A ℓ , we have κ(f g) = κ(f ) + κ(g).

Corollary 4-Let A = n≥0 A n be a commutative graded algebra. Let κ : A → C be a graded-additive map. Let d be a homogeneous derivation of A (there exists e ≥ 0 such that d A n ⊂ A n+e for any e ≥ 0). Then, the bracket defined on A by the bilinear extension of

b(f , g) = κ(f )f d(g) -κ(g)g d(f ) (f ∈ A k , g ∈ A ℓ )
is a Poisson bracket for which d is a Poisson derivation.

We turn on formal deformations of a commutative C-algebra A. Assume we have a family µ = (µ i ) i∈N of bilinear maps µ i : A× A → A such that µ 0 is the product. Let A[[ℏ]] be the commutative algebra of formal power series in one variable ℏ with coefficients in A. The family µ is a formal deformation of A if the non commutative product on A[[ℏ]] defined by extension of

f * g = j≥0 µ j (f , g)ℏ j (f , g ∈ A)
is associative. This condition is equivalent to

n r=0 µ n-r (µ r (f , g), h) = n r=0 µ n-r (f , µ r (g, h)) for all f , g, h ∈ A (4)
for all n ≥ 0. In this case, the product * is called a star product. If µ is a formal deformation and if moreover µ 1 is skew-symmetric and µ 2 is symmetric, then (A, µ 1 ) is a Poisson algebra.

Problems at issue. The first Rankin

-Cohen bracket RC 1 (f , g) = kf D(g) -D(f )ℓg (f ∈ M k , g ∈ M ℓ )
gives M * a structure of Poisson algebra. This is a consequence of Corollary 4. Cohen, Manin & Zagier [START_REF] Beazley Cohen | Automorphic pseudodifferential operators[END_REF] and Yao [START_REF] Yao | Autour des déformations de Rankin-Cohen[END_REF] or [START_REF] Rochberg | A survey on Rankin-Cohen deformations[END_REF] proved that the family of Rankin-Cohen brackets is a formal deformation of M * . In this case, the star product is called the Eholzer product. This subject has been widely studied then. See for example [START_REF] Olver | Transvectants, modular forms, and the Heisenberg algebra[END_REF][START_REF] Pevzner | Rankin-Cohen brackets and associativity[END_REF]. Can we construct formal deformations of M ≤∞ * ? In other words, can we construct suitable families (µ n ) n∈N of bilinear maps on M ≤∞ * that increase the weight by 2n, preserve the depth and define an analogue of Eholzer product? The brackets defined in (1) do not lead to a solution since RC 1 does not even provide M ≤∞ * with a Poisson structure. Our first step is to obtain admissible Poisson brackets on M ≤∞ * with the following definition.

Definition 1-A Poisson bracket b on M ≤∞ * is admissible if 1) b(f , g) = RC 1 (f , g) if f and g are in M * 2) it satisfies b M ≤s k , M ≤t ℓ ⊂ M ≤s+t k+ℓ+2 for all k, ℓ, s, t.
Remark-We could have replace condition 2 by the following one: there exists e ≥ 0 such that b M ≤s k , M ≤t ℓ ⊂ M ≤s+t k+ℓ+e for all k, ℓ, s, t. However, condition 1 implies that necessarily e = 2.

Equivalently, a Poisson bracket b on

M ≤∞ * is admissible if and only if b(E 4 , E 6 ) = -2∆ b(E 2 , E 4 ) ∈ M ≤∞ 8 , b(E 2 , E 6 ) ∈ M ≤∞ 10 b(E 2 , M * ) ⊂ M * E 2 +M * .
In order to classify the admissible Poisson brackets, we introduce the notion of Poisson modular isomorphism.

Definition 2-A Poisson isomorphism ϕ : (M ≤∞ * , b 1 ) → (M ≤∞ * , b 2 ) is called a Poisson mod- ular isomorphism if ϕ(M * ) ⊂ M * .
Indeed, if ϕ is a Poisson modular isomorphism then, its restriction to the subalgebra M * is the identity. This is a consequence of the following Proposition.

Proposition 5-The group of Poisson automorphisms of the Poisson algebra (M * , RC 1 ) is trivial.

