Universal Fourier expansions of modular forms

Loïc Merel

Introduction

Let X be the set of matrices ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ M_{2}(ℤ) such that a > b ≥ 0 and d > c ≥ 0.

Let us consider the following series in ℂ[M_{2}(ℤ)][[q]]:

∑_{M∈X} M q^{detM}

= ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ) q
+ (( \begin{matrix}2 & 0 \\ 0 & 1 \end{matrix} ) + ( \begin{matrix}1 & 0 \\ 0 & 2 \end{matrix} ) + ( \begin{matrix}2 & 1 \\ 0 & 1 \end{matrix} ) + ( \begin{matrix}1 & 0 \\ 1 & 2 \end{matrix} )) q^{2}
+ (( \begin{matrix}3 & 0 \\ 0 & 1 \end{matrix} ) + ( \begin{matrix}1 & 0 \\ 0 & 3 \end{matrix} ) + ( \begin{matrix}3 & 1 \\ 0 & 1 \end{matrix} ) + ( \begin{matrix}1 & 0 \\ 1 & 3 \end{matrix} ) + ( \begin{matrix}3 & 2 \\ 0 & 1 \end{matrix} ) + ( \begin{matrix}2 & 1 \\ 1 & 2 \end{matrix} ) + ( \begin{matrix}1 & 0 \\ 2 & 3 \end{matrix} )) q^{3}
+ ...

The aim of this paper is to establish that this series produces (in a sense that will be made precise in a moment) Fourier expansions at infinity of modular forms of integral weight ≥ 2 for congruence subgroups of SL_{2}(ℤ). This justifies the terminology “universal Fourier expansions of modular forms”. Here and in what follows “Fourier expansion” will always mean “Fourier expansion at infinity”.

Let k be an integer ≥ 2. Let N be an integer > 0. Let ℂ_{k−2}[X, Y] be the complex vector space of homogeneous polynomials in two variables and degree k − 2. Let Γ_{0}(N)

(resp. Γ_{1}(N)) be the group of matrixes ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ SL_{2}(ℤ) such that N|c (resp. N|c,

N|(a − 1)). Let χ be a Dirichlet character modulo N. We S_{k}(N) (resp. S_{k}(N, χ)) the complex vector space of cusp forms of weight k for Γ_{1}(N) (resp. for Γ_{0}(N) with multiplicative character χ, see section 2.5).

Let ℂ_{k−2}[X, Y][(ℤ/Nℤ)^{2}] be the vector space of linear combinations of elements of (ℤ/Nℤ)^{2} with coefficients in ℂ_{k−2}[X, Y]. If ϕ is a linear map ℂ_{k−2}[X, Y][(ℤ/Nℤ)^{2}] → ℂ and g = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ M_{2}(ℤ), we denote by ϕ_{|g} the linear map ℂ_{k−2}[X, Y][(ℤ/Nℤ)^{2}] → ℂ

defined by the formula

ϕ_{|g}(P (X, Y)[u, v]) = ϕ(P (aX + bY, cX + dY)[au + cv, bu + dv]),

P ∈ ℂ_{k−2}[X, Y], (u, v) ∈ (ℤ/Nℤ)^{2}. We denote by E_{N} the set of elements (u, v) of (ℤ/Nℤ)^{2} satisfying the relation ℤu + ℤv = ℤ/Nℤ.

Let us denote by P_{N} = ∪_{d|N}(ℤ/dℤ)^{∗}. By convention when d = 1, (ℤ/dℤ)^{∗} has one element. Let ℂ[P_{N}]_{k} be the quotient vector space of ℂ[P_{N}] modulo the vector space generated by the elements of the form [a] − (−1)^{k}[−a] = 0, a ∈ (ℤ/dℤ)^{∗}, d|N. If a ∈ ℤ/Nℤ, d|N and a invertible modulo d, we denote by [a]_{d} the image of [a (mod d)] in ℂ[P_{N}]_{k}.

Let b : ℂ_{k−2}[X, Y][E_{N}] → ℂ[P_{N}]_{k} be the ℂ-bilinear map which associates to P (X, Y)[u, v] the element P (1, 0)[v^{−1}]_{(u,N)} − P (0, 1)[u^{−1}]_{(v,N)}, where by abuse of notations v^{−1} is the inverse modulo (u, N) of v and (u, N) is the greater common divisor of u and N, i.e. the order of the sugroup of ℤ/Nℤ generated by u.

Theorem 1 Let ϕ be a linear map ℂ_{k−2}[X, Y][(ℤ/Nℤ)^{2}] → ℂ verifying the following equalities

ϕ + ϕ_{|σ} = ϕ + ϕ_{|τ} + ϕ_{|τ^{2}} = ϕ − ϕ_{|J} = 0,

where σ = ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ), τ = ( \begin{matrix}0 & −1 \\ 1 & −1 \end{matrix} ) and J = ( \begin{matrix}−1 & 0 \\ 0 & −1 \end{matrix} ), and ϕ(P [u, v]) = 0 if

(u, v) ∉ E_{N}, P ∈ ℂ_{k−2}[X, Y]. Let x ∈ ℂ_{k−2}[X, Y][E_{N}] such that b(x) = 0. Then

∑_{M∈X} ϕ_{|M}(x)q^{detM}

is the Fourier expansion of an element f of S_{k}(N). Furthermore all modular forms of such type can be produced by this method.

Let χ be a Dirichlet character ℤ/Nℤ → ℂ. Let us suppose that ϕ satisfies the additional condition

ϕ(P [λu, λv]) = χ(λ)ϕ(P [u, v])

((λ, u, v) ∈ (ℤ/Nℤ)^{3}, P ∈ ℂ_{k−2}[X, Y]). Then f belongs to S_{k}(N, χ).

If x does not satisfy the relation b(x) = 0, the series obtained should be, except for the constant term, the Fourier expansion of a holomorphic modular form of weight k for Γ_{1}(N) (see the remark in the section 3.2). This theorem can be refined to obtain only newforms (see sections 2.6 and 3.2).

We outline now the plan of the paper as well as the plan of the proof of the theorem 1. We recall in the first part the theory introduced by Manin and developed by Shokurov of modular symbols of arbitrary weight for subgroups of finite index of SL_{2}(ℤ) ([4], [12], [11], [14], [13]). This theory can be related to the Eichler-Shimura theory connected with the cohomology of subgroups of finite index of SL_{2}(ℤ). The Eichler-Shimura theory imbeds modular forms in a space of modular symbols; The Manin-Shokurov theory constructs pairings between modular symbols and modular forms. We construct a complex vector space (of modular symbols) M_{k}(N), which is a quotient of ℂ_{k−2}[X, Y][E_{N}] (see section 1.3). We denote by [P, x] the image of P [x] ∈ ℂ_{k−2}[X, Y][E_{N}] in M_{k}(N). There is a bilinear pairing of complex vector spaces between M_{k}(N) and S_{k}(N) ⊕ \overline{S_{k}(N)} (where \overline{S_{k}(N)} is the space of antiholomorphic cusp forms), see section 1.5. This pairing is given as follows:

(f_{1} + f_{2}, [P, x]) ↦ ∫_{0}^{∞} f_{|g}(z)P (z, 1) dz + ∫_{0}^{∞} f_{|g}(z)P (z̄, 1) dz̄,

where f_{1} ∈ S_{k}(N), f_{2} ∈ \overline{S_{k}(N)}, g is any element in SL_{2}(ℤ) such that π(g) = x and

f_{|g}(z) = (cz + d)^{−k}f(gz) and f_{|ḡ}(z) = (cz̄ + d)^{−k}f(gz).

We denote by S_{k}(N) the image in M_{k}(N) of the kernel of b. By a theorem of Shokurov, when restricted to S_{k}(N), the pairing defined above is nondegenerate (see section 1.5). We complete the theory of Shokurov by making explicit the relation between the modular symbols and the cohomology of SL_{2}(ℤ). Moreover the action of the complex conjugation on modular curves defines an involution ι^{∗} on M_{k}(N). This involution is studied in the section 1.6. In fact all the results in the first part of the paper are valid if Γ_{1}(N) is replaced by any subgroup of finite index of SL_{2}(ℤ).

The second part is devoted to the study of Hecke theory on M_{k}(N) (Hecke operators, Atkin-Lehner operators, old and new parts, degeneracy maps). We describe only here our main result. The Hecke operators operate by duality on S_{k}(N). In fact this action extends naturally to M_{k}(N). Let n be an integer > 0. We denote by M_{2}(ℤ)_{n} the set of matrices of M_{2}(ℤ) of determinant n. We say that an element ∑_{M} u_{M}M ∈ ℂ[M_{2}(ℤ)_{n}] satisfies the condition (C_{n}) if for all K ∈ M_{2}(ℤ)_{n}/SL_{2}(ℤ), we have in ℂ[ℙ^{1}(ℚ)]

∑_{M∈K} u_{M}([M∞] − [M0]) = [∞] − [0].

We denote by T_{n} the Hecke operator on M_{k}(N). Our main result about Hecke operators is as follows.

Theorem 2 Let P [u, v] ∈ ℂ_{k−2}[X, Y][E_{N}]. Let ∑_{M} u_{M}M ∈ ℂ[M_{2}(ℤ)_{n}] satisfying the condition (C_{n}). Then we have (the sum is taken with respect to the matrixes M =

( \begin{matrix}a & b \\ c & d \end{matrix} ))

T_{n}([P, (u, v)]) = ∑_{M} u_{M}[P (aX + bY, cX + dY), (au + cv, bu + dv)],

where the sum is restricted to the matrices M such that (au + cv, bu + dv) ∈ E_{N} (if n and N are coprimes this restriction is unnecessary).

We remark that the condition (C_{n}) depends neither on the level N nor on the weight k. This theorem can be understood as a formula on linear forms on modular forms without knowing anything about modular symbols (see the pairing defined above). We give an analogous formula for the action of Atkin-Lehner operators on M_{k}(N) (see section 2.4). Let X_{n} be the set of matrices of X of determinant n. Then

∑_{M∈X_{n}} M

satisfies the condition (C_{n}) (see the section 3.1). The theorem 2 extends some results obtained in my thesis for the weight 2 (see [8], [7], [9]). It extends also a result in [16] concerning only the modular forms for the full modular group SL_{2}(ℤ). We shall add that other elements of ℂ[M_{2}(ℤ)_{n}] satisfy the condition (C_{n}) (see part 3).

For x ∈ ℂ_{k−2}[X, Y][E_{N}] satisfying b(x) = 0 we can consider its image m(x) in S_{k}(N). A complex linear form ϕ on ℂ_{k−2}[X, Y][E_{N}] satisfying the conditions of the theorem 1 factorizes through a linear form on M_{k}(N), which induces a linear form ϕ_{m} on S_{k}(N). We obtain a linear form α(ϕ, x) on the Hecke algebra (i.e. the ℂalgebra generated by the Hecke operators T_{n} and T_{n,n} of End_{ℂ}(S_{k}(N)) for n ≥ 1) which associates to T the complex number ϕ_{m}(T m(x)). We then use the following fact: if α is a linear map from the Hecke algebra to ℂ, then ∑_{n=1}^{∞} α(T_{n})q^{n} is the Fourier expansion of an element of S_{k}(N). We apply this principle for α = α(ϕ, x) to obtain the theorem 1.

As an application of the theorem 1, we give a conditional method to construct bases of S_{k}(N) (see section 3.3).

In the course of the completion of this work, I was generously invited several times to the Institut für experimentelle Mathematik. I would like to thank this institution for its nice hospitality.

1 The theory of Manin-Shokurov

1.1 The algebraic description of modular symbols

Let Γ be a subgroup of finite index of SL_{2}(ℤ) and k an integer ≥ 2. Ik k is odd, we

impose the following condition: J = ( \begin{matrix}−1 & 0 \\ 0 & −1 \end{matrix} ) ∉ Γ (otherwise the following theory

is empty).

First we consider the torsion free abelian group M generated by the expressions {α, β} ((α, β) ∈ ℙ^{1}(ℚ)^{2}) with the following relations

{α, β} + {β, γ} + {γ, α} = 0 ((α, β, γ) ∈ ℙ^{1}(ℚ)^{3}).

So we have {α, β} = −{β, α} and {α, α} = 0. We consider now the complex vector space

M_{k} = ℂ_{k−2}[X, Y] ⊗ M,

where ℂ_{k−2}[X, Y] is the space of complex homogeneous polynomials in two variables of

degree k − 2. We define a linear action of g = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ GL_{2}(ℚ) on P ∈ ℂ_{k−2}[X, Y] and P ⊗ {α, β} ∈ M_{k} by the rules

P_{|g}(X, Y) = P (dX − bY, −cX + aY)

and

(P ⊗ {α, β})_{|g} = P_{|g} ⊗ {gα, gβ}.

For (g, h) ∈ GL_{2}(ℚ)^{2}, we have (P_{|g})_{|h} = P_{|hg}. Let M_{k}(Γ) be the quotient vector space of M_{k} obtained by imposing the relations x_{|γ} = x (x ∈ M_{k}, γ ∈ Γ). We will denote by P {α, β} the image of P ⊗ {α, β} ∈ M_{k} in M_{k}(Γ). The elements of M_{k}(Γ) will be called modular symbols of weight k for Γ.

Remark .- We can replace the field ℂ in this construction by any ring. In particular, the space M_{k}(Γ) carries a natural integral structure. The subgroup ℤ_{k−2}[X, Y] ⊗ M of M_{k} is stable by the action of GL_{2}(ℚ) ∩ M_{2}(ℤ). It makes sense to consider its image M_{k}(Γ, ℤ) in M_{k}(Γ). The complex vector space M_{k}(Γ) is canonically isomorphic to M_{k}(Γ, ℤ) ⊗ ℂ.

1.2 The Manin symbols

Let g ∈ SL_{2}(ℤ) and P ∈ ℂ_{k−2}[X, Y]. We introduce the Manin symbol [P, g] ∈ M_{k}(Γ) by the formula

[P, g] = P_{|g}{g0, g∞}.

We recall the notations of the introduction: σ = ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ), τ = ( \begin{matrix}0 & −1 \\ 1 & −1 \end{matrix} ) and

J = ( \begin{matrix}−1 & 0 \\ 0 & −1 \end{matrix} ). These matrices verify σ^{2} = J, J^{2} = ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ) and τ^{3} = ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ).

Proposition 1 Let g ∈ SL_{2}(ℤ) and P ∈ ℂ_{k−2}[X, Y]. The Manin symbol [P, g] depends only on the class Γg and on P. When g runs through SL_{2}(ℤ) and when P runs through ℂ_{k−2}[X, Y], the Manin symbols generate M_{k}(Γ). Furthermore they verify the following equalities:

[P, g] + [P_{|σ^{−1}}, gσ] = 0,
[P, g] + [P_{|τ^{−1}}, gτ] + [P_{|τ^{−2}}, gτ^{2}] = 0

and

[P, g] = [P_{|J}, −g].

