V Bosser

F Pellarin

Hyperdifferential properties of Drinfeld quasi-modular forms

. This last result is the analogue of a crucial property proved by Nesterenko [11] in characteristic zero in order to establish a multiplicity estimate.

Introduction

In [START_REF] Gekeler | On the coefficients of Drinfeld modular forms[END_REF], Gekeler introduced modular forms for GL 2 (F q [T ]), analogues in positive characteristic to the classical modular forms for SL 2 (Z); these functions are frequently called "Drinfeld modular forms". Both Drinfeld and classical modular forms have interesting differential properties but there is a slight imperfection, as the rings generated by them are not stable under derivation.

To overcome this problem in the classical case, one idea is to weaken a bit the definition of modular forms, allowing slightly more general functional equations, thus leading to the notion of quasi-modular forms as in [5, p. 166] and [START_REF] Martin | Formes modulaires et périodes[END_REF]Definition 113] (1 ).

The same can be done in the positive characteristic case, as we will see in this article: Drinfeld quasi-modular forms can be defined, and the ring 1 generated by these functions both contains Drinfeld modular forms and is stable under derivation.

To what extent do classical quasi-modular forms and Drinfeld quasimodular forms resemble each other? This question is the basis for our article. We will find that, apart from some differences arising from the unequal characteristics, the classical theory and Drinfeld's theory of quasi-modular forms bear striking similarities.

The classical framework. Before we describe our results, it is convenient to present the aspects of the classical theory whose analogues in positive characteristic will be discussed in this text.

A holomorphic function f : H → C (where H is the complex upper halfplane) is called quasi-modular form of weight w ∈ N and depth at most l ∈ N for Γ Q := SL 2 (Z) if there exist functions f 0 , . . . , f l , which are holomorphic both on H and "at infinity", and which satisfy the following condition: for all γ = a c b d ∈ Γ Q and all z ∈ H,

f az + b cz + d = (cz + d) w l r=0 f r (z) c cz + d r .
After Kaneko and Zagier (cf. [5, Proposition 1], see also [START_REF] Martin | Formes modulaires et périodes[END_REF]Proposition 124]), the ring generated by quasi-modular forms for Γ Q is also the ring of functions C[E 2 , E 4 , E 6 ], where E 2 , E 4 , E 6 are the classical Eisenstein series of weights 2, 4 and 6 respectively.

Recall that E 4 and E 6 are modular forms (hence, quasimodular forms of depth 0), but that E 2 is a non-modular quasi-modular form of depth 1. Since the derivative of a quasi-modular form of weight w and depth ≤ l is again a quasi-modular form, of weight w + 2 and depth ≤ l + 1, and since the functions E 2 , E 4 , E 6 are algebraically independent over C(q) by a classical result due to Rankin and Mahler (see the contribution of Bertrand in [START_REF] Yu | Introduction to algebraic independence theory[END_REF]), C[E 2 , E 4 , E 6 ] is thus isomorphic to a polynomial ring in three indeterminates, endowed with a structure of a differential, filtered (by the depth), graded ring.

Explicitly, if we set D = (2πi) -1 d/dz, the derivation on this ring is described by the following formulas (due to Ramanujan): 4 ).

   DE 2 = 1 12 (E 2 2 -E 4 ) DE 4 = 1 3 (E 2 E 4 -E 6 ) DE 6 = 1 2 (E 2 E 6 -E 2
(1)

Among several basic questions that we may ask about differential rings, one is to determine their D-differential prime ideals (recall that an ideal I is said D-differential if DI ⊂ I). In [START_REF] Pellarin | La structure différentielle de l'anneau des formes quasimodulaires pour SL 2 (Z)[END_REF], the D-differential prime ideals of C[E 2 , E 4 , E 6 ] were all determined applying elementary commutative algebra. There are many others, which we do not describe here, but they all contain ∆. This latter property (with a slight technical hypothesis that we skip here) was used by Nesterenko [START_REF] Yu | Modular functions and transcendence questions[END_REF] to prove a multiplicity estimate, which was a crucial tool in the proof of his famous theorem on the algebraic independence of values of Eisenstein series: if z ∈ H, at least three of the four numbers e 2πiz , E 2 (z), E 4 (z), E 6 (z) are algebraically independent over Q (see [START_REF] Yu | Modular functions and transcendence questions[END_REF], [START_REF] Yu | Introduction to algebraic independence theory[END_REF]Chapter 3], see also [START_REF] Bosser | Indépendance algébrique de valeurs de séries d'Eisenstein[END_REF]Théorème 1.1]).

Drinfeldian framework. After this survey on quasi-modular forms for the group Γ Q , we are ready to introduce the main results of this article.

Let T be an indeterminate, let A = F q [T ] be the polynomial ring over a finite field F q , and let K = F q (T ) be its field of fractions. Let K ∞ = F q ((1/T )) be the completion of K at the infinite prime, and let C be the completion of an algebraic closure of K ∞ . We will denote by Ω the set C \ K ∞ and, as usual, by p the characteristic of C.

It is well known that Ω is a connected admissible open subspace of the rigid analytic space P 1 (C). The group Γ K := GL 2 (A) acts discontinuously on Ω by homographies

γ = a c b d ∈ Γ K , z ∈ Ω, γ(z) = az + b cz + d .
The quotient space can also be given a rigid structure, and it is natural to consider "modular forms" with the straightforward definition.

The foundations of the theory of modular forms on Ω that we need here are essentially contained in [START_REF] Gekeler | On the coefficients of Drinfeld modular forms[END_REF]. Recall that since Γ K has the nontrivial character det : Γ K → F * q , modular forms naturally have a type (2 ), in addition to a weight. The subring of the ring of holomorphic functions on Ω generated by modular forms is C[g, h], where g and h are two special modular forms, which are defined in [START_REF] Gekeler | On the coefficients of Drinfeld modular forms[END_REF]. Here, g is the Eisenstein series of weight q -1 and type 0 normalized in a suitable way (3 ), and h is the Poincaré series P q+1,1 of weight q + 1 and type 1 (example (5.11) of [START_REF] Gekeler | On the coefficients of Drinfeld modular forms[END_REF]).

Our first task (Sections 2 to 4) will be to introduce Drinfeld quasimodular forms for the group Γ K and to study how they behave under the action of the so-called "divided derivatives". Drinfeld quasi-modular forms will have weights, types and depths, the zero depth corresponding to the case of modular forms. A useful result we will then prove is a structure theorem for these quasi-modular forms, similar to the one of Kaneko and Zagier quoted above. Let E be the "false" Eisenstein series as defined in [6, p. 686]. With our definition, this function will be a quasi-modular form of weight 2, depth 1 and type 1. Denote by M ≤l w,m the C-vector space of quasi-modular forms of weight w, type m and depth ≤ l, by M the ring generated by all quasi-modular forms (see § 2 for the precise definitions), and by C[E, g, h] ≤l w,m the subspace of C[E, g, h] generated by the monomials E α g β h γ satisfying the conditions 2α + β(q -1) + γ(q + 1) = w, α + γ ≡ m (mod q -1) and α ≤ l.

In Section 2 we prove:

Theorem 1
The functions E, g, h are algebraically independent over C, and we have

M ≤l w,m = C[E, g, h] ≤l w,m and M = C[E, g, h].
Before introducing the differential results of this paper, let us recall that Gekeler proved that the three functions E, g and h satisfy a system of differential equations, namely (4 ):

   D 1 E = E 2 D 1 g = -(Eg + h) D 1 h = Eh, (2)
where D 1 denotes the operator (-π) -1 d/dz, π being a fixed fundamental period of the Carlitz module. Thus, D 1 defines a derivation on the ring C [E, g, h]. At first glance, the system (2) seems to be an analogue of (1). But it turns out that the kernel of the derivation D 1 is far too "big" ( 5), and it is hopeless to get results about differential ideals analogous to those mentioned above in the classical case. For instance, the kernel of D 1 contains infinitely many irreducible elements, and so the differential ring C[E, g, h] has infinitely many D 1 -differential principal prime ideals, unlike the characteristic zero case.

The first important objective of this article is to show that in fact, the ring C[E, g, h] is not only stable under the derivation D 1 , but also under all the "divided derivations".

Let f : Ω → C be a holomorphic function and z ∈ Ω be a point. We define the divided derivatives or hyperderivatives (D n f )(z) of f at z (n ≥ 0) by the formula

f (z + ε) = n≥0 (D n f )(z)ε n , ε ∈ C, |ε| small.
This defines holomorphic functions D n f : Ω → C. In order to preserve Krationality properties, define further D n := 1/(-π) n D n (n ≥ 0). We then have the following result.

Theorem 2 Let n ≥ 0 be an integer. The operators

D n induce C-linear operators D n : C[E, g, h] → C[E, g, h]
such that, for all (w, l, m) ∈ N × N × Z/(q -1)Z,

D n K[E, g, h] ≤l w,m ⊂ K[E, g, h] ≤l+n w+2n,m+n . (3)
This Theorem will be proved in Section 3 as a consequence of Proposition 3.1. It is the hyperdifferential ring C[E, g, h] (that is, the ring equipped with the operators D 1 , D 2 , . . .) which should be interpreted as the analogue in the Drinfeldian framework of the differential ring

C[E 2 , E 4 , E 6 ].
By Theorem 2, any hyperderivative of a polynomial in the functions E, g, h is again a polynomial in these functions. However, the operators D n behave erratically and the problem of finding explicit formulas for D n E, D n g, and D n h when n ≥ p is not a trivial one, in contrast with the classical case, where it suffices to iterate the operator D (see for example the formula (25) of [START_REF] Swinnerton-Dyer | On ℓ-adic representations and congruences for coefficients of modular forms. Modular Functions of one Variable III[END_REF]). Some general results concerning the hyperderivatives D n E, D n g and D n h can nevertheless be obtained when n has a particular form. For example, in Proposition 3.7 we determine D p k E and D p k g for all k, up to a modular form (see Section 3.5 for other results in the same vein). Also, we give explicit formulas in Section 4 for D p k E, D p k g and D p k h when p k ≤ q 2 (Theorem 4.1). As we will see, these formulas allow, at least in principle, to compute D n f for any f ∈ C[E, g, h] and any n ≤ q 2 . For instance, in the particular case q = p, these formulas (which then determine

D n f even for all n ≤ p 3 -1) read    D p E = E p+1 + 1 d 1 h 2 D p g = E p g D p h = 2E p h -1 d 1 gh 2 , (4)
         D p 2 E = E p 2 +1 + 1 d p 1 g p-1 h p+1 + 1 d 2 g 2p h 2 D p 2 g = E p 2 g -d 1 d 2 g p+1 h p + d 2 -d p+1 1 d p-1 1 d 2 h 2p-1 D p 2 h = E p 2 h + 1 d p-1 1 E p g p-1 h p -1 d 2 g 2p+1 h 2 - d p+1 1 +d 2 d p 1 d 2 g p h p+1 , (5)
where

d 1 = [1] and d 2 = [2][1] q , the notation [i] being defined as usual by [i] = T q i -T .
The second aspect of the theory that we want to develop is, in view of Nesterenko's results quoted before, the study of the ideals which are stable by all the operators D n . If I is an ideal of the ring C[E, g, h], we say that

I is hyperdifferential, if it is stable under all the operators D n , that is, if D n (I) ⊂ I for all n ≥ 0. Hyperdifferential ideals in C[E, g, h] are natural analogues of D-differential ideals of C[E 2 , E 4 , E 6 ].
Non-trivial hyperdifferential ideals of C[E, g, h] exist: for example, in Proposition 3.6 we will see that the principal ideal (h) is hyperdifferential, in analogy with the ideal (∆) in the classical case. The next important result we will prove is that in fact, when q = 2, 3, this ideal is the only non-zero hyperdifferential principal prime ideal (again as in the classical case).