Proof. Let ϕ be a Poisson automorphism of M * . There exist two polynomials s and t in C[x, y] such that ϕ(E 4 ) = s(E 4 , E 6 ) and ϕ(E 6 ) = t(E 4 , E 6 ). By (2) we have

RC 1 (ϕ(E 4 ), ϕ(E 6 )) = jac(s, t)(E 4 , E 6 ) • RC 1 (E 4 , E 6 ).
Since ϕ is an automorphism, jac(s, t) is a non zero scalar, say λ. We get

ϕ (RC 1 (E 4 , E 6 )) = λ RC 1 (E 4 , E 6 ) hence s 3 -t 2 = λ(x 3 -y 2 ) in C[x, y].
(5)

We develop s and t into homogeneous components with respect to the weight:

s = m i=0 i 1 s 2i and t = n i=0 i 1 t 2i
where

s 2i = (a,b)∈N 2 2a+3b=i σ a,b x a y b and t 2i = (a,b)∈N 2 2a+3b=i τ a,b x a y b (σ a,b , τ a,b ∈ C)
for all i (where m = 0 or m ≥ 2 and n = 0 or n ≥ 2). Equation ( 5) implies that t(E 4 , E 6 ) 2s(E 4 , E 6 ) 3 has weight 12. Then only three cases are possible. 1) If 3m > 2n then m = 2 and so n ∈ {0, 2}. It implies that s = σ 00 + σ 10 x and t = τ 00 + τ 10 x. This contradicts jac(s, t) 0. 2) If 3m < 2n then n = 3 and m = 0. This contradicts jac(s, t) 0.

3) If 3m = 2n. We differentiate (5) with respect to x and y and get

3s 2 s x -2t t x = 3λx 2 , 3s 2 s y -2t t y = -2λy. This implies 2t = 3x 2 s y + 2y s x , 3s 2 = 3x 2 t y + 2y t x . ( 6
)
From the first differential equation of ( 6) we have

2t(E 4 , E 6 ) = 3E 2 4 s y (E 4 , E 6 ) + 2E 6 s x (E 4 , E 6 ).
The highest weight of the right hand side is less than or equal to 2m + 2. This implies n ≤ m + 1. From the second differential equation of (6) we have

3s 2 (E 4 , E 6 ) = 3E 2 4 t y (E 4 , E 6 ) + 2E 6 t x (E 4 , E 6 )
hence 2m ≤ n + 1. We deduce (m, n) ∈ {(0, 0), (2, 3)}. Since n = m = 0 would imply jac(s, t) = 0 we have n = 3 and m = 2. Then s = σ 00 + σ 10 x and t = τ 00 + τ 10 x + τ 01 y.

The first differential equation in (6) implies that τ 00 = τ 10 = 0 and τ 01 = σ 10 whereas the second one implies that σ 00 = 0 and σ 10 = 1. Finally ϕ(E 4 ) = s(E 4 , E 6 ) = E 4 and ϕ(E 6 ) = s(E 4 , E 6 ) = E 6 .

Since RC 1 (∆, f ) = (12 D(f ) -kf E 2 ) ∆ for any f ∈ M k , the first Rankin-Cohen bracket defines a derivation on M * called the Serre's derivative by linear extension of:

ϑf = RC 1 (∆, f ) 12∆ = D(f ) - k 12 f E 2 (f ∈ M k ). ( 7
)
This derivation is characterised by its values on the generators

ϑ E 4 = - 1 3 E 6 , ϑ E 6 = - 1 2 E 2 4 .
We shall need the following result.

Proposition 6-The kernel of Serre's derivative is the Poisson centraliser of ∆ for the first Rankin-Cohen bracket. This is C

[∆]. Proof. If f ∈ M k is in ker ϑ then kf D(∆) = 12∆ D(f ).
Solving the differential equation we find that 12 divides k and that f ∈ C∆ k/12 .

We note that, for any g ∈ M ℓ , we have

RC 1 (∆ m , g) = m∆ m (12 D(g) -ℓg E 2 )
and deduce that, for any f ∈ C[∆] and g ∈ M * we have

RC 1 (f , g) = 12ξ(f )ϑ(g) ( 8
)
where ξ is the Eulerian derivative on C[∆] defined by ξ = ∆ ∆ .