Proof .- The first assertion is a consequence of the construction of M_{k}(Γ). To prove the second assertion, we make use of a theorem of Manin which asserts that the elements {g0, g∞}, for g ∈ SL_{2}(ℤ) generate M (This is known as “Manin’s trick” [4], proposition 1.6. The proof of that assertion relies on continued fractions expansion). It follows quickly that the Manin symbols generate M_{k}(Γ). To prove the first two equalities we remark the following relations in ℙ^{1}(ℚ): ∞ = σ0 = σ^{2}∞ and ∞ = τ1 = τ^{2}0. We have

[P, g] + [P_{|σ^{−1}}, gσ] = P_{|g}{g0, g∞} + (P_{|σ^{−1}})_{|gσ}{gσ0, gσ∞}
= P_{|g}{g0, g∞} + P_{|gσσ^{−1}}{g∞, g0}
= 0

and

[P, g] + [P_{|τ^{−1}}, gτ] + [P_{|τ^{−2}}, gτ^{2}]
= P_{|g}{g0, g∞} + (P_{|τ^{−1}})_{|gτ} {gτ0, gτ∞} + (P_{|τ^{−2}})_{|gτ^{2}}{gτ^{2}0, gτ^{2}∞}
= P_{|g}({g0, g∞} + {g1, g0} + {g∞, g1}
= 0.

The third equality follows from the fact that the matrix J operates trivially on ℙ^{1}(ℚ).

Remark .- The Manin symbols can be written as complex linear combinations of Manin symbols of type [P, g], where P is a monomial of the form X^{q}Y^{k−2−q} (q ∈ {0, 1, ..., k − 2}). Since these monomials are in finite number and since Γ\SL_{2}(ℤ) is finite, M_{k}(Γ) is generated by a finite number of elements. So it is a finite dimensional complex vector space.

1.3 Comparison with the theory of Shokurov

We recall the theory of Shokurov (see [12], [11], [14], [13], especially the lemmadefinition 1.2 in the last paper). We use here the notations of Shokurov. Let ℌ be the Poincaré half-plane. Let Δ_{Γ} be the compactification of the Riemann surface Γ\ℌ. Let Π be the set of cusps of Δ_{Γ}. Shokurov considers a certain locally constant sheaf, which he denotes (R_{1}ϕ_{∗}ℚ)^{w}, on the modular curve Δ_{Γ} (where w = k − 2, see [13]). This sheaf is the usual sheaf on the modular curve Δ_{Γ} constructed from the space of modular forms of weight k for Γ; But there is no need to describe it here.

Let (α, β) ∈ (ℙ^{1}(ℚ))^{2} and (n, m) ∈ (ℤ^{k−2})^{2}. The modular symbol of Shokurov {α, β, n, m}_{Γ} (which we will call provisionnally Shokurov symbol) lies in the homology group H_{1}(Δ_{Γ}, Π; (R_{1}ϕ_{∗}ℚ)^{w}), which is a ℚ-vector space. The Shokurov symbols enjoy the following properties (see lemma-definition 1.2 of [13]):

• The Shokurov symbol {α, β, n, m}_{Γ} depends only on the polynomial ∏_{i=1}^{k−2}(n_{i}X + m_{i}Y) ∈ ℂ_{k−2}[X, Y], where n = (n_{1}, n_{2}, ..., n_{k−2}) and m = (m_{1}, m_{2}, ..., m_{k−2}).

• They generate the ℚ-vector space H_{1}(Δ_{Γ}, Π; (R_{1}ϕ_{∗}ℚ)^{w}).

• They satisfy the following relations

{α, β, n, m}_{Γ} + {β, γ, n, m}_{Γ} + {γ, α, n, m}_{Γ} = 0,

(α, β, γ) ∈ ℙ^{1}(ℚ)^{3}, (n, m) ∈ (ℤ^{k−2})^{2}.

• For γ = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Γ, we have

{γα, γβ, dn − cm, −bn + am}_{Γ} = {α, β, n, m}_{Γ},

(α, β) ∈ ℙ^{1}(ℚ)^{2}, (n, m) ∈ (ℤ^{k−2})^{2}.

Our present goal is to prove that the Shokurov symbols can be identified with our modular symbols. Because of the first property satisfied by the Shokurov symbols, we can denote by {α, β, P } the symbol {α, β, n, m}_{Γ}, where P = ∏_{i=1}^{n}(n_{i}X + m_{i}Y) ∈ ℤ_{k−2}[X, Y]. The last two properties can be rewritten

{α, β, P } + {β, γ, P } + {γ, α, P } = 0

and

{γα, γβ, P_{|γ}} = {α, β, P }.

We deduce from these properties that there exists a surjective homomorphism of complex vector spaces:

s_{1} : M_{k}(Γ) → H_{1}(Δ_{Γ}, Π; (R_{1}ϕ_{∗}ℚ)^{w}) ⊗_{ℚ} ℂ

which associates P {α, β} to {α, β, P } ⊗ 1, for P of the form ∏_{i=1}^{k−2}(n_{i}X + m_{i}Y) ∈ ℤ_{k−2}[X, Y], where n = (n_{1}, n_{2}, ..., n_{k−2}) ∈ ℤ^{k−2} and m = (m_{1}, m_{2}, ..., m_{k−2}) ∈ ℤ^{k−2}.

Proposition 2 The linear map s_{1} is an isomorphism of complex vector spaces.

Proof .- For j ∈ Γ\SL_{2}(ℤ), and q ∈ {0, 1, ..., k − 2}, Shokurov denotes by ξ(j, q) (and calls marked class) the element s_{1}([X^{q}Y^{k−2−q}]) of H_{1}(Δ_{Γ}, Π; (R_{1}ϕ_{∗}ℚ)^{w}) ⊗_{ℚ} ℂ, where g is any element of the class j.

After extending the scalars from ℚ to ℂ, the theorem 2.3 of [13] asserts that the kernel of the linear map

ℂ_{k−2}[X, Y][Γ\SL_{2}(ℤ)] → H_{1}(Δ_{Γ}, Π; (R_{1}ϕ_{∗}ℚ)^{w}) ⊗_{ℚ} ℂ

which associates to P [Γg] the element s_{1}([P, g]) is generated by the elements s_{1}([P, g])+ s_{1}([P_{|σ^{−1}}, gσ]), s_{1}([P, g])+s_{1}([P_{|τ^{−1}}, gτ])+s_{1}([P_{|τ^{−2}}, gτ^{2}]) and s_{1}([P, g])−s_{1}([P_{|J}, −g]), where P is a monomial, i.e. s_{1}([P, g]) is a marked class. It is a consequence of the proposition 1 that this linear map factorizes through

ℂ_{k−2}[X, Y][Γ\SL_{2}(ℤ)] → M_{k}(Γ) →^{s_{1}} H_{1}(Δ_{Γ}, Π; (R_{1}ϕ_{∗}ℚ)^{w}) ⊗_{ℚ} ℂ,

where the first linear map associates to P [Γg] the Manin symbol [P, g]. By considering the kernels of these linear maps, we obtain that s_{1} is injective.

Let ℂ_{k−2}[X, Y][Γ\SL_{2}(ℤ)]_{k} be the quotient vector space of ℂ_{k−2}[X, Y][Γ\SL_{2}(ℤ)] obtained by imposing the relations P [j] = P_{|J}[j(J)], j ∈ Γ\SL_{2}(ℤ), P ∈ ℂ_{k−2}[X, Y]. We define a linear action on the right of SL_{2}(ℤ) on ℂ_{k−2}[X, Y][Γ\SL_{2}(ℤ)]_{k} by the rule

(P [Γg])γ = P_{|γ^{−1}}[Γgγ],

P ∈ ℂ_{k−2}[X, Y], (g, γ) ∈ SL_{2}(ℤ)^{2}.

We denote by ℂ_{k−2}[X, Y][Γ\SL_{2}(ℤ)]_{k}^{σ} (resp. ℂ_{k−2}[X, Y][Γ\SL_{2}(ℤ)]_{k}^{τ}) the set of elements of ℂ_{k−2}[X, Y][Γ\SL_{2}(ℤ)]_{k} invariant under the action of σ (resp. τ).

Proposition 3 We have the following exact sequences of complex vector spaces

0 → ℂ_{k−2}[X, Y][Γ\SL_{2}(ℤ)]_{k}^{σ} × ℂ_{k−2}[X, Y][Γ\SL_{2}(ℤ)]_{k}^{τ} →^{i}

ℂ_{k−2}[X, Y][Γ\SL_{2}(ℤ)]_{k} →^{m_{k}} M_{k}(Γ) → 0,

if k > 2, and

0 → ℂ →^{ϵ} ℂ_{k−2}[X, Y][Γ\SL_{2}(ℤ)]_{k}^{σ} × ℂ_{k−2}[X, Y][Γ\SL_{2}(ℤ)]_{k}^{τ} →^{i}

ℂ_{k−2}[X, Y][Γ\SL_{2}(ℤ)]_{k} →^{m_{k}} M_{k}(Γ) → 0,

if k = 2, where m_{k} is the map which associates to P [Γg] the Manin symbol [P, g], i associates to (a, b) the element a + b, in the last exact sequence the linear map ϵ associates to λ the element (∑_{x∈E_{N}} λ[x], ∑_{x∈E_{N}} λ[x]).

Proof .- By a theorem of Shokurov (see theorem 2.3 of [13], this result has already been used in the proof of the proposition 2), the kernel of m_{k} is equal to the sum of the complex vector spaces ℂ_{k−2}[X, Y][Γ\SL_{2}(ℤ)]_{k}^{σ} and ℂ_{k−2}[X, Y][Γ\SL_{2}(ℤ)]_{k}^{τ}. The surjectivity of m_{k} is proved by the proposition 1. It remains to prove that the intersection of ℂ_{k−2}[X, Y][Γ\SL_{2}(ℤ)]_{k}^{σ} and ℂ_{k−2}[X, Y][Γ\SL_{2}(ℤ)]_{k}^{τ} is equal to {0} if k > 2 and is equal to the image of ϵ if k = 2. Let ∑_{j∈Γ\SL_{2}(ℤ)} P_{j}[j] ∈ ℂ_{k−2}[X, Y][Γ\SL_{2}(ℤ)]_{k}^{σ} ∩ ℂ_{k−2}[X, Y][Γ\SL_{2}(ℤ)]_{k}^{τ}. We have P_{j|σ} = P_{(jσ)} and P_{j|τ} = P_{(jτ)} for all j ∈ Γ\SL_{2}(ℤ). Since σ and τ generate SL_{2}(ℤ) (see [10]), we have P_{(jg)} = P_{j} = P_{j|g} for all j ∈ Γ\SL_{2}(ℤ) and all g ∈ SL_{2}(ℤ). This implies that P_{j} is a constant polynomial, i.e. P = 0 or k = 2. If k = 2, then P_{j} is a complex number independent of j (SL_{2}(ℤ) operates transitively on Γ\SL_{2}(ℤ)). So ∑_{j∈Γ\SL_{2}(ℤ)} P_{j}[j] is in the image of ϵ. This proves the proposition.

1.4 The boundary modular symbols

Let B be the abelian group generated by the elements {α} (α ∈ ℙ^{1}(ℚ)). Let B_{k} be the complex vector space ℂ_{k−2}[X, Y] ⊗ B. We define a linear action of g ∈ GL_{2}(ℚ) on P ⊗ {α, β} ∈ B_{k} by the formula

(P ⊗ {α})_{|g} = P_{|g} ⊗ {gα}.

We define B_{k}(Γ) as the complex vector space quotient of B_{k} obtained by imposing the relations

(P ⊗ {α})_{|γ} = P ⊗ {α},

for γ ∈ Γ.

For α ∈ ℙ^{1}(ℚ) and (n, m) ∈ (ℤ^{k−2})^{2}, Shokurov introduces the boundary symbol {α, n, m}_{Γ} which is an element of the ℚ-vector space H_{0}(Π; (R_{1}ϕ_{∗}ℚ)^{w}). This symbol

depends only on α and on ∏_{i=1}^{k−2}(n_{i}X + m_{i}Y) ∈ ℂ_{k−2}[X, Y]. For γ = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Γ, we

have

{γα, dn − cm, −bn + am}_{Γ} = {α, n, m}_{Γ}.

These symbols generate the ℚ-vector space H_{0}(Π; (R_{1}ϕ_{∗}ℚ)^{w}). See the lemma-definition 1.2 of [13] for all these properties. We will denote by {α, ∏_{i=1}^{k−2}(n_{i}X + m_{i}Y)} the boundary symbol {α, n, m}_{Γ}.

Let R_{k} be the equivalence relation in ℂ[Γ\ℚ^{2}] which identifies the element [Γ ( \begin{matrix}λu \\ λv \end{matrix} )]

with (\frac{λ}{|λ|})^{k}[Γ ( \begin{matrix}u \\ v \end{matrix} )] (λ ∈ ℚ − \{0\}). We denote by ℂ[Γ\ℚ^{2}]_{k} the finite dimensional com-

plex vector space ℂ[Γ\ℚ^{2}]/R_{k}. If k is even, this vector space is canonically isomorphic to ℂ[Γ\ℙ^{1}(ℚ)]. In any event its dimension is equal to |Γ\ℙ^{1}(ℚ)|, i.e. the number of cusps of the modular curve Δ_{Γ}.

Because of the properties satisfied by the boundary symbols of Shokurov there is an unique surjective linear map:

s_{2} : B_{k}(Γ) → H_{0}(Π; (R_{1}ϕ_{∗}ℚ)^{w}) ⊗_{ℚ} ℂ

which associates to P {α} the boundary symbol {α, P } ⊗ 1 for P a polynomial of the form ∏_{i=1}^{k−2}(n_{i}X + m_{i}Y) ∈ ℂ_{k−2}[X, Y], where n = (n_{1}, n_{2}, ..., n_{k−2}) ∈ ℤ^{k−2} and m = (m_{1}, m_{2}, ..., m_{k−2}) ∈ ℤ^{k−2}.

We consider now the complex linear map

μ : B_{k}(Γ) → ℂ[Γ\ℚ^{2}]_{k},

which associates to P \{\frac{u}{v}\} the element P (u, v)[Γ ( \begin{matrix}u \\ v \end{matrix} )] of ℂ[Γ\ℚ^{2}]_{k}. The fact that μ is

well defined is a part of the following proposition.

Proposition 4 The linear maps s_{2} and μ are isomorphisms of complex vector spaces.

