On coefficients of Poincaré series and single-valued periods of modular forms

We prove that the field generated by the Fourier coefficients of weakly holomorphic Poincaré series of a given level \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varGamma _0(N)$$\end{document}Γ0(N) and integral weight \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k\ge 2$$\end{document}k≥2 coincides with the field generated by the single-valued periods of a certain motive attached to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varGamma _0(N)$$\end{document}Γ0(N). This clarifies the arithmetic nature of such Fourier coefficients and generalises previous formulas of Brown and Acres–Broadhurst giving explicit series expansions for the single-valued periods of some modular forms. Our proof is based on Bringmann–Ono’s construction of harmonic lifts of Poincaré series.

For every integer m and every even integer k > 2, we define a Poincaré series (cf. [12,28]) where H denotes the Poincaré half-plane with its usual SL 2 (R)-action γ ·τ = (aτ +b)(cτ + d) −1 , j(γ , τ ) = cτ + d is the factor of automorphy, and Γ ∞ ≤ Γ 0 (N ) is the stabiliser of the cusp at infinity, given by the condition c = 0. The series (1) converges absolutely and uniformly on compact subsets of H and defines a weakly holomorphic modular form of weight k and level Γ 0 (N ), meaning that our discussion. Following Petersson [27], we shall also consider Poincaré series in weight k = 2, which are defined via a variation of Hecke's trick.
To a certain extent, the theory of cusp forms reduces to the study of Poincaré series of positive index m, as it is well known that P m,k,N , for m > 0, generate the finite-dimensional C-vector space of cusp forms S k (Γ 0 (N )). A natural question is then to understand the Fourier coefficients a n (P m,k,N ), defined by P m,k,N = n≥1 a n (P m,k,N )q n , q = e 2π iτ .
When the index m is positive, a classical computation yields the formulas a n (P m,k,N ) = δ m,n + 2π(−1) for every n ≥ 1, where I k−1 is an I-Bessel function of order k − 1.
The expressions (2) and (3) play a prominent role in questions involving growth estimates of Fourier coefficients of modular forms (see [32]), but are unsatisfying from an algebraic point of view. Although the real numbers a n (P m,k,N ), for positive m, are expected to be transcendental whenever they are not zero 1 , the Fourier coefficients of Poincaré series of negative index, a n (P −m,k,N ), can satisfy remarkable rationality properties.
The present work is based on the observation that all of the above rationality phenomena can be 'explained' by cohomology. The case of P −1,4,9 was proved by Bruinier et al. [10] using the theory of harmonic Maass forms, the key ingredient being the fact that the unique normalised newform in S 4 (Γ 0 (9)) has CM. In the end of [11], Candelori briefly discusses how to reformulate this result in terms of a certain cohomology group with local coefficients over the modular curve X 1 (9).
This hints to the fact that Fourier coefficients of Poincaré series, in general, are cohomological invariants. In this article, we show that a n (P m,k,N ), m ∈ Z \ {0}, are all rational expressions in periods of the modular motive H 1 cusp (Y 0 (N ), V k ) (see Sect. 0.2 below). These periods, in turn, are related to the classical periods ω + and ω − of cusp forms, and also to their 'quasi-periods' η + and η − , which have recently been introduced by Brown and Hain [8]. As we shall explain below, our result is actually more precise: it identifies the Q-extension generated by all the Fourier coefficients a n (P m,k,N ), for m ∈ Z \ {0}, with the Q-extension generated by a certain subclass of periods of H 1 cusp (Y 0 (N ), V k ), the single-valued periods, in the terminology of Brown [5,7]. A first manifestation of identities relating Fourier coefficients of Poincaré series and single-valued periods appears already in the work of Brown [6]. Subsequent work of Broadhurst and Acres and Broadhurst [1] provided many other explicit examples of such identities, under the guise of 'Rademacher sums' (see below). Our work can also be regarded as a generalisation of Brown's and Acres-Broadhurst's formulas.