More precisely, let us write F = C(E, g, h). The hyperderivatives D n (n = 1, 2, . . .) extend to F in a unique way, we denote these extensions by D n again. We have:

Theorem 3 Assume that q ∈ {2, 3}. Let f ∈ F × be such that for all r ≥ 0 (D r f )/f ∈ C[E, g, h]. Then, there exists n ∈ Z and c ∈ C × such that f = ch n .
Note that our proof of Theorem 3 really requires that q = 2, 3, and that we do not know if the result remains true when q ∈ {2, 3}. This Theorem is an analogue of Lemma 5.2 of [START_REF] Yu | Introduction to algebraic independence theory[END_REF]Chapter 10] (or Lemma 4.26 of [START_REF] Bosser | Indépendance algébrique de valeurs de séries d'Eisenstein[END_REF]), but its proof is different, and much more difficult. It requires two steps, contained in Section 5. The first step consists in showing that if f is as in Theorem 3, there exists a p-adic

integer σ = s 1 + s 2 p + • • • such that for all k ≥ 0, f h -(s 1 +s 2 p+•••+s k+1 p k ) ∈ ker D 1 ∩ • • • ∩ ker D p k .
Then, with the help of the expansions at infinity of the functions, we show that σ ∈ Z, which yields the result since

∩ k≥0 ker D p k = C.
Somewhat, this approach already appears in the paper [START_REF] Matzat | Iterative differential equations and the Abhyankar conjecture[END_REF] by Matzat and van der Put. But these authors work with linear hyperdifferential modules over a field, while in our context, the theory is essentially non-linear. This means that our proofs, even though initially inspired by the work [START_REF] Matzat | Iterative differential equations and the Abhyankar conjecture[END_REF], ultimately depend very slightly on it. We should also add that in the proof of Theorem 3, it is not enough to know the formulas (2) (or even the formulas of Section 4). Thus, in this part of our work, the formalism of Kaneko and Zagier becomes very helpful.

As in characteristic zero, one can use Theorem 3 to prove a result for arbitrary non-zero hyperdifferential prime ideals. One gets:

Corollary 4 Assume that q ∈ {2, 3}. If P is a non-zero hyperdifferential prime ideal of C[E, g, h], then h ∈ P.
Corollary 4 is proved in Section 5.7. It is an analogue of Proposition 5.1 of [START_REF] Yu | Introduction to algebraic independence theory[END_REF]Chapter 10] (see also Theorem 4.25 of [START_REF] Bosser | Indépendance algébrique de valeurs de séries d'Eisenstein[END_REF]). To prove it, we see from Theorem 3 that we only have to treat the case when the height of the prime ideal P is ≥ 2. In characteristic zero, to handle this case the original argument by Nesterenko was a generalisation of an idea of Siegel for the classification of the algebraic solutions of Bessel and Riccati differential equations. This method does not seem to work in our case; to avoid it, we apply a trick appealing to Rankin brackets as in [START_REF] Pellarin | La structure différentielle de l'anneau des formes quasimodulaires pour SL 2 (Z)[END_REF], which also works in positive characteristic.

Finally, in Section 6 (Theorem 6.1), we apply Theorem 3 and Corollary 4 to obtain a full description of all the hyperdifferential ideals of C[E, g, h], analogous to Theorem 1.2 of [START_REF] Pellarin | La structure différentielle de l'anneau des formes quasimodulaires pour SL 2 (Z)[END_REF].

Remarks about Nesterenko's Theorem. Since it was one of the first motivations of our work, we cannot finish this introduction without a discussion around a possible analogue in the Drinfeldian framework of the above quoted Nesterenko's Theorem. Let t(z) be the usual "parameter at infinity" (see Section 2 for the definition). It seems natural to state the following Conjecture (compare with [4, Conjecture 1]).

Conjecture. For all z ∈ Ω, at least three of the four numbers t(z), E(z), g(z), h(z) are algebraically independent over K.

In the direction of this Conjecture, an important progress would be to prove an analogue of Nesterenko's multiplicity estimate (see [START_REF] Yu | Modular functions and transcendence questions[END_REF]Theorem 3], see also [2, Théorème 2.9]). Here too, we could make a Conjecture but we skip it as it merely consists of substituting q and the q-expansions of E 2 , E 4 , E 6 with t and the t-expansions of E, g, h in Nesterenko's multiplicity estimate (see Section 2 for the precise definition of t-expansion).

As already mentioned, the analogues of Theorem 3 and Corollary 4 in characteristic zero play a decisive role in the proof of Nesterenko's estimate (see [2, Remarque 4.5 and Théorème 4.22]). However, although the results we obtain here are very similar to those in characteristic zero, they do not suffice to reach a multiplicity estimate: just as the proof of Theorem 3 is much more difficult and need new arguments, it seems that a proof of the multiplicity estimate in characteristic p will also require further ideas (we omit the details about the several occurring obstructions).

Consequently, the above Conjecture seems to be still out of reach. It is worth noting that such a situation is quite atypical: indeed, transcendence (and algebraic independence) theory in the Drinfeldian framework has now strongly been developed, and most of the classical transcendence results have a known analogue in positive characteristic. Recently, the theory has even gone far beyond the classical one in some cases ( [1], [START_REF] Papanikolas | Tannakian duality for Anderson-Drinfeld motives and algebraic independence of Carlitz logarithms[END_REF], [3]).

Let us notice that, just as for Nesterenko's Theorem, the Conjecture does not seem to follow from Grothendieck-like period conjectures in positive characteristic, but only from some variants of André's conjecture. Hence, it is unclear how it could be handled with the methods of [START_REF] Papanikolas | Tannakian duality for Anderson-Drinfeld motives and algebraic independence of Carlitz logarithms[END_REF].

The results we obtain in this text hopefully constitute a first step towards the Drinfeldian analogue of Nesterenko's multiplicity estimate. We hope taking others (and maybe all?) in a subsequent paper.

2 Basic properties of quasi-modular forms.

The aim of this Section is to introduce Drinfeld quasi-modular forms and to prove some of their fundamental properties, in particular Theorem 1.

Recall that in the previous Section we have denoted by π a fixed fundamental period of the Carlitz module. In the whole text, we will further denote by e C : C → C the Carlitz exponential, and by t : Ω → C the usual "parameter at infinity", that is, t(z) = 1/e C (πz). We have ker e C = πA and

e C (z) = n≥0 z q n d n , (6)
where

d i := [i][i -1] q • • • [1] q i-1 and [i] := T q i -T (see Section 4 of [6]).
Following the terminology of [START_REF] Gekeler | On the coefficients of Drinfeld modular forms[END_REF] and [START_REF] Fresnel | Rigid Analytic Geometry and its Applications[END_REF], we will call holomorphic function in Ω an analytic function on Ω in the rigid analytic sense. Holomorphic functions are thus locally analytic in the sense of [START_REF] Yu | Function field modular forms and higher derivations[END_REF] (but the converse is false).

The imaginary part of an element z ∈ C (see Section 5 of [START_REF] Gekeler | On the coefficients of Drinfeld modular forms[END_REF]) will be denoted by |z| i . If f : Ω → C is holomorphic in Ω and A-periodic (i.e. f (z + a) = f (z) for all a ∈ A and z ∈ Ω), we will say that it is holomorphic at infinity if f (z) is the value, for |z| i sufficiently large, of a converging series

f (z) = i≥0 c i t(z) i , (7)
where c 0 , c 1 , . . . are elements of C. These coefficients uniquely determine f and we will often write, by abuse of notation, f = i≥0 c i t i . The series [START_REF] Lucas | Sur les congruences des nombres eulériens et des coefficients différentiels des fonctions trigonométriques, suivant un module premier[END_REF] is the t-expansion of f . In the whole text, we will denote by O the ring of holomorphic functions f : Ω → C which are A-periodic and holomorphic at infinity. This ring contains in particular the modular forms for Γ K (by Definition 5.9 of [START_REF] Gekeler | On the coefficients of Drinfeld modular forms[END_REF]). If f is a non-zero element of O, we will denote by ν ∞ (f ) its order at infinity: this is the smallest integer i such that c i = 0 in the t-expansion [START_REF] Lucas | Sur les congruences des nombres eulériens et des coefficients différentiels des fonctions trigonométriques, suivant un module premier[END_REF]. The map ν ∞ extends to a map ν ∞ : Frac(O) \ {0} → Z in the obvious way. Definition 2.1 Let w ≥ 0 be an integer and m ∈ Z/(q-1)Z. A holomorphic function f : Ω → C is called quasi-modular form of weight w and type m if there exist functions f 0 , . . . , f l ∈ O such that, for all z ∈ Ω and γ = a b c d ∈ Γ K :

f (γ(z)) = (cz + d) w (det γ) -m l i=0 f i (z) c cz + d i . (8)
The type m of a quasi-modular form is a class in Z/(q -1)Z. However, to avoid heavy notations, we will sometimes use the same notation m for any representant of the class, thus considering m as an integer. This abuse of notation will not lead to confusion.

Remarks. (i).

Let f be a quasi-modular form of weight w and type m, and let f 0 , . . . , f l ∈ O be as in the definition. It is easy to see that if f = 0, then the weight w, the type m and the polynomial l i=0 f i X i ∈ O[X] are uniquely determined by f . Indeed, if the function f were also a quasimodular form of weight w ′ ≥ w and type m ′ , and if an identity analogous to (8) held with functions g 0 , . . . , g l ′ ∈ O at the place of f 0 , . . . , f l , then we would have

c w-w ′ (det γ) m ′ -m A z ( c cz + d ) -B z ( c cz + d ) = 0 (9)
for all z ∈ Ω and all γ ∈ Γ K with c = 0, where

A z = X w ′ -w l i=0 f i (z)X i ∈ C[X] and B z = l ′ i=0 g i (z)X i ∈ C[X]
.

Choosing now γ = 1 d -1 1 d ∈ Γ K in (9) (with d ∈ A arbitrary), we obtain A z = B z for all z ∈ Ω. By fixing an element z ∈ Ω such that A z = 0, choosing c ∈ A with deg c ≥ 1 such that A z (c/(cz + d))
= 0, and taking γ = λ 0 c 1 with λ ∈ F * q in the equation ( 9), we get

λ m ′ -m = c w ′ -w for all λ ∈ F * q .
This implies w = w ′ and m = m ′ in Z/(q -1)Z. The equality A z = B z for all z then yields the equality between the two polynomials l i=0 f i X i and l ′ i=0 g i X i , as claimed.

In virtue of this remark, if f is any non-zero quasi-modular form, the polynomial

P f := l i=0 f i X i of O[X]
is well defined. We call it the associated polynomial of f . The degree of P f is called the depth of f and is usually denoted by l; another notation is l(f ). If f = 0, we agree that it is a quasi-modular form of weight w, type m and depth l for all w, m, l, and we set P f := 0.

We denote by M l w,m (resp. by M ≤l w,m ) the set (resp. the C-vector space) of quasi-modular forms of weight w, type m and depth l (resp. depth ≤ l). Note that M l w,m is not a vector space for l ≥ 1. We will further denote by M w,m the C-vector space of quasi-modular forms of weight w and type m, and by M the C-algebra M = w,m M w,m . In fact, this sum is a direct sum (see Proposition 2.2 below).

(ii). If f is an element of M l w,m , then the constant term f 0 of the associated polynomial P f is necessarily equal to f , as follows from the formula (8) by choosing for γ the identity matrix.

(iii). It is clear from the definition that a modular form f of weight w and type m is an element of M 0 w,m with associated polynomial P f = f .

Conversely, Remark (ii) shows that any element f of M 0 w,m is a modular form of weight w and type m, thus we have:

M w,m = M 0 w,m , (10)
where M w,m denotes the C-vector space of Drinfeld modular forms of weight w and type m (equal to M m w , following the notation of [START_REF] Gekeler | On the coefficients of Drinfeld modular forms[END_REF]). (iv). If

f 1 ∈ M l 1 w 1 ,m 1 and f 2 ∈ M l 2 w 2 ,m 2 , it is obvious from the definitions that f 1 f 2 ∈ M l 1 +l 2 w 1 +w 2 ,m 1 +m 2 and that P f 1 f 2 = P f 1 P f 2 . Moreover, P λf 1 = λP f 1 for λ ∈ C, and if w 1 = w 2 , m 1 = m 2 , then f 1 + f 2 ∈ M ≤max{l 1 ,l 2 } w 1 ,m 1 with P f 1 +f 2 = P f 1 + P f 2 .
An important example of a quasi-modular form which is not a modular form is furnished by the "false" Eisenstein series E(z), which satisfies (see [6, formula (8.4)])

E(γ(z)) = (cz + d) 2 (det γ) -1 E(z) - c π(cz + d) (11)
for all z ∈ Ω and all γ ∈ Γ K . In virtue of the results of Section 8 of [START_REF] Gekeler | On the coefficients of Drinfeld modular forms[END_REF], we have

E ∈ O. Thus, E ∈ M 1 2,1 and P E = E -π -1 X. Proposition 2.2
We have:

M = w≥0 m∈Z/(q-1)Z M w,m .
Proof. The result can be proved exactly as in [10, proof of Lemma 16] and is thus left to the reader.