Proof .- Because of the way we constructed B_{k}(Γ), we can decompose it as a direct sum

B_{k}(Γ) = ⊕_{α}B_{k}(Γ)_{α},

where α runs through a set of representatives of Γ\ℙ^{1}(ℚ) and B_{k}(Γ)_{α} is the subspace of B_{k}(Γ) generated by the elements P {α} (P ∈ ℂ_{k−2}[X, Y]). For α ∈ ℙ^{1}(ℚ), we have a surjective linear map

ψ_{α} : ℂ_{k−2}[X, Y] → B_{k}(Γ)_{α}
P ↦ P {α}.

Its kernel is generated by the polynomials of the form P − P_{|g_{α}}, where g_{α} ∈ Γ verifies the equality g_{α}α = α. If we write α = \frac{u}{v} ((u, v) ∈ ℤ^{2}), we find (P − P_{|g_{α}})(u, v) = 0. This proves that μ is well defined. Since the kernel of ψ_{α} contains the kernel of a linear form, its image is at most of dimension 1. The dimension of B_{k}(Γ) is at most |Γ\ℙ^{1}(ℚ)|. Since the dimension of H_{0}(Π; (R_{1}ϕ_{∗}ℚ)^{w}) ⊗_{ℚ} ℂ is exactly |Γ\ℙ^{1}(ℚ)| (see [13], theorem 3.4), the map s_{2} must be bijective, and the dimension of B_{k}(Γ) must be |Γ\ℙ^{1}(ℚ)|. We deduce that the image of each map ψ_{α} is of dimension 1. So the linear map μ must be bijective.

Shokurov considers the canonical boundary map:

∂ : H_{1}(Δ_{Γ}, Π; (R_{1}ϕ_{∗}ℚ)^{w}) ⊗_{ℚ} ℂ → H_{0}(Π; (R_{1}ϕ_{∗}ℚ)^{w}) ⊗_{ℚ} ℂ.

We have ([13], lemma-definition 1.2)

∂({α, β, n, m} ⊗ 1) = {β, n, m} ⊗ 1 − {α, n, m} ⊗ 1.

The kernel of ∂ is canonically isomorphic to H_{1}(Δ_{Γ}; (R_{1}ϕ_{∗}ℚ)^{w}) ⊗ ℂ. By abuse of notations, we still denote by ∂ the linear map M_{k}(Γ) → B_{k}(Γ) obtained from ∂ by the identifications made. We denote by S_{k}(Γ) the kernel of ∂ in M_{k}(Γ). We have

∂(P {α, β}) = P {β} − P {α}.

Proposition 5 We have the exact sequences

0 → S_{k}(Γ) → M_{k}(Γ) →^{∂} B_{k}(Γ) → 0,

if k > 2, and

0 → S_{k}(Γ) → M_{k}(Γ) →^{∂} B_{k}(Γ) →^{θ} ℂ → 0,

if k = 2, where the linear form θ associates to λ{α} the complex number λ.

Proof .- Let us prove that ∂ is surjective if k > 2. Let β = \frac{u}{v} ∈ ℙ^{1}(ℚ). Let λ ∈ ℂ. Let α = \frac{u′}{v′} ∈ ℙ^{1}(ℚ) such that α ≠ β. Let P ∈ ℂ_{k−2}[X, Y] such that P (u, v) = λ and P (u′, v′) = 0 (since k > 2 such a polynomial exists). We have in ℂ[Γ\ℚ^{2}]_{k}

μ ∘ ∂(P {α, β}) = P (u, v)[Γβ] − P (u′, v′)[Γα] = λ[Γβ].

So ∂ is surjective.

If k = 2, the map θ is well defined and the image of ∂ is contained in the kernel of θ. The image of ∂ and the kernel of θ are both generated by elements of the type {β} − {α}. Since θ is surjective we deduce the validity of the second exact sequence.

We give now a formula for the boundary of Manin symbols.

Proposition 6 Let P ∈ ℂ_{k−2}[X, Y] and g ∈ SL_{2}(ℤ). We have in ℂ[Γ\ℚ^{2}]_{k}

μ ∘ ∂([P, g]) = P (1, 0)[Γg ( \begin{matrix}1 \\ 0 \end{matrix} )] − P (0, 1)[Γg ( \begin{matrix}0 \\ 1 \end{matrix} )].

Proof .- We set g = ( \begin{matrix}a & b \\ c & d \end{matrix} ). We have

∂([P, g]) = ∂(P_{|g}\{\frac{b}{d}, \frac{a}{c}\})
= P_{|g}\{\frac{a}{c}\} − P_{|g}\{\frac{b}{d}\}

and

μ ∘ ∂([P, g]) = P_{|g}(a, c)[Γ ( \begin{matrix}a \\ c \end{matrix} )] − P_{|g}(b, d)[Γ ( \begin{matrix}b \\ d \end{matrix} )]
= P (1, 0)[Γg ( \begin{matrix}1 \\ 0 \end{matrix} )] − P (0, 1)[Γg ( \begin{matrix}0 \\ 1 \end{matrix} )].

1.5 Pairings with modular forms

Let f be a map from the Poincaré half plane to ℂ. For g = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ GL_{2}(ℚ) and

z ∈ ℌ, we recall the usual notations

f_{|g}(z) = (cz + d)^{−k}f(gz)(detg)^{\frac{k}{2}}

and

f_{|ḡ}(z) = (cz̄ + d)^{−k}f(gz)(detg)^{\frac{k}{2}} ,

where the horizontal bar above complex numbers denotes the complex conjugation. We denote by S_{k}(Γ) (resp. \overline{S_{k}(Γ)}) the complex vector space of holomorphic (resp. antiholomorphic) cusp forms of weight k for the group Γ. There is a canonical isomorphism of real vector spaces between S_{k}(Γ) and \overline{S_{k}(Γ)} which associates to f the antiholomorphic modular form z ↦ \overline{f(z)}; we denote by f̄ this antiholomorphic modular form.

Shokurov defines a pairing ([13], lemma-definition 1.2)

(S_{k}(Γ) ⊕ \overline{S_{k}(Γ)}) × H_{1}(Δ_{Γ}, Π; (R_{1}ϕ_{∗}ℚ)^{w}) → ℂ

given by the formula

(f_{1} + f_{2}, {α, β, n, m}_{Γ}) ↦ ∫_{α}^{β} f_{1}(z) ∏_{i=1}^{k−2}(n_{i}z + m_{i}) dz + ∫_{α}^{β} f_{2}(z) ∏_{i=1}^{k−2}(n_{i}z̄ + m_{i}) dz̄

(f_{1} ∈ S_{k}(Γ), f_{2} ∈ \overline{S_{k}(Γ)}), where the path is, except for the extremities, contained in the upper half-plane. Using the identifications of section 1.3, we deduce from this a pairing of complex vector spaces

(S_{k}(Γ) ⊕ \overline{S_{k}(Γ)}) × M_{k}(Γ) → ℂ

given by the rule

(f_{1} + f_{2}, P {α, β}) ↦< f_{1} + f_{2}, P {α, β} >= ∫_{α}^{β} f_{1}(z)P (z, 1) dz + ∫_{α}^{β} f_{2}(z)P (z̄, 1) dz̄

f_{1} ∈ S_{k}(Γ), f_{2} ∈ \overline{S_{k}(Γ)}, (α, β) ∈ ℙ^{1}(ℚ)^{2}, P ∈ ℂ_{k−2}[X, Y]. In general, the pairing < ., . > is degenerate. But we have the following theorem, which is essentially a restatement of the theorem 0.2 of [14].

Theorem 3 The bilinear pairing < ., . > is nondegenerate if restricted to a pairing

(S_{k}(Γ) ⊕ \overline{S_{k}(Γ)}) × S_{k}(Γ) → ℂ.

Proof .- Shokurov proves([14], theorem 0.2) that the bilinear pairing between (S_{k}(Γ) ⊕ \overline{S_{k}(Γ)}) and H_{1}(Δ_{Γ}; (R_{1}ϕ_{∗}ℚ)^{w}) ⊗_{ℚ} ℂ is nondegenerate. The theorem 3 is deduced from the identification between H_{1}(Δ_{Γ}; (R_{1}ϕ_{∗}ℚ)^{w}) ⊗_{ℚ} ℂ and S_{k}(Γ).

Remark .- 1) Let f_{1} ∈ S_{k}(Γ) and f_{2} ∈ \overline{S_{k}(Γ)}. Let g ∈ SL_{2}(ℤ) and P ∈ ℂ_{k−2}[X, Y]. We have

< f_{1} + f_{2}, [P, g] >= ∫_{0}^{∞} f_{1|g}(z)P (z, 1) dz + ∫_{0}^{∞} f_{2|ḡ}(z)P (z̄, 1) dz̄.

So we find the formula given in the introduction.

2) The pairing < ., . > is degenerate in general. The space of elements m ∈ M_{k}(Γ) such that < f, m >= 0 for all f ∈ (S_{k}(Γ) ⊕ \overline{S_{k}(Γ)}) is the space of Eisenstein elements. By a theorem of Shokurov, it admits a basis in M_{k}(Γ, ℤ) ([13], theorem 4.3 and corollary 4.4, see the remark in the section 1.1 for the meaning of M_{k}(Γ, ℤ)). It would be interesting to find expressions in terms of Manin symbols of elements of such a basis.

3) The dimension of M_{k}(Γ) is equal to twice the dimension of S_{k}(Γ) plus the dimension of the space of Eisenstein series of weight k for Γ. Using the exact sequences appearing in the proposition 3, one can find the dimension of these spaces of modular forms.

1.6 The action of the complex conjugation

In this section we suppose that the matrix η = ( \begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix} ) normalizes the group Γ.

Proposition 7 The map ι which associates to f ∈ S_{k}(Γ) ⊕ \overline{S_{k}(Γ)} the function z ↦ f(−z̄) = f(ηz̄) = f_{|η}(z̄) = f_{|η̄}(z̄) is a complex linear involution of S_{k}(Γ) ⊕ \overline{S_{k}(Γ)} which exchanges S_{k}(Γ) and \overline{S_{k}(Γ)}.

The involution ι^{∗} on M_{k}(Γ) given by the rule

ι^{∗}(P {α, β}) = −P_{|η̃}{ηα, ηβ}

(P ∈ ℂ_{k−2}[X, Y], (α, β) ∈ ℙ^{1}(ℚ)^{2}) is adjoint to ι with respect to the pairing < ., . > of section 1.5. Moreover ι^{∗} acts as follows on Manin symbols

ι^{∗}([P, g]) = −[P_{|η̃}, ηgη^{−1}].

Proof .- Let f ∈ S_{k}(Γ). Let γ ∈ Γ. We have, since η^{−1}γη ∈ Γ,

ι(f)_{|γ̄} = f_{|\overline{γη}} = f_{|\overline{ηη^{−1}γη}} = f_{|η̄} = ι(f).

So ι(f) is an antiholomorphic modular from. The analogous result with respect to f ∈ \overline{S_{k}(Γ)} can be proved similarly.

Since η normalizes Γ, the definition of ι^{∗} is compatible with the construction of M_{k}(Γ). Let f ∈ S_{k}(Γ), P ∈ ℂ_{k−2}[X, Y] and (α, β) ∈ ℙ^{1}(ℚ)^{2}. We have

< ι(f), P {α, β} > = ∫_{α}^{β} f(−z̄)P (z̄, 1) dz̄
= − ∫_{−α}^{−β} f(z)P (−z, 1) dz
= < f, −P_{|η̃}{ηα, ηβ} >
= < f, ι^{∗}(P {α, β}) > .

The same equalities hold with respect to f ∈ \overline{S_{k}(Γ)}.

Finally we prove the formula with respect to the Manin symbols. We have

ι^{∗}([P, g]) = ι^{∗}(P_{|g}{g0, g∞})
= −(P_{|g})_{|η̃}{ηg0, ηg∞}
= −P_{|η̃η̃^{−1}gη̃}{ηgη^{−1}η0, ηgη^{−1}η∞}
= −(P_{|η̃})_{|η̃^{−1}gη̃}{ηgη^{−1}0, ηgη^{−1}∞}
= −(P_{|η̃})_{|η^{−1}gη}{ηgη^{−1}0, ηgη^{−1}∞}
= −[P_{|η̃}, ηgη^{−1}].

Let S_{k}(Γ)^{+} (resp. S_{k}(Γ)^{−}) be the subspace of S_{k}(Γ) constituted by the elements of S_{k}(Γ) invariant (resp. antiinvariant) under the action of ι^{∗}. We have a direct sum

S_{k}(Γ) = S_{k}(Γ)^{+} ⊕ S_{k}(Γ)^{−}.

Proposition 8 The bilinerar pairings induced by the pairing < ., . > on

S_{k}(Γ) × S_{k}(Γ)^{+} → ℂ and S_{k}(Γ) × S_{k}(Γ)^{−} → ℂ

respectively are nondegenerate.

Proof .- Let (S_{k}(Γ) ⊕ \overline{S_{k}(Γ)})^{+} be the subspace of S_{k}(Γ) ⊕ \overline{S_{k}(Γ)} invariant under the action of ι. The pairing < ., . > restricted to (S_{k}(Γ) ⊕ \overline{S_{k}(Γ)})^{+} × S_{k}(Γ)^{+} → ℂ is nondegenerate. Since the map

S_{k}(Γ) → (S_{k}(Γ) ⊕ \overline{S_{k}(Γ)})^{+}
f ↦ f + ι(f)

is an isomorphism, we deduce the proposition from the equality

< f + ι(f), x >=< f, x > + < f, ι^{∗}(x) >= 2 < f, x >

(f ∈ S_{k}(Γ), x ∈ S_{k}(Γ)^{+}). The assertion concerning S_{k}(Γ)^{−} can be proved in a similar way.

1.7 Connection with the cohomology of SL_{2}(ℤ)

The complex vector space ℂ_{k−2}[X, Y][Γ\SL_{2}(ℤ)]_{k} is endowed with an action on the right of SL_{2}(ℤ) given by the formula

(P [Γg])γ = P_{|γ̃}[Γgγ].

Proposition 9 We have an isomorphism of complex vector spaces

H^{1}(SL_{2}(ℤ), ℂ_{k−2}[X, Y][Γ\SL_{2}(ℤ)]) ≃ M_{k}(Γ).

Proof .- We note M = ℂ_{k−2}[X, Y][Γ\SL_{2}(ℤ)]. Let M^{σ} (resp. M^{τ}) be the subspace of M constituted by elements of M fixed under the action of σ (resp. τ). In view of the exact sequences of the proposition 5, we only have to prove that there is an isomorphism of complex vector spaces

H^{1}(SL_{2}(ℤ), ℂ_{k−2}[X, Y][Γ\SL_{2}(ℤ)]) ≃ M/(M^{σ} + M^{τ}).

We recall that Z^{1}(SL_{2}(ℤ), M) (resp. B^{1}(SL_{2}(ℤ), M)) is the complex vector space of functions ϕ : SL_{2}(ℤ) → M such that ϕ(gh) = ϕ(g)h + ϕ(h), (g, h) ∈ SL_{2}(ℤ)^{2} (resp. such that there exists m ∈ M with ϕ(g) = m − mg, g ∈ SL_{2}(ℤ)). We have H^{1}(SL_{2}(ℤ), M) = Z^{1}(SL_{2}(ℤ), M)/B^{1}(SL_{2}(ℤ), M).

