Structure and bases of modular spaces sequences (M2k(0(N)))kN*

The modular discriminant Δ is known to structure the sequence of modular forms of level 1 (M2k (SL2 (Z)))k∈N∗ . For any positive integer N , we define a strong modular unit ΔN of level N which enables us to structure the sequence (M2k (Γ0 (N )))k∈N∗ in an identical way. We then apply this novel result to the search of bases for each of the (M2k (Γ0 (N )))k∈N∗ spaces. Structure et bases des suites d’espaces modulaires (M2k (Γ0(N)))k∈N∗ Résumé Le discriminant modulaire Δ est connu pour structurer la famille de formes modulaires de niveau 1, (M2k (SL2 (Z)))k∈N∗ . Pour tout entier N , nous définissons une unité modulaire forte de niveau N notée ΔN , qui permet de structurer la famille (M2k (Γ0 (N )))k∈N∗ de manière identique. Nous appliquerons ce résultat à la recherche de bases pour chacun des espaces (M2k (Γ0 (N )))k∈N∗ . Introduction When studying modular forms, an important result relates to the structure of the sequence (M2k (SL2 (Z)))k∈N∗ obtained using the Δ function, and the opportunity to obtain an explicit basis for each subspace [11, p. 143–144]. Such a result appears to be missing for the sequences (M2k (Γ0 (N)))k∈N∗ , whenever N > 2. We propose in this paper an explicit decomposition of modular form spaces (M2k (Γ0 (N))) (k,N ) ∈N∗2 . As the formulae providing the dimension of these spaces [2, 12] hint towards, such a reduction cannot be simple. Nevertheless, we will show that for any given level N , there exists a function ΔN that will play for (M2k (Γ0 (N)))k∈N∗ the same rôle that Δ = Δ1 played in the study of (M2k (SL2 (Z)))k∈N∗ . More specifically, ρN being the weight of ΔN , we will prove that for any fixed positive integer N and any integer k: Knowing bases of M2k (Γ0 (N)) for 1 6 k 6 2 ρN + 1 leads to knowing bases of M2k (Γ0 (N)) for all k . What is more, for any N , this result is algorithmic. It allows us to derive the Fourier series of bases (B2k (N))k∈N∗ to any given accuracy level as soon as one has such series for 1 6 k 6 1 2 ρN + 1, which SAGE for example may provide.


Introduction
When studying modular forms, an important result relates to the structure of the sequence ( 2 (SL 2 (Z))) ∈N * obtained using the Δ function, and the opportunity to obtain an explicit basis for each subspace [11, p. 143-144].
Such a result appears to be missing for the sequences ( 2 (Γ 0 ( ))) ∈N * , whenever 2. We propose in this paper an explicit decomposition of modular form spaces ( 2 (Γ 0 ( ))) ( , ) ∈N * 2 . As the formulae providing the dimension of these spaces [2,12] hint towards, such a reduction cannot be simple. Nevertheless, we will show that for any given level , there exists a function Δ that will play for ( 2 (Γ 0 ( ))) ∈N * the same rôle that Δ = Δ 1 played in the study of ( 2 (SL 2 (Z))) ∈N * .
More specifically, being the weight of Δ , we will prove that for any fixed positive integer and any integer : Knowing bases of 2 (Γ 0 ( )) for 1 1 2 + 1 leads to knowing bases of 2 (Γ 0 ( )) for all .
What is more, for any , this result is algorithmic. It allows us to derive the Fourier series of bases ( 2 ( )) ∈N * to any given accuracy level as soon as one has such series for 1 First, the structure of families ( 2 (Γ 0 ( ))) ∈N * will be studied in Section 2 under the assumption of the existence of a strong modular unit Δ . This assumption will then be proven when is prime in Section 5. Finally, this result will be generalized to any in Section 7, on top of Sections 4 and 6 where modular units are constructed. Sections 8 and 9 will conclude. Sections 1 and 3 are primers on two essential tools: modular forms and Dedeking function, respectively.

Primer on modular forms
Let H = { ∈ C / Im( ) > 0} be the Poincaré half-plane. From now on, let be a complex variable belonging to H , and we define = 2 .
The Δ function is holomorphic and does not cancel on H , but since lim →∞ Δ( ) = 0, it vanishes at the infinite cusp.
Indeed, the mapping Φ ↦ → Φ.Δ −1 is an isomorphism between the space of modular forms of weight 2 vanishing at the infinite cusp (named cuspidal modular forms) and 2 −12 (Γ 0 (1)) [11]. It is this very result that we generalize from = 1 to ∈ N * . 2. Structure of ( 2 (Γ 0 ( ))) ∈N * spaces Let us define two natural ways to generalize the function Δ, which vanishes only at the infinite cusp with respect to Γ 0 (1). Definition 2.1. Let and be two positive integers, and Φ ∈ 2 (Γ 0 ( )). The function Φ is said to be a 2 strong modular unit with respect to Γ 0 ( ) (or equivalently "of level ") if and only if: (ii) The function Φ vanishes at the infinite cusp, (iii) The function Φ does not vanish at any other cusp with respect to Γ 0 ( ).
If we replace condition (iii) by ( iii) The function Φ vanishes at all rational cusps we are instead defining cuspidal modular forms.