Throughout this text, we will denote by C[E, g, h] ≤l w,m the space of elements of C[E, g, h] of weight w, type m and depth ≤ l, that is, the C-vector space of polynomials in E, g, h which are linear combinations with coefficients in C of monomials E α g β h γ with 2α + β(q -1) + γ(q + 1) = w, α + γ ≡ m (mod q -1) and α ≤ l.

To prove Theorem 1, we will follow a method similar to the one used in [START_REF] Martin | Formes modulaires et périodes[END_REF] in the complex case. We need four Lemmata. In the next Lemma, we will say that a polynomial

P ∈ C[Y 1 , Y 2 , Y 3 ] is isobaric of weight w ∈ N if it is the sum of monomials of the form λY α 1 Y β 2 Y γ 3 , where λ ∈ C and α, β, γ satisfy 2α+(q -1)β +(q +1)γ = w. Thus, an isobaric element of C[Y 1 , Y 2 , Y 3 ]
is nothing but a homogeneous element for the graduation defined associating to Y 1 , Y 2 , Y 3 the weights 2, q -1, q + 1.

If all the monomials above are moreover such that α+γ ≡ m (mod q-1), then we will say that P is of type m ∈ Z/(q -1)Z.

Lemma 2.3 Let P ∈ C[Y 1 , Y 2 , Y 3 ]
be a non-zero polynomial isobaric of weight w and type m. Then the function P (E, g, h) is a quasi-modular form of weight w, type m and depth l = deg

Y 1 P . Proof. Write P = l α=0 p α (Y 2 , Y 3 )Y α 1 ,
where p l = 0 and p α is an isobaric polynomial of weight w -2α and type

m -α (0 ≤ α ≤ l). Each function p α (g, h)E α is clearly an element of M α w,m . But it is plain that the sum of two non-zero elements of M w,m of different depths α 1 , α 2 is again an element of M w,m of depth max{α 1 , α 2 }. It follows at once that P (E, g, h) ∈ M deg Y 1 P w,m . Lemma 2.4
The functions E, g and h are algebraically independent over C.

Proof. Let P ∈ C[Y 1 , Y 2 , Y 3 ]
be such that P (E, g, h) = 0. We can write P = d w=0 m∈Z/(q-1)Z P w,m , where P w,m is an isobaric polynomial of weight w and type m. From P (E, g, h) = 0 and from Proposition 2.2 it follows that P w,m (E, g, h) = 0 for each pair (w, m). We fix w and m and write

P w,m = α≥0 p α (Y 2 , Y 3 )Y α 1 ,
where

p α ∈ C[Y 2 , Y 3
] is isobaric of weight w -2α and type m -α. Suppose that there is an index α 0 ≥ 1 such that p α 0 (g, h) = 0, and take α 0 maximal. We have

p α 0 (g, h)E α 0 = - 0≤α<α 0 p α (g, h)E α . (12)
Now, p α 0 (g, h)E α 0 is a non-zero quasi-modular form of depth exactly α 0 , and the right-hand side of ( 12) is a quasi-modular form of depth at most α 0 -1 (Lemma 2.3). By uniqueness of the depth, this is impossible. Hence we deduce that p α (g, h) = 0 for all α. Since now g and h are known to be algebraically independent over C, we get p α = 0 for all α, and thus P w,m = 0. This being true for all (w, m), we have P = 0, which completes the proof of the Lemma.

Lemma 2.5 Let f ∈ M l w,m , and let P f = l i=0 f i X i be its associated polynomial. Then, for all i, 0 ≤ i ≤ l, the function f i is an element of the space M ≤l-i w-2i,m-i , whose associated polynomial P f i is explicitly given by

P f i = l j=i j i f j X j-i .
In particular, we have

f l ∈ M w-2l,m-l .
Proof. The same proof as in [START_REF] Martin | Formes modulaires et périodes[END_REF]Lemma 119] works. First, an easy computation shows that for

N = α β γ δ ∈ Γ K , z ∈ Ω and i ∈ N with 0 ≤ i ≤ l,
the following equality holds:

f i (z) = l j=i j i (-γ) j-i (γ(z) + δ) i+j-w (det N ) m-j f j (N z).
Substituting z by L(z), where L = a b c d := N -1 , then gives the following transformation formula for f i (L(z)):

f i (L(z)) = (cz + d) w-2i (det L) -(m-i) l j=i j i f j (z) c cz + d j-i .
The assertions of the Lemma follow at once.

Lemma 2.6 For all integers w ≥ 0, l ≥ 0 and for all m ∈ Z/(q -1)Z, we have:

M ≤l w,m = 0≤i≤l M w-2i,m-i E i .
Proof. We already know from Lemma 2.4 that the sum 0≤i≤l is a direct sum, and that it is a subspace of M l w,m (Lemma 2.3). So it remains to prove the opposite inclusion. We argue by induction on l. If l = 0, the result is true (it is equality [START_REF] Martin | Formes modulaires et périodes[END_REF]). Suppose now that the result is true for l -1, where l ≥ 1. Let f ∈ M l w,m , f = 0, and let P f = l i=0 f i X i be its associated polynomial. Since P E = -π -1 X + E (see formula [START_REF] Yu | Modular functions and transcendence questions[END_REF]), we have P E l = (-π -1 X + E) l . Thus, since we have f l ∈ M w-2l,m-l by Lemma 2.5, we get

P f l E l = (-π) -l f l X l + terms of degree < l. Define now F := f -(-π) l f l E l .
The function F is a quasi-modular form of weight w and type m with associated polynomial

P F = P f -(-π) l P f l E l ,
whose degree is by construction at most l -1. Hence F ∈ M ≤l-1 w,m . By induction we can write now

F = l-1 i=0 F i E i , where F i ∈ M w-2i,m-i for all i, from which it follows f = F + (-π) l f l E l ∈ 0≤i≤l M w-2i,m-i E i . Proof of Theorem 1. Since C[E, g, h] ≤0 w,m = M w,m (see [6]), we obviously have C[E, g, h] ≤l w,m = 0≤i≤l M w-2i,m-i E i .
Therefore Theorem 1 follows from Lemmata 2.4 and 2.6.

Let L be a subfield of C containing K (recall that K = F q (T )). Denote by M ≤l w,m (L) the L-vector space of elements of M ≤l w,m having a t-expansion at infinity with coefficients in L, and define L[E, g, h] ≤l w,m similarly as before, i.e. as the L-vector space C[E, g, h] ≤l w,m ∩ L [E, g, h]. Theorem 1 gives rise to the following rationality result.

Corollary 2.7 Let L ⊂ C be any subfield of C containing K. Then M ≤l w,m (L) = L[E, g, h] ≤l w,m .
Proof. Since the functions E, g, h are elements of M ≤l w,m (L) (the coefficients of their t-expansions are in A, see [START_REF] Gekeler | On the coefficients of Drinfeld modular forms[END_REF]), we already have the inclusion

L[E, g, h] ≤l w,m ⊂ M ≤l w,m (L). ( 13
)
Extending now the scalars from L to C, we get

C[E, g, h] ≤l w,m ⊂ M ≤l w,m (L) ⊗ L C ⊂ M ≤l w,m (C) = M ≤l w,m .
By Theorem 1, these inclusions between C-vector spaces are in fact equalities, and thus

dim L M ≤l w,m (L) = dim C M ≤l w,m (L) ⊗ L C = dim C C[E, g, h] ≤l w,m = dim L L[E, g, h] ≤l w,m .
It then follows from ( 13) that L[E, g, h] ≤l w,m = M ≤l w,m (L).

Hyperderivatives and quasi-modular forms

In this Section, we study the hyperderivatives of quasi-modular forms. After recalling some basic properties on higher derivations and hyperderivatives in Section 3.1, we prove in Section 3.2 a very important proposition (Proposition 3.1), which gives the connection between the polynomial P f associated to a quasi-modular form f and the polynomial P Dnf associated to its nth hyperderivative D n f . Using this result, we can then prove Theorem 2 in Section 3.3, and the fact that the ideal (h) is hyperdifferential in Section 3.4. Finally, we determine in Section 3.5, up to a modular form, the hyperderivatives D p i E, D p i g for all integers i ≥ 0, and we prove a similar result for the

D p i h's.
In what follows, we put as usual

n i = 0 if 0 ≤ n < i and n i = (-1) i i-n-1 i if i ≥ 0, n < 0.
In particular, we have n 0 = 1 for all n ∈ Z. We also recall the following fact, which will be often used in the sequel: if r is any positive integer and n = n 0 + n 1 p r + . . . + n s p rs , i = i 0 + i 1 p r + . . . + i s p rs are two elements of N written in base p r , then one has in F p :

n i = n 0 i 0 n 1 i 1 • • • n s i s . (14)
The formula [START_REF] Pellarin | La structure différentielle de l'anneau des formes quasimodulaires pour SL 2 (Z)[END_REF] was first observed by Lucas in [START_REF] Lucas | Sur les congruences des nombres eulériens et des coefficients différentiels des fonctions trigonométriques, suivant un module premier[END_REF] (strictly speaking for r = 1) and can be easily derived by identifying the coefficient of X i in both sides of the following equality of polynomials of F p [X] (we expand of course each factor occurring in both sides with the help of the binomial formula):

(1 + X) n = (1 + X) n 0 (1 + X p r ) n 1 • • • (1 + X p rs ) ns .

Background on higher derivations and hyperderivatives

We begin by a short review on iterative higher derivations, following Matsumura [8, Section 27].

After [START_REF] Matsumura | Commutative ring theory[END_REF], a higher derivation (D n ) n∈N on a C-algebra R is a family of Clinear maps from R to itself, with D 0 the identity map, such that if f, g ∈ R, then the following Leibniz's rule holds

D i (f g) = i r=0 (D r f )(D i-r g). (15)
In particular, the following formula holds for f ∈ R and for all integers n, k ≥ 0 (see [8, p. 209])

D np k (f p k ) = (D n f ) p k . (16)
A higher derivation (D n ) n∈N is iterative [8, p. 209] if the following formula holds, for all integers i, j ≥ 0:

D i • D j = D j • D i = i + j i D i+j . (17)
It follows from ( 17) and ( 14) with r = 1 that the operators D n , n ≥ 0, are completely determined by the operators D p k for k ≥ 0. More precisely, if n = n s p s + • • • + n 1 p + n 0 is the representation of n in base p (so that 0 ≤ n i ≤ p -1 for all i), then the following formula holds:

D n = 1 n s ! • • • n 0 ! D ns p s • • • • • D n 1 p • D n 0 1 . (18)
We now review some basic properties of hyperderivatives. Let f : Ω → C be a locally analytic function and z ∈ Ω be a point. Recall that we have defined the hyperderivatives (D n f )(z) of f at z by the formula

f (z + ε) = n≥0 (D n f )(z)ε n , ε ∈ C, |ε| small (this corresponds to Definition 2.3 of [16]).
By Corollary 2.5 of [START_REF] Yu | Function field modular forms and higher derivations[END_REF], the family of operators (D n ) n∈N define a higher derivation on the C-algebra of locally analytic functions, and it is also wellknown (and easy to check) that this higher derivation is iterative.

Another important feature of the operators D n is their good behaviour under holomorphy: If f is holomorphic on Ω and n ∈ N, then the function D n f is also holomorphic on Ω by [START_REF] Yu | Function field modular forms and higher derivations[END_REF]Theorem 3.1], and if moreover f ∈ O, then it follows from [START_REF] Yu | Function field modular forms and higher derivations[END_REF]Lemma 3.6] that D n f ∈ O.

As explained in Section 1, it will be also convenient to work with the following operators

D n := 1 (-π) n D n (n ≥ 0). ( 19
)
The family (D n ) n is also an iterative higher derivation on O, and all the properties quoted above for the operators D n remain true for the operators D n .

Polynomial associated to a hyperderivative of a quasimodular form

We prove here Proposition 3.1. This proposition will be a crucial tool in the subsequent sections 3.3, 3.4 and 3.5.