Poisson structures on quasimodular forms

3.1. First family. This section is devoted to the proof of Proposition A. We fix λ ∈ C * and introduce in C[x, y, z] the three polynomials

r(x, y, z) = 1 3 (λy 2 -2xz) p(x, y, z) = -2(y 3 -z 2 ) q(x, y, z) = - 1 2 (λyz -2xy 2 ).
Since (p, q, r) • curl(p, q, r) = 0, we define a Poisson bracket on M ≤∞ * if we set

{E 4 , E 6 } λ = p(E 2 , E 4 , E 6 ) {E 2 , E 4 } λ = r(E 2 , E 4 , E 6 ) {E 6 , E 2 } λ = q(E 2 , E 4 , E 6 ).
Let us prove that { , } λ is not unimodular. If it were, we would have k ∈ C[x, y, z] such that curl(p, q, r) = (p, q, r) ∧ gradk. Identifying the first components would lead to

7 6 λy = 1 2 (-λyz + 2y 2 x) k z - 1 3 (λy 2 -2zx) k y that has no solution in C[x, y, z]. A Poisson modular isomorphism ϕ λ between M ≤∞ * , { , } λ and M ≤∞ * , { , } 1 is de- termined by ϕ λ (E 2 ) = λ E 2 , ϕ λ (E 4 ) = E 4 , ϕ λ (E 6 ) = E 6 .
Finally, we determine the Poisson center of the Poisson algebra M ≤∞ * , { , } 1 . Let us define a derivation on M * by σ = 2ϑ (see ( 7)) an a derivation on M * by linear extension of

δ(f ) = k 12 f E 4 (f ∈ M k ). We note that M ≤∞ * , { , } 1 is the Poisson-Ore extension C[E 4 , E 6 ][E 2 ] σ,δ Now, consider any f ∈ M ≤∞ * written as f = s i=0 f i E i 2 , f i ∈ M * .
We compute

{E 2 , f } 1 = δ(f 0 ) + s i=1 (σ(f i-1 ) + δ(f i )) E i 2 +σ(f s ) E s+1 2 . If {E 2 , f } 1 = 0 then δ(f 0 ) = 0 hence f 0 ∈ C and σ(f 0 ) = 0. We obtain inductively that f i ∈ C for all 0 ≤ i ≤ s so that the Poisson centraliser of E 2 is C[E 2 ]
. Suppose that the Poisson center contains a non scalar element. Then it is in the Poisson centraliser of E 2 and can be written

f = p j=0 α j E j 2 , p ≥ 1, α j ∈ C, α p 0.
We compute

{E 4 , f } 1 = p j=0 jα j E j-1 2 •{E 4 , E 2 } 1
and find that the coefficient of E p 2 is non zero. It follows that f is not in the Poisson center.

3.2. Second family. This section is devoted to the proof of Proposition B. We fix α ∈ C and introduce in C[x, y, z] the three polynomials r(x, y, z) = αxz p(x, y, z) = -2(y 3 -z 2 ) q(x, y, z) = -3 2 αxy 2 .

Since (p, q, r) • curl(p, q, r) = 0, we define a Poisson bracket on M ≤∞ * if we set

(E 4 , E 6 ) α = p(E 2 , E 4 , E 6 ) (E 2 , E 4 ) α = r(E 2 , E 4 , E 6 ) (E 6 , E 2 ) α = q(E 2 , E 4 , E 6 ).
Assume α 4. Let us prove that ( , ) α is not unimodular. If it were, we would have k ∈ C[x, y, z] such that curl(p, q, r) = (p, q, r) ∧ gradk. Identifying the second components would lead to

(4 -α)z = αxz k x + 2(y 3 -z 2 ) k z that has no solution in C[x, y, z].
If α = 4, then (p, q, r) = gradk 0 where k 0 = -2(y 3 -z 2 )x. As a consequence, the bracket ( , ) 4 provides M ≤∞ * with a Jacobian Poisson structure of potential

k 0 = -2∆ E 2 = -2 D(∆).
If ϕ : M ≤∞ * , ( , ) α → M ≤∞ * , ( , ) α ′ is a Poisson modular isomorphism, let us prove that α = α ′ . By Proposition 5, we have ϕ(E 4 ) = E 4 and ϕ(E 6 ) = E 6 . By surjectivity, it follows that ϕ(E 2 ) = η E 2 +F for some η ∈ C * and F ∈ M * . We compute