Single-valued periods of modular motives
Given a smooth algebraic variety X defined over a number field K ⊂ C and an integer i ≥ 0, there is a canonical comparison isomorphism [19] comp : given by 'integration of differential forms', where H i dR (X) denotes the ith algebraic de Rham cohomology with coefficients in K and H i B (X) denotes the ith Betti cohomology of X, defined as the dual Q-vector space of the ith singular homology H i (X(C); Q) = H i (X(C); Z) ⊗ Z Q of the complex manifold X(C). A period [23] of H i (X) is a complex number of the form σ , comp(α) = σ α ∈ C for some α ∈ H i dR (X) and σ ∈ H i (X(C); Q) = H i B (X) ∨ . When K ⊂ R, the continuous involution X(C) → X(C) given by complex conjugation induces a Q-linear involution sometimes called the 'real Frobenius'. By transport of structures, we obtain an involution sv := comp −1 • (F ∞ ⊗ id) • comp on H i dR (X) ⊗ K C, which can actually be shown to be defined over R; this is the single-valued involution: [7] of H i (X) is a real number of the form ϕ, sv(α) ∈ R for some α ∈ H i dR (X) and ϕ ∈ H i dR (X) ∨ . Single-valued periods can be understood in terms of periods as follows. Given a period matrix P of H i (X), that is, a matrix of comp in a K -basis of H i dR (X) and a Q-basis of H i B (X), a single-valued period matrix is given by The terminology 'single-valued period' has its origin in the theory of period functions (i.e. periods of families of algebraic varieties); we refer to [7] for further details and motivation.
Example 1 Consider the elliptic curve E over Q given by the equation A basis for its first de Rham cohomology . Given a oriented basis (γ 1 , γ 2 ) of H 1 (E(C); Z), we obtain a period matrix It does not depend on the choice of (γ 1 , γ 2 ), and necessarily satisfies S 2 = id and Tr S = 0. Numerically, we can compute a period lattice basis ω 1 , ω 2 in Sage [31] and obtain the quasi-periods η 1 , η 2 by integrating the Weierstrass ℘ function (here, (ω, η) = (dz, ℘(z)dz+ More generally, we can talk about periods and single-valued periods of motives, which can be thought as compatible systems of realisations (de Rham, Betti, étale, etc.) of 'geometric origin'. For instance, the classical periods ω + , ω − of a normalised Hecke cusp form f ∈ S k+2 (Γ 0 (1)) ( [12,25] Chapter 11) are defined as real numbers for which there is a decomposition with P + (resp. P − ) an even (resp. odd) homogeneous polynomial in X, Y . Here, K f = Q(a n (f ) ; n ≥ 1) is the totally real number field generated by the Fourier coefficients of f .
Up to a power of 2πi, the numbers ω + and ω − are periods of a pure motive H f , occurring in the cohomology of a Kuga-Sato variety [35].
In practice, instead of going to the Kuga-Sato variety, it is convenient to stay at the modular curve and to work with cohomology with local coefficients. For every integer N ≥ 1 we denote by Y 0 (N ) the moduli stack over Q classifying elliptic curves endowed with a cyclic subgroup of order N . Its analytification is isomorphic to the orbifold quotient Γ 0 (N )\\H, and its corresponding coarse moduli space is isomorphic to the usual open modular curve Y 0 (N ) over Q. Let E be the universal elliptic curve over Y 0 (N ).
Explicitly, there is a (mixed) motive H 1 (Y 0 (N ), V k ) over Q whose de Rham realisation is given by where V k is the vector bundle Sym k H 1 dR (E/Y 0 (N )) endowed with the Gauss-Manin connection. This cohomology group admits a description in terms of weakly holomorphic modular forms (cf. [8,11,13,22,34]): there is a canonical exact sequence of Q-vector spaces where M ! r (Γ 0 (N ); Q) denotes the space of weakly holomorphic modular forms of weight r and level Γ 0 (N ) with rational Fourier coefficients at infinity, and D k+1 is the 'Bol operator'. The Betti realisation of H 1 (Y 0 (N ), V k ) is isomorphic to the classical group cohomology of Γ 0 (N ) considered by Eichler and Shimura (see Sect. 6 below).
Motives of cusp forms are found in the weight k + 1 part of H 1 (Y 0 (N ), V k ), which is a polarisable pure motive H 1 cusp (Y 0 (N ), V k ) of Hodge type {(k + 1, 0), (0, k + 1)} whose realisations are given by cuspidal (or parabolic) cohomology. The single-valued periods of H 1 cusp (Y 0 (N ), V k ) are the main object of study of this paper. These include, up to a power of 2πi, the Petersson norms of modular forms of weight k + 2 and level Γ 0 (N ), but also some non-standard quantities, as we shall see below.

Single-valued periods and coefficients of Poincaré series
Our main result is the following.
Theorem 1 (cf. Theorem 8 below) Let k ≥ 0 and N ≥ 1 be integers, with k even, and let S = (s ij ) 1≤i,j≤r be a single-valued period matrix with respect to a Q-basis of H 1 cusp (Y 0 (N ), V k ) dR . Then, Q(s ij ; 1 ≤ i, j ≤ r) = Q(a n (P m,k+2,N ) ; for all n ≥ 1 and m = 0).
As an immediate corollary of the above theorem, we deduce that the field generated by Fourier coefficients of Poincaré series of positive and negative index Q(a n (P m,k+2,N ) ; n ≥ 1, m = 0) is finitely generated over Q. In the degenerate case where S k+2 (Γ 0 (N )) = 0 (or, equivalently, H 1 cusp (Y 0 (N ), V k ) = 0), there are no single-valued periods; in other words, S should be taken as the empty matrix, and the statement amounts to asserting that the Fourier coefficients of the corresponding Poincaré series are all rational. This is the case of P −1,2,1 considered above.
Apart from terminology, that Fourier coefficients of Poincaré series of positive index m can be written as rational expressions in single-valued periods of H 1 cusp (Y 0 (N ), V k ) is in fact a classical statement following from Petersson's formula: for every f ∈ S k+2 (Γ 0 (N )), The main novelty in Theorem 1 is the case of negative index.
In general, when dim S k+2 (Γ 0 (N )) = 1 (see Table 1 in Sect. 8 below), that is, when the motive H 1 cusp (Y 0 (N ), V k ) is of rank 2, we prove a refined version of Theorem 1. In this case, let us denote by Theorem 2 (cf. Proposition 8 below) With the above notation, for every integer m ≥ 1, there exists h m ∈ M ! −k (Γ 0 (N ); Q), depending on f and g, such that, for every n ≥ 1, a n (P m,k+2,N ) = − k! m k+1 a m (f )a n (f ) 1 s 21 and a n (P −m,k+2,N ) = k! m k+1 a m (f )a n (f ) where r m,n = k! m k+1 a m (f )a n (g) + n k+1 a n (h m ) ∈ Q.
From the relations S 2 = id and Tr S = 0, we conclude that the entries s 12 and s 22 of S are determined by s 11 and s 21 , so that the above result indeed refines Theorem 1 in the rank 2 case.
The above theorem also explains some rationality phenomena. For instance, when a m (f ) = 0, then the above formulas imply that P m,k+2,N vanishes identically and that a n (P −m,k+2,N ) is a rational number for all n ≥ 1. When the unique normalised cusp form in S k+2 (Γ 0 (N )) has CM, the motive H 1 cusp (Y 0 (N ), V k ) acquires extra endomorphisms after base change to a CM field ( [33] Chapter V), and we can deduce from this that the quotient of single-valued periods s 11 /s 21 is a rational number (cf. Example 4 below); thus, P −m,k+2,N has rational Fourier coefficients. Theorem 2 generalises Brown's Corollary 1.4 in [6], concerning level 1 and weight 12. It also provides a proof for the formulas in 'genus 1' of Acres and Broadhurst [1] found by numerical experimentation. To see this, one must remark that the 'Rademacher sums' of [1] are coefficients of Poincaré series after the action of the Fricke involution (cf. [24] Proposition 5.4). In general, Theorem 1 gives a conceptual explanation for the identities involving periods and quasi-periods of modular forms found by Broadhurst and Acres-Broadhurst (see also the formulas (39) below).