Proposition 3.1 Let f ∈ M ≤l w,m be a quasi-modular form of weight w, type m and depth ≤ l, and let P f = l i=0 f i X i be its associated polynomial. Then, for all n ≥ 0, D n f is an element of M ≤l+n w+2n,m+n , with associated polynomial

P Dnf = n+l j=0 n r=0 n + w + r -j -1 r D n-r f j-r X j ,
with the convention that f i = 0 if i < 0 or i > l.

In the sequel, we will adopt the following notations. For γ = a b c d ∈ Γ K and z ∈ Ω, we write

X γ (z) := c/(cz + d), J γ (z) := (cz + d) 2 /(det γ).
If f : Ω → C is a holomorphic function, we will also denote by f • γ the function z → f (γ(z)). The proof of Proposition 3.1 requires three Lemmata.

Lemma 3.2 Let M, N, W be integers with N ≥ 0. Then we have

N i=0 (-1) i M N -i W + i -1 i = M -W N = (-1) N W + N -M -1 N .
Proof. We have in F q [[X]]:

(1 + X) M = i≥0 M i X i and (1 + X) -W = i≥0 (-1) i W + i -1 i X i ,
so by multiplying we derive

(1 + X) M -W = N ≥0 N i=0 (-1) i M N -i W + i -1 i X N .
Since on the other hand (1+X)

M -W = N ≥0 M -W N
X N , the result follows.

Lemma 3.3 Let f : Ω → C be a holomorphic function. For all n ≥ 1, z ∈ Ω and γ ∈ Γ K , we have

(D n (f • γ))(z) = (-1) n X γ (z) n n i=1 (-1) i n -1 n -i (X γ (z)J γ (z)) -i (D i f )(γ(z)). (20)
Proof. Let ε ∈ Ω be sufficiently small. We compute easily

γ(z + ε) = γ(z) + a cz+d ε 1 + X γ (z)ε = γ(z) + H, where H = J γ (z) -1 j≥1 (-1) j-1 X γ (z) j-1 ε j .
Let us remark that H tends to zero when ε tends to zero. We have for i ≥ 1

H i = J γ (z) -i n≥i (-1) n-i X γ (z) n-i n -1 n -i ǫ n , which gives f (γ(z + ε)) = f (γ(z) + H) = f (γ(z)) + i≥1 (D i f )(γ(z))H i = f (γ(z)) + n≥1 (-1) n ε n 1≤i≤n (-1) i n -1 n -i J γ (z) -i X γ (z) n-i (D i f )(γ(z)).
(D n f )(γ(z)) = (cz + d) w+2n (det γ) m+n n j=0 w + n -1 n -j X γ (z) n-j (D j ϕ γ )(z). ( 21
)
Proof. We argue by induction on n. For n = 0 the Lemma is trivially true.

Let us now suppose that n ≥ 1 and that the Lemma has been proved for all integers < n. First of all, rewriting the formula (20) from Lemma 3.3, we obtain

(D n f )(γ(z)) = J γ (z) n (D n (f • γ))(z) + (-1) n-1 n-1 i=1 (-1) i n -1 n -i X γ (z) n-i J γ (z) n-i (D i f )(γ(z)).
Next, using Lemma 3.4 for (D i f )(γ(z)) (induction hypothesis) and exchanging the priority of the summations thus obtained, we get

(D n f )(γ(z)) = J γ (z) n (D n (f • γ))(z) + (cz + d) w (det γ) -m n-1 j=0 X γ (z) n-j J γ (z) n (D j ϕ γ )(z) × n-1 i=j (-1) n-1+i n -1 n -i w + i -1 i -j . (22)
In order to compute D n (f • γ) in this formula, we write

(f • γ)(z) = (det γ) -m (cz + d) w ϕ γ (z)
and apply Leibniz's rule [START_REF] Swinnerton-Dyer | On ℓ-adic representations and congruences for coefficients of modular forms. Modular Functions of one Variable III[END_REF]. We find

(D n (f • γ))(z) = D n (det γ) -m (cz + d) w ϕ γ (z) = (det γ) -m n j=0 D n-j (cz + d) w (D j ϕ γ )(z) = (det γ) -m (cz + d) w n j=0 w n -j X γ (z) n-j (D j ϕ γ )(z),
the last line coming from the following formula, which is easy to check:

D i (cz + d) w = w i X γ (z) i (cz + d) w (i ≥ 0).
Substituting in (22) the expression obtained for D n (f • γ), we derive

(D n f )(γ(z)) = (cz + d) w (det γ) m J γ (z) n n j=0 X γ (z) n-j (D j ϕ γ )(z) w n -j + (-1) n-1 n-1-j i=0 (-1) i+j n -1 n -i -j w + i + j -1 i ,
which gives, using Lemma 3.2 with N = n -j, M = n -1 and W = w + j,

(D n f )(γ(z)) = (cz + d) w+2n (det γ) m+n n j=0 w + n -1 n -j X γ (z) n-j (D j ϕ γ )(z).
This is the desired formula for n.

Proof of Proposition 3.1. Let γ be an element of Γ K and let ϕ γ be defined as in Lemma 3.4. Then we have, for all z ∈ Ω:

ϕ γ (z) = l i=0 f i (z)X γ (z) i . (23)
The following formula holds for all i, r ≥ 0, and is easy to check:

D r (X i γ ) = (-1) r i + r -1 r X i+r γ . (24)
Thus, Leibniz's formula [START_REF] Swinnerton-Dyer | On ℓ-adic representations and congruences for coefficients of modular forms. Modular Functions of one Variable III[END_REF] gives, for all j such that 0 ≤ j ≤ n,

D j (f i X i γ ) = j r=0 (-1) j-r i + j -r -1 j -r X i+j-r γ D r f i . (25)
Using now successively (21), ( 23) and (25), we obtain

(det γ) m+n (cz + d) w+2n (D n f )(γ(z)) = n j=0 k + n -1 n -j X γ (z) n-j l i=0 (D j (f i X i γ ))(z) = 0≤j≤n 0≤r≤j 0≤i≤l (-1) j-r k + n -1 n -j i + j -r -1 j -r X γ (z) n+i-r (D r f i )(z) = 0≤u≤n+l 0≤r≤n r≤j≤n (-1) j-r w + n -1 n -j u -n + j -1 j -r X γ (z) u (D r f u-n+r )(z).
This formula shows already that (det γ)

m+n (cz + d) -(k+2n) (D n f )(γ(z)) is a polynomial in X γ (z) of degree ≤ n + l with coefficients in O, whence D n f ∈ M ≤l+n w+2n,m+n
. Finally, we get the desired formula for P Dnf by making the change of summation index r ′ = n -r in the last formula, and then using Lemma 3.2 with

N = r ′ , M = w + n -1, W = u -r ′ .

Proof of Theorem 2

As a first Corollary of Proposition 3.1, we will now derive Theorem 2. We will need the following Lemma, which will also be useful later. Lemma 3.5 Let f : Ω → C be an element of the ring O having the texpansion

f (z) = n≥0 a n t n , (26)
where t = t(z). Then, for all integers i ≥ 1, one has

(D i f )(z) = n≥2 b i,n t n , where b i,n = n-1 r=1 (-1) i+r n -1 r i 1 ,...,ir≥0 q i 1 +•••+q ir =i 1 d i 1 • • • d ir a n-r .
Proof. Fix z ∈ Ω with |z| i sufficiently large, so that f has a t-expansion (26), and let ε be any element of the field K ∞ . Then |z + ε| i = |z| i , and thus

f (z + ε) = n≥0 a n t(z + ε) n . (27)
Now, putting as usual t := t(z) and using the formula (6), we have, for |ε| sufficiently small:

t(z + ε) n = 1 e C (πz) + e C (πε) n = t n 1 + e C (πε)t n = r≥0 (-1) r r + n -1 r (e C (πε)) r t r+n = r≥0 (-1) r r + n -1 r t r+n i≥0 π q i ε q i d i r = r≥0 (-1) r r + n -1 r t r+n i≥0 α r,i π i ε i , (28)
where α r,i = i 1 ,...,ir≥0

q i 1 +•••+q ir =i 1 d i 1 • • • d ir if r ≥ 1 and i ≥ 0,
and where α 0,i = 1 for i = 0 0 for i = 0.

Substituting (28) in ( 27) and exchanging the order of the summations, we get

f (z + ε) = i≥0 r≥0 n≥0 (-1) r a n α r,i r + n -1 r t r+n π i ε i .
Since on the other hand we have (by virtue of (19))

f (z + ε) = i≥0 (D i f )(z)(-1) i π i ε i ,
we obtain

(D i f )(z) = r≥0 n≥0 (-1) r+i r + n -1 r α r,i a n t r+n (i ≥ 0). ( 29
)
Let now i ≥ 1. For r = 0 we have α r,i = 0, and for r ≥ 1 and n = 0 we have r+n-1 r = 0. Thus, when i ≥ 1, we can take r ≥ 1 and n ≥ 1 in the formula (29). This gives the Lemma.

We are now in position to prove Theorem 2.

Proof of Theorem 2. By Corollary 2.7, it suffices to prove that D n ( M ≤l w,m (K)) ⊂ M ≤l+n w+2n,m+n (K). But if f ∈ M ≤l w,m (K), then the function D n f is an element of M ≤l+n w+2n,m+n by Proposition 3.1, and the coefficients of its t-expansion all lie in K after Lemma 3.5. Hence f ∈ M ≤l+n w+2n,m+n (K) as claimed.

The principal ideal (h) is hyperdifferential.

As a second application of Proposition 3.1, we prove here the following Proposition, which will be very useful in this text.

Proposition 3.6 The ideal (h) is hyperdifferential.

Proof. Applying Proposition 3.1 with f = h ∈ M q+1,1 we see that, for all n ≥ 0, the polynomial P Dnh associated to D n h is:

P Dnh = n j=0 n + q j (D n-j h)X j . (30)
Now, the function D n-j h vanishes at infinity if 0 ≤ j < n after Lemma 3.5, and of course the same holds for j = n because ν ∞ (h) = 1. Since h does not vanish on Ω (by formula (5.14) of [START_REF] Gekeler | On the coefficients of Drinfeld modular forms[END_REF]), the functions f j := D n-j h/h (0 ≤ j ≤ n) are all holomorphic on Ω and at infinity; consequently, they belong to O. We deduce from (30) that for all n ≥ 0, z ∈ Ω and γ ∈ Γ K :

D n h h (γ(z)) = (cz + d) 2n (det γ) -n n j=0 n + q j f j (z)X γ (z) j , with f j ∈ O.
In other words, the function D n h/h is a quasi-modular form (of weight 2n, type n and depth ≤ n), and therefore belongs to the polynomial ring C[E, g, h] by virtue of Theorem 1.

Hyperderivatives of E, g, h.

We end this Section 3 by establishing Proposition 3.7 below, which is another consequence of Proposition 3.1. Roughly speaking, this Proposition determines, up to a modular form, the functions D p i E, D p i g and D p i h for all integers i ≥ 0. As we will see in Section 4, this result, together with some technical computations, yields a practical way to compute explicitly D p i E, D p i g and D p i h when p i ≤ q 2 . Proposition 3.7 will also be needed in the proof of Lemma 6.2. Proposition 3.7 (i) For every integer i ≥ 0, we have

D p i E -E p i +1 ∈ M 2p i +2,p i +1 .
(ii) For every integer i ≥ 1, we have

D p i g = 0 if p i < q D p i g -E p i g ∈ M 2p i +q-1,p i if p i ≥ q.
(iii) For every integer i ≥ 0, we have

D p i h -E p i h -E q (D p i -q h) ∈ M 2p i +q+1,p i +1 ,
where we have used the convention D n h = 0 when n < 0.

To prove Proposition 3.7, we will need three Lemmata.

Lemma 3.8 For every integer i ≥ 0, we have

D p i -1 E = E p i .
Proof. To prove this Lemma, we use the following formula (see [6, page 686]), valid for every z ∈ Ω:

E(z) = 1 π a∈A a monic b∈A a az + b .
Since the series are here uniformly convergent (in the above order) on every open ball contained in Ω, we find, by [16, Proposition 2.6] and by using the formula

D n a az + b = (-1) n a az + b n+1 (n ≥ 0) which follows from (24), (D p i -1 E)(z) = 1 π a monic b∈A (-1) p i -1 a az + b p i = (-π) p i -1 E(z) p i .
Thus, we get D p i -1 E = E p i as annouced. Lemma 3.9 We have D n g = 0 for every integer n such that 2 ≤ n < q.