ϕ ((E 2 , E 4 ) α ) = αη E 6 E 2 +α E 6 F and (ϕ(E 2 ), ϕ(E 4 )) α ′ = α ′ η E 6 E 2 +(F, E 4 ) α ′ . Since (F, E 4 ) α ′ ∈ M * we get α ′ = α.
Finally, we determine the Poisson center of the Poisson algebra M ≤∞ * , ( , ) α . We note that M ≤∞ * , ( ,

) α is the Poisson-Ore extension C[E 4 , E 6 ][E 2 ] σ,δ where σ = -3αϑ (see (7)) and δ = 0. Let f = s j=0 f j E j 2 , f j ∈ M * .
We have

(E 2 , f ) α = s j=0 σ(f j ) E j+1 2
hence f is in the Poisson centraliser of E 2 if and only each f j is in the Poisson centraliser of ∆ for RC 1 . By Proposition 6 we deduce that the centraliser of

E 2 is C[∆, E 2 ]. Let f = s j=0 f j (∆) E j 2 ∈ C[∆, E 2 ].
We use (8) to compute

(f , E 4 ) α = s j=0 -4ξ(f j ) + jαf j E 6 E j 2 and (f , E 6 ) α = 3 2 s j=0 -4ξ(f j ) + jαf j E 2 4 E j 2 .
We deduce that f is in the Poisson center of ( , ) α if and only if

ξ(f j ) = jα 4 f j
for all j, i.e. if and only if any f j is of the form f j = λ j ∆ m j for some λ j ∈ C and m j ∈ N such that jα = 4m j . If α Q or if α < 0 then j = 0 and m j = 0 hence f = f 0 ∈ C. If α = p/q with p ≥ 1, q ≥ 1 and (p, q) = 1 then λ∆ m j E j 2 is in the Poisson center if and only if pj = 4qm j . The result follows by obvious arithmetical consideration. Finally, if α = 0, then ( , ) 0 is the trivial bracket and its Poisson center is C[E 2 ].

3.3. Third family. In this section, we study the third family, i.e. we prove Proposition C.

For any µ ∈ C, let us introduce

k µ = -2∆ E 2 +µ E 2 4 E 6 . Then jac(E 4 , E 6 , k µ ) = µ k E 2 = -2 E 3 4 + E 2 6 jac(E 2 , E 4 , k µ ) = µ k E 6 = 4 E 6 E 2 +µ E 3 4 jac(E 2 , E 6 , k µ ) = - µ k E 4 = 6 E 2 4 E 2 -2µ E 4 E 6 .
The third family of Poisson bracket is then defined by f , g µ = jac(f , g, k µ ). With the notation of Proposition B, we have in particular f , g 0 = (f , g) 4 .

For any µ ∈ C * , we define a Poisson modular isomorphism ϕ µ between M ≤∞ * , , µ and M ≤∞ * , , 1 by setting ϕ µ (E 2 ) = µ E 2 , ϕ µ (E 4 ) = E 4 and ϕ µ (E 6 ) = E 6 . Since the degree in E 2 of k µ as a polynomial in E 2 , E 4 , E 6 is 1, Lemma 1 implies that the Poisson center of

M ≤∞ * , , µ is C[k µ ].
3.4. Classification. This section is devoted to the proof of Theorem C. Let { , } be an admissible bracket on M ≤∞ * . By Definition 1 and Theorem 2 , there exist a Poisson derivation σ of M * and a Poisson σ-derivation δ of M * such that

{E 2 , f } = σ(f ) E 2 +δ(f ) (f ∈ M * ).
By definition, σ(M k ) ⊂ M k+2 and δ(M k ) ⊂ M k+4 for any k. The admissible bracket { , } is then defined by the four scalars α, β, γ and ε such that

σ(E 4 ) = α E 6 , δ(E 4 ) = β E 2 4 , σ(E 6 ) = γ E 2
4 , and δ(E 6 ) = ε E 4 E 6 . The condition that σ is a Poisson derivation imposes the condition {σ(E 4 ), E 6 } + {E 4 , σ(E 6 )} = -2σ(E 3 4 -E 2 6 ) or equivalently 3α = 2γ. The condition that δ is a Poisson σ-derivation imposes

δ ({E 4 , E 6 }) = (2β + ε) E 4 {E 4 , E 6 } + αε E 4 E 2 6 -βγ E 4 4 or equivalently 4β + (α -2)ε = 0 (3α -4)β + 4ε = 0.
Either β = ε = 0 is the only solution, or α ∈ {-2/3, 4} and ε = 4 2α β.

