Structure and bases of modular spaces sequences <span class="mathjax-formula">$(M_{2k}(\Gamma _0(N)))_{k\in \protect \mathbb{N}^*}$</span>

ANNALES MATHÉMATIQUES

BLAISE PASCAL

Jean-Christophe Feauveau

Structure and bases of modular spaces sequences

Volume 27, n^o2 (2020), p. 181-206.

<http://ambp.centre-mersenne.org/item?id=AMBP_2020__27_2_181_0>

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Structure and bases of modular spaces sequences ( 𝑀

(Γ

( 𝑁 )))

Abstract

The modular discriminant Δis known to structure the sequence of modular forms of level 1 (𝑀_2𝑘(SL2(Z)))_𝑘∈N∗. For any positive integer𝑁, we define a strong modular unitΔ𝑁of level𝑁which 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(𝑀_2𝑘(Γ₀(𝑁)))𝑘∈N^∗spaces.

Structure et bases des suites d’espaces modulaires(𝑀_2𝑘(Γ₀(𝑁)))𝑘∈N^∗ Résumé

Le discriminant modulaireΔest connu pour structurer la famille de formes modulaires de niveau 1, (𝑀_2𝑘(SL2(Z)))_𝑘∈N∗. Pour tout entier𝑁, nous définissons une unité modulaire forte de niveau𝑁notée Δ𝑁, qui permet de structurer la famille(𝑀_2𝑘(Γ₀(𝑁)))𝑘∈N^∗de manière identique. Nous appliquerons ce résultat à la recherche de bases pour chacun des espaces(𝑀_2𝑘(Γ₀(𝑁)))_𝑘∈N∗.

Introduction

When studying modular forms, an important result relates to the structure of the sequence (𝑀_2𝑘(SL2(Z)))𝑘∈N^∗ obtained using the Δfunction, 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 (𝑀_2𝑘(Γ₀(𝑁)))𝑘∈N^∗, when- ever 𝑁 > 2. We propose in this paper an explicit decomposition of modular form spaces (𝑀_2𝑘(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 level𝑁, there exists a functionΔ𝑁 that will play for(𝑀_2𝑘(Γ₀(𝑁)))𝑘∈N^∗

the same rôle thatΔ = Δ₁played in the study of(𝑀_2𝑘(SL2(Z)))𝑘∈N^∗.

More specifically,𝜌_𝑁 being the weight ofΔ𝑁, we will prove that for any fixed positive integer𝑁and any integer𝑘:

Knowing bases of𝑀_2𝑘(Γ₀(𝑁))for16𝑘6 ¹₂𝜌_𝑁 +1leads to knowing bases of𝑀_2𝑘(Γ₀(𝑁))for all𝑘.

What is more, for any𝑁, this result is algorithmic. It allows us to derive the Fourier series of bases(𝐵_2𝑘(𝑁))𝑘∈N^∗to any given accuracy level as soon as one has such series for 16𝑘6 ¹₂𝜌_𝑁 +1, which SAGE for example may provide.

Keywords:modular forms, modular units, Dedekind eta function.

2020Mathematics Subject Classification:11F11, 11G16, 11F33, 33E05.

First, the structure of families(𝑀_2𝑘(Γ₀(𝑁)))𝑘∈N^∗will be studied in Section 2 under the assumption of the existence of a strong modular unitΔ𝑁. This assumption will then be proven when𝑁is prime in Section 5. Finally, this result will be generalized to any𝑁 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.

1.

Primer on modular forms

Let H = {𝜏 ∈ C/Im(𝜏) > 0}be the Poincaré half-plane. From now on, let 𝜏be a complex variable belonging toH, and we define𝑞=𝑒^{2𝑖 𝜋 𝜏}.

For(𝑁 , 𝑘) ∈N^∗2, let𝑀_2𝑘(Γ₀(𝑁))be the space of modular forms of weight 2𝑘with respect toΓ₀(𝑁), and let𝑑_2𝑘(𝑁)be the dimension of𝑀_2𝑘(Γ₀(𝑁)). For a primer on these spaces (definitions, theorems on cusps or cuspidal modular forms . . . ), one can read [1, 2].

For𝑘>2 and𝜏∈ H, the normalized Eisenstein series are defined as the following modular forms:

𝐸_2𝑘(𝜏)= 1 2𝜁(2𝑘)

∑︁

(𝑚, 𝑛) ∈Z² (𝑚, 𝑛)≠(0,0)

1

(𝑚 𝜏+𝑛)^2𝑘 =1+𝑂(𝑞).

It is easy to show that 𝐸_2𝑘 ∈ 𝑀_2𝑘(SL2(Z)), which ensures the non-triviality of this space. It is nevertheless the function Δ ∈ 𝑀₁₂(SL2(Z)) that will structure the sequence(𝑀_2𝑘(SL2(Z)))𝑘∈N^∗:

∀𝜏 ∈ H, Δ(𝜏)=𝑞

+∞

Ö

𝑛=1

(1−𝑞^𝑛)²⁴=𝑞+𝑂(𝑞).

TheΔfunction is holomorphic and does not cancel onH, but since lim𝜏→∞Δ(𝜏)=0, it vanishes at the infinite cusp.

Lastly, let us recall the well-known structural result of modular forms with respect to SL2(Z)= Γ₀(1):

∀𝑘>6, 𝑀_2𝑘(Γ₀(1))=span(𝐸_2𝑘) ⊕Δ. 𝑀_2𝑘−12(Γ₀(1)).

Indeed, the mappingΦ↦→Φ.Δ⁻¹is an isomorphism between the space of modular forms of weight 2𝑘 vanishing at the infinite cusp (named cuspidal modular forms) and𝑀_2𝑘−12(Γ₀(1))[11]. It is this very result that we generalize from𝑁 =1 to𝑁∈N^∗.

2.

Structure of

spaces

Let us define two natural ways to generalize the functionΔ, which vanishes only at the infinite cusp with respect toΓ₀(1).

Definition 2.1. Let 𝑘 and 𝑁 be two positive integers, and Φ ∈ 𝑀_2𝑘(Γ₀(𝑁)). The functionΦis said to be a 2𝑘strong modular unit with respect toΓ₀(𝑁)(or equivalently

“of level𝑁”) if and only if:

(i) The functionΦdoes vanish onH,

(ii) The functionΦvanishes at the infinite cusp,

(iii) The functionΦdoes not vanish at any other cusp with respect toΓ₀(𝑁).