Remark 1
In higher rank, after base change to an appropriate number field K , we can split H 1 cusp (Y 0 (N ), V k ) by using Hecke operators; this leads to simple formulas (with coefficients in K ) akin to the rank 2 case. This is worked out in Sect. 10 in the case where there are no old forms, such as in level 1.
Our proof method relies on the theory of harmonic Maass forms and in its relation with the single-valued involution as explained by Brown [6]. We summarise this relation in the following theorem, which reformulates and generalises Brown's results (cf. Candelori [11]).

Theorem 3 (cf. Corollary 4 below)
The following diagram of C-vector spaces commutes: where the vertical maps are induced by (5), and c dR denotes complex conjugation on coefficients.
Here, H ! −k (Γ 0 (N )) denotes the space of harmonic Maass forms of manageable growth of weight −k and level Γ 0 (N ) (see Sect. 7 below), and ξ −k and D k+1 are the differential operators considered in [10]. Let us remark that Theorem 3 clarifies a number of constructions in the theory of weakly holomorphic modular forms and harmonic Maass forms. For instance, using the above diagram one can show that the 'regularised Petersson inner product' of [3] is induced by the usual Hermitian pairing on H 1 dR (Y 0 (N ), V k )⊗ Q C given by the polarisation of its Hodge structure (see Remark 11 below); this immediately implies, among other properties, the compatibility of the regularised inner product with Hecke operators.
By combining Theorem 3 with a theorem of Bringmann and Ono on the harmonic lifts of Poincaré series [2], we deduce the following result, which is the main ingredient in the proof of Theorems 1 and 2.

Organisation of the article
In Sects. 0 and 1 we outline the basic formalism concerning single-valued periods. As in [7], we avoid explicit mention of a category of motives by working in a suitable category of realisations H, which is sufficient for our purposes. We then focus on the case of pure polarisable objects of H of Hodge type {(n, 0), (0, n)}, as 'motives of modular forms' are of this type. Our main result here is Proposition 2, which explicitly describes the algebraic relations the single-valued period matrix of such an object must satisfy. Sections 2 and 3 concern the theory of weakly holomorphic modular forms; their main purpose is to set up notation. We first introduce several spaces of weakly holomorphic modular forms for the group Γ 0 (N ), especially S !,∞ r (Γ 0 (N )), the space of weakly holomorphic cusp forms which are holomorphic at every cusp different from ∞. Then, we recall their geometric interpretation in terms of the moduli stack Y 0 (N ), and the q-expansion principle. A key result is Lemma 4, which will allow us to control the field of definition of certain Fourier coefficients.
We then proceed to an explicit description of the objects H 1 cusp (Y 0 (N ), V k ) of H in Sects. 4 and 5. This is well known to experts, but we could not find a reference with all the properties, in the particular setting and degree of generality we need, spelled out. After describing the cuspidal de Rham cohomology group H 1 dR,cusp (Y 0 (N ), V k ) (see Corollary 2), we show how to express it as the de Rham realisation of an object in H (see Theorem 5); in particular, it is endowed with a single-valued involution.
In Sect. 6, we 'compute' the single-valued involution of H 1 cusp (Y 0 (N ), V k ) in terms of harmonic lifts of modular forms in Theorem 7. Using a theorem of Bringmann and Ono, this allows us to understand the action of the single-valued involution on the cohomology classes defined by Poincaré series (Proposition 7).
Finally, in Sects. 7, 8 and 9, we employ the previous results to prove our main theorems relating the single-valued periods of H 1 cusp (Y 0 (N ), V k ) and the Fourier coefficients at infinity of the Poincaré series P m,k+2,N ; see Proposition 8, Theorems 8 and 9.

Terminology and notation
Our notations for modular forms are standard. The Poincaré upper half-plane is denoted by H = {τ ∈ C | τ > 0}, and the left action of SL 2 (R) on H ∪ P 1 (R) by If f : H → C is a function, g ∈ SL 2 (R), and r is an integer, we set j(g, τ ) = cτ + d, and we denote If N ≥ 1 is an integer, the Hecke congruence subgroup of level N is the subgroup of SL 2 (Z) defined by The set of cusps of Γ 0 (N ) is the quotient Γ 0 (N )\P 1 (Q). The stabiliser of a cusp p is denoted by Γ 0 (N ) p . For g = a b c d ∈ SL 2 (Z), the cusp determined by g is the class of (a : c) in Γ 0 (N )\P 1 (Q).
We denote where q = e 2π iτ .

The basic formalism of single-valued periods
Let K be a subfield of R, and H(K ) the 'category of realisations' considered in [7] (see also [16,17]). Its objects are given by triples and a decreasing (Hodge) filtration F dR , and -comp : commutes, where c B and c dR denote the action of complex conjugation on coefficients.

Example 3 (Tate objects)
For every n ∈ Z, we define an object Q(−n) of H(K ) as follows: The commutativity of the diagram (7) implies that the C- Note that a morphism ϕ : H → H in H(K ) commutes with the single-valued involutions: where ϕ dR ⊗ id : We can express the action of the Betti complex conjugation id ⊗ c B on the de Rham side in terms of the single-valued involution as follows.

Lemma 1 For any object H of
where ω ∈ H dR , ϕ ∈ H ∨ dR := Hom K (H dR , K ), and , denotes the natural duality pairing.
If r denotes the dimension of the K -vector space H dR , and b dR : K ⊕r ∼ → H dR is a K -basis of H dR , then there exists a unique S ∈ GL r (R), the matrix of sv in the basis b dR , such that The coefficients of S are single-valued periods of H and generate the K -linear span of all single-valued periods of H. Since sv is an involution, single-valued periods always satisfy the relations Single-valued periods may also satisfy other relations: Example 4 An endomorphism ϕ of H in H(K ) induces a K -linear relation between its single-valued periods. Keeping the above notation, let M ∈ GL r (K ) be the matrix of ϕ dR in the basis b dR , so that ϕ dR • b dR = b dR • M. By (9), and by definition of M and S, we get We remark that, if K is a real number field and L ⊂ C is a finite extension of K , then we can also define a category of realisations H(L) (see [7] 2.1.2), and we get similar linear relations over L, with M ∈ GL r (L), if ϕ is now an endomorphism of H ⊗ K L in H(L).
Let us recall how the notion of single-valued periods connects with the notion of periods (cf. [7] 2.4). The periods of an object H of H(K ) are complex numbers of the form where ω ∈ H dR and σ ∈ H ∨ B := Hom Q (H B , Q). We can write single-valued periods in terms of periods as follows.
The coefficients of P generate the K -linear span of all the periods of H. Using Lemma 1 and the definitions of S and P, we get Remark 2 Beware that a period of an object H of H(K ) as defined above is only a period in the sense of Kontsevich and Zagier [23] if K is a number field and H comes from a mixed motive over K , e.g. H = H i (X) for some K -algebraic variety X.