• The case β = ε = 0 leads to the second family: { , } = ( , ) α .

• The case α = -2/3 and ε = 3β/2 0 leads to the first family: { , } = { , } 3β .

• The case α = 4 and ε = -2β 0 leads to the third family: { , } = , β .

Using Propositions B and C we conclude that the only admissible Poisson brackets, up to Poisson modular-isomorphisms are { , } 1 , , 1 and ( , ) α for any α ∈ C. Looking at the centres, it is clear that the Poisson algebras M ≤∞ * , , 1 and M ≤∞ * , { , } 1 are not Poisson modular isomorphic. Suppose that there exists a Poisson modular isomorphism ϕ from M ≤∞ * , ( , ) α to M ≤∞ * , { , } 1 . We know (see § 3.2) that ϕ(E 4 ) = E 4 , ϕ(E 6 ) = E 6 , and ϕ(E 2 ) = η E 2 +F for some η ∈ C * and F ∈ M * . From ϕ ((E 2 , E 4 ) α ) = {ϕ(E 2 ), ϕ(E 4 )} 1 we obtain

αη E 6 E 2 +αF E 6 = - 2 3 η E 6 E 2 + 1 3 η E 2 4 +{F, E 4 } 1 hence α = - 2 3 , 1 3 η E 2 4 = - 2 3 F E 6 -{F, E 4 } 1 = - 2 3 F E 6 +2(E 3 4 -E 2 6 )
F E 6 by (2). We get a contradiction. Replacing { , } 1 by , 1 we get

α = 4, η E 2 4 = 4F E 6 -2(E 3 4 -E 2 6 )
F E 6 and again, we get a contradiction.

Star products on quasimodular forms

4.1. Extension of the first family. This section is devoted to the proof of Theorem A.

We will use the following result of Zagier [Zag94, Example 1]. Let A =

A k be a commutative graded algebra with a derivation d homogeneous of degree 2 (i.e. d(A k ) ⊂ A k+2 ). Let us define, for any f ∈ A k , g ∈ A ℓ , r ≥ 0:

[f , g] d,r = r i=0 (-1) i k + r -1 r -i ℓ + r -1 i d i (f ) d r-i (g) ∈ A k+ℓ+2r . ( 9
)
Then A equipped with these brackets is a Rankin-Cohen algebra, that means that all algebraic identities satisfied by the usual Rankin-Cohen brackets on modular forms are also satisfied, in particular those expressing of the associativity of the corresponding star product. We obtain the following result.

Theorem 7-The star product defined by

f #g = n≥0 [f , g] d,n ℏ n
defines a formal deformation on A.

In particular, we recover the fact, given by Corollary 4, that [ , ] d,1 is a Poisson bracket. Note also that this Theorem can be obtained from Connes & Moscovici's result cited below (see § 4.2).

Let a ∈ C and d a be the homogeneous derivation of degree 2 on M ≤∞ * defined by

d a (E 2 ) = 2a E 2 2 - 1 12 E 4 , d a (E 4 ) = 4a E 4 E 2 - 1 3 E 6 , d a (E 6 ) = 6e E 6 E 2 - 1 2 E 2 4 .
A direct computation proves that the two Poisson brackets [ , ] d a ,1 and { , } 1 coincide on generators hence are equal on M ≤∞ * .

Remark-A derivation d on M ≤∞ * is complex-like if d M ≤s k ⊂ M ≤s+1 k+2 for all k and s. Let π be the derivation on M ≤∞ * defined by linear extension of π(f ) = kf E 2 for all f ∈ M ≤∞ k . The set of complex-like derivations d such that [ , ] d,1 = 0 is the vector space of dimension 1 over C generated by π. Let us define w on M ≤∞ * by w

(f ) = {∆, f } 1 12∆ .
Then, d a = w +aδ.