If we replace condition (iii) by

(eiii) The functionΦvanishes at all rational cusps we are instead defining cuspidal modular forms.

Definition 2.2. An integer𝑛is said to be the valuation of a modular formΦif Φ(𝜏)=𝑎 𝑞^𝑛+𝑂(𝑞^𝑛+1)

with𝑎≠0 and we write𝜈(Φ)=𝑛. Of particular interest is the case𝑎=1, in which case the functionΦis said to be unitary. A basis

B₂𝑘(Γ₀(𝑁))=(𝐸^(𝑟)

2𝑘 , 𝑁)₀₆𝑟6𝑑_2𝑘(𝑁)−1

of the space𝑀_2𝑘(Γ₀(𝑁))that verifies𝜈(𝐸^(𝑟)

2𝑘 , 𝑁)< 𝜈(𝐸^(𝑟+1)

2𝑘 , 𝑁)for all 06𝑟6𝑑_2𝑘(𝑁) −2 is said to be upper triangular, or in echelon form. If the elements ofB₂𝑘(Γ₀(𝑁))are also unitary, we say that this basis is unitary upper triangular.

Lemma 2.3. For any positive integers𝑁and𝑘, the space𝑀_2𝑘(Γ₀(𝑁))has a unitary upper triangular basis. Moreover, the sequence of integers(𝜈(𝐸₂𝑘 , 𝑁(𝑟)))₀₆𝑟6𝑑_2𝑘(𝑁)−1

is independent of the choice of such a basis(𝐸^(𝑟)

2𝑘 , 𝑁)₀₆𝑟6𝑑_2𝑘(𝑁)−1.

Proof. Existence comes directly from a Gaussian elimination. The result on valuations is

straightforward.

Theorem 2.4. Let𝑁be a positive integer such that there exists a strong modular unit of level𝑁. LetΦ₀be such a strong modular unit of level𝑁and of minimal weight2𝑘₀. Other strong modular units of the family(𝑀_2𝑘(Γ₀(𝑁)))𝑘∈N^∗are then exactly of the form𝛼Φ^𝑛₀ with𝛼∈C^∗and𝑛∈N^∗.

Proof. LetΦbe a modular unit of weight 2𝑘with𝑘>𝑘₀. By Euclidean division 𝑘=𝑞 𝑘₀+𝑟 , 06𝑟 < 𝑘₀.

The inequality𝜈(Φ) < 𝑞 .𝜈(Φ₀)would lead toΦ^𝑞₀Φ⁻¹ ∈ 𝑀_−2𝑟(Γ₀(𝑁)). This function would then vanish at the infinite cusp and would therefore be null, which is impossible.

The inequality𝑞 .𝜈(Φ₀) < 𝜈(Φ)would lead toΦΦ^−𝑞₀ ∈ 𝑀_2𝑟(Γ₀(𝑁))being a strong modular unit, which would contradict the minimality of𝑘₀.

Therefore𝑞 .𝜈(Φ₀)=𝜈(Φ)andΦΦ⁻₀^𝑞does not cancel onHnor 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(𝑀_2𝑘(Γ₀(𝑁)))𝑘∈N^∗ under the assumption that a strong modular unit exists (which is always the case, as will be shown later).

Theorem 2.5. Let𝑁be a positive integer andΦa strong modular unit in𝑀_2ℓ(Γ₀(𝑁)), withℓ ∈N^∗. For𝑘 ∈N^∗, let(𝐸^(𝑠)

2𝑘 , 𝑁)₀₆𝑠6𝑑_2𝑘(𝑁)−1be a unitary upper triangular basis of𝑀_2𝑘(Γ₀(𝑁)). Then for all integer𝑘>ℓ,

𝑀_2𝑘(Γ₀(𝑁))= Φ. 𝑀_2𝑘−2ℓ(Γ₀(𝑁)) ⊕span 𝐸^(𝑠)

2𝑘 , 𝑁 /𝜈(𝐸^(𝑠)

2𝑘 , 𝑁)< 𝜈(Φ) . Therefore, if𝑘 ∈N^∗and𝑘=𝑞ℓ+𝑟with16𝑟6ℓ,

𝑀_2𝑘(Γ₀(𝑁))= Φ^𝑞. 𝑀_2𝑟(Γ₀(𝑁))

𝑞−1

Ê

𝑛=0

Φ^𝑛span 𝐸^(𝑠)

2𝑘−2𝑛ℓ , 𝑁 /𝜈(𝐸^(𝑠)

2𝑘−2𝑛ℓ , 𝑁) < 𝜈(Φ) . Proof. Just like in the𝑁=1 case, the result stems from the isomorphism

𝜑: span 𝐸^(𝑠)

2𝑘 , 𝑁 /𝜈(𝐸^(𝑠)

2𝑘 , 𝑁)>𝜈(Φ) −→𝑀_2𝑘−2ℓ(Γ₀(𝑁))

Ψ↦−→Ψ/Φ.

Our primary goal is to provideconcrete and computable results. Theorem 2.5 does not meet these criteria until we know how to compute the elements of

𝐸^(𝑠)

2𝑘 , 𝑁/𝜈(𝐸^(𝑠)

2𝑘 , 𝑁) <

𝜈(Φ) . In particular, we need to prove the existence ofΦ once and for all instead of assuming it.

To construct the strong modular units, the central tool will be Dedekind𝜂function.

For clarity, we first recall the properties of this function.

3.

Primer on Dedekind

function

Together with (Weierstrass or Jacobi) elliptic functions, the Dedekind𝜂 function is a must-have tool to construct modular functions and forms. Rademacher [10] first proposed modular functions (of weight 0) with respect toΓ₀(𝑝), for𝑝prime, by constructing them

on top of the𝜂 function. But it was Newman [7, 8] who first constructed a (weakly) modular function with respect toΓ0(𝑁)for any𝑁, also starting from𝜂. More studies followed, extending these results to the modular forms with respect toΓ₀(𝑁)[5, 9], leading to the results used here [1, 4].