Definition 2.2. An integer is said to be the valuation of a modular form Φ if
with ≠ 0 and we write (Φ) = . Of particular interest is the case = 1, in which case the function Φ is said to be unitary. A basis of the space 2 (Γ 0 ( )) that verifies ( ( ) 2 , ) < ( ( +1) 2 , ) for all 0 2 ( ) − 2 is said to be upper triangular, or in echelon form. If the elements of B 2 (Γ 0 ( )) are also unitary, we say that this basis is unitary upper triangular. Theorem 2.4. Let be a positive integer such that there exists a strong modular unit of level . Let Φ 0 be such a strong modular unit of level and of minimal weight 2 0 . Other strong modular units of the family ( 2 (Γ 0 ( ))) ∈N * are then exactly of the form Φ 0 with ∈ C * and ∈ N * . Proof. Let Φ be a modular unit of weight 2 with 0 . By Euclidean division . This function would then vanish at the infinite cusp and would therefore be null, which is impossible. The inequality . (Φ 0 ) < (Φ) would lead to ΦΦ − 0 ∈ 2 (Γ 0 ( )) being a strong modular unit, which would contradict the minimality of 0 .
Therefore . (Φ 0 ) = (Φ) and ΦΦ − 0 does not cancel on H nor at any cusp, which is a well-known characteristic of constant modular forms.
The following result provides the structure of the sequence of modular forms spaces ( 2 (Γ 0 ( ))) ∈N * under the assumption that a strong modular unit exists (which is always the case, as will be shown later). Theorem 2.5. Let be a positive integer and Φ a strong modular unit in 2ℓ (Γ 0 ( )), with ℓ ∈ N * . For ∈ N * , let ( ( ) 2 , ) 0 2 ( )−1 be a unitary upper triangular basis of 2 (Γ 0 ( )). Then for all integer ℓ, Proof. Just like in the = 1 case, the result stems from the isomorphism Our primary goal is to provide concrete and computable results. Theorem 2.5 does not meet these criteria until we know how to compute the elements of ( ) 2 , / ( ( ) 2 , ) < (Φ) . In particular, we need to prove the existence of Φ once and for all instead of assuming it.
To construct the strong modular units, the central tool will be Dedekind function. For clarity, we first recall the properties of this function.

Primer on Dedekind function
Together with (Weierstrass or Jacobi) elliptic functions, the Dedekind function is a must-have tool to construct modular functions and forms. Rademacher [10] first proposed modular functions (of weight 0) with respect to Γ 0 ( ), for prime, by constructing them on top of the function. But it was Newman [7,8] who first constructed a (weakly) modular function with respect to Γ 0 ( ) for any , also starting from . More studies followed, extending these results to the modular forms with respect to Γ 0 ( ) [5,9], leading to the results used here [1,4].
As noted in [4], the behavior of Φ at the cusp − / only depends on . We can restrict the analysis even further: given that gcd( , ) = gcd(gcd( , ), ) for any divisor of , it is enough to check the condition ord(Φ, ) 0 at the cusps = 1/ for the divisors of , 1 . For 1 − 1, condition ord(Φ, 1 ) = 0 indicates the non-nullity of Φ at all rational cusps. The ord(Φ, 1 ) > 0 condition indicates that Φ vanishes at the infinite cusp, because 2 = 1 0 0 1 and 1 0 1 are two representatives of the Γ 0 ( ) class. This leads to the following result.
The function Φ is then a strong modular unit of level and of weight (Φ).
Proof. For such a function Φ, condition (ii) of Theorem 3.2 results from condition (ii) above, and condition (iii) of Theorem 3.2 is derived from condition (iii) above. The same goes for condition (iv). Condition (ii) shows that Φ vanishes at the infinite cusp and provides the order of Φ at infinity (i.e. its valuation). Finally, condition (iii) indicates that Φ does not vanish at any cusp other than the infinite cusp.
We will use Theorem 3.5 to construct in Section 4 a modular unit Δ when the level is prime. This will give, in Section 5, a more precise and operational version of Theorem 2.5. The results obtained for prime will be extended in Sections 6 and 7 to any level 1.