Single-valued periods in the presence of a polarisation
Let H be an object of H(K ) such that (H B , W B , comp(F dR )) is a pure Q-Hodge structure of weight n ∈ Z. This means that We say that H is pure of weight n; the Hodge type of H is the set {(p, q) ∈ Z 2 | H p,q = 0}.
in H(K ) inducing a polarisation on the pure Hodge structure (H B , comp(F dR )) ([15] 2.1.15). This means that the C-linear extension of , B to H B ⊗ Q C is (−1) n -symmetric and satisfies the classical 'Hodge-Riemann relations': It follows from the commutativity of and from Example 3 that (2πi) n ω, η dR = comp(ω), comp(η) B (11) for every ω, η ∈ H dR ⊗ K C.

Remark 3
The second Hodge-Riemann relation implies that the Q-bilinear pairing , B , and hence also the K -bilinear pairing , dR by (11), is non-degenerate.
The de Rham pairing , dR is compatible with the single-valued involution as follows.
Proof Since the de Rham pairing is defined over K ⊂ R, it suffices to prove the above formula for ω, η ∈ H dR . In this case, we have (sv ⊗ c dR )(ω) = sv(ω) and similarly for η. The equality sv(ω), sv(η) dR = (−1) n ω, η dR then follows from (9) and from the fact that the single-valued involution of Q(−n) is (−1) n (Example 3). Alternatively, we may apply Lemma 1 together with the formula (11), and use that The presence of a polarisation imposes additional algebraic relations on single-valued periods. For future reference, we spell out these relations in the particular case where H is pure of Hodge type {(n, 0), (0, n)} for some n ∈ Z.
In this case, let d : of H dR such that Note that ω i , ω j dR = 0 because F n dR is isotropic for , dR by the first Hodge-Riemann relation. Such bases always exist by a simple variation of the Gram-Schmidt process (cf. Remark 3).
Let us write the matrix S of the single-valued involution sv : Proposition 2 Let H be a polarisable pure object of H(K ), pure of Hodge type {(n, 0), (0, n)} for some n ∈ Z. With the above notation, we have C ∈ GL d (R), and In particular, the field K (S) generated by all of the coefficients of S is equal to the field K (A, C), generated by the coefficients of A and C.
In coordinate-free parlance, the field of rationality of the single-valued involution sv : It follows from Lemma 1 and from the Hodge decomposition It follows from the definition of the basis b dR , and from Lemma 2, that or, equivalently, using that S −1 = S, Finally, the equation S 2 = 1 gives Remark 4 With the above notation, set R := AC −1 . Then, R t = (−1) n+1 R, C t = C, and we have We finish this section with an alternative way of encoding the single-valued period matrix, in terms of the Hermitian form defined by the polarisation.
Proof That ( , ) dR is a Hermitian form (i.e. (zω, wη) dR = zw(ω, η) dR for z, w ∈ C) follows from formula (11) and from Lemma 1. The positive definiteness on F n dR ⊗ K C is a consequence of the second Hodge-Riemann relation.

Consider a de Rham basis
and, similarly, ( 1 4 ) Note that we proved in Proposition 2 that C is invertible; using the above formulas, we see that this is equivalent to the positive definiteness of ( , ) dR over F n dR ⊗ K C (Lemma 3).

Spaces of weakly holomorphic modular forms
Let r and N ≥ 1 be integers.

Definition 3 A weakly holomorphic modular form of weight r and level
for every γ ∈ Γ 0 (N ), and meromorphic at all cusps: for every g ∈ SL 2 (Z), there exists ρ > 0 such that as τ → +∞.
The C-vector space of weakly holomorphic modular forms of weight r and level ) admits a Fourier expansion at infinity, which we denote by f = n −∞ a n (f )q n where q = e 2π iτ , and a n (f ) ∈ C. For every n ∈ Z, we can regard a n : M ! r (Γ 0 (N )) → C, f → a n (f ) as a linear functional on M ! r (Γ 0 (N )). If K is any subfield of C, we denote by M ! r (Γ 0 (N ); K ) the K -subspace of M ! r (Γ 0 (N )) consisting of those f such that a n (f ) ∈ K for every n ∈ Z.
Definition 4 For f ∈ M ! r (Γ 0 (N ); K ), the (extended) principal part of f at the cusp at infinity is defined by the finite sum P f = n≤0 a n (f )q n ∈ K [q −1 ].
There are also Fourier expansions at other cusps. Let If w denotes the smallest positive integer such that 1 w 0 1 ∈ g −1 Γ 0 (N )g (the 'width of the cusp determined by g'), then we can write f | g,r = n −∞ a n,g (f )q n w for some a n,g (f ) ∈ C. This is a 'Fourier expansion of f at the cusp [(a : c)] ∈ Γ 0 (N )\P 1 (Q)', but it really depends on g: for j ∈ Z, we have a n,gT j = e 2π i j w a n,g , where T = 1 1 0 1 . Note, however, that the vanishing of the nth Fourier coefficient at a given cusp is a well defined property.