Let us define the Dedekind function, of weight ¹₂ [1]:

∀𝜏∈ H, 𝜂(𝜏)=𝑒^{𝑖 𝜋 𝜏/12}

+∞

Ö

𝑛=1

(1−𝑞^𝑛).

Definition 3.1. Let𝑁be a positive integer. We call𝜂-quotient of level𝑁any function of the form

∀𝜏∈ H, Φ(𝜏)= Ö

𝑚|𝑁

𝜂(𝑚 𝜏)^𝑎𝑚 (3.1)

where(𝑎₁, . . . , 𝑎_𝑁)is a sequence of integers indexed on the divisors of𝑁. The relation 3.1 shows that ifΦis modular, its weight is necessarily 2𝑘= ¹₂Í

𝑚|𝑁𝑎_𝑚 and in this case𝜈(Φ)= ₂₄¹ Í

𝑚|𝑁𝑚 𝑎_𝑚∈N^∗.

The following results are handy since they remove lots of calculations from future proofs. They are derived from the modular properties of the𝜂function and are found in the literature under various forms. The initial sources are [7, Theorem 1], [5, Proposition 3.2.1]

and finally [4, Corollary 2.3], as well as [9, Theorem 1.64] and [3, Theorem 1].

Theorem 3.2. LetΦ(𝜏)=Î

𝑚|𝑁 𝜂(𝑚 𝜏)^𝑎𝑚be an𝜂-quotient of level𝑁. For𝑚a divisor of𝑁, we define𝑚⁰=𝑁/𝑚. If the functionΦsatisfies the following four conditions

(i) Î

𝑚|𝑁𝑚^0𝑎𝑚 ∈Q² (ii) ₂₄¹ Í

𝑚|𝑁𝑚 𝑎_𝑚∈Z (iii) ₂₄¹ Í

𝑚|𝑁𝑚⁰𝑎_𝑚 ∈Z (iv) ¹₂Í

𝑚|𝑁 𝑎_𝑚∈2N^∗

thenΦis weakly modular with respect toΓ₀(𝑁)and of weight2𝑘= ¹₂Í

𝑚|𝑁 𝑎_𝑚. Definition 3.3. For𝑟=−^𝑑_𝑐 ∈Qwith gcd(𝑐, 𝑑)=1, the vanishing order of

Φ(𝜏)= Ö

𝑚|𝑁

𝜂(𝑚 𝜏)^𝑎𝑚 at the𝑟cusp is defined by

ord(Φ, 𝑟)= 𝑁 24

∑︁

𝑚|𝑁

gcd(𝑐, 𝑚)² 𝑚

𝑎_𝑚.

Theorem 3.4. The functionΦhas a limit at the cusp𝑟if and only iford(Φ, 𝑟)>0, and Φvanishes at this cusp if and only iford(Φ, 𝑟) >0. Therefore, under assumptions(i), (ii),(iii)and(iv)of Theorem 3.2, if for any cusp𝑟 =−^𝑑_𝑐 ∈Qwe haveord(Φ, 𝑟) >0, thenΦ∈𝑀_2𝑘(Γ₀(𝑁)).

As noted in [4], the behavior ofΦat the cusp−𝑑/𝑐only depends on𝑐. We can restrict the analysis even further: given that gcd(𝑐, 𝑚)=gcd(gcd(𝑐, 𝑁), 𝑚)for any divisor𝑚 of 𝑁, it is enough to check the condition ord(Φ, 𝑟) > 0 at the cusps𝑟 =1/𝑐for the divisors𝑐of𝑁, 16𝑐6𝑁.

For 1 6 𝑐6 𝑁−1, condition ord(Φ,¹

𝑐) = 0 indicates the non-nullity ofΦat all rational cusps. The ord(Φ, ¹

𝑁)>0 condition indicates thatΦvanishes at the infinite cusp, because𝐼₂ = ^{1 0}_{0 1}

and _𝑁^{1 0}₁

are two representatives of theΓ₀(𝑁)class. This leads to the following result.

Theorem 3.5. LetΦ(𝜏)=Î

𝑚|𝑁 𝜂(𝑚 𝜏)^𝑎𝑚be an𝜂-quotient of level𝑁such that:

(i) 𝑃(Φ)=Î

𝑚|𝑁𝑚^0𝑎𝑚∈Q² (ii) ord(Φ,∞)= ₂₄¹ Í

𝑚|𝑁 𝑚 𝑎_𝑚 ∈N^∗ (iii) ∀𝑐∈ {1, . . . 𝑁−1,}, ord(Φ,¹

𝑐)= ₂₄¹ Í

𝑚|𝑁

gcd(𝑐 , 𝑚)² 𝑚 𝑎_𝑚=0 (iv) 𝑊(Φ)= ¹₂Í

𝑚|𝑁𝑎_𝑚∈2N^∗

The functionΦis then a strong modular unit of level𝑁and of weight𝑊(Φ).

Proof. For such a functionΦ, 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 thatΦvanishes at the infinite cusp and provides the order ofΦat infinity (i.e. its valuation). Finally, condition (iii) indicates thatΦdoes not vanish at any

cusp other than the infinite cusp.

We will use Theorem 3.5 to construct in Section 4 a modular unitΔ𝑝when the level𝑝is prime. This will give, in Section 5, a more precise and operational version of Theorem 2.5.

The results obtained for𝑝prime will be extended in Sections 6 and 7 to any level𝑁>1.

4.

Strong modular units

,

prime

It is simpler to start by constructing strong modular units of minimum weight for 𝑝=2 and𝑝=3, these cases being exceptions.

The

case

It is well known that the space𝑀₂(Γ₀(2))is one-dimensional and is generated by a form 𝐸⁽⁰⁾

2,2(𝜏)=1+𝑂(𝑞). This excludes the existence of a strong modular form of weight 2 which must vanish at the infinity cusp.

Theorem 4.1. The function

Δ₂(𝜏)=𝜂(2𝜏)¹⁶𝜂(𝜏)⁻⁸=𝑞

+∞

Ö

𝑛=1

(1−𝑞^2𝑛)¹⁶ (1−𝑞^𝑛)⁸

belongs to𝑀₄(Γ₀(2)). It is a strong modular unit of level2with minimal weight.