Since SL_{2}(ℤ) is the free product of the groups generated respectively by σ and by τ, the map

Z^{1}(SL_{2}(ℤ), M) → ker(1 + σ + σ^{2} + σ^{3}) × ker(1 + τ + τ^{2})
ϕ ↦ (ϕ(σ), ϕ(τ))

is an isomorphism of complex vector spaces (ker(1 + σ + σ^{2} + σ^{3}) and ker(1 + τ + τ^{2}) denote the kernels in M of the multiplications on the right by (1 + σ + σ^{2} + σ^{3}) and (1 + τ + τ^{2}) respectively). Since M is a complex vector space and since σ^{4} = τ^{3} = 1, we have ker(1 + σ + σ^{2} + σ^{3}) = M(1 − σ) and ker(1 + τ + τ^{2}) = M(1 − τ). We have an obvious isomorphism between M/M^{σ} and M(1 − σ) (resp. between M/M^{τ} and M(1 − τ)). So there is an isomorphism of complex vector spaces Λ : Z^{1}(SL_{2}(ℤ), M) → M/M^{σ} × M/M^{τ}. Since SL_{2}(ℤ) is generated by σ and τ, B^{1}(SL_{2}(ℤ), M) is the set of functions ϕ : SL_{2}(ℤ) → M such that there exists m ∈ M with ϕ(σ) = m(1 − σ) and ϕ(τ) = m(1 − τ). The image by Λ of B^{1}(SL_{2}(ℤ), M) ⊂ Z^{1}(SL_{2}(ℤ), M) is the diagonal image of M in M/M^{σ} × M/M^{τ}. The map

M/M^{σ} × M/M^{τ} → M/(M^{σ} + M^{τ})
(m_{1} + M^{σ}, m_{2} + M^{τ}) ↦ m_{1} − m_{2} + M^{σ} + M^{τ}

is surjective. We only have to prove that its kernel is precisely Λ(B^{1}(SL_{2}(ℤ), M)) in order to prove the proposition. Let (a, b) ∈ M^{2} such that a − b ∈ M^{σ} + M^{τ}. There exists (a′, b′) ∈ M^{σ} × M^{τ} such that a − b = a′ − b′. So we have a − a′ = b − b′ = c. We have a + M^{σ} = c + M^{σ} and b + M^{τ} = c + M^{τ}. So (a + M^{σ}, b + M^{τ}) belongs to the diagonal image of M in M/M^{σ} × M/M^{τ} and the proposition has been proved.

The space S_{k}(Γ) ⊕ \overline{S_{k}(Γ)} is imbedded, by the theory of Eichler-Shimura combined with the Shapiro lemma, in H^{1}(SL_{2}(ℤ), ℂ_{k−2}[X, Y][Γ\SL_{2}(ℤ)]). This theory can be extended to include, not only cups forms, but also Eisenstein series. This point of view has been adopted by Wang ([15]).

1.8 Some comments on the the whole space of modular forms

This paper is concerned mainly with cusp forms and does not fully take care of the Eisenstein series. The vector space M_{k}(Γ) does not correspond to the full space of holomorphic modular forms of weight k for Γ just as S_{k}(Γ) corresponds S_{k}(Γ). To summarize this one might say that the space of cusp forms is reflected in M_{k}(Γ) with multiplicity two and the space of Eisenstein series with multiplicity one. We explain informally in this section how to construct a more complete theory.

Let us consider the torsion free abelian group M̃ generated by the expressions { \atop \{α, β\}} ((α, β) ∈ (ℙ^{1}(ℚ))^{2}) with the relations

{ \atop \{α, β\}} − { \atop \{γ, β\}} + { \atop \{γ, δ\}} − { \atop \{α, δ\}} = 0,

(α, β, γ, δ all in ℙ^{1}(ℚ)). We set M̃^{0} = \{∑ λ_{α,β}{ \atop \{α, β\}} ∈ M̃/ ∑ λ_{α,β} = 0\}; It is a

subgroup of M̃ of ℤ-corank 1. Let us notice that in M̃ we do not have the relation { \atop \{α, α\}} = 0. There is an obvious exact sequence

0 → ℤ[ℙ^{1}(ℚ)] → M̃ → M → 0,

where the injection associates to [α] the element { \atop \{α, α\}} and the surjection associates to the element { \atop \{α, β\}} of M̃ the element \{α, β\} of M.

We continue the construction just as in section 1.1. We use the notations M̃_{k} = ℂ_{k−2}[X, Y] ⊗ M̃ and M̃_{k}^{0} = ℂ_{k−2}[X, Y] ⊗ M̃^{0}. There is a linear action of GL_{2}(ℚ) on M̃_{k} and M̃_{k}^{0} given by the formula (P ⊗ { \atop \{α, β\}})_{|g} = P_{|g} ⊗ { \atop \{gα, gβ\}}. So we may consider the quotient groups M̃_{k}(Γ) and M̃_{k}^{0}(Γ) obtained by quotienting M̃_{k} and M̃_{k}^{0} by the action of Γ. We denote by P { \atop \{α, β\}} the image in M_{k}(Γ) of the element P ⊗{ \atop \{α, β\}} of M_{k}. When k > 2, there is actually a canonical isomorphism between M̃_{k}(Γ) and M̃_{k}^{0}(Γ); When k = 2, M̃_{k}^{0}(Γ) can be identified with a subspace of M̃_{k}(Γ) of codimension one. Moreover one has the exact sequence

0 → B_{k}(Γ) → M̃_{k}(Γ) → M_{k}(Γ) → 0

deduced from the previous exact sequence. Notice that the dimension of the complex vector space M_{k}^{0}(Γ) is twice the dimension of the full space of holomorphic modular forms of weight k for Γ.

We introduce the Manin symbol { \atop [P, g]} ∈ M̃_{k}(Γ) as the image of P ⊗ { \atop \{g0, g∞\}} in M̃_{k}(Γ), where P ∈ ℂ_{k−2}[X, Y] and g ∈ SL_{2}(ℤ). For Λ = ∑_{h} u_{h}[h] ∈ ℤ[SL_{2}(ℤ)], we denote by { \atop [P, g]} • Λ the linear combination of Manin symbols ∑_{h} u_{h}{ \atop [P_{|h^{−1}}, gh]} (see the notations of section 1.2). We use the notation

θ = ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} )−( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} )+( \begin{matrix}0 & 1 \\ −1 & 1 \end{matrix} )−( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} )+( \begin{matrix}1 & 1 \\ −1 & 0 \end{matrix} )−( \begin{matrix}1 & 0 \\ −1 & 1 \end{matrix} ) ∈ ℤ[SL_{2}(ℤ)].

We have following relations for all Manin symbols in M_{k}(Γ):

{ \atop [P, g]} • θ = 0

and

{ \atop [P, g]} − { \atop [P, g]} • (J) = 0.

Let us consider again the vector space ℂ_{k−2}[X, Y][Γ\SL_{2}(ℤ)]_{k} of the section 1.3. Let us denote by ℂ_{k−2}[X, Y][Γ\SL_{2}(ℤ)]_{k}^{θ} the image of ℂ_{k−2}[X, Y][Γ\SL_{2}(ℤ)]_{k} under the action of the right of θ (there is a unique action of ℤ[SL_{2}(ℤ)] on ℂ_{k−2}[X, Y][Γ\SL_{2}(ℤ)]_{k} which extends by ℤ-linearity the action of SL_{2}(ℤ) defined in the section 1.3). We expect then to have an exact sequence of complex vector spaces:

0 → ℂ_{k−2}[X, Y][Γ\SL_{2}(ℤ)]_{k}^{θ} → ℂ_{k−2}[X, Y][Γ\SL_{2}(ℤ)]_{k} → M̃_{k}(Γ) → 0,

where the injection is the inclusion and the surjection associates to the class of P [g] the Manin symbol { \atop [P, g]}.

We consider now the generalisation of the theorem 2 to M̃_{k}(Γ), when Γ = Γ_{1}(N). In that case the Manin symbols can be written under the form { \atop [P, x]}, where P ∈ ℂ_{k−2}[X, Y] and x ∈ E_{N} (see the section 2.2). Let n be an integer > 0. There is an obvious action of Hecke operators T_{n} on M̃_{k}(N) which extends the action on M_{k}(Γ). This action is given by the formula

T_{n}(P { \atop \{α, β\}}) = ∑_{δ∈R} P_{|δ}{ \atop \{δα, δβ\}},

where R is a set of representatives of Γ_{1}(N)\Δ_{n} ; Δ_{n} is the set of matrixes ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈

M_{2}(ℤ)_{n} such that N|c and N|(a − 1). Let ∑_{M} u_{M}[M] ∈ ℤ[M_{2}(ℤ)_{n}] satisfying the following condition (C̃_{n}) : For all class K ∈ M_{2}(ℤ)_{n}/SL_{2}(ℤ), we have the relations ∑_{M∈K} u_{M}[M∞] = [∞] and ∑_{M∈K} u_{M}[M0] = [0] in ℤ[ℙ^{1}(ℚ)]. This condition is stronger than the condition (C_{n}) given in the introduction.

We have then the following formula for the action of Hecke operators on Manin symbols

T_{n}{ \atop [P, (u, v)]} = ∑_{M} u_{M} { \atop [P (aX + bY, cX + dY), (au + cv, bu + dv)]},

where the sum is taken with respect to the matrixes M = ( \begin{matrix}a & b \\ c & d \end{matrix} ) and restricted to

the terms for which (au + cv, bu + dv) belongs to E_{N}, P ∈ ℂ_{k−2}[X, Y] and (u, v) ∈ E_{N}.

This formula can be proved easily by extending the techniques used in the sections 2.1 and 2.3. We indicate a family of elements of ℤ[M_{2}(ℤ)_{n}] satisfying the condition (C̃_{n}) in the section 3.4.

Moreover we expect the existence of a nondegenerate pairing between M̃_{k}(Γ)^{0} and M_{k}(Γ) ⊕ \overline{M_{k}(Γ)}, where M_{k}(Γ) (resp. \overline{M_{k}(Γ)}) is the space of holomorphic (resp. antiholomorphic) modular forms of weight k for Γ and the existence of an alternate bilinear pairing M̃_{k}^{0}(Γ) × M̃_{k}^{0}(Γ) → ℂ. The first of these pairings should extend the pairing considered in the section 1.6. To make the picture complete one expects a nondegenerate self-pairing extending the Petersson scalar product on M_{k}(Γ). When Γ = SL_{2}(ℤ), such a pairing has been constructed by Zagier ([17]).

2 Hecke theory on modular symbols

2.1 The general principle for the action of linear operators

We keep in this section the notations of the first part. Let Δ ⊂ GL_{2}(ℚ) such that ΓΔ = ΔΓ and such that Γ\Δ is finite.

Let R be a set of representatives of Γ\Δ. There is a well defined linear map

T_{Δ} : M_{k}(Γ) → M_{k}(Γ)

which associates to P {α, β} the element ∑_{δ∈R} P_{|δ}{δα, δβ}. This map does not depend on the choice of R.

For g = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ M_{2}(ℚ), we note g̃ = ( \begin{matrix}d & −b \\ −c & a \end{matrix} ) = g^{−1}detg and Δ̃ = \{g ∈

GL_{2}(ℚ)/g̃ ∈ Δ}.

Let ϕ be a map

Δ̃SL_{2}(ℤ) → SL_{2}(ℤ)

satisfying the following three conditions (more properly the conditions are satisfied by the pair (Δ, ϕ))

1. For all γ ∈ Δ̃SL_{2}(ℤ), and g ∈ SL_{2}(ℤ) we have Γϕ(γg) = Γϕ(γ)g.

2. For all γ ∈ Δ̃SL_{2}(ℤ), we have γϕ(γ)^{−1} ∈ Δ̃ (or equivalently ϕ(γ)γ̃ ∈ Δ).

3. The map Γ\Δ → Δ̃SL_{2}(ℤ)/SL_{2}(ℤ) which associates to Γδ the element δ̃SL_{2}(ℤ) is injective (it is necessarily surjective).

Let ∑ u_{M}M ∈ ℂ[M_{2}(ℤ)]. We will say that ∑ u_{M}M satisfies the condition (C_{Δ}) if and only if, for all K ∈ Δ̃SL_{2}(ℤ)/SL_{2}(ℤ) we have the following equality in ℂ[ℙ^{1}(ℚ)]:

∑_{M∈K} u_{M}([M∞] − [M0]) = [∞] − [0].

Theorem 4 Let P ∈ ℂ_{k−2}[X, Y] and g ∈ SL_{2}(ℤ). Let ∑ u_{M}M ∈ ℂ[M_{2}(ℤ)] satisfying the condition (C_{Δ}). We have in M_{k}(Γ)

T_{Δ}([P, g]) = ∑_{M,gM∈Δ̃SL_{2}(ℤ)}u_{M}[P_{|M̃}, ϕ(gM)].

Proof .- We start from the right side of the equality. Let S be a set of representatives of g^{−1}Δ̃SL_{2}(ℤ)/SL_{2}(ℤ). We have

∑_{M,gM∈Δ̃SL_{2}(ℤ)}u_{M}[P_{|M̃}, ϕ(gM)]

= ∑_{s∈S} ∑_{M∈sSL_{2}(ℤ)}u_{M}P_{|ϕ(gM)M̃}{ϕ(gM)0, ϕ(gM)∞}
= ∑_{s∈S} ∑_{M∈sSL_{2}(ℤ)}u_{M}P_{|ϕ(gss^{−1}M)M̃}{ϕ(gss^{−1}M)0, ϕ(gss^{−1}M)∞}
= ∑_{s∈S} ∑_{M∈sSL_{2}(ℤ)}u_{M}P_{|ϕ(gs)s^{−1}MM̃}{ϕ(gs)s^{−1}M0, ϕ(gs)s^{−1}M∞}.

The last equality is a consequence of the property 1 satisfied by ϕ. Since s and M have the same determinant we have s^{−1}MM̃ = s̃. We make use now of the condition (C_{Δ}) and of the properties of modular symbols to obtain the equalities:

∑_{M,gM∈Δ̃SL_{2}(ℤ)}u_{M}[P_{|M̃}, ϕ(gM)] = ∑_{s∈S} P_{|ϕ(gs)s̃}{ϕ(gs)s^{−1}0, ϕ(gs)s^{−1}∞}

= ∑_{s∈S} P_{|ϕ(gs)s̃g̃g}{ϕ(gs)s̃g̃g0, ϕ(gs)s̃g̃g∞}.