Proof. By [6, formula (6.8)] we have, for all z ∈ Ω:

g(z) = π 1-q (T q -T ) (a,b)∈A 2 (a,b) =0 1 (az + b) q-1 .
Using the formula (24), we deduce that for all n ≥ 0 the following holds:

(D n g)(z) = π 1-q (T q -T )(-1) n n + q -2 n (a,b)∈A 2 (a,b) =0 a n (az + b) n+q-1 .
Since now for 2 ≤ n ≤ q -1 we have in F p (for instance by ( 14) with p r = q) n+q-2 n = 0, we get the Lemma.

Lemma 3.10 For every integer i ≥ 0 such that p i ≥ q, we have

D p i -q h ∈ M 2p i -q+1,p i .
Proof. Applying Proposition 3.1 we get

P D p i -q h = p i -q j=0 p i j (D p i -q-j h)X j .
But in F p we have (see [START_REF] Pellarin | La structure différentielle de l'anneau des formes quasimodulaires pour SL 2 (Z)[END_REF])

p i j = 1 if j = 0 0 if 0 < j < p i .
Therefore, we obtain P D p i -q h = D p i -q h. Thus, deg(P D p i -q h ) = 0 and it follows that D p i -q h is a modular form, which proves the Lemma.

Proof of Proposition 3.7. (i) For i = 0 the assertion (i) is clearly true by the relations (2), so we may assume i ≥ 1. Since P E = -π -1 X +E, we first note from Proposition 3.1 that the polynomial P D p i E associated to D p i E equals

P D p i E = - 1 π X p i +1 + p i j=0 p i + 1 j (D n-j E)X j .
But in F p we have

p i + 1 j = 1 if j = 0, j = 1 or j = p i 0 if 1 < j < p i .
Therefore, by Lemma 3.8,

P D p i E = - 1 π X p i +1 + EX p i + (D p i -1 E)X + D p i E = - 1 π X p i +1 + EX p i + (-π) p i -1 E p i X + D p i E.
Arguing now as in the proof of Lemma 2.6, if we define f = (-π) -p i D p i E -E p i +1 , we get:

P f = (-π) -p i P D p i E -(- 1 π X + E) p i +1 = (-π) -p i P D p i E -((-π) -p i X p i + E p i )(- 1 π X + E) = f.
Since deg P f = 0, it follows that f is a modular form, which proves (i).

(ii) Let i ≥ 1 be an integer. If p i < q, then the assertion (ii) is true by Lemma 3.9, so we may now assume p i ≥ q. Applying proposition 3.1, we have

P D p i g = p i j=0 p i + q -2 j (D p i -j g)X j . ( 31
)
For every integer j such that 0 ≤ j ≤ p i , write p i -j = αq + β with α ≥ 0 and 0 ≤ β ≤ q -1. If β ≥ 2, then D β g = 0 by Lemma 3.9, and thus

D p i -j g = D αq+β g = αq + β β D αq+β g = D αq (D β g) = 0. If β = 1, one has p i + q -2 j = p i + q -2 αq + q -1 = 0,
and if β = 0, one finds easily (recall that q is a power of p)

p i + q -2 j = p i + q -2 αq + q -2 = 1 if α = 0 or α = p i /q 0 if 0 < α < p i /q.
Putting these remarks together, (31) leads to

P D p i g = gX p i + D p i g.
Thus, by the same argument as before, D p i g -E p i g is a modular form, and so is an element of M 2p i +q-1,p i .

(iii) For every integer i ≥ 0 we have, by Proposition 3.1,

P D p i h = p i j=0 p i + q j (D p i -j h)X j . ( 32
)
In the case p i < q one easily computes

p i + q j = 0 if 0 < j < p i 1 if j = 0 or j = p i . If p i = q, then p i + q j =    1 if j = 0 2 if j = p i 0 if 0 < j < p i ,
and if p i > q, we find

p i + q j = 1 if j ∈ {0, q, p i } 0 if 0 < j < p i , j = q.
Hence, we deduce in the three cases from (32):

P D p i h = D p i h + (D p i -q h)X q + hX p i .
If we now set f = (-π) -p i D p i h -E p i h, then we have

P f = (-π) -p i P D p i h -(- 1 π X + E) p i h = (-π) -p i D p i h + (D p i -q h)X q + hX p i -(-π) -p i hX p i -E p i h = f + (-π) -p i (D p i -q h)X q .
Since the function D p i -q h is a modular form by Lemma 3.10, it follows that f -E q (D p i -q h) is also modular. This means that

D p i h -E p i h -E q (D p i -q h) ∈ M 2p i +q+1,p i +1
and part (iii) of Proposition 3.7 is proved.

Remark. Proposition 3.7 may be applied to give another proof of Proposition 3.6. Due to the importance of the latter result in this text, we give here all the details.

Alternative proof of Proposition 3.6. By Proposition 3.7 Part (iii), the function

D p i h-E p i h-E q (D p i -q h
) is a modular form for every i ≥ 0. By Lemma 3.5, this modular form vanishes at infinity, hence belongs to the ideal (h).

From this we deduce:

D p i h ∈ (h) for p i < q D p i h -E q D p i -q h ∈ (h) for p i ≥ q.
By induction on i ≥ 0, we now show that D n h ∈ (h) for all 0 ≤ n < p i . The first property above, together with the formulas ( 15) and ( 18), easily imply that D n h ∈ (h) for all n < q, so the assertion is true for i such that 1 ≤ p i ≤ q. Supposing now that the property holds for an integer i such that p i ≥ q, we have in particular that D p i -q h ∈ (h), which implies D p i h ∈ (h). By ( 15), ( 18) and the induction hypothesis, we then get D n h ∈ (h) for all 0 ≤ n < p i+1 , i.e. the assertion for i + 1.

Explicit formulas for D

i when 1 ≤ i ≤ q 2
In this (more technical) Section, we compute explicitly D n E, D n g and D n h as polynomials in E, g, h, when n ≤ q 2 is a power of p. The formulas obtained allow to compute (at least in principle) D n f for any f ∈ C[E, g, h] and any n such that 0 ≤ n ≤ q 2 , thanks to Leibniz's rule [START_REF] Swinnerton-Dyer | On ℓ-adic representations and congruences for coefficients of modular forms. Modular Functions of one Variable III[END_REF] and the formula (18).

We summarise the main results in the following Theorem. Here, e is the integer such that q = p e . Theorem 4.1 Let i, n be integers. The following formulas hold.

(i) D n E = E n+1 if 0 ≤ n < q. (ii) D 1 g = -(Eg + h) and D n g = 0 if 2 ≤ n < q. (iii) D n h = E n h if 0 ≤ n < q. (iv) D p i E = E p i +1 + 1 d p i-e 1 g p i-e -1 h p i-e +1 if q ≤ p i < q 2 . (v) D p i g = E p i g if q ≤ p i < q 2 . (vi) D p i h = E p i h+ 1 d p i-e -1 1 E q g p i-e -1 h p i-e -1 d p i-e 1 g p i-e h p i-e +1 if q ≤ p i < q 2 . (vii) D q 2 E = E q 2 +1 + 1 d q 1 g q-1 h q+1 + 1 d 2 g 2q h 2 . (viii) D q 2 g = E q 2 g -d 1 d 2 g q+1 h q + ( 1 d q-1 1 - d 2 1 d 2 )h 2q-1 . (ix) D q 2 h = E q 2 h + 1 d q-1 1 E q g q-1 h q -1 d 2 g 2q+1 h 2 -( d 1 d 2 + 1 d q 1
)g q h q+1 .

The proof of Theorem 4.1 requires several technical Lemmata. Lemma 4.2 Let i, j be integers such that q < p i < q 2 and q < p j ≤ q 2 . We have the following t-expansions.

(i) E = t + t q 2 -2q+2 + • • • (ii) g = 1 -d 1 t q-1 -d 1 t q 3 -2q 2 +2q-1 + • • • (iii) h = -t -t q 2 -2q+2 + • • • (iv) D q E = 1 d 1 t 2 + t q+1 + • • • (v) D q g = t q + • • • (vi) D q h = -1 d 1 t 2 -t q+1 + • • • (vii) D p i E = 1 d p i-e +1 1 t p i-e +1 + • • • (viii) D p i g = O(t p i-e +1 ) (ix) D p i h = -1 d p i-e +1 1 t p i-e +1 + • • • (x) D p j -q h = -1 d p j-e -1 1 t p j-e + • • • (xi) D q 2 E = 1 d 2 t 2 + 1 d q 1 t q+1 + • • • (xii) D q 2 g = d 1 d 2 t q -1 d q-1 1 t 2q-1 + • • • (xiii) D q 2 h = -1 d 2 t 2 -1 d q 1 t q+1 + • • • (iii) M 2q 2 +2,q 2 +1 =    g q-1 h q+1 , g 2q h 2 if q ≥ 4 g 2 h 4 , g 6 h 2 , g 10 if q = 3 gh 3 , g 4 h 2 , g 7 h, g 10 if q = 2.
(iv) If q ≤ p i < q 2 , then

M 2p i +q-1,p i = g p i-e +1 h p i-e if (p, q, i) = (2, 2, 1) g 5 , g 2 h if (p, q, i) = (2, 2 , 1).
(v) M 2q 2 +q-1,q 2 = g 2q+2 h, g q+1 h q , h 2q-1 .

(vi) If p ≤ p i < q, then M 2p i +q+1,p i +1 = {0} if (p, q, i) = (2, 4, 1) g 3 if (p, q, i) = (2, 4 , 1)
(vii) If q ≤ p i < q 2 , then

M 2p i +q+1,p i +1 =        g 7 , g 2 h 3 if (p, q, i) = (2, 4, 3) g 5 , gh 2 if (p, q, i) = (3, 3, 1) g 7 , g 4 h, gh 2 if (p, q, i) = (2, 2, 1) g p i-e h p i-e +1 otherwise. (viii) M 2q 2 +q+1,q 2 +1 =    g q h q+1 , g 2q+1 h 2 if q ≥ 4 g 3 h 4 , g 7 h 2 , g 11 if q = 3 g 2 h 3 , g 5 h 2 , g 8 h, g 11 if q = 2. (ix) M 2p i -q+1,p i = g p i-e -1 h p i-e if q < p i < q 2 g q-1 h q , g 2q h if p i = q 2 .
Proof. We know that a basis of M w,m consists of monomials g α h β , where α and β are non-negative integers satisfying α(q -1) + β(q + 1) = w β ≡ m (mod q -1).

It is now a simple exercise to solve this system of equations for each of the cases (i) to (ix). We leave it to the reader.

Proof of Theorem 4.1.

(1) We first prove the formulas involving the function E, and we begin by showing (i). Let n be an integer such that 0 ≤ n < q, and suppose first that n has the form n = p i . If p ≤ p i < q and (p, q, i) = (2, 4, 1), Proposition 3.7 (i) and Lemma 4.3 (i) yield immediately

D p i E = E p i +1 . If (p, q, i) = (2, 4 , 1)
, the same Proposition and the same Lemma now imply the

existence of λ ∈ C such that D p i E -E p i +1 = λg 2 . But since D p i E -E p i +1
vanishes at infinity (Lemma 3.5), whereas g does not, we have λ = 0 and D p i E = E p i +1 again. Thus, the formula (i) is proved when n has the form n = p i (for i = 0 this follows from ( 2)). It is easy to deduce from this, by induction on α, that the formula (i) also holds when n = αp i with 0 ≤ α ≤ p -1 and 1 ≤ p i < q. For if the formula is true for (α -1)p i (where α > 1), then

αD αp i E = D p i • D (α-1)p i E = D p i (E (α-1)p i • E) = E (α-1)p i D p i E + D p i (E (α-1)p i )E = E αp i +1 + (D 1 E α-1 ) p i E = E αp i +1 + ((α -1)E α-2 (D 1 E)) p i E = αE αp i +1 .
Now, if n is arbitrary in the range 1 ≤ n < q, we argue by induction on n as follows. Define i ≥ 0 as the integer such that p i ≤ n < p i+1 , and write n = αp i + β, where 1 ≤ α ≤ p -1 and 0 ≤ β < p i . By induction hypothesis we may assume that D β E = E β+1 , from which we deduce

D n E = D β (D αp i E) = D β (E αp i +1 ) = E αp i D β E = E αp i +β+1 = E n+1 .
To prove (iv), we use Proposition 3.7 (i) and Lemma 4.3 (ii). In the case (p, q, i) ∈ {(2, 4, 3), (3, 3, 1), (2, 2, 1)}, we find that there exists λ ∈ C such that D p i E -E p i +1 = λg p i-e -1 h p i-e +1 . Since by Lemma 4.2

D p i E = 1
d p i-e +1 1 t p i-e +1 + • • • and g = 1 + • • • , h = -1 + • • • , it follows that λ = 1/d p i-e 1
, which gives (iv). If (p, q, i) = (2, 4, 3), the same method shows that there exist λ, µ ∈ C such that D p i E -E p i +1 = λg 6 + µgh 3 . The comparison of the t-expansions in both sides implies µ = 0 and λ = 1/d p i-e 1 , from which (iv) follows again. The other cases are similar.