Proof. The functionΔ₂ is an𝜂-quotient of level 𝑁 = 2. Its divisors are 𝑚 ∈ {1,2}, providing two coefficients𝑎₁ =−8 and𝑎₂=16.

The functionΔ₂ is of weight𝑊(Δ₂)= ¹₂(𝑎₁+𝑎₂)=4∈2N^∗and satisfies the other hypotheses of Theorem 3.5:

Ö

𝑚|2

𝑚^0𝑎𝑚=2⁻⁸∈Q², 1 24

∑︁

𝑚|2

𝑚 𝑎_𝑚=1∈N^∗ and ∑︁

𝑚|2

𝑎_𝑚 𝑚

=0∈2N^∗.

The

case

The space𝑀₂(Γ₀(3))is also one-dimensional, generated by the modular form𝐸⁽⁰⁾

2,3(𝜏)= 1+12𝑞+𝑂(𝑞²). Hence, as in the case𝑝 =2, there is no strong modular unit in this space.

The 𝑀₄(Γ₀(3)) space is 2-dimensional and does not contain any strong modular unit. Indeed, we can choose𝐸⁽⁰⁾

4,3 = [𝐸⁽⁰⁾

2,3]² =1+24𝑞+𝑂(𝑞²), but we also know an element of 𝑀₄(Γ₀(3)) constructed from the Eisenstein series 𝐸₄, namely 𝐸₄(3𝜏) = 1+240𝑞³+𝑂(𝑞⁶).

We deduce from these two linearly independent modular forms that𝐸⁽¹⁾

4,3is 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(𝑀₆(Γ₀(3))) =2 which is false, the space being of dimension 3. This leads us to the following result.

Theorem 4.2. The function

Δ₃(𝜏)=𝜂(3𝜏)¹⁸𝜂(𝜏)⁻⁶=𝑞²

+∞

Ö

𝑛=1

(1−𝑞^3𝑛)¹⁸ (1−𝑞^𝑛)⁶

belongs to𝑀₆(Γ₀(3)). It is the strong modular unit of level3of minimal weight.

Proof. The functionΔ₃is an𝜂-quotient of level𝑁=3, having as divisors𝑚 ∈ {1,3}

with𝑎₁ =−6 and𝑎₃=18.

The functionΔ₃ is of weight𝑊(Δ₃)= ¹₂(𝑎₁+𝑎₃)=6∈2N^∗and satisfies the other hypotheses of Theorem 3.5:

Ö

𝑚|3

𝑚^0𝑎𝑚 =3⁻⁶ ∈Q², and 1 24

∑︁

𝑚|3

𝑚 𝑎_𝑚=2∈N^∗. Finally, for𝑐=1 and𝑐=2, we findÍ

𝑚|3

gcd(𝑐 , 𝑚)²

𝑚 𝑎_𝑚=Í

𝑚|3 𝑎_𝑚

𝑚 =0.

The

prime case

We can now derive a general formula for strong modular units of level𝑝>5 prime. Let us first define the functionΔ𝑝onHfor any prime number𝑝>5 as:

Δ𝑝(𝜏)=𝜂(𝑝 𝜏)^2𝑝𝜂(𝜏)⁻² =𝑞^(𝑝

2−1)/12 +∞

Ö

𝑛=1

(1−𝑞^{𝑝 𝑛})^2𝑝

(1−𝑞^𝑛)² . (4.1) Theorem 4.3. TheΔ𝑝 function is a𝑀_𝑝−1(Γ₀(𝑝))modular unit.

Notice that equalityΔ𝑝(𝜏)¹²= Δ(𝑝 𝜏)^𝑝Δ(𝜏)⁻¹indicates a modular property ofΔ𝑝for the weight 𝑝−1.

Also, note that if 𝑝 > 5 is prime, then ^𝑝₁₂²⁻¹ ∈ N. More generally, if 𝑁 > 5 is an integer such that 𝑁 ≡ 1 (mod 6) or 𝑁 ≡ 5 (mod 6), which is the case for 𝑝 > 5 prime, then ^𝑁₁₂²⁻¹ ∈N. Indeed, if𝑁 =6𝑘+1 then ^𝑁₁₂²⁻¹ =3𝑘²+𝑘 and if𝑁 =6𝑘+5 then ^𝑁₁₂²⁻¹ =3𝑘²+𝑘+2.

Proof. The function Δ𝑝 is an 𝜂-quotient of level 𝑝, with 𝑝 divisors 𝑚 ∈ {1, 𝑝} corresponding to the coefficient 𝑎₁ = −2 and 𝑎_𝑝 = 2𝑝. The function Δ𝑝 is of weight𝑊(Δ𝑝)=𝑝−1∈2N^∗and satisfies the other hypotheses of Theorem 3.5:

Ö

𝑚|𝑝

𝑚^0𝑎𝑚= 𝑝⁻²∈Q² and 1 24

∑︁

𝑚|𝑝

𝑚 𝑎_𝑚= 𝑝²−1 12 ∈N^∗. Finally, for𝑐∈ {1, . . . , 𝑝−1}, we deriveÍ

𝑚|𝑝

gcd(𝑐 , 𝑚))²

𝑚 𝑎_𝑚=Í

𝑚|𝑝 𝑎_𝑚

𝑚 =0.

5.

Structure and bases of

,

prime

Our goal is to construct a family of unitary upper triangular bases (B₂𝑘(Γ₀(𝑝)))𝑘∈N^∗

of(𝑀_2𝑘(Γ₀(𝑝)))_𝑘∈

N^∗, using the generic notationB₂𝑘(Γ₀(𝑝))=(𝐸^(𝑠)

2𝑘 , 𝑝)₀₆𝑠6𝑑_2𝑘(𝑝)−1. Let us start with the special cases𝑝=2 and𝑝 =3 that need to be treated separately.

This also allows us to get the gist of the coming algorithm producing bases.

The

case

2,2 be the unit generator of 𝑀₂(Γ₀(2)) which is of valuation 0. It is therefore possible to choose 𝐸⁽⁰⁾

2𝑘 ,2 = [𝐸⁽⁰⁾

2,2]^𝑘 as the first vector of the unitary upper triangular basisB₂𝑘(Γ₀(2)). Since functionΔ₂has a weight of 4 and a valuation of 1, Theorem 2.5 gives the following result.