Because of the property 2 satisfied by ϕ, ϕ(gs)s̃g̃ belongs to Δ. Because of the property 3 satisfied by ϕ, for s ≠ s′ we have Γϕ(gs)s̃g̃ ≠ Γϕ(gs′)s̃′g̃. We deduce that ϕ(gs)s̃g̃ runs through a set of representatives of Γ\Δ when s runs through S. This concludes the proof of the theorem.

We denote by T_{Δ}^{∗} the linear operator on S_{k}(Γ) (resp. \overline{S_{k}(Γ)}) defined by the rule

f ↦ ∑_{δ∈R}(detδ)^{\frac{k}{2} −1}f_{|δ}

(resp.

f ↦ ∑_{δ∈R}(detδ)^{\frac{k}{2} −1}f_{|δ̄}).

This operator is independant of the choice of the set of representatives R of Γ\Δ made at the beginning of the section.

Proposition 10 The operators T_{Δ}^{∗} and T_{Δ} are adjoint with respect to the bilinear pairing < ., . > defined in the section 1.5.

Proof .- Let (α, β) ∈ ℙ^{1}(ℚ)^{2} and P ∈ ℂ_{k−2}[X, Y]. Let f ∈ S_{k}(Γ). For g = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈

GL_{2}(ℚ), we use the otation j(g, z) = (cz + d). We have

< T_{Δ}^{∗}(f), P {α, β} > = ∫_{α}^{β} T_{Δ}^{∗}(f)P (z, 1) dz

= ∑_{δ∈R} ∫_{α}^{β}(detδ)^{k−1}f(δz)j(δ, z)^{−k}P (z, 1) dz

We make now the change of variables u = δz. We obtain

< T_{Δ}^{∗}(f), P {α, β} > = ∑_{δ∈R} ∫_{δα}^{δβ}(detδ)^{k−1}f(u)j(δ, δ^{−1}u)^{−k}P (δ^{−1}u, 1) dδ^{−1}u
= ∑_{δ∈R} ∫_{δα}^{δβ}(detδ)^{k−1}f(u)j(δ̃, u)^{k}(detδ)^{−k}P (δ̃u, 1)\frac{detδ du}{(j(δ̃, u))^{2}}
= ∑_{δ∈R} ∫_{δα}^{δβ} f(u)P_{|δ}(u, 1) du
= < f, T_{Δ}(P {α, β}) > .

The similar equality holds for f ∈ \overline{S_{k}(Γ)}. This proves the proposition. Proposition 11 The operator T_{Δ} maps S_{k}(Γ) into itself. Proof .- Let T_{Δ}^{∂} be the endomorphism of B_{k}(Γ) which associates to P {α} the element

∑_{δ∈R} P_{|δ}{δα}.

It does not depend on the choice of R. The proposition follows from the equality ∂ ∘ T_{Δ} = T_{Δ}^{∂} ∘ ∂.

Remark .- 1) If Δ ⊂ M_{2}(ℤ), then T_{Δ} maps M_{k}(Γ, ℤ) into itself. As we will see in the section 2.3, the Hecke operators preserve the integral structure of M_{k}(Γ).

2) If η = ( \begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix} ) normalizes Γ and Δ (see section 1.6) then we have

T_{Δ} ∘ ι = ι ∘ T_{Δ}.

In particular this applies to Hecke operators.

2.2 Manin symbols for Γ_{1}(N)

Let N be an integer > 0. From now, we apply the results previously obtained to Γ = Γ_{1}(N). We use thereafter the notations M_{k}(N) = M_{k}(Γ_{1}(N)), S_{k}(N) = S_{k}(Γ_{1}(N)), B_{k}(N) = B_{k}(Γ_{1}(N)) ... We remark that the matrix η = ( \begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix} ) normalizes Γ_{1}(N) (see section 1.6). The surjection

π : SL_{2}(ℤ) → E_{N} ⊂ (ℤ/Nℤ)^{2}
( \begin{matrix}a & b \\ c & d \end{matrix} ) ↦ (c, d)

defines a bijection between Γ_{1}(N)\SL_{2}(ℤ) and E_{N}. Let λ be a section of π. Let x ∈ E_{N}. The Manin symbol [P, λ(x)] depends only on Γ_{1}(N)λ(x) and P (see proposition 1). So it depends only on P and x. By abuse of notations we denote it by [P, x]. We define

an action of M = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ M_{2}(ℤ) on x = (u, v) ∈ (ℤ/Nℤ)^{2} by the formula

xM = (au + cv, bu + dv).

Proposition 12 Let x = (u, v) ∈ E_{N} and P ∈ ℂ_{k−2}[X, Y]. We have

ι^{∗}([P, x]) = −[P_{|η̃}, xη] = −[P (−X, Y), (−u, v)].

Proof .- Let g = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ SL_{2}(ℤ) such that π(g) = x. By application of the

proposition 7, we have

ι^{∗}([P, x]) = ι^{∗}([P, g]) = −[P_{|η̃}, ηgη^{−1}].

We have

π(ηgη^{−1}) = π(( \begin{matrix}a & −b \\ −c & d \end{matrix} )) = (−u, v).

The proposition follows.

There is a bijection between Γ\(ℚ^{2} − {0})/ℚ_{+}^{∗} and P_{N} (see the introduction for the

definition of P_{N}) which associates to Γ_{1}(N) ( \begin{matrix}u \\ v \end{matrix} ) ℚ_{+}^{∗} the element u (mod ()v, N) ∈

(ℤ/(v, N)ℤ)^{∗} ⊂ P_{N}. This bijection defines an isomorphism of complex vector spaces between ℂ[Γ_{1}(N)\ℚ^{2}]_{k} and ℂ[P_{N}]_{k}. The image of ∂([P, x]) (P ∈ ℂ_{k−2}[X, Y], x ∈ E_{N}) by this isomorphism is equal to b([P, x]) (see the introduction).

2.3 Hecke operators

We use the notations already introduced, especially those of section 2.2. Let n be an

integer ≥ 1. Let Δ_{n} be the set of matrices ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ M_{2}(ℤ), of determinant n, such

that N|c and N|(a − 1). We have Γ_{1}(N)Δ_{n} = Δ_{n}Γ_{1}(N) = Δ_{n}. The set Γ_{1}(N)\Δ_{n} is finite. We denote by T_{n} the Hecke operator T_{Δ_{n}} on M_{k}(N) (see section 1.1). We remark that ηΔ_{n} = Δ_{n}η. So the Hecke operators commute with the involution ι^{∗} defined in the section 1.6.

We denote by M_{2}(ℤ)_{n} the set of matrices of M_{2}(ℤ) of determinant n. We recall (see the introduction) that ∑_{M} u_{M}M ∈ ℂ[M_{2}(ℤ)_{n}] satisfies the condition (C_{n}) if for all classes K ∈ M_{2}(ℤ)_{n}/SL_{2}(ℤ) we have the following equality in ℂ[ℙ^{1}(ℚ)]:

∑_{M∈K} u_{M}([M∞] − [M0]) = [∞] − [0].

We prove now the theorem 2.

Proof .- We will prove this theorem as a special case of the theorem 4. The set Δ̃_{n}SL_{2}(ℤ)

is the set of matrices ( \begin{matrix}a & b \\ c & d \end{matrix} ) of M_{2}(ℤ) of determinant n such that (c, d, N) are globally

coprime (i.e. cℤ/Nℤ + dℤ/Nℤ = ℤ/Nℤ). Let ϕ_{n} be a map

Δ̃_{n}SL_{2}(ℤ) → SL_{2}(ℤ)

such that π(ϕ_{n}(( \begin{matrix}a & b \\ c & d \end{matrix} ))) = (c, d) ∈ E_{N}. In particular we have π(ϕ_{n}(M)) = (0, 1)M.

Lemma 1 The map ϕ_{n} and the set Δ_{n} satisfy the conditions 1., 2. and 3. of the section 2.1.

Proof .- Let ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Δ̃_{n} and g ∈ SL_{2}(ℤ), we have

π(ϕ_{n}(( \begin{matrix}a & b \\ c & d \end{matrix} ) g)) = (c, d)g = π(ϕ_{n}(( \begin{matrix}a & b \\ c & d \end{matrix} ))g).

This equality proves that the pair (Δ_{n}, ϕ_{n}) satisfies the property 1.

An element g ∈ Δ̃_{n}SL_{2}(ℤ) belongs to Δ̃_{n} if and only if (0, 1)g = (0, 1) in E_{N}. Let

g = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Δ̃_{n}SL_{2}(ℤ). We have (0, 1)g = (0, 1)ϕ_{n}(g) and so (0, 1)gϕ_{n}(g)^{−1} =

(0, 1). We deduce that gϕ_{n}(g)^{−1} belongs to Δ̃_{n}. So we have the property 2.

Let (δ, δ′) = (( \begin{matrix}a & b \\ c & d \end{matrix} ) , ( \begin{matrix}a′ & b′ \\ c′ & d′ \end{matrix} )) ∈ Δ_{n}^{2} such that

δ′δ^{−1} = ( \begin{matrix}\frac{da′−b′c}{n} & \frac{−ba′+ab′}{n} \\ \frac{dc′−d′c}{n} & \frac{−bc′+ad′}{n} \end{matrix} ) ∈ SL_{2}(ℤ).

Because of the definition of Δ_{n} we have n ≡ d ≡ d′ (mod N). So we have N|\frac{dc′−d′c}{n}. The matrices δ, δ′ and δ′δ^{−1} are all trigonal matrices modulo N. Since the upper left entry of δ and δ′ are both 1 modulo N, the upper left entry of δ′δ^{−1} must be 1 modulo N. So δ′δ^{−1} belongs to Γ_{1}(N). We deduce that the set Δ_{n} satisfies the property 3.

Let g ∈ SL_{2}(ℤ) such that π(g) = x. We have gM ∈ Δ̃_{n}SL_{2}(ℤ) if and only if xM belongs to E_{N}.

Since ∑_{M} u_{M}M satisfies the condition (C_{n}) it satisfies the condition (C_{Δ_{n}}) of the section 2.1. By application of the theorem 4, we obtain the theorem 2.

2.4 Atkin-Lehner operators

Let N′ be a positive integer dividing N such that N and N/N′ are coprimes. Let Δ′_{N′}

be the set of matrices ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ M_{2}(ℤ), of determinant N′, such that N|c, N′|a,

N′|d, N′|(b − 1), (N/N′)|(a − 1). We have Γ_{1}(N)Δ′_{N′} = Δ′_{N′}Γ_{1}(N) = Δ′_{N′}. We have ιΔ′_{N′} = Δ′_{N′}ι.

The operator T_{Δ′_{N′}} (see section 2.1) is an Atkin-Lehner operator. It will be denoted by W_{N′}.

Theorem 5 Let x ∈ E_{N} and P ∈ ℂ_{k−2}[X, Y]. Let g ∈ SL_{2}(ℤ) such that π(g) = x. Let ∑_{M} u_{M}M ∈ ℂ[M_{2}(ℤ)_{N′}] satisfying the condition (C_{N′}). We have

W_{N′}([P, x]) = ∑_{M,xM=(0,0) (mod N′)} u_{M}[P_{|M̃}, ϵ_{N′}(gM)],

where ϵ_{N′}(gM) is the unique element of E_{N} congruent to (1, 0)gM modulo N′ and congruent to (0, 1)gM = xM modulo N/N′.

Proof .- We will apply the theorem 4 for Δ = Δ′_{N′}. The set Δ̃′_{N′}SL_{2}(ℤ) is the set

of matrices ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ M_{2}(ℤ) of determinant N′ such that N′|c and N′|d (we have

necessarily aℤ/Nℤ + bℤ/Nℤ = ℤ/Nℤ).

Let δ_{0} ∈ Δ′_{N′}. We consider the map

ϕ′_{N′} : Δ̃′_{N′}SL_{2}(ℤ) → SL_{2}(ℤ)
δ̃_{0}g ↦ g.

Since Δ̃′_{N′} = δ_{0}Γ_{1}(N), the pair (Δ′_{N′}, ϕ′_{N′}) satisfies the conditions 1 and 2. Let δ =

( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Δ̃′_{N′} and g = ( \begin{matrix}a′ & b′ \\ c′ & d′ \end{matrix} ) ∈ SL_{2}(ℤ), such that δg = ( \begin{matrix}aa′ + bc′ & ab′ + bd′ \\ ca′ + dc′ & cb′ + dd′ \end{matrix} ) ∈

Δ̃′_{N′}. We have the congruences

( \begin{matrix}0 & 1 \\ 0 & 0 \end{matrix} ) ≡ δg ≡ ( \begin{matrix}c′ & d′ \\ 0 & 0 \end{matrix} ) (mod N′)
( \begin{matrix}1 & ∗ \\ 0 & ∗ \end{matrix} ) ≡ δg ≡ ( \begin{matrix}a′ + bc′ & ∗ \\ dc′ & d′d \end{matrix} ) (mod N/N′).

We obtain the congruences

dd′ ≡ d (mod N/N′)
c′d ≡ 0 (mod N/N′)
c′ ≡ 0 (mod N′)
d′ ≡ 1 (mod N′).

Since d is prime to N/N′, we have g ∈ Γ_{1}(N). This implies that the pair (Δ′_{N′}, ϕ′_{N′}) satisfies the condition 3 of the section 2.1.

We remark that π(ϕ′_{N′}(gM)) = ϵ_{N′}(gM). By application of the theorem 4, the theorem 5 follows.

2.5 Modular forms with characters

Let n be an integer prime to N. Let D_{n} = ( \begin{matrix}n & 0 \\ 0 & n \end{matrix} ). Let Δ_{n,n} be the set of matrices

( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ D_{n}SL_{2}(ℤ) such that N|c and N|(a−1). We have Γ_{1}(N)Δ_{n,n} = Δ_{n,n}Γ_{1}(N).

By definition the operator T_{n,n} is the operator T_{Δ_{n,n}} in the sense of the section 2.1. The group Γ_{1}(N) operates transitively by multiplication on the right on Δ_{n,n}.

Proposition 13 Let x = (u, v) ∈ E_{N} and P ∈ ℂ_{k−2}[X, Y]. We have

T_{n,n}([P, x]) = [P_{|D_{n}}, xD_{n}] = n^{k−2}[P, (nu, nv)].