Definition 5
A weakly holomorphic cusp form of weight r and level Γ 0 (N ) is a weakly holomorphic modular form f ∈ M ! r (Γ 0 (N )) such that a 0,g (f ) = 0 for every g ∈ SL 2 (Z).
We shall also consider weakly holomorphic modular forms which are holomorphic at all finite cusps. Namely, we define a subspace M !,∞ r (Γ 0 (N )) of M ! r (Γ 0 (N )) consisting of f such that a n,g (f ) = 0 for every n < 0 and g = a b c d ∈ SL 2 (Z) with c ≡ 0 (mod N ). In other words, M !,∞ r (Γ 0 (N )) is the space of weakly holomorphic modular forms with no pole at any cusp different from ∞, but which are allowed to have poles at ∞. We also define S !,∞ r (Γ 0 (N )) := S ! r (Γ 0 (N )) ∩ M !,∞ r (Γ 0 (N )) and, for a subfield K of C, M !,∞ r (Γ 0 (N ); K ) := M !,∞ r (Γ 0 (N )) ∩ M ! r (Γ 0 (N ); K ) and . We then have the following diagram of K -vector spaces, where all the arrows are the natural inclusions Note that M r (Γ 0 (N ); K ) (resp. S r (Γ 0 (N ); K )) is the subspace of M !,∞ r (Γ 0 (N ); K ) of weakly holomorphic modular forms with constant (resp. vanishing) principal part at the cusp at infinity.
We conclude this subsection with the example of Poincaré series. Let Γ ∞ ≤ Γ 0 (N ) be the stabiliser of ∞ = (1 : 0) ∈ P 1 (Q). The proofs of the following propositions are well known in weight k ≥ 4; see, for instance, [12] Chapter 8. For Poincaré series of weight 2, a precise exposition can be found in [30] 5.7.

Geometric interpretation; the q-expansion principle
We work over the moduli stacks Y 0 (N ) and X 0 (N ) over Spec Q. By definition, for every Q-scheme S, the fibre of Y 0 (N ) at S is given by pairs (E, C), where E is an elliptic curve over S, and C is a cyclic S-subgroup scheme of E of order N (i.e. locally for the fppf topology on S, the subgroup C admits a generator of order N ; see [21] 3.4). The 'compactified' moduli stack X 0 (N ) has a similar definition, but now E is allowed to be a generalised elliptic curve, as in [18].
One can prove that X 0 (N ) is a proper smooth Deligne-Mumford stack over Spec Q containing Y 0 (N ) as an open substack. More precisely, it follows from [18], Thérème 3.4, that Y 0 (N ) is the complement of a closed substack Z N ⊂ X 0 (N ), the cuspidal locus of X 0 (N ), which is finite étale over Spec Q (therefore representable by a Q-scheme).
In the category of complex analytic spaces, let E be the elliptic curve over H whose fibre at τ ∈ H is the complex torus

Then, the map
induces an isomorphism where the double backslash stands for the 'orbifold quotient' (or 'stacky quotient') of H by the left action of Γ 0 (N ). For g ∈ SL 2 (Z), and w the width of the cusp p determined by g, a coordinate chart of the 'compactification' X 0 (N ) an C of Y 0 (N ) an C at a neighbourhood of p is induced by Remark 7 The moduli stacks Y 0 (N ) and X 0 (N ) are never representable by schemes. Indeed, any subgroup of an elliptic curve E is preserved by multiplication by −1; thus, any object of Y 0 (N ) admits non-trivial automorphisms. If p is an odd prime not dividing N , the moduli stack Y 0 (N ) p classifying elliptic curves with a cyclic subgroup of order N and a full level p structure is representable by a smooth affine Q-scheme Y with a GL 2 (F p )action (cf. [18] Théorème 2.7). The natural map Y → Y 0 (N ) is finite étale and induces an isomorphism Y 0 (N ) ∼ = Y //GL 2 (F p ). Similar considerations also hold for the compactified modular curve X 0 (N ).
Let F be the Hodge line bundle over Y 0 (N ). By definition, if π : E → Y 0 (N ) denotes the universal elliptic curve over Y 0 (N ), then F := π * Ω 1 E/Y 0 (N ) . Concretely, for a k-point y : Spec k → Y 0 (N ) corresponding to a pair (E, C) over a field k, the fibre of F at y is given by F(y) = H 0 (E, Ω 1 E/k ). The Hodge line bundle can be extended to a line bundle F over X 0 (N ) by setting F := π * Ω 1 E/X 0 (N ) (log π −1 (Z N )), where π : E → X 0 (N ) denotes the universal generalised elliptic curve over X 0 (N ).

Example 5
The pullback of F an to H via the uniformisation map (15) is trivialised by the global section ω ∈ H 0 (H, F an ) whose fibre at τ ∈ H is the holomorphic 1-form ω τ := 2πi dz on the complex torus E τ = C/(Z + τ Z). Note that every γ ∈ Γ 0 (N ) defines an automorphism of E by so that it acts on F an by pullback. Explicitly, γ * ω = j(γ , τ ) −1 ω.
Let r be an integer. If f ∈ M ! r (Γ 0 (N )), then it follows from the above example that the section f ω ⊗r ∈ H 0 (H, (F an ) ⊗r ) is Γ 0 (N )-invariant, so that it descends to a global section ω f in H 0 (Y 0 (N ) an C , (F an ) ⊗r ).
sketch Let p be an odd prime not dividing N , so that X 0 (N ) p (resp. Y 0 (N ) p ) is representable by a Q-scheme X (resp. Y ) (see Remark 7). By the growth condition of f at the cusps, ω f extends to a meromorphic global section of (F an ) ⊗r over X an C . Since F is an algebraic coherent sheaf, and since X C is projective, Serre's GAGA implies that the pullback of ω f to X an C is given by a rational global section of F ⊗r over X C ; since it is regular over Y C , it belongs to H 0 (Y C , F ⊗r ). By considering GL 2 (F p )-invariants, we conclude that where D * denotes the punctured complex unit disc. The section ω over H also descends to D * , so that the pullback of ω f to D * is simply f (q)ω ⊗r , where f (q) denotes the Fourier expansion of f at infinity. We now remark that D * → Y 0 (N ) an C 'descends' to an algebraic morphism Spec Q((q)) → Y 0 (N ), given by the Tate curve (see [18] VII.4). From this, we conclude that ω f ∈ H 0 (Y 0 (N ), F ⊗r ) if and only if f (q) ∈ Q((q)).