This implies in particular that if a complex-like derivation d satisfies [ , ] d,1 = { , } 1 then d = d a for some a ∈ C.

Point ii) of Theorem A is obtained by a direct application of Theorem 7. We prove now iii). The term of highest degree with respect to E The condition that [E 4 , E 6 ] K δ α,b ,2 has to be a modular form implies b = 0 or α = -1/3. For α = -1/3 condition (4) for µ r = [ , ] K δ α,b ,r and n = 3 is not satisfied (this can be done by using computer assistance, for example with Sage [S + 13]). We assume then that b = 0.

2 in [E 2 , E 4 ] d a ,2 is 8a 2 E 4 E 3 2 . This forces a = 0. Conversely, if a = 0 then d 0 M ≤∞ * ⊂ M * . For any f = f i E i 2 with f i ∈ M * we have d 0 (f ) = d 0 (f i ) E i 2 - 1 12 if i E 4 E i-1 2 hence deg E 2 d 0 (f ) ≤ deg E 2 f
Connes & Moscovici [CM04, Remark 14] (see also [Yao07, § II.2] for a nice presentation of this result) proved that, if E and H are two derivations of an algebra R such that HE -EH = E, then the applications µ n : R × R → R defined by Hence a direct application of formula (10) gives formula (9).

  y, z] is said to have a Jacobian Poisson structure (JPS) of potential (or Casimir function) k. The Poissonian triple (p, q, r) is said then to be exact. The Poisson center (or zeroth Poisson cohomology group) of a Poisson algebra A is the Poisson subalgebra HP 0 (A) = {g ∈ A: b(f , g) = 0, ∀f ∈ A} . The Poisson center is contained in the Poisson centraliser of any element in the algebra: let f ∈ A, its Poisson centraliser is {g ∈ A: b(f , g) = 0}. The following lemma computes the Poisson center of polynomial algebras in three variables equipped with a Jacobian Poisson structure. It allows one to recover for example [Pic06b, Poposition 4.2] in the particular case where the potential is a weight homogeneous polynomial with an isolated singularity.

  If A and B are two Poisson algebras, with respective Poisson brackets b A and b B , a map ϕ : A → B is a morphism of Poisson algebras when it is a morphism of algebras that satisfies ϕ (b A (f , g)) = b B (ϕ(f ), ϕ(g)) for any f and g in A. Two Poisson-isomorphic Poisson algebras have isomorphic Poisson centres. We detail now a canonical way to extend a Poisson structure from an algebra A to a polynomial algebra A[x]. This construction is due to Sei-Qwon Oh [Oh06]. A Poisson derivation of A is a derivation σ of A satisfying σ (b(f , g)) = b (σ(f ), g) + b (f , σ(g)) for all f and g in A. If σ is a Poisson derivation of A, a Poisson σ-derivation is a derivation δ of A such that δ (b(f , g)) = b (δ(f ), g) + b (f , δ(g)) + σ(f )δ(g)δ(f )σ(g) for all f and g in A. Theorem 2 (Oh)-Let (A, b A ) be a Poisson algebra. Let σ and δ be linear maps on A. The polynomial ring A[x] becomes a Poisson algebra with Poisson brackets b defined by b

  Then i) the map b is bilinear skew-symmetric and satisfies Leibniz rule. ii) If (δ, d) is solvable, then b satisfies Jacobi identity and so becomes a Poisson bracket. iii) Under the hypothesis in ii), d is a Poisson derivation for b. Proof of Proposition 3. Point i) is immediate. Point ii) is a consequence of the following computation. If (δ, d) is solvable with δ dd δ = α d and if B :