Proof .- Let g ∈ SL_{2}(ℤ) such that π(g) = x. Let γ ∈ Δ_{n,n}. We can write γ = D_{n}γ_{0} with

γ_{0} = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Γ_{0}(N). We have d ≡ n (mod N). So we have π(γ_{0}g) = (du, dv) =

xD_{n}. The proposition is proved by the following equalities

T_{n,n}([P, x]) = P_{|D_{n}γ_{0}g}{D_{n}γ_{0}g0, D_{n}γ_{0}g∞}
= P_{|D_{n}γ_{0}g}{γ_{0}g0, γ_{0}g∞}
= [P_{|D_{n}}, xD_{n}]

Let χ be a Dirichlet character ℤ/Nℤ → ℂ. We say that f ∈ S_{k}(N) (resp. f ∈ \overline{S_{k}(N)}) belongs to S_{k}(N, χ) (resp. \overline{S_{k}(N, χ)}) if we have f_{|γ} = χ(d)f (resp. f_{|γ̄} = χ(d)f)

for all γ = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Γ_{0}(N). This condition is equivalent to T_{n,n}^{∗}(f) = n^{k−2}χ(n)f for

all n > 0 inversible modulo N, where T_{n,n}^{∗} is the operator dual to T_{n,n} in the sense of the section 2.1.

We define M_{k}(N, χ) as the quotient vector space of M_{k}(N) by the equivalence relation which identify χ(n)x and n^{2−k}T_{n,n}(x) for all n > 0 inversible modulo N and all x ∈ M_{k}(N). Equivalently M_{k}(N, χ) is the quotient vector space of M_{k}(N) by the equivalence relation which identify the Manin symbol [P, (λu, λv)] ((λ, u, v) ∈ (ℤ/Nℤ)^{3}, P ∈ ℂ_{k−2}[X, Y]) in M_{k}(N, χ) to χ(λ)[P, (u, v)].

We denote by B_{k}(N, χ) the image of ∂ in B_{k}(N) and S_{k}(N, χ) the kernel of ∂ in M_{k}(N, χ). We can state the following refinement of the theorem 3.

Proposition 14 The pairing

(S_{k}(N, χ) ⊕ \overline{S_{k}(N, χ)}) × S_{k}(N, χ) → ℂ

deduced from the pairing < ., . > of the section 1.5 is nondegenerate.

2.6 The new and old parts of M_{k}(N)

We do not give the detailed proofs of some statements in this section, since these statements parallel the classical theory of old and newforms (see for instance [3]). Let

N′ be a positive divisor of N. Let t be a divisor of N/N′. The matrix T = ( \begin{matrix}1 & 0 \\ 0 & t \end{matrix} )

satisfies T^{−1}Γ_{1}(N) ⊂ Γ_{1}(N′)T^{−1}. So there are well defined linear maps

ϵ_{t} : M_{k}(N) → M_{k}(N′)
P (X, Y){α, β} ↦ P (X, tY){tα, tβ} = P_{|T̃}{T̃α, T̃β}

and

ϵ′_{t} : M_{k}(N′) → M_{k}(N)
P {α, β} ↦ ∑_{γ} P_{|γ}{γα, γβ},

where γ runs through a set of representatives of Γ_{1}(N)\Γ_{1}(N′)T. We have ϵ_{t} ∘ ϵ′_{t} = multiplication by a nonzero scalar. The linear maps ϵ_{t} and ϵ′_{t} are respectively surjective and injective.

The intersection of the kernels of ϵ_{t} when N′ runs through all the positive divisors of N and t runs through all the divisors of N/N′ is by definition the new part M_{k}(N)^{n} of M_{k}(N). The space generated by the images of ϵ′_{t} when N′ runs through all the positive divisors of N and t runs through all the divisors of N/N′ is by definition the old part M_{k}(N)^{o} of M_{k}(N). We have a direct sum

M_{k}(N) = M_{k}(N)^{n} ⊕ M_{k}(N)^{o}.

This decomposition respects the decomposition of S_{k}(N) in old and new part, i.e. M_{k}(N)^{n} (resp. M_{k}(N)^{o}) is orthogonal to the old (resp. new) part of S_{k}(N), when one considers the bilinear pairing < ., . >.

The two following propositions are not, properly speaking, applications of the general principle of the section 2.1. Nevertheless we use the same techniques to prove them.

Proposition 15 Let x ∈ E_{N}. Let P ∈ ℂ_{k−2}[X, Y]. Let ∑_{M} u_{M}M ∈ ℂ[M_{2}(ℤ)_{t}] satisfying the condition (C_{t}). We have in M_{k}(N)

ϵ_{t}([P, x]) = ∑ u_{M}[P_{|M̃}, \frac{1}{t}xM],

where the sum is restricted to the matrices M such that xM ∈ tE_{N}, and where the multiplication by \frac{1}{t} is well defined from E_{N} to E_{N′}.

Proof .- Let g ∈ SL_{2}(ℤ) such that π(g) = x. We use the notation T = ( \begin{matrix}1 & 0 \\ 0 & t \end{matrix} ).

Let M ∈ M_{2}(ℤ)_{t} such that xM ∈ tE_{N}. Then we have π(T^{−1}gM) = \frac{1}{t}xM. So two matrices M and M′ of M_{2}(ℤ)_{t} satisfying xM ∈ tE_{N} and xM′ ∈ tE_{N} are in the same class modulo multiplication on the right by an element of SL_{2}(ℤ). If y ∈ tE_{N} and h ∈ SL_{2}(ℤ) then we have yh ∈ tE_{N}. So the sum in the formula in the proposition is taken with respect to all elements belonging to only one class of M_{2}(ℤ)_{t}/SL_{2}(ℤ). We prove now the validity of the formula. We begin with the right side of the equality:

∑ u_{M}[P_{|M̃}, \frac{1}{t}xM] = ∑ u_{M}P_{|T^{−1}gMM̃}\{T^{−1}gM0, T^{−1}gM∞\}
= ∑ u_{M}P_{|T̃g}{T^{−1}gM0, T^{−1}gM∞}.

We use the condition (C_{t}) and the fact that the sum is taken over only one class of M_{2}(ℤ)_{t}/SL_{2}(ℤ). We have

∑ u_{M}P_{|T̃g}{T^{−1}gM0, T^{−1}gM∞} = P_{|T̃g}{T^{−1}g0, T^{−1}g∞}
= P_{|T̃g}{tg0, tg∞}
= ϵ_{t}([P, x]).

The proposition is proved.

Proposition 16 Let g ∈ SL_{2}(ℤ). Let H be a system of representatives in Γ_{1}(N′) of Γ_{1}(N)\Γ_{1}(N′)g. Let ∑_{M} u_{M}M ∈ ℤ[M_{2}(ℤ)_{t}] satisfying the condition (C_{t}). We have

ϵ′_{t}([P, π(g)]) = ∑_{h∈H} ∑_{M} u_{M}[P_{|M̃}, π(( \begin{matrix}\frac{1}{t} & 0 \\ 0 & 1 \end{matrix} ) hM)],

where the second sum is restricted to the matrices M such that hM ∈ ( \begin{matrix}t & 0 \\ 0 & 1 \end{matrix} ) SL_{2}(ℤ).

Proof .- First we remark that, given h ∈ H, the set of matrices M ∈ M_{2}(ℤ)_{t} such

that hM ∈ ( \begin{matrix}t & 0 \\ 0 & 1 \end{matrix} ) SL_{2}(ℤ) is exactly a class of M_{2}(ℤ)_{t}/SL_{2}(ℤ). We use the notation

T = ( \begin{matrix}t & 0 \\ 0 & 1 \end{matrix} ). We have

∑_{h∈H} ∑_{M} u_{M}[P_{|M̃}, π(( \begin{matrix}\frac{1}{t} & 0 \\ 0 & 1 \end{matrix} ) hM)] = ∑_{h∈H} ∑_{M} u_{M}P_{|T̃^{−1}hMM̃}\{\frac{1}{t}hM0, \frac{1}{t}hM∞\}
= ∑_{h∈H} ∑_{M} u_{M}P_{|T̃h}\{\frac{1}{t}hM0, \frac{1}{t}hM∞\}
= ∑_{h∈H} P_{|T̃h}\{\frac{1}{t}h0, \frac{1}{t}h∞\}
= ϵ′_{t}([P, π(g)]).

The proposition is proved.

To determine the new part of M_{k}(N) it is useful to consider the following proposition (which is a counterpart of a classical statement in the theory of modular forms, see [3]).

Proposition 17 Let x ∈ M_{k}(N). This element belongs to M_{k}(N)^{n} if and only if x and W_{N}x belong to the kernel of ϵ_{1} : M_{k}(N) → M_{k}(N/t) for all divisors t of N.

3 Families of elements satisfying the condition (C_{n})

3.1 The Manin-Heilbronn family

Let S (resp. S′) be the set of matrices ( \begin{matrix}x & −y \\ y′ & x′ \end{matrix} ) ∈ M_{2}(ℤ) of determinant > 0 and

satisfying one at least of the following three conditions

• x > |y|, x′ > |y′|, yy′ > 0

• y = 0, |y′| < \frac{x′}{2}

• y′ = 0, |y| < \frac{x}{2} (resp. one of the following two conditions

• y = 0, |y′| = \frac{x′}{2}

• y′ = 0, |y| = \frac{x}{2}).

Let n be an integer > 0. Let S_{n} (resp. S′_{n}) be the set of elements of S (resp. S′) of determinant n. Proposition 18 The element

∑_{M∈S_{n}} M + \frac{1}{2} ∑_{M∈S′_{n}} M

of ℂ[M_{2}(ℤ)_{n}] satisfies the condition (C_{n}). Proof .- We use several lemmas. Lemma 2 The set of elements ( \begin{matrix}d & b \\ 0 & \frac{n}{d} \end{matrix} ) ∈ M_{2}(ℤ) (resp. ( \begin{matrix}\frac{n}{d} & 0 \\ b & d \end{matrix} )), with d > 0 and −\frac{d}{2} < b ≤ \frac{d}{2} is a set of representatives of M_{2}(ℤ)_{n}/SL_{2}(ℤ). The same assertion holds if we replace the condition −\frac{d}{2} < b ≤ \frac{d}{2} by −\frac{d}{2} ≤ b < \frac{d}{2}. These elements are the only elements ( \begin{matrix}x & −y \\ y′ & x′ \end{matrix} ) ∈ S_{n} ∪ S′_{n} such that y′ = 0 (resp. y = 0).

Proof .- We prove the lemma with respect to the set of matrices ( \begin{matrix}d & b \\ 0 & \frac{n}{d} \end{matrix} ) ∈ M_{2}(ℤ) with d > 0 and −\frac{d}{2} < b ≤ \frac{d}{2}. The other situations can be dealt with similarly. Let g ∈ M_{2}(ℤ)_{n}. Since SL_{2}(ℤ) operates transitively on ℙ^{1}(ℚ), there exists γ ∈ SL_{2}(ℤ) such that gγ∞ = ∞. So there exists an element g′ = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ gSL_{2}(ℤ) such that g′∞ = ∞. So we have c = 0. We can multiply by ( \begin{matrix}−1 & 0 \\ 0 & −1 \end{matrix} ) to obtain a > 0. Two matrices of M_{2}(ℤ) of the type ( \begin{matrix}d & b \\ 0 & \frac{n}{d} \end{matrix} ) and ( \begin{matrix}d′ & b′ \\ 0 & \frac{n}{d′} \end{matrix} ), with d > 0 and d′ > 0 are congruent modulo SL_{2}(ℤ) if and only if d = d′ and b ≡ b′ (mod d). This proves the first statement of the lemma.

The second statement follows immediately from the definition of the sets S_{n} and S′_{n}.

Lemma 3 Let g = ( \begin{matrix}x & −y \\ y′ & x′ \end{matrix} ) ∈ S_{n} such that y′ ≠ 0 (resp. y ≠ 0). There exists

an unique element g_{0} = ( \begin{matrix}x_{0} & −y_{0} \\ y′_{0} & x′_{0} \end{matrix} ) ∈ S_{n} ∩ gSL_{2}(ℤ) such that ϵ(y′) ( \begin{matrix}x \\ y′ \end{matrix} ) = ( \begin{matrix}−y_{0} \\ x′_{0} \end{matrix} )

(resp. ( \begin{matrix}−y \\ x′ \end{matrix} ) = ϵ(y′_{0}) ( \begin{matrix}x_{0} \\ y′_{0} \end{matrix} )) where ϵ(y′) (resp. ϵ(y′_{0})) is the sign of y′ (resp. y′_{0}); We have g∞ = g_{0}0 (resp. g0 = g_{0}∞).

Proof .- We prove the lemma with respect to the hypothesis y′ ≠ 0. The other assertion can be proved similarly. We consider two cases.

First we suppose that \frac{x′}{y′} is not integral. Let q be the unique integer such that |q + \frac{x′}{y′} | < 1 and (qy′ + x′) > 0. Since |y′| < x′, we have ϵ(y′)q < 0. The matrix

( \begin{matrix}−qϵ(y′ & ϵ(y′) \\ −ϵ(y′) & 0 \end{matrix} ) belongs to SL_{2}(ℤ) and we have

g_{0} = g ( \begin{matrix}−qϵ(y′) & ϵ(y′) \\ −ϵ(y′) & 0 \end{matrix} ) = ( \begin{matrix}−(qx − y)ϵ(y′) & ϵ(y′)x \\ −(qy′ + x′)ϵ(y′) & ϵ(y′)y′ \end{matrix} ) .

The nondiagonal entries of this matrix are nonzero. Since |y| < x, we have (qx − y)ϵ(y′) > | − ϵ(y′)x| = x. Because of the construction of q we have |(qy′ + x′)ϵ(y′)| < ϵ(y′)y′. We have (qy′ + x′)ϵ(y′)(−ϵ(y′)x) = −(qy′ + x′)x < 0. We have proved that g_{0} belongs to S_{n}. This proves the existence of the matrix. We now prove the unicity. Let g_{0} ∈ M_{2}(ℤ) satisfying the conditions of the lemma. Let γ ∈ SL_{2}(ℤ) such that gγ = g_{0}.

We have gγ0 = g_{0}0 = g∞. So we have γ0 = ∞ and γ is of the form ( \begin{matrix}−qϵ & ϵ \\ −ϵ & 0 \end{matrix} ), with

q ∈ ℤ and ϵ ∈ {1, −1}. We have

g_{0} = g ( \begin{matrix}−qϵ & ϵ \\ −ϵ & 0 \end{matrix} ) = ( \begin{matrix}−(qx − y)ϵ & ϵx \\ −(qy′ + x′)ϵ & ϵy′ \end{matrix} ) .

Since \frac{x′}{y′} is not integral, we have qy′ +x′ ≠ 0. So we have ϵy′ > 0 and ϵ = ϵ(y′). We have also |(qy′+x′)ϵ| < ϵy′, that implies |q+ \frac{x′}{y′} | < 1. Moreover the inequality ϵx(qy′+x′)ϵ > 0 implies qy′ + x′ > 0. So we have proved the unicity of q and consequently of g_{0}.