Corollary 1 For every integer r and every subfield K of C, the map
is an isomorphism of K -vector spaces. In particular, It follows from the explicit description of the coordinates at the cusps of X 0 (N ) an C given in (16) that, if f ∈ M r (Γ 0 (N )), then ω f extends holomorphically to every cusp. Thus, for every subfield K ⊂ C. Similarly, M !,∞ r (Γ 0 (N )) corresponds to those elements of H 0 (Y 0 (N ) C , F ⊗r ) which extend holomorphically at every cusp different from ∞, and the space S ! r (Γ 0 (N )) is given by the elements of H 0 (Y 0 (N ) C , F ⊗r ) with trivial residue at every cusp.
The following technical lemma will be helpful throughout this paper.
In other words, for any weakly holomorphic modular form of negative or zero weight, its field of definition is determined by its principal part.
Proof Since there are no nonzero modular forms of negative weight, and the only modular forms in weight 0 are constant (Remark 5) 2 , the C-linear map M !,∞ −k (Γ 0 (N )) → C[q −1 ], f → P f , given by the principal part at the cusp at infinity is injective. It follows from Corollary 1 that, for any subfield K of C, this injection is the C-linear extension of the

De Rham cohomology with coefficients on modular curves
Let (V, ∇) be the de Rham bundle over Y 0 (N ), with its Gauss-Manin connection. Formally, the induced symplectic pairing on V k . By abuse, we continue to denote by ∇ the induced connection on V k . Consider the Q-vector space Since Y 0 (N ) admits an étale cover by a smooth affine scheme (Remark 7), we can identify Recall that there is a Kodaira-Spencer isomorphism Note that the pullback of ω f to the Poincaré half-plane H via (15) gets identified with f ω ⊗k ⊗ 2πi dτ . Given the natural inclusion of F ⊗k into V k , we obtain from (17) and (18) a map Recall that we have the holomorphic derivation D = 1 Theorem 4 (cf. [8,11,13,22,34]) The sequence of Q-vector spaces is exact.
Note that M k+2 (Γ 0 (N ); Q) maps injectively into H 1 dR (Y 0 (N ), V k ). The above theorem appears to have been rediscovered by several authors. For later reference, we recall the main steps its proof.
Sketch We work over Y , an affine Q-scheme representing Y 0 (N ) p for some odd prime p not dividing N (Remark 7); our statement is then obtained by taking GL 2 (F p )-invariants.
Consider the Hodge filtration The key fact is that Gauss-Manin connection ∇ induces an isomorphism of O Y -modules This is a consequence of the Kodaira-Spencer isomorphism, for it implies that ∇ induces the successive isomorphisms The surjectivity of the natural map follows immediately from (19): , so that α and β map to the same class in H 1 dR (Y, V k ). Its kernel is given by the image of the O Y -morphism where P denotes a section of V k , [P] its class in V k /F 1 V k , and Q the unique section of F 1 V k such that ∇P − ∇Q lies in F k V k ⊗ Ω 1 Y /Q ; the existence and uniqueness of Q is guaranteed by the isomorphism (19).

Remark 8
The spectral sequence associated to the Hodge filtration induces an exact sequence (cf. proof of Theorem 2.7 in [34]) In particular, Theorem 4 (and the exact sequence (21)) can be refined: we only need to consider weakly holomorphic cusp forms whose poles are concentrated at the cusp ∞. This follows from the next lemma. Proof Let p be an odd prime not dividing N , and X (resp. Y ) be a Q-scheme representing the étale cover X 0 (N ) p (resp. Y 0 (N ) p ) of X 0 (N ) (resp. Y 0 (N ). Recall that Y is affine, and X is the projective compactification of Y (Remark 7).
We denote by Z the cuspidal locus, so that X \ Z = Y . Consider the affine curve X * = X \{∞} containing Y . We can identify H 0 (X * , F ⊗k ⊗ Ω 1 X/Q (log Z)) as a Q-subspace of H 0 (Y, F ⊗k ⊗ Ω 1 Y /Q ). By taking GL 2 (F p )-invariants, our statement will follow from the surjectivity of the natural map . As (V k , ∇)| X * is Deligne's canonical extension of (V k , ∇) to X * , we have a natural isomorphism ; in particular, we obtain a surjective map The surjectivity of its restriction to the subspace H 0 (X * , F ) is then proved in the same way as the surjectivity of (20).

Corollary 2 The map M
We finish this section with a brief discussion of the de Rham pairing Explicitly, we can compute , dR (defined in terms of the cup product in de Rham cohomology) via the usual recipe involving residues at the cusps (cf. [13] Theorem 5.2). In fact, by Lemma 5, we only need to consider the cusp ∞. Let ϕ : Spec Q((q)) → Y 0 (N ) be the morphism given by the Tate curve (as in the proof of Proposition 5). For f ∈ S !,∞ k+2 (Γ 0 (N ); Q) and g ∈ M !,∞ k+2 (Γ 0 (N ); Q), we have Solving for F , we obtain the explicit formula Note that, since g is meromorphic at infinity, the above sum is finite.

Realisations of modular motives
Next, we construct an object H 1 (Y 0 (N ), V k ) of H(Q) whose de Rham realisation is as defined in the last section. Throughout, k ≥ 0 is an even integer. For the Betti realisation, let π : E → Y 0 (N ) be the universal elliptic curve over Y 0 (N ), and consider the Q-local system V := R 1 π an C, * Q E an C over Y 0 (N ) an C , whose stalk at a point y of Y 0 (N ) an C , corresponding to a pair (E, C) defined over C, is the Betti cohomology with rational coefficients V y = H 1 (E an , Q). Setting V k := Sym k V, we define The Betti realisation has a concrete description in terms of group cohomology. The pullback of V to H via the uniformisation map H → Y 0 (N ) an C (15) is trivialised by the global sections where σ 1 , σ τ ∈ H 0 (H, V ∨ ) are the constant families of 1-cycles given by 1 and τ under the identification H 1 (E τ , Q) ∨ = Q + τ Q. Thus, V can be regarded as the vector space V = QX ⊕ QY endowed with the right Γ 0 (N )-action In particular, if we set V k := Sym k V k , then we have a canonical identification Fixing an arbitrary τ 0 ∈ H, the comparison isomorphism can be explicitly given by where [f ] denotes the class of some f ∈ M ! k+2 (Γ 0 (N )) (see Sect. 5), and C f : Γ 0 (N ) → V k is the 1-cocycle defined by This follows from the fact that the 'relative comparison isomorphism' comp : (V an C , ∇ an ) satisfies The weight and Hodge filtrations on H 1 dR (Y 0 (N ), V k ) are explicitly defined by and