Strong modular units Δ , prime
It is simpler to start by constructing strong modular units of minimum weight for = 2 and = 3, these cases being exceptions.
It is well known that the space 2 (Γ 0 (2)) is one-dimensional and is generated by a form belongs to 4 (Γ 0 (2)). It is a strong modular unit of level 2 with minimal weight.
We deduce from these two linearly independent modular forms that (1) 4,3 is of valuation 1, and unique if we require it to be unitary. This function could still be a strong modular unit, but by division, we would then derive dim( 6 (Γ 0 (3))) = 2 which is false, the space being of dimension 3. This leads us to the following result.

Theorem 4.2. The function
belongs to 6 (Γ 0 (3)). It is the strong modular unit of level 3 of minimal weight.
∈ 2N * and satisfies the other hypotheses of Theorem 3.5: Finally, for = 1 and = 2, we find

5, prime case
We can now derive a general formula for strong modular units of level 5 prime. Let us first define the function Δ on H for any prime number 5 as: Notice that equality Δ ( ) 12 = Δ( ) Δ( ) −1 indicates a modular property of Δ for the weight − 1.
Proof. These are direct consequences of Theorem 2.5.
Of course, a similar result stands for = 1 and leads to unitary upper triangular bases structured by Δ, instead of the usual result obtained with the generators 4 and 6 .

The = 3 case
The strong modular form Δ 3 has a weight of 6, a valuation of 2. Applying Theorem 2.5 gives us a useful corollary: We then have a basis of 2 (Γ 0 (3)) for 1: Proof. Once again, the first equality is a direct application of Theorem 2.5. The second equality comes from a recursion. We know that dim( 2 (Γ 0 (3))) = 1, dim( 4 (Γ 0 (3))) = 2 with ( (0) Similarly, we can choose for any 3, (1) It is easy to check that relation (5.2) produces a basis for = 1 and = 2, and assume the result holding true to the order − 1 2. Given the above, the relation (5.1) shows that ) gives a basis B 2 (Γ 0 (3)). We can then see that Proof. The second equality is known. The first is in fact a special case of Theorem 7.2, valid for any , which will be proven in Section 7. The central element of this proof is an explicit formula providing the dimension of the space 2 (Γ 0 ( )) as a function of and . See [12].
Moreover, we can deduce from Theorem 2.5 the following equality, for any ∈ N * : 12 , from which we derive the following theorem.
This result is important: it shows that the new elements appearing in B 2 (Γ 0 ( )) have regularly spaced valuations, with the remaining elements coming from Δ .B 2 −( −1) (Γ 0 ( )). We still need to characterize these new elements. Let us first prove the following result and its corollary: Theorem 5.5. For any integer 2, 2 (Γ 0 ( )) has elements of valuation 0.
These elements are spread evenly (without jumps) and unitary in 2 (Γ 0 ( )). As such, they are potential candidates to be the first 2 −1 2 elements of B 2 (Γ 0 ( )). We can now give a more precise version of Theorem 2.5: Theorem 5.7. Let 5 be a prime number. Then for all ∈ N * such that In order to get a unitary upper triangular basis B 2 (Γ 0 ( )), 1, Theorem 2.5 is now operational since the knowledge of all bases is reduced to the knowledge of the finite family of bases (B 2 (Γ 0 ( ))) 1 +1 2 . 6. Strong modular units Δ , 1 In Section 5, we derived structured bases of ( 2 (Γ 0 ( ))) ∈N * when is prime. The central tool, which reduced the search of an infinity of unitary upper triangular bases to the search of a finite number of bases, was the existence of a strong modular form Δ . The next logical step is thus to establish the existence of strong modular forms Δ in the general case 1. With this in mind, the Definition 4.1 of Δ , for 5 prime, lead to defining the family of functions : Additionally, the empirical search of strong modular units (Δ ) 1 10 (of minimal weight) lead to the following notations: We can see that Λ is an -product of level and that the two definitions of Λ coincide when is its own radical. The weight of Λ is given by: ( − 1). Table 6.1 presents the minimal strong modular units of level for 1 10 empirically found.
As suggested above, we will show, for ∈ N * , that there exists ∈ Z * such that Λ is a strong modular unit of level . To that end, we will systematically apply Theorem 3.5 then Λ is a strong modular unit of level and of weight (−1) 2 =1 ( − 1).
We can already notice that the structure of the Λ functions leads to the automatic satisfaction of hypothesis (iii) of Theorem 3.5.
be the radical of and = . Following the notations of Theorem 3.5, Λ is an -product of level with = 0 except if = where | . In this case, = .
(1) First, we have By symmetry, it is enough to study 1 .
We deduce the equivalence between (i) and (1) for the functions Λ .
(2) Then, ord(Λ , ∞) = 1 24 We deduce the equivalence between (ii) and (2) The two terms are therefore equal, leading to the last equality needed to finish the proof: The strong modular units Δ will be expressed using Λ functions. The following result reduces the general case to the case = ( ).
Proof. Verifying that Λ satisfies the assumptions of Theorem 6.3 is enough: This proves that Λ is a strong modular unit of level .
When = 3, for all 2, When 5 prime, for all 1, Proof. Let us handle the various subcases separately.
We can unify the previous two results by saying that is a strong modular unit for all prime numbers 1 ≠ 2 . However, the relation (6.2) allows to divide by 2 the weight of the strong modular unit selected when 2 is not one of the prime factors, which will be useful when searching for bases, for example. Additionally, relation (6.1) provides the valuation of Δ 2 1 2 while relation (6.2) provides the valuation of Δ 1 These two numbers are always integers. Let us give two examples.
Proof. Given Corollary 6.4, where = 1 . . . , we need only establish that Δ = Λ (−1) is a strong modular unit of level and of weight 1 2 ( 1 − 1) . . . ( − 1). To this end, let us check that Λ (−1) satisfies the hypotheses of Theorem 6.3. For any ∈ {1, . . . , }, is even since there is at least one prime factor other than 2 in the product. As a result, Moreover, because one of the factors, calling it , is greater than or equal to 5, making 1 24 ( 2 − 1) an integer. Finally, the missing piece comes straightforwardly: Thus, Δ is a strong modular unit of level , which finishes the proof.
Notation 6.8. Let us call the weight of .
Before moving on to the last piece of the proof, Table 6.2 summarizes the characteristics of Δ and its representations as functions of Λ and . 7. Structure and bases of ( 2 (Γ 0 ( ))) ∈N * , positive integer Let us first remind of the explicit formula for the dimension of 2 (Γ 0 ( )) in the general case. Once more, we refer to [2,6,12].
To prove this result, we could make direct use of equation (7.1) but that modus operandi would require studying several cases according to the divisibility of by 3 and 4. A more pleasant approach follows a lemma analogous to Corollary 5.6: Lemma 7.3. Let and be integers larger or equal to 2. If ( ( ) 2 , ) 0 2 ( )−1 is an upper triangular basis of 2 (Γ 0 ( )), then ( (1) 2 , ) = 1. Proof. Let us reuse some elements of the proof of Theorem 5.5.