Corollary 5.1. For all𝑘>3,

𝑀_2𝑘(Γ₀(2))=span{𝐸⁽⁰⁾

2𝑘 ,2} ⊕Δ₂. 𝑀_2𝑘−4(Γ₀(2)) Moreover, for all𝑘>1,

B₂𝑘(Γ₀(2))= [𝐸⁽⁰⁾

2,2]^𝑎Δ^𝑏₂, with(𝑎, 𝑏) ∈N²such as𝑎+2𝑏=𝑘

is a unitary upper triangular basis of𝑀_2𝑘(Γ₀(2)).

Proof. These are direct consequences of Theorem 2.5.

Of course, a similar result stands for𝑁=1 and leads to unitary upper triangular bases structured byΔ, instead of the usual result obtained with the generators𝐸₄and𝐸₆.

The

case

The strong modular formΔ₃has a weight of 6, a valuation of 2. Applying Theorem 2.5 gives us a useful corollary:

Corollary 5.2. For all𝑘>4,

𝑀_2𝑘(Γ₀(3))=span{𝐸⁽⁰⁾

2𝑘 ,3, 𝐸⁽¹⁾

2𝑘 ,3} ⊕Δ₃. 𝑀_2𝑘−6(Γ₀(3)). (5.1) We then have a basis of𝑀_2𝑘(Γ₀(3))for𝑘>1:

B₂𝑘(Γ₀(3))= [𝐸⁽⁰⁾

2,3]^𝑎.Δ^𝑏₃, (𝑎, 𝑏) ∈N²/𝑎+3𝑏=𝑘

∪ 𝐸⁽¹⁾

4,3.[𝐸⁽⁰⁾

2,3]^𝑎.Δ₃^𝑏, (𝑎, 𝑏) ∈N²/𝑎+3𝑏=𝑘−2

. (5.2) 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(𝑀₂(Γ₀(3)))=1, dim(𝑀₄(Γ₀(3)))=2 with𝜈(𝐸⁽⁰⁾

2,3)=0,𝜈(𝐸⁽⁰⁾

4,3)= 0 and𝜈(𝐸⁽¹⁾

4,3)=1. Therefore,𝐸⁽⁰⁾

4,3 =[𝐸⁽⁰⁾

2,3]², and more generally,𝐸⁽⁰⁾

2𝑘 ,3=[𝐸⁽⁰⁾

2,3]^𝑘can be chosen as the first element of𝑀_2𝑘(Γ₀(3))unitary upper triangular basis.

Similarly, we can choose for any𝑘>3,𝐸⁽¹⁾

2𝑘 ,3=𝐸⁽¹⁾

4,3[𝐸⁽⁰⁾

2,3]^𝑘−2.

It is easy to check that relation (5.2) produces a basis for𝑘=1 and𝑘=2, and assume the result holding true to the order𝑘−1>2. Given the above, the relation (5.1) shows

that

[𝐸⁽⁰⁾

2,3]^𝑘, 𝐸⁽¹⁾

4,3[𝐸⁽⁰⁾

2,3]^𝑘−2

∪Δ₃B_2𝑘−4(Γ₀(3)) gives a basisB₂𝑘(Γ₀(3)). We can then see that

[𝐸⁽⁰⁾

2,3]^𝑎.Δ^𝑏₃, (𝑎, 𝑏) ∈N²/𝑎+3𝑏=𝑘

= [𝐸⁽⁰⁾

2,3]^𝑘

∪Δ₃ [𝐸⁽⁰⁾

2,3]^𝑎.Δ₃^𝑏, (𝑎, 𝑏) ∈N²/𝑎+3𝑏 =𝑘−3 and

𝐸⁽¹⁾

4,3.[𝐸⁽⁰⁾

2,3]^𝑎.Δ₃^𝑏, (𝑎, 𝑏) ∈N²/𝑎+3𝑏 =𝑘−2

= 𝐸⁽¹⁾

4,3[𝐸⁽⁰⁾

2,3]^𝑘−2

∪Δ₃ 𝐸⁽¹⁾

4,3.[𝐸⁽⁰⁾

2,3]^𝑎.Δ₃^𝑏, (𝑎, 𝑏) ∈N²/𝑎+3𝑏 =𝑘−5

which, by induction, completes the proof.

The

case,

prime

Lemma 5.3. For all𝑘∈N^∗,

dim(𝑀_{2𝑘+𝑝−1}(Γ₀(𝑝))) −dim(𝑀_2𝑘(Γ₀(𝑝)))=𝜈(Δ𝑝)= 𝑝²−1 12 .

Proof. The second equality is known. The first is in fact a special case of Theorem 7.2, valid for any𝑁, which will be proven in Section 7. The central element of this proof is an explicit formula providing the dimension of the space𝑀_2𝑘(Γ₀(𝑁))as a function of𝑘

and𝑁. See [12].

Moreover, we can deduce from Theorem 2.5 the following equality, for any𝑘∈N^∗: dim(𝑀_{2𝑘+𝑝−1}(Γ₀(𝑝)))=dim(𝑀_2𝑘(Γ₀(𝑝))) +card 𝑠/𝜈(𝐸^(𝑠)

2𝑘+𝑝−1, 𝑁)<

𝑝²−1

12 .

As a result card({𝑠/𝜈(𝐸^(𝑠)

2𝑘+𝑝−1, 𝑁)< ^𝑝

2−1

12 }) = ^𝑝2₁₂⁻¹, from which we derive the following theorem.

Theorem 5.4. For any𝑝>5prime and any integer𝑘>1, let(𝐸^(𝑠)

2𝑘 , 𝑝)₀₆𝑠6𝑑_2𝑘(𝑁)−1be a unitary upper triangular basis of𝑀_2𝑘(Γ₀(𝑝)). Then,

∀𝑘> 𝑝+1

2 , ∀𝑠∈

0, . . . , 𝑝²−1

12 −1

, 𝜈(𝐸⁽

𝑠) 2𝑘 , 𝑝)=𝑠.

This result is important: it shows that the new elements appearing in B₂𝑘(Γ₀(𝑝)) have regularly spaced valuations, with the remaining elements coming from Δ𝑝.B₂𝑘−(𝑝−1)(Γ₀(𝑝)). We still need to characterize these new elements. Let us first prove the following result and its corollary:

Theorem 5.5. For any integer𝑁>2,𝑀₂(Γ₀(𝑁))has elements of valuation0.