Theorem 5 With the above definitions, the triple
is an object of the category of realisations H(Q). Moreover, the weight k + 1 part of Note that the Hodge filtration F k+1 Proof The commutativity of the diagram (7) can be readily verified from the above explicit definitions of comp and F ∞ . That (H 1 (Y 0 (N ), V k ) B , W B , comp(F dR )) is a Q-mixed Hodge structure follows from a theorem of Zucker [37] (Sections 12 and 13). Finally, that H 1 cusp (Y 0 (N ), V k ) is of Hodge type {(k + 1, 0), (0, k + 1)} is the content of the classical Eichler-Shimura isomorphism (see also [37] Section 12).
The pure object H 1 dR,cusp (Y 0 (N ), V k ) of H(Q) admits a canonical polarisation whose de Rham realisation , dR was defined at the end of Sect. 5. The Betti realisation , B is defined in terms of the cup product in the same way; here, we use the Q-linear pairing of local systems , : V × V → Q Y 0 (N ) an C whose fibres are induced by the usual intersection pairing on H 1 (E an , Z), where E is a complex elliptic curve. Explicitly, over H, we have Let us now consider the de Rham Hermitian form ( , ) dR on the vector space 2. The next proposition shows that its restriction to S k+2 (Γ 0 (N )) is, up to an explicit factor, the usual Petersson inner product of cusp forms. 5. For every f, g ∈ S k+2 (Γ 0 (N )), we have

Proposition 6 Consider the injection f
Proof We work in C ∞ de Rham cohomology. Note that the real structure coming from Betti cohomology coincides with the real structure coming from real valued C ∞ de Rham cohomology, so that, by Lemma 1, sv ⊗ c dR acts as the usual complex conjugation of complex valued C ∞ differential forms. Since [f ] is represented by f ω k ⊗ (2πidτ ), and similarly for [g], it follows from the construction of the pairing , dR in terms of the de Rham cup product that We compute the de Rham pairing ω, ω in terms of the Betti pairing on V by using the relative comparison isomorphism (27) and Eq. (28): Thus, ω k , ω k k = (−4πy) k and the statement follows.
Remark 10 Although we do not work explicitly with Betti realisations in this paper, let us remark that, under the identification (24), one can relate the Betti pairing , B with a generalisation of Harberland's inner product. Then, the Haberland formula (see [26] Theorem 3.2), relating this inner product to the Petersson inner product, follows immediately from the commutativity of (10) and from the above proposition. Let us recall that the above formula implies in particular that the Poincaré series P m,k+2,N with m ≥ 1 generate the (finite-dimensional) vector space S k+2 (Γ 0 (N )); this will be used throughout in Sect. 9.

Harmonic Maass forms and the single-valued involution
Next, we relate the single-valued involution on the objects H 1 (Y 0 (N ), V k ) of H(Q) constructed in Sect. 6 with the theory of harmonic Maass forms of integral weight, which we now briefly recall.
The hyperbolic Laplacian of weight r ∈ Z is the differential operator on H defined by where τ = x + iy.
Definition 6 (cf. [4] 4.1) A harmonic Maass form of manageable growth of weight r and level Γ 0 (N ) is a C ∞ function F : H → C which is modular of weight r for Γ 0 (N ): for every γ ∈ Γ 0 (N ), harmonic: and has 'manageable growth' at all cusps: for every g ∈ SL 2 (Z), there exists ρ > 0 such that as τ → +∞.
The space of harmonic Maass forms of manageable growth of weight r and level Γ 0 (N ) is denoted by H ! r (Γ 0 (N )). If the last condition above is replaced by "for every g ∈ SL 2 (Z), there exists P ∈ C[q −1 ] and ρ > 0 such that as τ → +∞", then we say that F is a harmonic Maass form of weight r and level Γ 0 (N ). The subspace of harmonic Maass forms is denoted by H r (Γ 0 (N )) ⊂ H ! r (Γ 0 (N )); it contains the space M ! r (Γ 0 (N )) of weakly holomorphic modular forms of weight r and level Γ 0 (N ). The following result summarises the main properties of harmonic Maass forms of manageable growth with respect to the differential operators ξ −k and D k+1 , defined on a smooth function F : H → C by Theorem 6 ([4,9,10] Chapter 5) Let k ≥ 0 be an even integer, and F ∈ H ! −k (Γ 0 (N )). Then, ξ −k F and D k+1 F belong to M ! k+2 (Γ 0 (N )). Moreover, F ∈ H −k (Γ 0 (N )) if and only if ξ −k F ∈ S k+2 (Γ 0 (N )). Finally, the following sequence of C-vector spaces To state our next theorem, we introduce the following notion, which will also be used in the following sections.