Putting theory into practice
Theorem 7.5 reveals the structure of classical modular forms spaces with respect to Γ 0 ( ).
To obtain unitary upper triangular bases of these spaces, it remains to determine partial bases B 2 (Γ 0 ( )) = ( ( ) 2 , ) 0 2 ( )−1 , for 1 2 , as well as the first elements of B +2 (Γ 0 ( )): ( ( ) +2, ) 0 (Δ )−1 . This is no easy task, but many modular forms are identified in the literature; one can for example consult [4] for a broad study on the subject. We have checked that this work can be carried out, essentially thanks to Weierstrass elliptic functions, for between 1 and 10.
Noteworthily, the computational approach can benefit directly from the results of previous sections. The knowledge of the unitary upper triangular bases for 1 2 +2 with a precision of terms in the development in powers of enables one to directly obtain unitary upper triangular bases for any weight 2 > + 2, still with a precision of terms.

Conclusion
Let us conclude this study with a few words to better put the Δ functions back into the context of previous works. Products of functions have been studied by Rademacher [10] who introduced the functions ( ) = ( )/ ( ) in order to establish that, if 5 was prime and an even integer, then would be a weakly modular function of weight 0 with respect to Γ 0 ( ). This result was extended by Newmann [7,8] who constructed, also starting from functions, weakly modular functions with respect to Γ 0 ( ), for any this time, and thus of weight 0. Theorem 3.5, stating that functions Δ are strong modular units, was essentially proven by Ligozat [5] in his study of elliptical modular curves. From then on, mathematicians essentially looked for -quotients in their quest for cuspidal modular forms. Perhaps therein lies the reason why the notion of strong modular units did not pan out, having been overshadowed by the highly-justified importance given to cuspidal forms that followed from Hecke's seminal work.
By introducing the Δ functions, we were able to clarify the structure of the sequences of modular spaces ( 2 (Γ 0 ( ))) ∈N * , and provide an effective tool to provide bases for each of these spaces. The reader will certainly appreciate that, in a similar way, the strong modular unit Δ also makes it possible to structure sequences of cuspidal modular spaces ( 2 (Γ 0 ( ))) ∈N * , and to give explicit bases for each of these spaces.