Now we consider the second case: \frac{x′}{y′} is integral. We first prove the existence

property. We find that the matrix g ( \begin{matrix}−qϵ & ϵ \\ −ϵ & 0 \end{matrix} ), with q = −\frac{x′}{y′} and ϵ = ϵ(y′), is equal

to ( \begin{matrix}\frac{nϵ}{y′} & ϵx \\ 0 & ϵy′ \end{matrix} ). We only have to prove that |x| < \frac{nϵ}{2y′}, i.e. that |xy′| < \frac{n}{2}. Since y′|x′

and |y′| < x′, we have |\frac{x′}{y′} | ≥ 2. Since xx′ + yy′ = n, we have 0 < xx′ < n, except perhaps if y = 0. If y ≠ 0, we obtain the inequalities

|xy′| ≤ \frac{1}{2}|xx′| < \frac{n}{2}.

If y = 0, we have |y′| < \frac{1}{2}x′ (because g ∈ S_{n}) and the inequality

|xy′| < \frac{1}{2}|xx′| ≤ \frac{n}{2}.

So the existence is proved. To prove the unicity we proceed as in the first case.

Lemma 4 Let d be an even positive divisor of n. The elements of the set

\{( \begin{matrix}d & \frac{d}{2} \\ 0 & \frac{n}{d} \end{matrix} ) , ( \begin{matrix}d & −\frac{d}{2} \\ 0 & \frac{n}{d} \end{matrix} ) , ( \frac{d}{2} 0_{\frac{n}{d} \frac{2n}{d}} ) , ( \begin{matrix}\frac{d}{2} & 0 \\ −\frac{n}{d} & \frac{2n}{d} \end{matrix} )\}

belongs to the same class modulo SL_{2}(ℤ). Moreover these sets form a partition of S′_{n} which respects the classes modulo SL_{2}(ℤ) when d runs through the positive even divisors of n.

Proof .- The first assertion is proved by straightforward calculations. The second assertion is obtained by comparing with the definition of S′_{n} and by using the lemma 2.

Lemma 5 We have

S_{n}SL_{2}(ℤ) ∩ S′_{n}SL_{2}(ℤ) = ∅.

Proof .- Let g = ( \begin{matrix}x & −y \\ y′ & x′ \end{matrix} ) ∈ S_{n} ∩ S′_{n}SL_{2}(ℤ). We can not have y′ = 0 because of

the lemma 2. Since g ∈ S_{n} and y′ ≠ 0, we use the lemma 3 to prove the existence of

an element g_{0} = ( \begin{matrix}x_{0} & −y_{0} \\ y′_{0} & x′_{0} \end{matrix} ) ∈ S_{n} ∩ gSL_{2}(ℤ) = S_{n} ∩ S′_{n}SL_{2}(ℤ) with |y′_{0}| < y′. By

repeated use of the lemma 3 we obtain a sequence of matrices g_{m} = ( \begin{matrix}x_{m} & −y_{m} \\ y′_{m} & x′_{m} \end{matrix} ) ∈

S_{n} ∩ S′_{n}SL_{2}(ℤ) with decreasing |y_{m}|. For some m we must have y_{m} = 0. So we obtain a contradiction.

We turn now to the proof of the proposition. The equation det ( \begin{matrix}x & −y \\ y′ & x′ \end{matrix} ) =

xx′ + yy′ = n has only a finite number of integral solutions with xx′ ≥ 0 and yy′ ≥ 0. So the set S_{n} is finite. Let K ∈ M_{2}(ℤ)_{n}/SL_{2}(ℤ).

If K ⊂ S′_{n}SL_{2}(ℤ), then the lemma 4 tells us that there exists a unique positive divisor d of n such that

K ∩ S′_{n} = \{( \begin{matrix}d & \frac{d}{2} \\ 0 & \frac{n}{d} \end{matrix} ) , ( \begin{matrix}d & −\frac{d}{2} \\ 0 & \frac{n}{d} \end{matrix} ) , ( \frac{d}{2} 0_{\frac{n}{d} \frac{2n}{d}} ) , ( \begin{matrix}\frac{d}{2} & 0 \\ −\frac{n}{d} & \frac{2n}{d} \end{matrix} )\}.

We have in ℂ[ℙ^{1}(ℚ)]

∑_{M∈K∩S′_{n}}([M∞] − [M0]) = [∞] − [\frac{d^{2}}{2n}] + [∞] − [−\frac{d^{2}}{2n}] + [\frac{d^{2}}{2n}] − [0] + [−\frac{d^{2}}{2n}] − [0]
= 2([∞] − [0]).

If K ⊂ S_{n}SL_{2}(ℤ), we have in ℂ[ℙ^{1}(ℚ)] (the sums are taken with respect to the

matrices M = ( \begin{matrix}x & −y \\ y′ & x′ \end{matrix} ))

∑_{M∈K∩S_{n}}[M∞] − [M0] = ∑_{M∈K∩S_{n}}[\frac{x}{y′}] − ∑_{M∈K∩S_{n}}[−\frac{y}{x′}]

Because of the lemma 3, all the terms cancel each other except those corresponding to y′ = 0 or y = 0. We obtain

∑_{M∈K∩S_{n}}[M∞] − [M0] = ∑_{M∈K∩S_{n},y′=0}[\frac{x}{y′}] − ∑_{M∈K∩S_{n},y=0}[−\frac{y}{x′}]

We now use the lemma 2 and we obtain that the last member is equal to [∞] − [0]. The proposition is proved.

This result is very close to one obtained in [8], where the element appearing in the proposition 18 is called the Manin-Heilbronn element for n odd (if n is even it is called the Manin-Heilbronn element symmetrized by the complex conjugation). The phenomenon underlying the proposition 18 was discovered by Heilbronn in a work about continued fractions ([2]) and it was later used by Manin in the theory of modular symbols ([4], [5], [6]).

Let us remark that the lemmas 2 and 3 enable us to construct systematically the set S_{n}. The set S′_{n} is given directly by the lemma 4.

In [8], using a result of Heilbronn we gave an estimate for the cardinality of S_{n}: We have the following asymptotic formula when n → ∞

|S_{n}| ∼ \frac{12 log 2}{π^{2}}σ_{1}(n) log n,

where σ_{1}(n) is the sum of the positive divisors of n.

We can now introduce another universal expansion of modular forms, which seems for practical purposes (i.e. explicit construction of bases of spaces of modular forms), to be more useful than the series considered in the introduction.

We call the following series in ℂ[M_{2}(ℤ)][[q]] the Manin-Heilbronn expansion :

∑_{M∈S} Mq^{detM} + \frac{1}{2} ∑_{M∈S′} Mq^{detM}

= ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ) q
+ (( \begin{matrix}2 & 0 \\ 0 & 1 \end{matrix} ) + ( \begin{matrix}1 & 0 \\ 0 & 2 \end{matrix} ) + \frac{1}{2} ( \begin{matrix}2 & 1 \\ 0 & 1 \end{matrix} ) + \frac{1}{2} ( \begin{matrix}2 & −1 \\ 0 & 1 \end{matrix} ) + \frac{1}{2} ( \begin{matrix}1 & 0 \\ 1 & 2 \end{matrix} ) + \frac{1}{2} ( \begin{matrix}1 & 0 \\ −1 & 2 \end{matrix} ))q^{2}
+ (( \begin{matrix}3 & 0 \\ 0 & 1 \end{matrix} ) + ( \begin{matrix}1 & 0 \\ 0 & 3 \end{matrix} ) + ( \begin{matrix}3 & 1 \\ 0 & 1 \end{matrix} ) + ( \begin{matrix}1 & 0 \\ 1 & 3 \end{matrix} ) + ( \begin{matrix}3 & −1 \\ 0 & 1 \end{matrix} ) + ( \begin{matrix}1 & 0 \\ −1 & 3 \end{matrix} ))q^{3} + ...

3.2 The family related to Γ(2)

Let n be an odd integer > 0. Let

U_{n} = \{( \begin{matrix}x & −y \\ y′ & x′ \end{matrix} ) ∈ M_{2}(ℤ)/ xx′ + yy′ = n, x ∈ (1 + 4ℤ), x and x′ odd,
y and y′ even, x > |y|, x′ > |y′|}

and

V_{n} = \{( \begin{matrix}x & −y \\ y′ & x′ \end{matrix} ) ∈ M_{2}(ℤ)/ xx′ + yy′ = n, x ∈ (1 + 4ℤ), x and x′ odd,
y and y′ even, y > |x|, y′ > |x′|}.

Proposition 19 The element

θ_{n} = ∑_{M∈U_{n}} M − ∑_{M∈V_{n}} M

satisfies the condition (C_{n}). See [7] for the proof. These elements θ_{n} are somewhat more canonical than the Manin-Heilbronn elements. Let us mention that they satisfy the following relations in ℤ[M_{2}(ℤ)] ([7])

θ_{n}θ_{n′} = θ_{nn′}

if n and n′ are coprimes odd integers, and

θ_{p^{q}} = θ_{p}θ_{p^{q−1}} − p ( \begin{matrix}p & 0 \\ 0 & p \end{matrix} ) θ_{p^{q−2}}

if p is an odd prime number and q an integer ≥ 2.

3.3 The set X

Let n be an integer > 0. Proposition 20 The element

∑_{a>b≥0,d>c≥0,ad−bc=n} ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ ℂ[M_{2}(ℤ)_{n}]

satisfies the condition (C_{n}). Proof .- The proof is similar to the proof of the proposition 18. Let

X_{n} = \{( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ M_{2}(ℤ)_{n}/a > b ≥ 0, d > c ≥ 0, ad − bc = n\}.

Lemma 6 The set of elements ( \begin{matrix}d & b \\ 0 & \frac{n}{d} \end{matrix} ) ∈ M_{2}(ℤ) (resp. ( \begin{matrix}\frac{n}{d} & 0 \\ b & d \end{matrix} )), with d > 0 and 0 ≤ b < d is a set of representatives of M_{2}(ℤ)_{n}/SL_{2}(ℤ). These elements are the only elements ( \begin{matrix}α & β \\ γ & δ \end{matrix} ) ∈ X_{n} such that γ = 0 (resp. β = 0). Proof .- The proof of this lemma is similar to the proof of the lemma 2.

Lemma 7 Let g = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ X_{n} such that c ≠ 0 (resp. b ≠ 0). There exists an unique element g_{0} = ( \begin{matrix}a_{0} & b_{0} \\ c_{0} & d_{0} \end{matrix} ) ∈ X_{n} ∩ gSL_{2}(ℤ) such that ( \begin{matrix}a \\ c \end{matrix} ) = ( \begin{matrix}b_{0} \\ d_{0} \end{matrix} ) (resp. ( \begin{matrix}b \\ d \end{matrix} ) = ( \begin{matrix}a_{0} \\ c_{0} \end{matrix} )). We have g∞ = g_{0}0 (resp. g0 = g_{0}∞).

Proof .- We prove the assertion corresponding to c ≠ 0. We prove first the existence of g_{0}. Let m be the smallest integer ≥ d/c. We have m ≥ 2. We have ma − b > a > 0 and c > mc − d ≥ 0 by construction of m. So the matrix

g ( \begin{matrix}m & 1 \\ −1 & 0 \end{matrix} ) = ( \begin{matrix}ma − b & a \\ mc − d & c \end{matrix} )

belongs to X_{n} and the existence is proved. Now we consider the problem of the unicity. We have

g^{−1}g_{0} = ( \begin{matrix}\frac{a_{0}d−bc_{0}}{n} & \frac{b_{0}d−bd_{0}}{n} \\ \frac{ac_{0}−a_{0}c}{n} & \frac{ad_{0}−b_{0}c}{n} \end{matrix} ) = ( \begin{matrix}\frac{a_{0}d−bc_{0}}{n} & 1 \\ −1 & 0 \end{matrix} ) ∈ SL_{2}(ℤ).

We use the notation m = \frac{a_{0}d−bc_{0}}{n}. We have

g_{0} = g ( \begin{matrix}m & 1 \\ −1 & 0 \end{matrix} ) = ( \begin{matrix}ma − b & a \\ mc − d & c \end{matrix} ) .

Since g_{0} belongs to X_{n}, we must have c > mc − d ≥ 0. We deduce that m must be the smallest integer ≥ d/c. The unicity is proved.

Lemma 8 The set X_{n} is finite.

Proof .- We have to prove that the set of quadruples (a, b, c, d) ∈ ℤ^{4} solutions of the equation n = ad − bc, with a > b ≥ 0 and d > c ≥ 0, is finite. We have the inequality n = ad − bc ≥ ad − (a − 1)(d − 1) = a + d − 1. Therefore the four numbers a,,b, c and d are all positive and smaller than n.

We finish now the proof of the proposition 20. Let K ∈ M_{2}(ℤ)_{n}/SL_{2}(ℤ). We have

in ℂ[ℙ^{1}(ℚ)] (the sums are taken with respect to the matrices M = ( \begin{matrix}a & b \\ c & d \end{matrix} ))

∑_{M∈K∩X_{n}}[M∞] − [M0] = ∑_{M∈K∩X_{n}}[\frac{a}{c}] − ∑_{M∈K∩X_{n}}[\frac{b}{d}].

Because of the lemma 7, all the terms destroy each other except those corresponding to c = 0 or b = 0. We obtain

∑_{M∈K∩X_{n}}[M∞] − [M0] = ∑_{M∈K∩X_{n},c=0}[\frac{a}{c}] − ∑_{M∈K∩X_{n},d=0}[\frac{b}{d}].

We now use the lemma 6 and we obtain that the last member is equal to [∞] − [0]. The proposition is proved.

Remark .- The linear combination of matrices appearing in the proposition 20, satisfies a stronger condition than the condition (C_{n}): for all K ∈ M_{2}(ℤ)_{n}/SL_{2}(ℤ) we have in ℤ[ℚ_{+}^{∗}\ℚ^{2}]

∑_{M∈K} u_{M}([M ( \begin{matrix}1 \\ 0 \end{matrix} )] − [M ( \begin{matrix}0 \\ 1 \end{matrix} )]) = [( \begin{matrix}1 \\ 0 \end{matrix} )] − [( \begin{matrix}1 \\ 0 \end{matrix} )].

This is an easy consequence of the lemmas used in the proof of the proposition 20.

3.4 An additional family

Let n be an integer > 0. Let us mention an other family of elements of ℤ[M_{2}(ℤ)_{n}] satisfying the condition (C_{n}), and even the condition (C̃_{n}) of the section 1.8.

Proposition 21 The element

∑_{a>b≥0,d>−c≥0,ad−bc=n} ( \begin{matrix}a & b \\ c & d \end{matrix} ) − ∑_{b≥a>0,−c≥d>0,ad−bc=n} ( \begin{matrix}a & b \\ c & d \end{matrix} )

of ℤ[M_{2}(ℤ)_{n}] satisfies the condition (C_{n}).

The proof of this proposition is based on the same kind of arguments than those used in the proofs of the propositions 18, 19 and 20. We leave this to the reader.

Remark .- The same proposition holds if we exchange simultaneously the signs of b and c in the two sums of the proposition 21.