Definition 7
We say that f, g ∈ M ! k+2 (Γ 0 (N )) are Betti conjugate if Our terminology is justified by Lemma 1, which implies that f and g are Betti conjugate if and only if their classes in Betti cohomology are complex conjugate: Theorem 7 Let k ≥ 0 be an even integer, and f, g ∈ M ! k+2 (Γ 0 (N )) be Betti conjugate. Then, there exists F ∈ H ! −k (Γ 0 (N )) such that If k = 0, then F is unique up to an additive constant; otherwise, F is unique.
This theorem is essentially equivalent to the results of Brown [6] when N = 1, and of Candelori [11] when N ≥ 5 and f is a cusp form. We prove it below for completeness; our approach is similar to Candelori's.
Proof Recall that working over Y 0 (N ) an C amounts to working over H equivariantly under the left action of Γ 0 (N ). This will be implicit in what follows.
After the Kodaira-Spencer identification F ⊗2 ∼ = Ω 1 Y 0 (N )/Q , we have and similarly for ω g . As the real structure on the complex analytic de Rham cohomology H 1 dR (Y 0 (N ) an C , V an k ) coming from real valued C ∞ de Rham cohomology coincides with the real structure given by the Betti cohomology H 1 (Y 0 (N ) an C , V k ) ⊗ Q R after the natural comparison isomorphisms (cf. proof of Proposition 6), the image of the C ∞ differential form with coefficients in V an where ∇ denotes the Gauss-Manin connection on V an k . Since ω and ω trivialise V an over H, we can uniquely write where A r,s are C ∞ functions on H. We set To check that F satisfies ξ −k F = (4π) k+1 f and D k+1 F = k!g, it is convenient to use Brown's terminology and notation in [6]. Note first that, by Γ 0 (N )-invariance of A, the C ∞ functions A r,s are modular of weights (r, s) for Γ 0 (N ). In particular, since 2L := −4π τ is modular of weights (−1, −1), the function F is modular of weights (−k, 0).
By definition of A, we have Using the formulas which can be checked by computing the periods along the 1-cycles σ 1 and σ τ , we obtain from (29) the equations where ∂ r := 2LD + r. Thus, ([6] Lemmas 3.2 and 3.3) Finally, we check that F is a harmonic Maass form of manageable growth of weight −k and level Γ 0 (N ). The modularity property has already been remarked above. That F is harmonic follows from the formula Δ −k = −∂ −k−1 ∂ 0 and from the fact that ξ −k F = (4π) k+1 f is holomorphic. The growth condition at the cusps follows from those of f and g, and from the equations D k+1 F = k!g and ξ −k F = (4π) k+1 f ; this can be seen either via the Fourier expansion of F as in [9] (Section 3), or via an explicit expression of F in terms of f and g as in [6] (Section 5.3).

Corollary 4
The following diagram of C-vector spaces commutes: where the vertical arrows are the quotient maps f → [f ] of Theorem 4.
In particular, P m vanishes identically if and only if a m (f ) = 0, and in this case a n (P −m ) ∈ Q for every n ≥ 0.
It also follows from the above proposition that Q(sv) := Q(a, b, c, d) = Q(c, ρ) coincides with the field Q(a n (P m ) ; n ≥ 1, m = 0) generated by the coefficients of all the weakly holomorphic Poincaré series of level N and weight k + 2.
Proof Let m be a positive integer and, since dim S k+2 (Γ 0 (N )) = 1, write In particular, we obtain the following general formula for the Fourier coefficients at infinity of P m in terms of the single-valued period c: a n (P m ) = − k! m k+1 a m (f )a n (f )c −1 .
Note that P m vanishes identically if and only if a m (f ) = 0. Now, set Since f and f are Betti conjugate (Definition 7), and P −m and −P m are Betti conjugate (Proposition 7), it follows from (30) that −λf and P −m map to the same class in Thus, there exists h m ∈ M !,∞ −k (Γ 0 (N ); R) such that P −m = −λf + D k+1 h m .
By considering principal parts at the cusp at infinity, we get 1 q m = −λP f + P D k+1 h m (32) = −λcP g + D k+1 P h m = k! m k+1 a m (f )P g + D k+1 P h m . (34) As both f and g have rational Fourier coefficients at infinity, we get D k+1 P h m ∈ Q[q −1 ]. Thus, P h m ∈ Q[q −1 ], and Lemma 4 implies that h m ∈ M !,∞ −k (Γ 0 (N ); Q). Finally, by taking Fourier coefficients at infinity in (33) and applying (31), we get a n (P −m ) = k! m k+1 a m (f )a n (f )ρ + k! m k+1 a m (f )a n (g) + n k+1 a n (h m ).
When m = 1, we can take h m to be zero. Indeed, in the above cases we always have a 1 (f ) = 0, so we can normalise a 1 (f ) = 1. Moreover, we can assume that ord ∞ (g) ≥ −1, so that P g = (k!) −1 q −1 by the condition [f ], [g] dR = 1. Equation (34) in the above proof then implies that P h 1 = 0, which yields h 1 = 0.
Corollary 5 For f and g as above, we have a n (P −1 ) = k!(a n (f )ρ + a n (g)).
Finally, since λ i ∈ Q, by taking principal parts at the cusp at infinity, we deduce from Proposition 3 and Lemma 4 that h ∈ M !,∞ −k (Γ 0 (N ); Q).
Proof Consider Fourier coefficients at infinity in the statement of Proposition 9.
We prove first that Q(P) ⊂ Q(sv). For this, let m 1 , . . . , m d ≥ 1 be integers such that (P m 1 , . . . , P m d ) is an R-basis of S k+2 (Γ 0 (N ); R), and write for the matrix of the single-valued involution sv in the basis b dR . By Corollary 6, it suffices to prove that a n (P m i ), a n (P −m i ) ∈ Q(S) for every integer n ≥ 1 and 1 ≤ i ≤ d. Let Λ = (Λ ij ) i,j ∈ GL d (R) be such that By applying Corollary 3 in the above formula, we obtain where we used the identity (13) in the last equality above. Set Q := ( k! m k+1 j a m j (f r )) r,j ∈ GL d (Q).
The above equations amount to the matrix identity in GL d (R) Since Q has rational coefficients, this proves already that a n (P m j ) ∈ Q(C) ⊂ Q(S) for every n ≥ 1 and 1 ≤ j ≤ d.
By definition of S, f j is Betti conjugate to f j . Then, it follows from Proposition 7 and from (36) that both P −m j and − d i=1 Λ ij f i are Betti conjugate to −P m j . Thus, by Corollary 2, there exists h j ∈ M !,∞ −k (Γ 0 (N )) such that By taking principal parts at the cusp at infinity in the above equation, and by applying Lemma 4 and (37), we get h j ∈ M !,∞ −k (Γ 0 (N ); Q(Λ)) = M !,∞ −k (Γ 0 (N ); Q(C)).
of H f j is determined by ω + j , ω − j , ρ j , since one always has the relation det P j ∈ Q × (2πi) k+1 . As a result, we can obtain 'universal' expressions for the quasi-periods of Hecke eigenforms (in the terminology of [8]) in terms of explicit series involving Kloosterman sums and special values of Bessel functions.