To establish (vii), we argue exactly in the same way, but using now Lemmata 4.3 (iii) and 4.2 (xi).

(2) Let us now prove the formulas involving the function g. The formulas (ii) have already been proved (see [START_REF] Bosser | Indépendance algébrique de valeurs de séries d'Eisenstein[END_REF] and Lemma 3.9), so we have to prove (v) and (viii). The same method as the one used in the proof of (iv) above applies: We simply use Part (ii) of Proposition 3.7 instead of Part (i), Lemma 4.2 for the t-expansions, and Parts (iv), (v) of Lemma 4.3 to know a basis of the space M 2p i +q-1,p i .

(3) Finally, we prove the formulas (iii), (vi) and (ix). Let n be an integer such that 0 ≤ n < q, and first suppose that n has the form n = p i . If p ≤ p i < q and (p, q, i) = (2, 4, 1), then Proposition 3.7 (iii) and Lemma 4.3 (vi) yield at once the formula (iii) for n = p i . If (p, q, i) = (2, 4, 1), arguing similarly and using the fact that D p i h -E p i h vanishes at infinity, we obtain again (iii). Thus, (iii) is proved when n = p i (if p i = 1 it follows from (2)). Arguing now as in the proof of (i), we first deduce from this that (iii) remains valid when n = αp i < q with 0 ≤ α ≤ p -1, and then, as before, that the formula is true for all n with 1 ≤ n < q.

To prove (vi) and (ix), we apply Proposition 3.7 again: The function

D p i h -E p i h -E q (D p i -q h)
is thus a modular form belonging to M 2p i +q+1,p i +1 for all i ≥ e. But this function has a t-expansion which is clearly an O(t 2 ) by Lemma 3.5. So, using Lemma 4.3 (parts (vii) and (viii)), we see that there exist elements λ, µ ∈ C such that

D p i h -E p i h -E q (D p i -q h) = λg p i-e h p i-e +1 + µg 2q+1 h 2 ,
where µ = 0 if p i < q 2 . With the help of Lemma 4.2, we can compute the first terms of the t-expansions on both sides. We find λ = -1/d p i-e 1 if p i < q 2 and, if

p i = q 2 , λ = -( d 1 d 2 + 1 d q 1
), µ = -1/d 2 . To complete the proof, it remains to express D p i -q h as a polynomial in E, g, h. If p i = q, we have D p i -q h = h and we get immediately (vi). If p i > q, then D p i -q h = O(t 2 ) and therefore, by Lemma 4.3 (ix) and Lemma 3.10, there exists ν ∈ C such that D p i -q h = νg p i-e -1 h p i-e . The t-expansion of D p i -q h given by Lemma 4.2

(x) now yields ν = 1/d p i-e -1 1
, thus proving (vi) and (ix) .

5 Proofs of Theorem 3 and Corollary 4.

We consider the subfields of F ( 6)

F k = k i=0 Ker(D p i ),
with k ∈ N and we set F -1 = F . Clearly, F k-1 ⊃ F k for all k ∈ N and by [START_REF] Swinnerton-Dyer | On ℓ-adic representations and congruences for coefficients of modular forms. Modular Functions of one Variable III[END_REF], one sees that

D p k (f g) = (D p k f )g + f (D p k g) for f ∈ F k-1 , g ∈ F .
In particular, the restriction of D p k to F k-1 is a derivation and thus, if k ≥ 0, x, y ∈ F k-1 and y = 0, then:

D p k x y = y(D p k x) -x(D p k y) y 2
(see Proposition 2.2 of [START_REF] Matzat | Iterative differential equations and the Abhyankar conjecture[END_REF] for further general properties of these subfields).

We also need to work with two families of subsets of F × . If k ∈ N, we write

Ψ k = {f ∈ F × such that (D p j f )/f ∈ M for all 0 ≤ j ≤ k},
and we define

G k := Ψ k ∩ F × k-1 . We also write Ψ -1 = G -1 = F × . Clearly, Ψ k-1 ⊃ Ψ k for all k ∈ N and G k ⊂ Ψ k for all k ≥ -1. Moreover, G k-1 ⊃ F × k-1 ⊃ G k for all k ∈ N.
In Lemma 5.1 of Section 5.1 we will prove that the G k 's and the Ψ k 's are multiplicative groups.

Plan of Section 5. In order to describe the content of this part, it is worth outlining the structure of proof of Theorem 3 we intend to develop.

By Proposition 3.6, h ∈ ∩ ∞ k=0 Ψ k , hence C × h Z ⊂ ∩ ∞ k=0 Ψ k .
We also observe that Theorem 3 is equivalent to:

∞ k=0 Ψ k = C × h Z .
We will prove that for all k ≥ 0:

Ψ k = F × k h Z . (33)
This result, combined with the results of Section 5.2 (where we study texpansions in F × ), will easily yield Theorem 3.

The proof of the equalities (33) is by induction on k ≥ -1; the case k = -1 being trivial, let us assume that Ψ k-1 = F × k-1 h Z and let f be a quasi-modular form in Ψ k . We know by induction hypothesis that there exists n ∈ Z such that f h n is in F × k-1 so that to explain the idea behind the induction step we may restrict our attention to

f ∈ G k = Ψ k ∩ F × k-1 . Since h s ∈ ∩ ∞ k=0 Ψ k , for all s ∈ Z we have D p k (f h -sp k ) = a s f h -sp k
with a s a quasi-modular form of weight 2p k in F × k-1 . The depth of a s is in general ≤ p k . In Section 5.3 we determine some properties of the associated polynomials P Dnf ; by means of them, we observe that since a s ∈ F × k-1 , its depth is divisible by p k . Hence, l(a s ) ∈ {0, p k }.

In section 5.4 we characterise the integers s such that l(a s ) < p k ; their set is non-empty. For those integers s, a s is a modular form because then l(a s ) = 0.

In Section 5.5, we observe that modular forms in F × k-1 are very easy to describe (contrarily to quasi-modular forms). Thanks to these considerations, we see that if s is such that a s is modular (and we will find such integers), then a s = 0 hence proving (33) for the index k. Finally, Sections 5.6 and 5.7 serve to end the proofs of the Theorem and Corollary in question.

This sketch of proof will hopefully help the reader to access the present part of the text, but it must be considered carefully, as several technical problems arise while taking at the place of f quasi-modular, a more general element of F × .

Preliminaries.

We begin with the following Lemma. Proof. This is clear for k = -1. Let us write Θ n = {f ∈ F × such that (D j f )/f ∈ M for all 0 ≤ j ≤ n}, so that Θ p j = Ψ j for all j ∈ N. We will prove by induction on n that for all n ≥ 0, Θ n is a subgroup of F × (this will imply the required property for the Ψ k 's). It is obviously true for n = 0, so we suppose now that n ≥ 1 and that Θ j is a subgroup of F × for all j ≤ n -1. We will show that if x, y ∈ Θ n , then xy -1 ∈ Θ n .

Let x, y be elements of Θ n . For all j = 0, . . . , n we have D j x = α j x and D j y = β j y, with α j , β j ∈ M . Now, since Θ j is a subgroup of F × for all j = 0, . . . , n -1, we have xy -1 ∈ Θ j , and therefore there exists γ j ∈ M with D j (xy -1 ) = γ j xy -1 . Thus, it only remains to show that D n (xy -1 )/(xy -1 ) ∈ M . But we have

α n x = D n x = D n (xy -1 • y) = yD n (xy -1 ) + n-1 j=0 D j (xy -1 )D n-j y = yD n (xy -1 ) + n-1 j=0 γ j β n-j x,
from which it follows that

D n (xy -1 )/(xy -1 ) = α n - n-1 j=0 γ j β n-j ∈ M .
Hence, xy -1 ∈ Θ n and Θ n is a multiplicative subgroup of F × for all n. The property for G k follows because

G k = Ψ k ∩ F × k-1 for all k ≥ 0.

t-expansions.

We obviously have an embedding F ⊂ C((t)). More generally, we have the following Lemma.

Lemma 5.2 For all k ≥ 0 we have an embedding:

F k-1 ⊂ C((t p k )).
Proof. We check that for all k ≥ 0, each element x of F k-1 has a convergent t-expansion (in a neighborhood of 0):

x = m≥m 0 c m t p k m , c m ∈ C.
We argue by induction on k, the case k = 0 being clear. Let k be a strictly positive integer and let x be in F k-1 . We then have x ∈ F k-2 , and by induction hypothesis:

x = m≥m 0 c m t p k-1 m . (34)
By ( 16) and since D 1 t = t 2 , we have m+1) . Thus, we get

D p k-1 (t p k-1 m ) = (D 1 t m ) p k-1 = mt p k-1 (
D p k-1 x = m≥m 0 mc m t (m+1)p k-1 ,
and D p k-1 x = 0 if and only if mc m = 0 for all m. This implies p | m if c m = 0, and hence x ∈ C((t p k )) by (34).

Lemma 5.3 Let f be a non-zero element of F , and let us suppose that there exists a p-adic integer

s 1 + s 2 p + • • • + s k+1 p k + • • • and a sequence (f k ) k∈N ⊂ F × such that, for all k, f k ∈ F k and f = h s 1 +s 2 p+•••+s k+1 p k f k . ( 35
)
Then there exists n ∈ Z and c ∈ C × with f = ch n .

Proof. Since F k ⊂ C((t p k+1 )) (Lemma 5.2), we can write

f k = b k t l k p k+1 + ( terms of higher degree in t ), k = -1, 0, . . . ,
with b k ∈ C × and l k ∈ Z. By (35), we have

f k = f k-1 h -s k+1 p k , k ≥ 0.
Since ν ∞ (h) = 1, we obtain that:

l k p k+1 = l k-1 p k -s k+1 p k , k ≥ 0,
from which we deduce

l k = l k-1 p - s k+1 p , k ≥ 0. (36)
First, let us suppose that ν ∞ (f ) = 0. Then l -1 = 0, and (36) implies l 0 = l 1 = • • • = 0 because for all k, l k and s k are integers, and 0 ≤ s k+1 /p < 1 by hypothesis. This implies

s 1 = s 2 = • • • = 0 and f = f 0 = f 1 = f 2 = • • • . Hence: f ∈ ∞ k=0 F k = C, which implies f = b -1 = c ∈ C × .
Let us now suppose that ν ∞ (f ) = 0. Since ν ∞ (h) = 1, if we set m = -ν ∞ (f ) and f = h m f , then we have ν ∞ ( f ) = 0. Let us now observe that there exist two sequences of integers (a i ) i≥1 and (m i ) i≥0 unique with the property that for all k ≥ 0, a k ∈ {0, . . . , p -1} and

m + s 1 + s 2 p + • • • + s k+1 p k = a 1 + a 2 p + • • • + a k+1 p k + m k p k+1 .
The expression on the right hand side is nothing but the p-adic expansion of the p-adic integer m + ∞ k=0 s k+1 p k , truncated to the k-th digit. Hence, for all k ≥ 0:

f = h m f = h m+s 1 +s 2 p+•••+s k+1 p k f k = h a 1 +a 2 p+•••+a k+1 p k (h m k p k+1 f k ) = h a 1 +a 2 p+•••+a k+1 p k fk , with fk := h m k p k+1 f k . Since h m k p k+1 ∈ F × k and f k ∈ F × k , we get fk ∈ F × k
for all k. Now, since ν ∞ ( f ) = 0, we can use the first part of the proof and we obtain f = h m f = c ∈ C × , whence f = ch -m .

Computing depths of derivatives.