Proof. This is a well-known result and is usually obtained thanks to Eisenstein series𝐺₂ (see [2, p. 18] or [6]). Let us define

𝐺₂(𝜏)= ∑︁

𝑚∈Z

∑︁

𝑛∈Z⁰𝑚

1

(𝑚 𝜏+𝑛)² =2𝜁(2) −8𝜋²

+∞

∑︁

𝑛=1

𝜎(𝑛)𝑞^𝑛

whereZ^0𝑚=Z− {0}ifZ^0𝑚=0 andZ^0𝑚=Zotherwise. Then, some calculations allow us to derive that𝐺_{2, 𝑁}(𝜏)=𝐺₂(𝜏) −𝑁 𝐺₂(𝑁 𝜏)belongs to𝑀₂(Γ₀(𝑁)). Moreover,

𝜏→+∞lim

𝐺_{2, 𝑁}(𝜏)=2(1−𝑁)𝜁(2)≠0,

which concludes the proof.

Corollary 5.6. Let 𝑁 > 2be an integer. If (𝐸^(𝑠)

2𝑘 , 𝑝)₀₆𝑠6𝑑_2𝑘(𝑁)−1 is a unitary upper triangular basis of𝑀_2𝑘(Γ₀(𝑁)), then𝜈(𝐸⁽⁰⁾

2𝑘 , 𝑝)=0and we can choose𝐸⁽⁰⁾

2𝑘 , 𝑝 =[𝐸⁽⁰⁾

2, 𝑝]^𝑘. Theorem 5.5 and Corollary 5.6 enable an algorithmic construction of structured bases.

Indeed, for𝑘> ^𝑝+1₂ , we can choose 𝐸^(𝑠)

2𝑘 , 𝑝 =𝐸^(𝑠)

𝑝+1, 𝑝[𝐸⁽⁰⁾

2, 𝑝]^{𝑘−𝑝+12} , 06𝑠 <

𝑝²−1 2 .

These elements are spread evenly (without jumps) and unitary in𝑀_2𝑘(Γ₀(𝑝)). As such, they are potential candidates to be the first ^𝑝2₂⁻¹ elements ofB₂𝑘(Γ₀(𝑝)). We can now give a more precise version of Theorem 2.5:

Theorem 5.7. Let𝑝 >5be a prime number. Then for all𝑘∈N^∗such that𝑘> ^𝑝−1₂ , 𝑀_2𝑘(Γ₀(𝑝))= Δ𝑝. 𝑀_{2𝑘−(𝑝−1)}(Γ₀(𝑝)) ⊕span

𝐸^(𝑠)

𝑝+1, 𝑝[𝐸⁽⁰⁾

2, 𝑝]^{𝑘−𝑝+12} /06𝑠 <

𝑝²−1 12

. Therefore, if𝑘 ∈N^∗is such that𝑘=𝑞^𝑝−1

2 +𝑟with16𝑟 6 ^𝑝−1₂ , 𝑀_2𝑘(Γ₀(𝑝))

= Δ^𝑞𝑝. 𝑀_2𝑟(Γ₀(𝑝))

𝑞−1

Ê

𝑛=0

Δ^𝑛𝑝.span

𝐸^(𝑠)

𝑝+1, 𝑝[𝐸⁽⁰⁾

2, 𝑝]^{𝑘−(𝑛+1)𝑝−12 −1}/06𝑠 <

𝑝²−1 12

.

In order to get a unitary upper triangular basisB₂𝑘(Γ₀(𝑝)),𝑘 >1, Theorem 2.5 is now operational since the knowledge of all bases is reduced to the knowledge of the finite family of bases(B₂𝑘(Γ₀(𝑝)))_16𝑘6𝑝+1

2 . 6.

Strong modular units

,

In Section 5, we derived structured bases of (𝑀_2𝑘(Γ₀(𝑝)))𝑘∈N^∗ when𝑝is 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 formΔ𝑝. The next logical step is thus to establish the existence of strong modular formsΔ𝑁 in the general case𝑁>1. With this in mind, the Definition 4.1 ofΔ𝑝, for𝑝>5 prime, lead to defining the family of functions𝜂_𝑘:

Notation 6.1. For any𝑘∈N^∗,

∀𝜏∈ H, 𝜂_𝑘(𝜏)=𝜂(𝑘 𝜏)^𝑘.

Additionally, the empirical search of strong modular units(Δ𝑁)₁₆𝑁610(of minimal weight) lead to the following notations:

Notation 6.2. Let𝑁 ∈N^∗be an integer, with𝑁 = 𝑝^𝑟1

1 . . . 𝑝^𝑟_𝑛^𝑛,(𝑟₁, . . . , 𝑟_𝑛) ∈ (N^∗)^𝑛its prime factors decomposition. Let𝑅=𝑅(𝑁)=𝑝₁. . . 𝑝_𝑛be the radical of𝑁. We can now define

Λ𝑅(𝜏)=Ö

𝑚|𝑅

𝜂(𝑚 𝜏)^{𝑚 𝜇𝑚}=Ö

𝑚|𝑅

𝜂_𝑚(𝜏)^𝜇𝑚 and

Λ𝑁(𝜏)= Λ𝑅

𝑁 𝑅 𝜏

= Λ𝑅(𝑝

𝑟₁−1

1 . . . 𝑝^𝑟_𝑛^𝑛−1𝜏)=Ö

𝑚|𝑅

𝜂_𝑚(𝑝

𝑟₁−1

1 . . . 𝑝^𝑟_𝑛^𝑛−1𝜏)^𝜇𝑚, where𝜇denotes the Möbius function and𝜇_𝑚=𝜇(𝑚), for𝑚∈N^∗.

We can see thatΛ𝑁 is an𝜂-product of level 𝑁and that the two definitions ofΛ𝑁

coincide when𝑁is its own radical. The weight ofΛ𝑁 is given by:

1 2

∑︁

𝑑|𝑅

𝑑𝜇_𝑑 = 1 2

∑︁

(𝜀1, ... 𝜀𝑛) ∈ {0,1}^𝑛

(−𝑝₁)^𝜀1. . .(−𝑝_𝑛)^𝜀𝑛 = (−1)^𝑛 2

𝑛

Ö

𝑖=1

(𝑝_𝑖−1). Table 6.1 presents the minimal strong modular units of level𝑁for 16𝑁610 empirically found.