If we set A = \{( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ M_{2}(ℤ)/a > b ≥ 0, d > −c ≥ 0\}, and B = \{( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈

M_{2}(ℤ)/b ≥ a > 0, −c ≥ d > 0} then we obtain an other universal Fourier expansion:

∑_{M∈A} Mq^{detM} − ∑_{M∈B} Mq^{detM}

= ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ) q
+ (( \begin{matrix}2 & 0 \\ 0 & 1 \end{matrix} ) + ( \begin{matrix}1 & 0 \\ 0 & 2 \end{matrix} ) + ( \begin{matrix}2 & 1 \\ 0 & 1 \end{matrix} ) − ( \begin{matrix}1 & 1 \\ −1 & 1 \end{matrix} ) + ( \begin{matrix}1 & 0 \\ −1 & 2 \end{matrix} ))q^{2}
+ (( \begin{matrix}3 & 0 \\ 0 & 1 \end{matrix} ) + ( \begin{matrix}1 & 0 \\ 0 & 3 \end{matrix} ) + ( \begin{matrix}3 & 1 \\ 0 & 1 \end{matrix} ) − ( \begin{matrix}1 & 1 \\ −2 & 1 \end{matrix} ) + ( \begin{matrix}1 & 0 \\ −2 & 3 \end{matrix} )

+ ( \begin{matrix}3 & 2 \\ 0 & 1 \end{matrix} ) − ( \begin{matrix}1 & 2 \\ −1 & 1 \end{matrix} ) + ( \begin{matrix}1 & 0 \\ −1 & 3 \end{matrix} ))q^{3} + ...

It would be interesting to find other elements satisfying the condition (C_{n}). It seems that Zagier has recently found such elements closely related to the trace formula of Hecke operators.

4 Construction of modular forms

4.1 Modular forms as linear maps on the Hecke algebra

Let M be any of the following vector spaces: S_{k}(N), S_{k}(N, χ), any of the previous ones but extending spaces of cusp forms to spaces of holomorphic modular forms (i.e. including Eisenstein series), any of the previous ones but restricting to spaces of newforms, any of the previous ones but considering antiholomorphic modular forms instead.

Let 𝕋 be the corresponding Hecke algebra, i.e. the complex commutative subalgebra of End_{ℂ}(M) generated by the Hecke operators T_{n}^{∗} and T_{n,n}^{∗} (see section 2.3 and 2.5, n is an integer > 0) which we will denote simply by T_{n} and T_{n,n}.

The statement and the proof of the following theorem is modeled on [1], page 306.

Theorem 6 Let α be a linear map 𝕋 → ℂ. Then

∑_{n=1}^{∞} α(T_{n})q^{n}

is, except for the constant coefficient, the Fourier expansion of an element of M.

Proof .- We denote by a_{n} the linear form on M which associates to a modular form its n-th Fourier coefficient. We have a_{n} = a_{1} ∘ T_{n}.

Lemma 9 The bilinear pairing of complex vector spaces

M × 𝕋 → ℂ
(f, T) ↦ a_{1}(T f)

is nondegenerate.

Proof .- Let T ∈ 𝕋. Suppose that for all f ∈ M we have a_{1}(T f) = 0. We have then for all n ≥ 1

0 = a_{1}(T T_{n}f) = a_{1}(T_{n}T f) = a_{n}(T f).

So we have T f = 0 for all f ∈ M. We deduce the equality T = 0.

Conversely, let f ∈ M. Suppose that for all T ∈ 𝕋, we have a_{1}(T f) = 0. Then for all n ≥ 1, we have

a_{1}(T_{n}f) = a_{n}(f) = 0.

So we have f = 0.

We turn now to the proof of the theorem. It follows from the lemma that there exists f ∈ M such that α(T) = a_{1}(T f) (T ∈ 𝕋). Then we have the equalities

∑_{n=1}^{∞} α(T_{n})q^{n} = ∑_{n=1}^{∞} a_{1}(T_{n}f)q^{n} = ∑_{n=1}^{∞} a_{n}(f)q^{n}.

The later expression is the Fourier expansion of f. This concludes the proof of the theorem.

4.2 Proof of the main theorem

We prove now the theorem 1 by putting together all the parts of this paper. We write

x = ∑_{λ∈E_{N}} P_{λ}[λ].

We consider m(x) = ∑_{λ∈E_{N}} [P_{λ}, λ] ∈ M_{k}(N). The equality b(x) = 0 is equivalent to m(x) ∈ S_{k}(N) (proposition 6 combined with the fact that S_{k}(N) is the kernel of ∂). Because of the relations

ϕ + ϕ_{|σ} = ϕ + ϕ_{|τ} + ϕ_{|τ^{2}} = ϕ − ϕ_{|J} = 0,

the linear map ϕ factorizes through a linear map on M_{k}(N) (see proposition 3), which induces a linear map ϕ_{m} on S_{k}(N). It is a consequence of the theorem 3 and the proposition 10 that the Hecke algebra on S_{k}(N) is canonically isomorphic to the complex algebra generated by the operators T_{n} on S_{k}(N). The map

𝕋 → ℂ
T ↦ ϕ_{m}(T m(x))

is a linear map on the Hecke algebra. We use the theorem 2 and the proposition 20 to establish the following equalities

ϕ_{m}(T_{n}m(x)) = ∑_{λ∈E_{N}} ϕ_{m}( ∑_{M∈X_{n}}[P_{λ}_{|M̃}, λM])
= ∑_{λ∈E_{N}} ϕ( ∑_{M∈X_{n}} P_{λ}_{|M̃}[λM])
= ∑_{M∈X_{n}} ϕ_{|M}(x).

Because of the property ϕ(P [x]) = 0, if x ∉ E_{N}, we do not have to make the restriction λM ∈ E_{N}. By application of the theorem 6 we obtain that

∑_{n=1}^{∞} ϕ_{m}(T_{n}m(x))q^{n} = ∑_{n=1}^{∞}( ∑_{M∈X_{n}} ϕ_{|M}(x))q^{n}
= ∑_{M∈X} ϕ_{|M}(x)q^{detM}

is the Fourier expansion of an element of S_{k}(N).

Let A be an element of S_{k}(N) such that for all f ∈ S_{k}(N), we have a_{1}(f) =< f, A > (We can take the element A_{1} of section 3.3).

Let f ∈ S_{k}(N). Let ϕ_{f} be the unique linear form S_{k}(N) → ℂ such that ϕ_{f}(y) =< f, y > for all y ∈ S_{k}(N). By considering the linear form ϕ equal to ϕ_{f} composed with the canonical surjection from the kernel of b to S_{k}(N) and by choosing an element x ∈ ℂ_{k−2}[X, Y][(ℤ/Nℤ)^{2}] such that b(x) = 0 and of image A in S_{k}(N) as parameters, we obtain the following Fourier expansion of f

∑_{n=1}^{∞} ϕ(T_{n}x)q^{n} = ∑_{n=1}^{∞} ϕ_{f}(T_{n}A)q^{n} = ∑_{n=1}^{∞} a_{1}(T_{n}f)q^{n} = ∑_{n=1}^{∞} a_{n}(f)q^{n}.

This proves that all modular forms can be obtained by this method.

If ϕ satisfies the additional condition

ϕ(P [λu, λv] = χ(λ)ϕ(P [u, v])

((λ, u, v) ∈ (ℤ/Nℤ)^{3}, P ∈ ℂ_{k−2}[X, Y]), then f belongs to S_{k}(N, χ). This follows from the proposition 14 and the theorem 6.

Remark .- 1) We will outline now the proof of the statement following the theorem 1 in the introduction. The Hecke operators operate on B_{k}(N) (see the operator T_{Δ}^{∂} of the section 2.1). One can construct an isomorphism of complex vector spaces between the space of Eisenstein series of weight k for Γ_{1}(N) and ∂(M_{k}(N)) ⊂ B_{k}(N). Moreover this isomorphism is compatible with the action of Hecke operators. This can be seen through the Eichler-Simura isomorphism. Since we have the exact sequence

0 → S_{k}(N) → M_{k}(N) → B_{k}(N),

the Hecke algebra for M_{k}(N) is isomorphic to the Hecke algebra for the space S_{k}(N)⊕ Eisenstein series. By application of the theorem 6 we should then prove the following result. If the element x in the theorem 1 does not verify b(x) = 0, then the series

∑_{M∈X} ϕ_{|M}(x)q^{detM}

is, except for the constant term, the Fourier expansion at infinity of a modular form of weight k for Γ_{1}(N) which is not necessarily parabolic.

2) If x belongs to the new part of M_{k}(N), then the Fourier expansion is the Fourier expansion of a newform. This follows from an application of the theorem 6.

3) As noticed in the introduction, we can replace the series connected to X by the Manin-Heilbronn series: With the hypotheses of the theorem 1, the series

∑_{M∈S} ϕ_{|M}(x)q^{detM} + \frac{1}{2} ∑_{M∈S′} ϕ_{|M}(x)q^{detM}

is the Fourier expansion of an element of S_{k}(N). The possibility of producing modular forms using the sets S and S′ was already noticed by Manin (see [4] and [6]).

4.3 Construction of bases of S_{k}(N)

Proposition 22 Let (ψ_{i})_{i∈I} be a basis of the complex vector space Hom_{ℂ}(S_{k}(N)^{+}, ℂ) (resp. Hom_{ℂ}(S_{k}(N)^{−}, ℂ)). Let A ∈ S_{k}(N)^{+} (resp. S_{k}(N)^{−}) such that 𝕋A = S_{k}(N)^{+} (resp. 𝕋A = S_{k}(N)^{−}). Then, for i ∈ I, the modular form f_{i} of Fourier expansion

∑_{n=1}^{∞} ψ_{i}(T_{n}A)q^{n}

runs through a basis of S_{k}(N) when i runs through I.

Proof .- The fact that f_{i} (i ∈ I) is a modular form is a consequence of the theorem 6 and of the theorem 3. We suppose that the family of modular forms (f_{i})_{i∈I} is linearly dependant, i.e. there exists a family of complex numbers (λ_{i})_{i∈I} such that ∑_{i∈I} λ_{i}f_{i} = 0. We set ψ = ∑_{i∈I} λ_{i}ψ_{i}. We have ψ(T_{n}A) = 0 for all n ≥ 1. Since 𝕋A = S_{k}(N)^{+}, the elements T_{n}A generate S_{k}(N)^{+} when n runs through the integers ≥ 1. So we have ψ = 0. Since (ψ_{i})_{i∈I} is a basis of Hom_{ℂ}(S_{k}(N)^{+}, ℂ), we have λ_{i} = 0 for all i ∈ I. Because of the equality of the dimensions of S_{k}(N)^{+} and S_{k}(N) the proposition is proved with respect to S_{k}(N)^{+} (for S_{k}(N)^{−} the proof is of course similar).

Remark .- It is in fact difficult to find an element A in S_{k}(N)^{+} expressed as a linear combination of Manin symbols and satisfying 𝕋A = S_{k}(N)^{+}. But the set of such elements is S_{k}(N)^{+} minus the union of a finite number of proper subspaces.

We can restate the proposition 22 as follows.

Corollary 1 Let A be as in the proposition 20. Let x ∈ ℂ_{k−2}[X, Y][E_{N}] such that the image of x in M_{k}(N) is equal to A. Let (ϕ_{i})_{i∈I} be a linearly free family in the space of linear forms ϕ ∈ Hom_{ℂ}(ℂ_{k−2}[X, Y][E_{N}], ℂ) satisfying

ϕ + ϕ_{|σ} = ϕ + ϕ_{|τ} + ϕ_{|τ^{2}} = ϕ − ϕ_{|−I} = 0,

and

ϕ = ϕ_{|η}.

We suppose also that the space generated by the family (ϕ_{i})_{i∈I} is in direct sum with the space of elements of Hom_{ℂ}(ℂ_{k−2}[X, Y][E_{N}], ℂ) which factorize through the linear map

b : ℂ_{k−2}[X, Y][E_{N}] → ℂ[Γ_{1}(N)\ℚ^{2}]_{k}.

Moreover we suppose that the family (ϕ_{i})_{i∈I} is maximal with respect to the above properties (i.e. the cardinality of I is equal to the dimension of S_{k}(N)). Then

∑_{M∈X} ϕ_{i|M}(x)q^{detM}

runs through the Fourier expansion of a basis of S_{k}(N) when i runs through I.

Proof .- This is a direct reformulation of the proposition 22, obtained by considering the results obtained in the section 1.3, 1.4, 2.2 and 4.2. The list of equalities satisfied by ϕ_{i} (i ∈ I) expresses that ϕ_{i} factorizes through M_{k}(N)/(1 − ι^{∗})M_{k}(N). Because of the direct sum, the family of linear forms induced by (ϕ_{i})_{i∈I} on S_{k}(N)/((1 − ι^{∗})M_{k}(N) ∩ S_{k}(N)) ≃ S_{k}(N)^{+}) is linearly free. So the family of linear forms on S_{k}(N)^{+} defined by (ϕ_{i})_{i∈I} is linearly free. It is a basis because of the maximality condition. We are in position to apply the proposition 22, with (ψ_{i})_{i∈I} equal to the family of linear forms on S_{k}(N)^{+} defined by (ϕ_{i})_{i∈I}.

Let A_{1} be the unique modular symbol in S_{k}(N) such that for all f ∈ S_{k}(N), we have a_{1}(f) =< f, A_{1} > and a_{1}(f) =< ι(f), A_{1} >. Let A_{1}^{+} + A_{1}^{−} be the decomposition of A_{1} with respect to the direct sum

S_{k}(N) = S_{k}(N)^{+} ⊕ S_{k}(N)^{−}.

Proposition 23 The modular symbols A_{1}^{+} ∈ S_{k}(N)^{+} and A_{1}^{−} ∈ S_{k}(N)^{−} verify

𝕋A_{1}^{+} = S_{k}(N)^{+} and 𝕋A_{1}^{−} = S_{k}(N)^{−}.

Proof .- Because of the proposition 7, we only have to prove that no modular form f ∈ S_{k}(N) is orthogonal to 𝕋A_{1}^{+}. We have for f ∈ S_{k}(N) and n an integer ≥ 1

< f, T_{n}A_{1}^{+} > = < T_{n}f, \frac{1}{2}(A_{1} + ι^{∗}(A_{1})) >
= \frac{1}{2} < T_{n}f + ι(T_{n}f), A_{1} >
= \frac{1}{2}(a_{1}(T_{n}f) + a_{1}(ι(ι(T_{n}f))))
= a_{1}(T_{n}f)
= a_{n}(f).

If f is orthogonal to 𝕋A_{1}^{+}, we have a_{n}(f) = 0 for all n. So f = 0. The proposition is proved.

Remark .- The modular symbol A_{1} defined above is a canonical element of M_{k}(N). It would be interesting to obtain an expression as a linear combination of Manin symbols of A_{1}.

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