Terminology. In the following, we will say that a function

Φ : Γ K × Ω → C is an error term of degree ≤ l if for all γ = a c b d ∈ Γ K and z ∈ Ω: Φ(γ, z) = l j=0 i∈Z/(q-1)Z f i,j (z) det(γ) -i X γ (z) j ,
where the f i,j 's are holomorphic functions on Ω which do not depend on γ.

To simplify matters, the notation R(l) is reserved to designate any function which is an error term of degree ≤ l. Hence, two different functions may be denoted by the same symbol.

Let us denote by M l w, * the subset of M whose elements are linear combinations of quasi-modular forms f (i) ∈ M ≤l w,i (i ∈ Z/(q -1)Z) such that there exists i with 0 = f (i) ∈ M l w,i ; notice that this is not a C-vector space. By Theorem 1, M l w, * is also the subset of C[E, g, h] of the polynomials P which are isobaric of weight w, such that deg E (P ) = l. If P is an element of C[E, g, h], we will also write l(P ) = deg E (P ), and we will refer to it as to the depth of P . Proposition 5.4 Let f be a quasi-modular form of weight w, depth l and type m (satisfying (8)), let n ≥ 1 be an integer. Then, for γ ∈ Γ K , we have:

(D n f )(γ(z)) = (cz + d) w+2n det(γ) -m-n × w -l + n -1 n f l (z)X γ (z) l+n + R(l + n -1) .
Let f be a non-zero element of M l w, * . Then:

(D n f )(γ(z)) = (cz + d) w+2n (det γ) -n × w -l + n -1 n Π γ (z)X γ (z) l+n + R(l + n -1) ,
where for all γ ∈ Γ K , Π γ is a non-zero holomorphic function Ω → C which does not depend on n.

The depth of D n f is < l + n if and only if

w -l + n -1 n ≡ 0 (mod p). (37)
Proof. The first formula of the Proposition is a simple application of Proposition 3.1. We now prove the second part of the Proposition and we begin by explaining how to construct Π γ . Let us consider a non-zero element f of M l w, * . We can write f = i∈Z/(q-1)Z

f (i) (38)
with f (i) quasi-modular of weight w, depth l i and type i. For γ ∈ Γ K we have:

f (i) (γ(z)) = (cz + d) w det(γ) -i l i j=0 f (i) j (z)X γ (z) j with f (i)
j quasi-modular for all i, j (Lemma 2.5). Thus, we can write:

f (γ(z)) = (cz + d) w X γ (z) l i * det(γ) -i f (i) l (z) + (cz + d) w q-2 i=0 det(γ) -i min{l i ,l-1} j=0 f (i) j (z)X γ (z) j = (cz + d) w X γ (z) l i * det(γ) -i f (i) l (z) + R(l -1) ,
where the sum * runs over the indexes i ∈ Z/(q -1)Z such that l(f (i) ) = l(f ) = l (by hypothesis, this sum is non-empty). By setting

Π γ (z) = * det(γ) -i f (i) l (z) (39) we get, for all γ = a c b d ∈ Γ K : f (γ(z)) = (cz + d) w X γ (z) l Π γ (z) + R(l -1) . (40)
5.5 Modular forms in F k .

We need some information about the modular forms contained in F k .

Lemma 5.7 Let f ∈ C[g, h] be a modular form. If D 1 f = D p f = • • • = D p k f = 0
, then there exists a modular form f such that f = f p k+1 .

Proof. Let us consider the case k = 0 first. By assumption, f is an element of M w,m for some w ∈ N and m ∈ Z/(q -1)Z. Let ∂ : C[g, h] → C[g, h] be the derivation defined by ∂ := -h ∂ ∂g . It is easily seen that the restriction of ∂ to M w,m coincides with the operator ∂ w : M w,m → M w,m (of weight 2 and type 1) defined on p. 687 of [START_REF] Gekeler | On the coefficients of Drinfeld modular forms[END_REF], that is, we have ∂f = D 1 f -wEf . Since ∂f is a modular form and D 1 f = 0, we have thus ∂f = -wEf = 0, so that w ≡ 0 (mod p). On the other hand, we have ∂f = -h∂f /∂g = 0, so f is of the form f = i, j such that j ≡ 0 (mod p) λ i,j h i g j .

(44)

Since the weight of h is q + 1 and w ≡ 0 (mod p), we see that λ i,j = 0 in (44) implies i, j ≡ 0 (mod p), and f is the p-th power of another modular form.

We continue the proof of the Lemma by induction on k: let us consider a modular form f

∈ C[g, h] such that D 1 f = D p f = • • • = D p k f = 0, with k > 0. Since D 1 f = D p f = • • • = D p k-1 f = 0,
we see from the induction hypothesis that there exists r ∈ C[g, h] such that f = r p k . Now, from [START_REF] Yu | Function field modular forms and higher derivations[END_REF] immediately follows:

0 = D p k f = D p k (r p k ) = (D 1 r) p k
and r is a p-th power after the first part of the Lemma: r = f p ; thus f = f p k+1 . Lemma 5.8 Let us suppose that q ∈ {2, 3}. For all k ≥ 0 there does not exist a non-zero element of M l 2p k , * ∩ F k-1 with l < p k .

Proof. The Lemma for k = 0 is clear: there does not exist a non-zero modular form of weight 2, because q ≥ 4.

Let us suppose by contradiction that for k > 0 there exists a non-zero isobaric element f of M ∩ F k-1 of weight 2p k and depth l < p k . Since f ∈ F k-1 we have, after Lemma 5.5, p k (2 -α) = l with α ∈ N, and this condition implies α = 2 and l = 0. Thus, f is a modular form in F k-1 . After Lemma 5.7, it is a p k -th power of another modular form f , which must have weight 2, and which is zero after the first part of the Lemma; this implies a contradiction.

Remark. For q = 2, 3, Lemma 5.8 is false: g has weight 2 for q = 3 and g 2 has weight 2 for q = 2; Lemma 5.8 is the only tool of the proof of Theorem 3 which needs the hypothesis q ∈ {2, 3}.

Proof of Theorem 3.

We need the next Lemma. Lemma 5.9 For all k ≥ 0 the following property holds. Let us consider f ∈ G k : there exists a unique s ∈ {0, . . . , p -1} such that

h -p k s f ∈ F × k .
In particular,

G k = h p k Z F × k . Proof. Write f = a/b with a, b ∈ M non-zero. We have b p k f = ab p k -1 with f ∈ F × k-1
(by hypothesis) and b p k ∈ F k-1 (by ( 16)), so x := ab p k -1 ∈ M ∩ F k-1 and y := b p k ∈ M ∩ F k-1 . Thus we have f = x/y with x, y ∈ M ∩ F × k-1 . Now, since D 1 h/h ∈ M and D 1 (h p i ) = 0 for i ≥ 1, we easily deduce from (16) that h p k ∈ G k . Hence, since G k is a group by Lemma 5.1, we find that for all s ∈ Z, there exists a = a s ∈ M such that:

D p k (h -sp k f ) = ah -sp k f.
We claim that a is isobaric of weight 2p k . To prove this, we have of course to consider only the case a = 0. Since D p k is a derivation on F k-1 and h sp k , x, y ∈ M ∩ F k-1 , we have:

h sp k y(D p k x) -xD p k (h sp k y) = axyh sp k .
Let us write: with x i , y j , a l ∈ M isobaric of weights i, j, l, and x α , x β , y γ , y δ , a µ , a ν nonzero. We have: (45)

If a non-vanishing isobaric summand y j h sp k (D p k x i ) -x i D p k (y j h sp k ) on the left-hand side of ( 45) is non-zero, then its weight r := 2p k + sp k (q + 1) + i + j satisfies 2p k + sp k (q + 1) + α + γ ≤ r ≤ 2p k + sp k (q + 1) + β + δ. Since h sp k a µ x α y γ and h sp k a ν x β y δ are two non-vanishing isobaric summands in the right-hand side of (45), whose weights are respectively sp k (q+1)+µ+α+γ ≤ sp k (q + 1) + ν + β + δ, we deduce that there exist two non-zero isobaric summands in the sum on the left-hand side. We find:

2p k + sp k (q + 1) + α + γ ≤ sp k (q + 1) + µ + α + γ ≤ sp k (q + 1) + ν + β + δ ≤ 2p k + sp k (q + 1) + β + δ, which implies 2p k ≤ µ ≤ ν ≤ 2p k , that is, µ = ν = 2p k . Hence a ∈ M is isobaric of weight 2p k , as claimed.

From this we deduce:

h sp k y δ (D p k x β ) -x β D p k (h sp k y δ ) = ax β y δ h sp k . ( 46
)
After Lemma 5.6, there exists exactly one s ∈ {0, . . . , p -1} such that the term on the left-hand side of (46) has depth < p k + l(x β ) + l(y δ ). Thus we see that the depth of ax β y δ h sp k is < p k + l(x β ) + l(y δ ). Since the depth of h is zero, we deduce that a has depth < p k . Since a = D p k (h -p k s f )/(h -p k s f ) and h -p k s f, D p k (h -p k s f ) ∈ F k-1 , we have a ∈ F k-1 . Thus, a ∈ M ∩F k-1 is isobaric of weight 2p k and depth < p k . After Lemma 5.8, a = 0; this implies D p k (h -p k s f ) = 0 and h -p k s f ∈ F k . We deduce the equality of multiplicative groups G k = h p k Z F × k as well.

We can terminate the proof of Theorem 3. Let f ∈ n Ψ n : thus, f ∈ G 0 = Ψ 1 . After Lemma 5.9, there exists (a unique) s 1 ∈ {0, . . . , p -1} such that f 0 := h -s 1 f ∈ F × 0 ; we continue by induction on k.

Proof. We closely follow the proof of Lemma 2.3 of [START_REF] Pellarin | La structure différentielle de l'anneau des formes quasimodulaires pour SL 2 (Z)[END_REF]. Let P be a nonprincipal isobaric prime ideal of C [E, g, h]. Eliminating E, we see that P contains a non-zero isobaric element of C[g, h], say f . We choose f so that its total degree deg f is minimal. Let us define f g = ∂f ∂g and f h = ∂f ∂h . By minimality of deg f , f is not a p-th power in C[g, h]. It follows that f g = 0 or f h = 0.

We first suppose that f g = 0. By minimality of deg f , we have f g ∈ P. Now, by Lemma 5.10, we have d h (f ) ∈ P and

d h (f ) = f g d h (g) + f h d h (h) = f g d h (g).
Since P is prime, it follows that d h (g) ∈ P. But by (2), d h (g) = h 2 . Thus, we get h ∈ P.

If f g = 0 then f h = 0. By using the same arguments as before, we find that f h / ∈ P, and then d g (h) = -h 2 ∈ P, which implies h ∈ P.

Lemma 5.12 Let I be an ideal of C[E, g, h] and let us denote by Ĩ the ideal generated by the isobaric elements of I. Let us suppose that I is hyperdifferential. Then, we have the following properties Proof. We have to show that for every monomial f in E and g, and for any n ∈ N, we have

D n (f ) ≡ w(f ) -l(f ) + n -1 n E n f (mod h). (48)
First of all, we notice that if this formula holds for f 1 and f 2 (and all n), then it also holds for the product f 1 f 2 : this follows from Leibniz formula [START_REF] Swinnerton-Dyer | On ℓ-adic representations and congruences for coefficients of modular forms. Modular Functions of one Variable III[END_REF] and Lemma 3.2 with N = n, W = w(f ) -l(f ) and M = l(g) -w(g) (notice also that (-1) i M n-i = (-1) N N -M -i-1

N -i
). Thus, it suffices to prove the lemma for f = E and for f = g.

Next, we show that it suffices to prove the formula (48) when n has the form n = p k , k ≥ 0. Indeed, assuming the formula for n = p k , an easy induction on m first shows that for all m ∈ N,

D m p k f ≡   1≤i≤m w(f ) -l(f ) + ip k -1 p k   E mp k f (mod h)
(here we need the fact that

f 1 ≡ f 2 (mod h) implies D n f 1 ≡ D n f 2 (mod h)). Since 1≤i≤m w(f ) -l(f ) + ip k -1 p k = w(f ) -l(f ) + mp k -1 mp k 1≤i≤m ip k p k = m! w(f ) -l(f ) + mp k -1 mp k
by [START_REF] Pellarin | La structure différentielle de l'anneau des formes quasimodulaires pour SL 2 (Z)[END_REF], we find, using D mp k = 1 m! D m p k (for 0 ≤ m ≤ p -1), that the formula (48) also holds for n = mp k with 0 ≤ m ≤ p -1. If now n is arbitrary, write m = n 0 + • • • + n s p s in base p (0 ≤ n i ≤ p -1). Then formula (48) for n i p i together with (18) gives

D n f =   1≤i≤s w(f ) -l(f ) + n 0 + • • • + n i p i -1 n i p i   E n 0 +•••+nsp s f.
But the expression in brackets is also equal to

w(f ) -l(f ) + n -1 n 1≤i≤s n 0 + • • • + n i p i n i p i ,
which is equal to w(f )-l(f )+n-1 n by [START_REF] Pellarin | La structure différentielle de l'anneau des formes quasimodulaires pour SL 2 (Z)[END_REF]. So (48) holds for all n.