  and deg E 2 d j 0 (f ) ≤ deg E 2 f for any f ∈ M ≤∞ * and j ≥ 0. This implies that [M ≤s k , M ≤t ℓ ] d 0 ,n ⊂ M ≤s+t k+ℓ+2n . 4.2. Extension of the second family. The aim of this section is to prove Theorem B. The proof of i) is similar to the proof of i) in Theorem A. Let K : M ≤∞ * → C a gradedadditive map. For any integer n ≥ 0, we define a bilinear application [ , ] K d,n by bilinear extension of [f , g] K d,n = n r=0 (-1) r K(f ) + n -1 nr K(g) + n -1 r d r f d n-r g.By Corollary 4, we know that [f , g] K d,1 is a Poisson bracket. Let us fix K α to be the linear extension onM ≤∞ * = k s M s k of K α (f ) = (k -(3α + 2)s) (f ∈ M s k ).Let π α be the derivation on M ≤∞ * defined by π α (f ) = K α (f )f E 2 for all f ∈ M ≤∞ * . The set of complex-like derivations such that [ , ] K α d,1 = 0 is the vector space of dimension 1 over C generated by π α . Define derivations v and δ α,b on M ≤∞ * by v(f ) = (∆, f ) α 12∆ and δ α,b = v +bπ α . Note that v does not depend on α. By comparing the values on the generators, it is immediate that ( , ) α = [ , ] K α δ α,b ,1 . Remark-Direct computations show that if d is a homogeneous derivation of degree 2 and K is such that ( , ) α = [ , ] K d,1 then we necessarily have K = K α and d = δ α,b for some b ∈ C.

  µ n (f , g) = n r=0 (-1) r r!(nr)! [E r • (2H + r) <n-r> (f )] • [E n-r • (2H + nr) <r> (g)](10)define a formal deformation on R with the notationF <m> = F • (F + 1) • (F + 2) • • • • • (F + m -1).Let ̟ be the derivation defined on M ≤∞ * by ̟(f ) = K(f )f . Then we have̟ • δ α,0δ α,0 • ̟ = 2δ α,0 .We use Connes & Moscovici's result with E = δ α,0 and H = ̟/2 to obtainµ n (f , g) = n r=0 (-1) r k -(3α + 2)s + n -1 nr ℓ -(3α + 2)t + n -1 r δ r α,0 (f )δ n-r α,0 (g).This implies Theorem B.Remark-We could have apply Connes & Moscovici's result to extend the first family. Indeed Zagier's result is a consequence of Connes & Moscovici's one. Let d be a derivation homogeneous of degree 2 of the commutative graded algebra A = A k . It is obvious that the linear map defined on each A k by H(f ) = k 2 f is a derivation of A. It is also clear that it satisfies H • dd • H = d. In particular, for any f ∈ A k and g ∈ A ℓ we calculate (2H + r) <n-r> (f ) = (k + n -1)! (k + r -1)! f and (2H + (nr)) <r> (g) = (ℓ + n -1)! (ℓ + nr -1)! g.

  The Poisson algebras M ≤∞ * , , µ and M ≤∞ * , , µ ′ are Poisson modular isomorphic for all µ and µ ′ in C * . iii) For any µ ∈ C, the Poisson center of M ≤∞

* , , µ is the polynomial algebra C[k µ ].

  , RC 1 ) and is indeed studied by this work. If αβ = 0, the Poisson algebras are no more Poisson isomorphic (they do not have the same group of automorphisms). This degenerate case deserves another study.

	Theorem C-Up to Poisson modular isomorphism, the only distinct admissible Poisson
	brackets on the algebra of quasimodular forms are { , } 1 , , 1 and the family ( , ) α for any α ∈ C.
	Remark-We could endow the algebra of modular forms with another Poisson struc-
	ture b. If we require b (M k , M ℓ ) ⊂ M k+ℓ+2 , then b is necessarily defined by b(E 4 , E 6 ) = α E 3 4 +β E 2 6 for some complex numbers α and β. If αβ 0 then (M * , b) is Poisson iso-morphic to (M

*

4.3. Extension of the third family. We do not extend the third family since, for µ 0, the bracket , µ has not the shape of a Rankin-Cohen bracket. More precisely, if there exists a function κ : M ≤∞ * → C and a complex-like derivation δ of M ≤∞ * such that

for all f and g in M ≤∞ * , then µ = 0. Indeed, assume κ and δ exist, then

for some complex numbers A, B, C, D, E and F. Since we know the values of , µ on the generators, we get a system relying A, B, C, D, E, F, κ(E 2 ), κ(E 4 ) and κ(E 6 ). It is not difficult to prove that this system has a solution if and only if µ = 0.