As suggested above, we will show, for𝑁 ∈N^∗, that there exists𝛼∈Z^∗such thatΛ^𝛼_𝑁 is a strong modular unit of level𝑁. To that end, we will systematically apply Theorem 3.5

Table 6.1. Empirical table of minimal strong modular units of level𝑁 for 16𝑁610

Δ₁(𝜏)= Λ²⁴₁ (𝜏)=𝜂(𝜏)²⁴=𝑞

+∞

Ö

𝑖=1

(1−𝑞^𝑖)²⁴

Δ₂(𝜏)= Λ⁻₂⁸(𝜏)=𝜂(𝜏)⁻⁸𝜂(2𝜏)¹⁶=𝑞

+∞

Ö

𝑖=1

(1−𝑞^2𝑖)¹⁶ (1−𝑞^𝑖)⁸ Δ₃(𝜏)= Λ⁻⁶₃ (𝜏)=𝜂(𝜏)⁻⁶𝜂(3𝜏)¹⁸=𝑞²

+∞

Ö

𝑖=1

(1−𝑞^3𝑖)¹⁸ (1−𝑞^𝑖)⁶

Δ₄(𝜏)= Λ⁻₄⁴(𝜏)=𝜂(2𝜏)⁻⁴𝜂(4𝜏)⁸ =𝑞

+∞

Ö

𝑖=1

(1−𝑞^4𝑖)⁸ (1−𝑞^2𝑖)⁴

Δ₅(𝜏)= Λ⁻²₅ (𝜏)=𝜂(𝜏)⁻²𝜂(5𝜏)¹⁰=𝑞²

+∞

Ö

𝑖=1

(1−𝑞^5𝑖)¹⁰ (1−𝑞^𝑖)²

Δ₆(𝜏)= Λ²₆(𝜏)=𝜂(𝜏)²𝜂(2𝜏)⁻⁴𝜂(3𝜏)⁻⁶𝜂(6𝜏)¹²=𝑞²

+∞

Ö

𝑖=1

(1−𝑞^𝑖)²(1−𝑞^6𝑖)¹² (1−𝑞^2𝑖)⁴(1−𝑞^3𝑖)⁶

Δ₇(𝜏)= Λ⁻²₇ (𝜏)=𝜂(𝜏)⁻²𝜂(7𝜏)¹⁴=𝑞⁴

+∞

Ö

𝑖=1

(1−𝑞^7𝑖)¹⁴ (1−𝑞^𝑖)² Δ8(𝜏)= Λ⁻⁴₈ (𝜏)=𝜂(4𝜏)⁻⁴𝜂(8𝜏)⁸ =𝑞²

+∞

Ö

𝑖=1

(1−𝑞^8𝑖)⁸ (1−𝑞^4𝑖)⁴

Δ₉(𝜏)= Λ⁻₉²(𝜏)=𝜂(3𝜏)⁻²𝜂(9𝜏)⁶ =𝑞²

+∞

Ö

𝑖=1

(1−𝑞^9𝑖)⁶ (1−𝑞^3𝑖)²

Δ₁₀(𝜏)= Λ²₁₀(𝜏)=𝜂(𝜏)²𝜂(2𝜏)⁻⁴𝜂(5𝜏)⁻¹⁰𝜂(10𝜏)²⁰=𝑞⁶

+∞

Ö

𝑖=1

(1−𝑞^𝑖)²(1−𝑞^10𝑖)²⁰ (1−𝑞^2𝑖)⁴(1−𝑞^5𝑖)¹⁰ whose assumptions generate exceptions that should be treated separately, when𝑛∈ {1,2}

and𝑝∈ {2,3}. Let us now translate Theorem 3.5 for functionsΛ^𝛼_𝑁. Theorem 6.3. Let𝑁 ∈N^∗be an integer,𝑁=𝑝^𝑟1

1 . . . 𝑝^𝑟_𝑛^𝑛,(𝑟₁, . . . , 𝑟_𝑛) ∈ (N^∗)^𝑛its prime factors decomposition, and𝛼∈Z^∗. IfΛ^𝛼_𝑁 satisfies the three conditions

(1) 𝑃(Λ^𝛼_𝑁)=𝑃(Λ𝑁)^𝛼 =Î^𝑛

𝑖=1𝑝

𝛿_𝑖

𝑖 ∈Q², with𝛿_𝑖 =−𝛼Î

16𝑗6𝑛,𝑗≠𝑖(1−𝑝_𝑗)for 16𝑖6𝑛,

(2) ord(Λ^𝛼_𝑁,∞)=𝛼ord(Λ𝑁,∞)=𝛼 ^𝑁

𝑅(𝑁) (−1)^𝑛

24

Î

16𝑖6𝑛(𝑝²

𝑖 −1) ∈N^∗,

(3) 𝑊(Λ^𝛼_𝑁)=(−1)^{𝑛 𝛼}₂ Î^𝑛

𝑖=1(𝑝_𝑖−1) ∈2N^∗

thenΛ^𝛼_𝑁 is a strong modular unit of level𝑁and of weight(−1)^{𝑛 𝛼}₂ Î^𝑛

𝑖=1(𝑝_𝑖−1).

We can already notice that the structure of theΛ^𝛼_𝑁 functions leads to the automatic satisfaction of hypothesis (iii) of Theorem 3.5.

Proof. Let 𝑅= 𝑝₁. . . 𝑝_𝑛be the radical of𝑁 and𝑀 = ^𝑁_𝑅. Following the notations of Theorem 3.5,Λ_𝑁^𝛼 is an𝜂-product of level𝑁with𝑎_𝑚=0 except if𝑚=𝑀 𝑑where𝑑|𝑅. In this case,𝑎_𝑚=𝛼 𝜇_𝑑𝑑.