Part (iii). Taking into account the t-expansions (i), (ii) and (iii) of Lemma 4.2, we first notice that P 0 = (E, h) is also the ideal generated by the quasimodular forms which vanish at the infinity. After Lemma 3.5, it contains the images of all the operators D n (n ≥ 1). Thus, we have D n (g -c) ∈ (E, h) for all n ≥ 1, c ∈ C (by the way, this argument gives an alternative way to check that P 0 is hyperdifferential). Since now P 0 is hyperdifferential after part (i), it follows that for all c ∈ C, the maximal ideal (E, g -c, h) is hyperdifferential.

Let P = (E -c 1 , g -c 2 , h -c 3 ) be a hyperdifferential maximal ideal. In particular, D 1 E, D 1 g, D 1 h belong to P, that is, E 2 , -Eg -h, Eh ∈ P, after the formulas [START_REF] Bosser | Indépendance algébrique de valeurs de séries d'Eisenstein[END_REF]. Thus E ∈ P, which implies c 1 = 0, and h ∈ P, which implies c 3 = 0.

Remark. There is also another way to check that P ∞ is hyperdifferential. By Theorem 1, P ∞ is the ideal generated by the quasi-modular forms f which have weights w and depths l satisfying the inequality l < w/2. By Theorem 2, this property holds for D n f , for all n ≥ 1 and for each quasimodular form f ∈ P ∞ . Hence, D n P ∞ ⊂ P ∞ for all n.

Lemma 3 . 4 Γ

 34 Let f be a non-zero element of M l w,m , and let γ = K . Define ϕ γ : Ω → C by ϕ γ (z) := (det γ) m (cz + d) -w f (γ(z)). For n ≥ 0 and z ∈ Ω, we have

Lemma 5 . 1

 51 For all k ≥ -1, Ψ k and G k are multiplicative subgroups of F × .

  h sp k (D p k x i ) -x i D p k (y j h sp k )) = = h sp k β+δ+ν w=α+γ+µ i+j+l=w a l x i y j .

More implicitly, Quasi-modular forms already appear in several previous works by Ramanujan and Rankin.

It is a class in Z/(q -1)Z.

In[START_REF] Gekeler | On the coefficients of Drinfeld modular forms[END_REF], the notation g changes in the course of the text. In our text we use the notation g for the function gnew of[START_REF] Gekeler | On the coefficients of Drinfeld modular forms[END_REF] (p. 683).

See[START_REF] Gekeler | On the coefficients of Drinfeld modular forms[END_REF]: the last equation corresponds to the definition of E (p. 686), while the first relation corresponds to(8.6) and the second relation is equivalent to Theorem 9.1.

It can be proved that it is equal to the ringC[E p , g p , h p , Eg + h, Eh p-1 , E 2 h p-2 , . . . , E p-1 h].

By[START_REF] Matsumura | Commutative ring theory[END_REF] Theorem 27.2], the iterative derivation (Dn) n≥0 extends to an iterative higher derivation on F in a unique way. We denote this extension by (Dn) n≥0 again.

Proof. The formulas (i), (ii) and (iii) follow from [START_REF] Gekeler | On the coefficients of Drinfeld modular forms[END_REF], Corollaries (10.5), (10.11) and (10.4). To determine the other t-expansions, we apply Lemma 3.5. Define, for r ≥ 1 and i ≥ 1,

One computes painlessly, for all integers i, j such that q < p i < q 2 and q < p j ≤ q 2 ,

Thanks to these expressions and the expansions (i), (ii), (iii) already obtained, Lemma 3.5 now yields Lemma 4.2 without difficulty.

In the next Lemma, if v 1 , . . . , v n are elements of a vector space, we denote by v 1 , . . . , v n the subspace they generate. Lemma 4.3 Let i, n be integers. The following holds:

(ii) If q ≤ p i < q 2 , then

By Lemma 2.5, if f (i) l = 0, it is a modular form of weight w-2l and type i-l. For all γ, the function Π γ : Ω → C is not identically zero because, as follows from Proposition 2.2 (see also Theorem 5.13 of [START_REF] Gekeler | On the coefficients of Drinfeld modular forms[END_REF]), non-zero modular forms of the same weight but with different types are C-linearly independent.

Let us go back to the expression (38) for f ; we have the identity (40) with Π γ (z) as in (39).

More explicitly we have, for γ = a c b d ∈ Γ K and for i ∈ Z/(q -1)Z:

). After the first part of the Proposition:

and the second part of the Proposition follows.

In particular, since Π γ is not identically zero for all γ, the depth of D n f is < l + n if and only if (37) holds.

Further properties of elements in

Proof. We prove the Lemma by induction on k. Let f be as in the hypotheses of the Lemma. For k = -1 the result is trivial, so let us now suppose that k ≥ 0, and that we have already proved the Lemma for the integer k -1. As F k ⊂ F k-1 , we have f ∈ F k-1 and by induction hypothesis we get w -l = α k-1 p k with α k-1 ∈ N. Moreover, by condition (37) of Proposition 5.4,

taking into account [START_REF] Pellarin | La structure différentielle de l'anneau des formes quasimodulaires pour SL 2 (Z)[END_REF] we see that (β+1)p k -1

We thus obtain α k-1 = α k p with α k ∈ N, and w -l = α k p k+1 .

Lemma 5.6 Let k ≥ 0 be an integer and let us consider elements

For all s ∈ N we have

and there exists a unique integer s ∈ {0, . . . , p -1} such that:

Proof. The first property of the Lemma easily follows from Theorem 1. Since f 1 , f 2 ∈ F × k-1 by hypothesis, we have w 1 -l 1 = β 1 p k and w 2 -l 2 = β 2 p k with integers β 1 , β 2 ≥ 0 (Lemma 5.5). For all γ ∈ Γ K we have, after Proposition 5.4 or identity (40):

where Π γ,j (j = 1, 2) is defined as in the equality (39). Proposition 5.4

Hence,

This implies that for γ ∈ Γ K :

where

after (41). Since the product:

is not identically zero over Γ K × Ω, we have that l(f ) < l 1 + l 2 + p k if and only if s ≡ β 1 -β 2 (mod p), and there exists only one such integer in {0, . . . , p -1}.

Remark. In particular, by choosing f 1 = 1 in Lemma 5.6, we find the following property. Let k ≥ 0 be an integer and let f ∈ M l w, * ∩ F k-1 be a non-zero quasi-modular form. There exists a unique integer s ∈ {0, . . . , p-1} such that l(D p k (h

Let us suppose that

for k > 0 and s 1 , . . . , s k ∈ {0, . . . , p -1}. Proposition 3.6 and Lemma 5.1 imply that h

After Lemma 5.9, there exists a unique s k+1 ∈ {0, . . . , p -1} such that

Thanks to this inductive process, for all k ≥ 0, we can construct a sequence of elements f k ∈ F × k and a p-adic integer s 1 +s 2 p+• • •+s k+1 p k +• • • such that for all k, the equality (35) is satisfied. Lemma 5.3 implies that there exists n ∈ Z and c ∈ C × , with f = ch n .

Proof of Corollary 4.

If U, V are isobaric polynomials of C[E, g, h] of weights w(U ), w(V ), their Rankin bracket [U, V ] is defined by:

In the following Lemma we collect the properties of these brackets that we need.

Lemma 5.10 Let M be an isobaric element of C[E, g, h].

• The map d M : X → [X, M ] defined on the set of isobaric elements of C[E, g, h] satisfies

The proof of this Lemma is easy and left to the reader. For the following Lemma, we recall that an ideal is isobaric if it is generated by isobaric elements of C[E, g, h]. Lemma 5.11 Every isobaric non-principal prime ideal P such that D 1 P ⊂ P contains h.

Classification of hyperdifferential ideals.

The aim of this Section is to provide a full description of the hyperdifferential ideals of C [E, g, h]. Let us write:

These ideals are all prime, and we have the following diagram of inclusions (here c and d vary in C × ):

In the following Theorem, we prove (under the condition q = 2, 3) that all the ideals of the diagram (47) are the only non-zero hyperdifferential prime ideals of C[E, g, h]. Theorem 6.1 Let us assume that q = 2, 3.

(i) If P is a principal hyperdifferential non-zero prime ideal of C[E, g, h],

then P = (h).

(ii) For all d ∈ C ∪ {∞}, the ideal P d is hyperdifferential. If P is a nonzero hyperdifferential prime ideal of C[E, g, h] of height 2, then there exists d ∈ C ∪ {∞} such that P = P d .

(iii) For all c ∈ C, the ideal (E, g -c, h) is hyperdifferential. If P is a maximal ideal which also is hyperdifferential, then there exists c ∈ C such that P = (E, g -c, h).

Before going on with the proof of this Theorem, we will need two Lemmata. In the next Lemma, we use the notation

It follows from the above remarks that we only need to prove the Lemma when n = p k (k ≥ 0) and for the functions f = E and f = g. But in this case this follows from Proposition 3.7, Parts (i) and (ii). Indeed, by Lemma 3.5, the differences D p k E -E p k +1 and D p k g -E p k g are modular form which vanish at infinity, hence multiples of h.

The next Lemma, easy, will also be needed in the Proof of Theorem 6.1. Lemma 6.3 Let r, r ′ be two rational integers such that for all n ≥ 0:

Then, r = r ′ .

Proof. Let X be an indeterminate. Let us suppose by contradiction that two distinct integers r, r ′ exist, such that for all n, the congruence of the Lemma holds. Since

], and a similar identity holds for (1 -X) -r ′ , we have that (1 -

], which means that (1 -X) s = 1 with s = r ′ -r = 0; this is impossible.

Proof of Theorem 6.1: Part (i). This follows from Proposition 3.6 and Theorem 3.

Part (ii). We introduce a new graduation on the ring C[E, g], namely the one defined by assigning to E the degree 1 and to g the degree q -1. In what follows, the word "homogeneous" will then refer to this new graduation.

Let P be a homogeneous polynomial in C[E, g] of weight r. By Lemma 6.2 we see that for all n,

Thus, for such a polynomial P , the ideal (P, h) is hyperdifferential. In particular, the property above is fulfilled for the polynomial E with r = 1, and for the polynomials E q-1 -dg (d ∈ C × ) and g, with r = q -1. Hence the ideals P 0 , P d (with d ∈ C × ) and P ∞ are hyperdifferential.

Let us now consider a hyperdifferential prime ideal P of height 2. By Theorem 4, h ∈ P. Thus, there exists an irreducible polynomial P ∈ C[E, g] such that P = (P, h); without loss of generality, we can suppose that P is not proportional to E (otherwise, P = (E, h) = P 0 and we are done). By assumption, for all n ≥ 0 there exists A n ∈ C[E, g] such that

g] is homogeneous of degree n by Lemma 6.2, it is easy to see that A n must be a linear combination of homogeneous terms of weight ≤ n.

We can write:

where P s = 0 and P r ∈ C[E, g] is homogeneous of weight r for all r. We have, explicitly by Lemma 6.2, (n ≥ 0, 0 ≤ r ≤ s)

Thus, E n divides A n P , which implies that A n = λ n E n for some λ n ∈ C. We get: By Lemma 6.3, we find that the sum s r=0 P r has exactly one nonvanishing term, P s . Therefore, P is homogeneous of degree s, and we have:

for integers µ, ν, a, µ 1 , . . . , µ a not all zero, and non-zero elements λ 1 , . . . , λ a ∈ C. Now, P is irreducible. Hence, considering each type of factor in the product on the right hand side of (49), we get P = E (case which is excluded) or P = g or P = E q-1 -dg for d ∈ C × . If P = g we get P = P ∞ , and if P = E q-1 -dg for some d ∈ C × , then we get P = P d .