(1) First, we have 𝑃(Λ^𝛼_𝑁)=Ö

𝑑|𝑅

𝑁 𝑀 𝑑

^{𝛼𝑑 𝜇}𝑑

=Ö

𝑑|𝑅

(𝑑⁰)^{𝛼 𝑑 𝜇𝑑} =𝑃(Λ^𝛼_𝑅)=

𝑛

Ö

𝑖=1

𝑝^𝛿𝑖

𝑖 . By symmetry, it is enough to study𝛿₁.

𝛿₁ =−𝛼

∑︁

(𝜀₂, ... 𝜀_𝑛) ∈ {0,1}^𝑛−1

(−𝑝₂)^𝜀2. . .(−𝑝_𝑛)^𝜀𝑛 =−𝛼

𝑛

Ö

𝑖=2

(1−𝑝_𝑖).

We deduce the equivalence between (i) and (1) for the functionsΛ^𝛼_𝑁. (2) Then,

ord(Λ^𝛼𝑁,∞)= 1 24

∑︁

𝑑|𝑅

(𝑀 𝑑)𝛼 𝜇_𝑑𝑑

= 1 24𝛼 𝑀

∑︁

𝑑|𝑅

𝑑²𝜇_𝑑

= 1 24𝛼 𝑀

∑︁

(𝜀₁, ... 𝜀_𝑛) ∈ {0,1}^𝑛

(−𝑝²

1)^𝜀1. . .(−𝑝²_𝑛)^𝜀𝑛

= (−1)^𝑛 24 𝛼 𝑀

𝑛

Ö

𝑖=1

(𝑝²

𝑖 −1).

We deduce the equivalence between (ii) and (2) for the functionsΛ^𝛼_𝑁.

(3) Let us check assumption (iii) is satisfied for all functionsΛ^𝛼_𝑁. For𝑐∈ {1, . . . , 𝑁−1},

24 ord

Λ^𝛼_𝑁,1 𝑐

= ∑︁

𝑚|𝑁

gcd(𝑐, 𝑚)² 𝑚

𝑎_𝑚=𝛼 𝑅 𝑁

∑︁

𝑑|𝑅

gcd

𝑐, 𝑁 𝑅 𝑑

2

𝜇_𝑑.

We can then write𝑐=e𝑐 𝑝^𝑠1

1 . . . 𝑝^𝑠_𝑛^𝑛 with gcd(e𝑐, 𝑅) =1, showing there exists𝑖, with 16𝑖6𝑛, such that𝑠_𝑖 < 𝑟_𝑖. Let us assume, for example, that𝑠₁< 𝑟₁and define𝐷₀={𝑑/𝑑|𝑝₂. . . 𝑝_𝑛}and𝐷₁={𝑝₁𝑑/𝑑|𝑝₂. . . 𝑝_𝑛}that together form a partition of𝐷={𝑑/𝑑|𝑝₁. . . 𝑝_𝑛}.

For𝑑 =𝑝^𝜀2

2 . . . 𝑝_𝑛^𝜀𝑛 ∈𝐷₀,(𝜀₂, . . . , 𝜀_𝑛) ∈ {0,1}^𝑛−1, we notice that gcd

𝑐,

𝑁 𝑅 𝑑

=gcd(𝑝^𝑠1

1 . . . 𝑝^𝑠𝑛

𝑛 , 𝑝^𝑟1−1

1 𝑝^{𝑟2−1+𝜀2}

2 . . . 𝑝^{𝑟𝑛−1+𝜀𝑛}

𝑛 )

= 𝑝^𝑠1

1 gcd(𝑝^𝑠2

2 . . . 𝑝^𝑠𝑛

𝑛 , 𝑝^{𝑟2−1+𝜀2}

2 . . . 𝑝^{𝑟𝑛−1+𝜀𝑛}

𝑛 ),

Furthermore, gcd

𝑐,

𝑁 𝑅

𝑝₁𝑑

=gcd(𝑝

𝑠₁

1 . . . 𝑝^𝑠_𝑛^𝑛, 𝑝

𝑟₁ 1 𝑝

𝑟₂−1+𝜀₂

2 . . . 𝑝^𝑟_𝑛^{𝑛−1+𝜀𝑛})

=𝑝^𝑠1

1 gcd(𝑝^𝑠2

2 . . . 𝑝^𝑠𝑛

𝑛 , 𝑝^{𝑟2−1+𝜀2}

2 . . . 𝑝^{𝑟𝑛−1+𝜀𝑛}

𝑛 ).

The two terms are therefore equal, leading to the last equality needed to finish the proof:

∑︁

𝑑|𝑅

gcd

𝑐, 𝑁 𝑅 𝑑

2

𝜇_𝑑 = ∑︁

𝑑∈𝐷₀

gcd

𝑐, 𝑁 𝑅 𝑑

2

𝜇_𝑑+ ∑︁

𝑑∈𝐷₀

gcd

𝑐, 𝑁 𝑅

𝑝₁𝑑 2

𝜇_𝑝

1𝑑=0.

The strong modular unitsΔ𝑁 will be expressed usingΛ^𝛼_𝑁 functions. The following result reduces the general case to the case𝑁=𝑅(𝑁).

Corollary 6.4. Let𝑁 ∈Nbe an integer,𝑁>2with𝑁=𝑝^𝑟1

1 . . . 𝑝^𝑟_𝑛^𝑛,(𝑟₁, . . . , 𝑟_𝑛) ∈ (N^∗)^𝑛 its prime factors decomposition,𝑅the radical of𝑁and𝛼∈Z^∗. If the functionΛ_𝑅^𝛼satisfies the hypotheses of Theorem 6.3, thenΛ^𝛼_𝑁 is a strong modular unit of the same weight with respect toΓ₀(𝑁).

Proof. Verifying thatΛ^𝛼_𝑁 satisfies the assumptions of Theorem 6.3 is enough:

𝑃(Λ_𝑁^𝛼)=𝑃(Λ_𝑅^𝛼) ∈Q², ord(Λ^𝛼𝑁,∞)= 𝑁

𝑅 ord(Λ𝑅^𝛼,∞) ∈N^∗ and 𝑊(Λ^𝛼_𝑁)=𝑊(Λ_𝑅^𝛼) ∈2N^∗.

This proves thatΛ^𝛼_𝑁 is a strong modular unit of level𝑁.

The case

,

prime and

As always, we need to separate cases𝑝=2, 𝑝=3 and𝑝>5. The result is as follows: