Period functions associated to real-analytic modular forms

We define L-functions for the class of real-analytic modular forms recently introduced by F. Brown. We establish their main properties and construct the analogue of period polynomial in cases of special interest, including those of modular iterated integrals of length one.

Against this contrasting picture, Brown [3][4][5] identified a "hybrid" space, which, though infinite dimensional and consisting of non-holomorphic functions, does appear to be "motivic". This is the space of modular iterated integrals, denoted by MI ! . This, in turn, sits naturally in the broader class of real-analytic modular forms. The definitions are given in Sect. 2, but examples of modular iterated integrals include E 2,0 E 0,2 + 1 2 E 2 1,1 , where E r,s (z) (r, s ∈ N) is the real-analytic Eisenstein series of Sect. 2.2, as well as other linear combinations of products of E r,s and powers of Im(z). A genuinely new function in the space of modular iterated integrals is the function F (1) 1,1 which satisfies (Δ + 2)F (1) 1,1 (z) = 32π 3 Im(z) 3 G 4 (z)G 4 (z) where Δ is the Laplacian (5) and G 4 the holomorphic Eisenstein series with constant term 1 240 . Conjecturally, F 1,1 provides an explicit expression of an important modular graph function, whose theory was another motivation for defining real-analytic modular forms (see [3], §9.3).
Brown provided evidence for the motivic nature of modular iterated integrals by proving interesting structural theorems and algebraicity and rationality results for Fourier coefficients of elements of MI ! . In this paper, we give further credence to the motivic nature of the space of modular iterated integrals by defining L-functions in MI ! (and, in fact, in the full space of real-analytic modular forms) and period functions.
The precise construction of L-functions in the space M ! r,s of real-analytic modular forms of weight r, s is found in Sect. 3.1, but in the special case of "cuspidal" f , it is given by (it)t r+s−w−1 dt wheref andf stand for the "pieces" of f that are, respectively, exponentially decreasing and exponentially increasing at infinity. The integral from 1 to −∞ is explained in (14). It is not obvious how to define period functions in M ! r,s appropriately. This is not surprising because period functions are normally supposed to reflect arithmeticity and the full space M ! r,s is too large to be of arithmetic nature in its entirety. However, here we show that, in the class MI ! 1 of modular iterated integrals of length 1, it is possible to define period functions and, indeed, polynomials. The definition (Sect. 4) requires some preparation, so we will illustrate the construction here under a simplifying assumption that does not hold in general.
Fix s, r ∈ N of the same parity. For an integer k ≤ s, r, a ζ ∈ C and a smooth g : H ∪H → C, where H (resp.H) denote the upper (resp. lower) half-plane, we define the differential 1-form where z = x + iy. Let now F ∈ M ! r,s which is a modular iterated integral (of length 1). As shown in [5], it can be uniquely decomposed as F = F 0 + · · · + F min(r,s) where F k ∈ M ! r,s is an eigenfunction of the Laplacian with eigenvalue (k − 1)(r + s − k). Our simplifying assumption, for the purposes of the introduction, is that each F k has a vanishing "constant term" (see Sect. 2). We then set We can now state Definition 1.2 Let F ∈ M ! r,s which is a modular integral. With the above notation, the period function P(ζ ) of F is given by We will briefly explain why this period function induces a cocycle which is consistent with the Eichler cocycle of standard modular forms: For each fixed k ∈ {0, . . . , min(r, s)}, let Γ := SL 2 (Z) act on the space P r+s−2k (C) of polynomials of degree ≤ r + s − 2k, via: (The reason for the unusual notation || will become clearer later). With this action, the Eichler cohomology group is the group H 1 (SL 2 (Z), P r+s−2k (C)). For each fixed k ∈ {0, . . . , min(r, s)}, let the map σ k : SL 2 (Z) → P r+s−2k (C) be given by We have P(ζ ) = σ 1 (S)(ζ ) + · · · + σ min(r,s) (S)(ζ ). Further, defineσ k : SL 2 (Z) → P r+s−2k (C) asσ By comparison, the Eichler cocycle of a weight k cusp form f for Γ is σ (γ ) = with ω(f ; ζ ) = f (z)(z − ζ ) k−2 dz. Further, the cocycle σ k induced by our P(ζ ) is also consistent with the Eichler cocycleσ k in the following sense: The maps σ k ,σ k define 1-cocycles in P r+s−2k (C). Their classes in the Eichler cohomology H 1 (SL 2 (Z), P r+s−2k (C)) coincide.
The connection with our L-function is provided by the following theorem (see Theorem 5.3).

Theorem 1.4
Assume that r ≡ s mod 4. Let P (ζ ) denote the polynomial obtained from P(ζ ) upon removing its constant and leading terms. Then, for some explicit a k,l ∈ Q(i).
Finally, as an additional evidence that our definition of L-function is the"right" one, we prove the following: This differs from the full Manin's Periods Theorem in that it does not say anything about the values at 1 and k − 1. As confirmed by numerical experiments, the lack of K fproportionality of L * f (1) (and of L * f (k − 1)) with the other odd "critical values" seems to be genuine and not just due to any incompleteness of our proof.
The precise statement of Theorem 1.5 is Theorem 6.3. To prove it, we use an analogous identity to Theorem 1.4 and the algebraic de Rham theory of weakly holomorphic modular forms [7]. K. Bringmann has shown us how we can use an identity of [2] to deduce a statement which, combined with Theorem 1.4, implies Theorem 1.5.

Real-analytic modular forms
We start by recalling the definition of real-analytic modular forms. Let Γ =SL 2 (Z) and set S = 0 −1 1 0 , T = 1 1 0 1 and U = TS = 1 −1 1 0 . Suppose that r, s are positive integers of the same parity. For a smooth g : H → C and γ ∈ Γ , we define a function f || We extend the action to C[Γ ] by linearity. With this notation, we can make the following definition.
We denote the space of real analytic modular forms of weights (r, s) for Γ by M ! r,s . We set M ! = ⊕ r,s M ! r,s . This class of functions was introduced by Brown [3-5] whose initial motivation was related, on the one hand, to some non-holomorphic modular forms originating from iterated extensions of pure motives and with coefficients that are periods. On the other hand, he was motivated by open questions about modular graph functions appearing in string perturbation theory.
The space M ! contains or intersects various previously studied classes of important modular objects and the point of view we adopt here is to consider real-analytic forms as a unifying tool for those classes. For example, when s = 0, an element f of M ! r,0 which is holomorphic in H is a standard weakly holomorphic modular form of weight r for Γ . We denote their space by M ! r and set M ! = ⊕ r M ! r . The space M ! r contains, of course, the space M r (resp. S r ) of classical modular (resp. cusp) forms of weight r.
When r = 0 we are similarly led to the spaceM r of weakly anti-holomorphic modular forms. Another subspace is M r,s , which is obtained upon imposing the condition that a (k) m,n should vanish when m or n are negative. It was defined and studied in [3]. Their direct sum over all r, s is denoted by M.
The relation with Maass forms is more complicated. On the one hand, the definition of M ! allows for forms which are not eigenfunctions of the Laplacian, but, on the other, it requires a more restrictive form of a Fourier expansion than that of Maass forms. We will exploit this relation in the sequel in order to define some of our main objects, and, in particular, we will see that constructions from the theory of Maass cusp forms will be the basis for period functions of certain elements of M ! .
The Fourier expansion (1) can be uniquely decomposed into a sum of an "principal part" f , an exponentially decaying partf and the "constant term" f 0 defined as follows:

Eigenforms for the Laplacian
It induces a bigraded operator Δ of bidegree (0, 0) on M ! . An operator which is essentially equivalent to Δ r,s but which is more convenient for some computations in the sequel is When working with Ω k , the following version of the "stroke" operator will be more appropriate to work with than || r,s . Specifically, for γ ∈ Γ and f : H → C, the function f | k γ is defined by We extend the action to C[Γ ] by linearity. To move between the Δ to the Ω formalism, the following lemma will be useful:

Lemma 2.2
If, for some r, s ∈ Z of the same parity, F is an element of M ! r,s such that Δ r,s F = λF for some λ ∈ R, then F 1 := y r+s 2 F satisfies Proof We observe that and that, for each smooth f : H → C, An easy computation then implies the lemma.
For λ ∈ R, set Also set HM ! := ⊕ λ HM ! (λ). The following lemma summarises some of the special features of the Fourier expansions of f ∈ HM ! (λ).
Furthermore, the constant term has the form f 0 (z) = ay k 0 + by 1−r−s−k 0 for some a, b ∈ C. Finally, f h , f a , y k 0 and y 1−r−s−k 0 are eigenfunctions of Δ r,s with eigenvalue λ.
Proof Lemma 4.3 of [5] together with the remarks following it.

Real analytic Eisenstein series
An example of an element of M ! r,s , and, indeed, of M r,s , which is, in addition, an eigenform for the Laplacian is the non-holomorphic Eisenstein series. Specifically, in a normalisation and notation which is more appropriate for the theory of real-analytic forms, a real analytic Eisenstein series E r,s is given for r, s ∈ N and z ∈ H by where B is the subgroup of translations. This series converges absolutely and belongs to M r,s It further has a meromorphic continuation to the entire complex plane and is an eigenfunction of Δ with eigenvalue −r − s.
Its Fourier expansion has been computed explicitly in Th. 3.1. of [11] and in Prop. 11.2.16 of [9]. We summarise it here and will see how it fits with Lemma 2.3. With the notation of that lemma, Here a b with a < 0 are defined in accordance with the convention that, if a < 0 and j ≥ 0, then a+j which is consistent with Lemma 2.3.

Modular iterated integrals of length one
In [5], the space of modular iterated integrals is defined. We will consider only the special case of length one: The space MI ! 1 of modular iterated integrals of length one is defined to be the largest subspace of M ! which satisfies A characterisation of this space is provided in [5]: This, in particular, implies that the value of the invariant k 0 (see Lemma 2.3) for F k is We will interpret the functions F k of the last proposition in the setting of the last section.

Proposition 2.5 Let F be an element of MI ! 1 with weights r, s and let F k (k ∈ {0, . . . , min(r, s)}) be as in Proposition 2.4. Then y r+s 2 F k (z) is Γ -invariant under the action of | r−s and is an eigenfunction of Ω r−s with eigenvalue
Proof By Lemma 2.2 and Proposition 2.4, y To establish the eigen-properties (10), we apply the last statement of Lemma 2.3 to Let P be the space of polynomials of y over C. By (2.22) of [3], we have Δ r,s (P · q mqn ) ⊂ P · q mqn (12) for each m, n ∈ Z. Therefore, the LHS of (11) will be a polynomial in q −1 with coefficients in P, and thus, if not identically 0, it will have exponential growth as y → ∞. This is impossible because, by (12), the RHS of (11) decays exponentially as y → ∞. Therefore, the LHS vanishes and We similarly see that −r Since y k−r−s , y 1−k are also eigenfunctions of Δ r,s with eigenvalue λ k , we deduce (with Lemma 2.2) the desired eigen-properties of y (s+r)/2 y k−r−s and y (s+r)/2 y 1−k . The eigenproperties of y (s+r)/2F k , y (s+r)/2 y k−r−s , y (s+r)/2 y 1−k , just proved, together with the eigenproperty of y (s+r)/2 F k then imply the eigen-property of y (s+r)/2F k .

L-functions
The obstacles to extending the definition of L-functions of standard modular forms to HM ! are due to the potentially exponential growth of functions in M ! combined with the lack of holomorphicity. To tackle the former, we can give a definition that is based on the expression of standard L-functions through Mellin transforms. This will, in fact, allow us to define L-functions on the entire M ! .

L-functions in M !
Let f ∈ M ! r,s with an expansion (1). We let the implied logarithm take the principal branch of the logarithm and we set, for w = −j, r + s + j (|j| ≤ M), The rigorous meaning of the first integral from 1 to −∞ is where Γ (r, z) denotes the incomplete Gamma function For z = 0, this has an analytic continuation to the entire r-plane, and therefore, (14) is welldefined for all values of w by the analytic continuation of incomplete Gamma function. By contrast, the real integral as written in (13) is not convergent at 0 unless (w) > 1 + M. We interpret likewise the second integral from 1 to −∞ in (13). The reason we preferred to write formally those terms as integrals was to stress the symmetry with the other terms and hint at the origin of the definition in a "regularisation" introduced in [6].
Since, in addition,f decays exponentially at infinity, all integrals in (13) are well-defined. As mentioned above, the above construction was inspired by "regularisation" introduced in [6], Sect. 4. (See [10], for another application of this idea.) The definition immediately implies the following: . The L-function of f is meromorphic with finitely many poles and satisfies for all w away from the poles.
In such generality, the definition is somewhat formal and would be unlikely to lead to arithmetic insight for all f ∈ M ! r,s . To obtain more refined information, we restrict to subspaces of M ! r,s . We first note that, in the subspace M r,s of f ∈ M ! r,s with moderate growth at infinity, our definition coincides with that of Sect. 9.4 of [3]. Specifically, in that case,f = 0 and the Fourier coefficients of f have polynomial growth. For Re(w) 0, the change of variable t → 1/t in the third integral of (13) together with the transformation law of f implies that L * f (w) coincides with the function (f, w) of Sect. 9.4. of [3]. See also, Section 9.4 of [4] where this construction is applied to the important subclass of M r,s consisting of modular analogues of the single-valued polylogarithms.

L-functions in HM ! (λ) and in MI ! 1
Let f ∈ HM ! (λ) ∩ M ! r,s . Using the Fourier expansion of f provided by Lemma 2.3, the general definition of L * f (w) we gave above leads to an expression as a series. This is more natural because it is reminiscent of the original definition of L-series of standard modular forms and because, in the case of weakly holomorphic modular forms, it coincides with the L-functions already associated with such forms ([1] and references therein).
To ensure that the series we will eventually obtain converges absolutely, we need an analogue of the "trivial bound" about the Fourier coefficients. As in the case of weakly holomorphic forms ( [8], Lemma 3.2), the growth is, in general, exponential. Although the proof parallels that of [8], there are some complications because of the presence of two weights and of the powers of y, so we present a full proof.
Suppose that −s > k 0 . Then (15) implies The RHS will be a sum of products of e 2π ny , polynomials in y and n and for j ∈ {0, . . . , −s − k 0 }. Now, we note that, for all k, l, and, since y −1 ∈ M ! 1,1 , we have ∂ ∂y M ! ⊂ M ! . Thus, We will use this to bound (18) with the help of this lemma: Arguing as above for the second term, and using the bound for a (−s) n we proved above, we deduce the bound for j = −s − 1. Continuing in this way, we deduce the result for all j.
It is clear from the argument (essentially by interchanging the roles of s and k 0 ), that it remains valid when −s ≤ k 0 .
To prove the bound for b (j) n , we work in the same way but based on (16), instead of (15).
We are now ready to use the Fourier expansion given in Lemma 2.3 to express the L-function of an f ∈ HM ! (λ) as a series. For compactness of notation, we set, for each j ∈ Z, c m ) are taken to be 0 if j or m is outside the range of j-or m-summation in (6) (resp. (7)). Then, by substituting the Fourier expansion of f into (13), we deduce, for w = −k 0 , −k 0 + 1, k 0 + r + s − 1, k 0 + r + s, where P(w) denotes Because of Proposition 3.2 and the asymptotics Γ (r, x) ∼ e −x x r−1 as x → ∞, we see that this series converges absolutely for all w ∈ C.

Example: L-function of a weakly holomorphic modular form
In the special case of a weakly holomorphic form, this formula coincides with the earlier definition of an L-function for such forms. Indeed, an f ∈ M ! k can be considered as an which coincides with, say, (6.1) of [1].

L-functions of modular iterated integrals of length 1
Using Proposition 2.4, we can now express the L-function of a function F in the broader class MI ! 1 of modular iterated integrals of length 1, in terms of the L-function in HM ! (λ) for varying λ. Indeed, let F be an element of MI ! 1 of weights r, s. Then, for each s ∈ C, we have where F k are elements of HM ! of weight r, s such that F = F 0 + · · · + F min(r,s) as in Proposition 2.4.

L-functions in M r,s
We now consider the case that f is of polynomial growth at the cusps, i.e., f ∈ M r,s . Then, (it)t w−1 dt converges, and therefore, we can make the change of variables t → 1/t in the third integral of (13) to derive Here we used the transformation law for f and the formula for the antiderivative of f 0 . This coincides with Brown's definition of L-functions of f ∈ M r,s given in [3] (Sect. 9.4). There, up to a different normalisation, the L-function is actually defined, for Re(w) 0, by where, with the notation of (1), The equivalence of this with our definition is established in the proof of Th. 9.7 of [3].

L-functions in HM ! (λ) ∩ M r,s
When, in addition, f ∈ M r,s is an eigen-function of the Laplacian, then by computing Mellin transforms as usual, L * f (w) obtains a more familiar form which, however, is valid . Then for (w) 0, (22) [or, directly, (13)] becomes

Example: L-function of the real-analytic Eisenstein series
We can use the above representation of L * f (s) and (8) to compute explicitly the L-function of the real-analytic Eisenstein series. For (w) 0, we have has been computed as in the case of L-functions of the usual Eisenstein series.) The last expression also gives the meromorphic continuation to the entire w-plane.

Maass-Selberg forms
In [14], the authors extend the classical theory of period polynomials of (holomorphic) cusp forms by assigning a period function to Maass cusp forms of weight 0. Mühlenbruch [16] later generalised that to Maass cusp forms of real weight. One of the ways to define the period function, in both [14,16], is based on a differential form called Maass-Selberg form. We recall its definition and some of its properties.
Let Let k ∈ 2Z. The Maass-Selberg form is then defined by (We normalise slightly differently from [16] because we use Brown's version of the Maass operators instead of the operators E + 2k = 2∂ k and E − 2k = 2∂ −k used in [16].) The next lemma summarises the properties of Maass-Selberg form we will be needing. If η k (f, g) • γ denotes the pull-back of the differential form η k (f, g) by the map z → γ z (z ∈ U), then we have Then η k (f, g) is closed.

A Maass-Selberg form associated to modular iterated integrals of length one
To define the Maass-Selberg form that we will associate to modular iterated integrals of length one, we need a function R n,ν (n ∈ 2Z, ν ∈ C) we now define. For z ∈ H ∪H and ζ = z,z, it is given by For each ζ ∈ C, this gives a well-defined real-analytic function of z if we restrict z to the complement in H of some path joining ζ andζ and then choose an appropriate branch for the implied logarithm. Likewise, for a suitable subset ofH.
For the specific values of n, ν we will use the function R n,ν , it can be defined for all ζ ∈ C and z ∈ H. Specifically, for n = s − r and μ k = −k + (r + s + 1)/2 with k as in Proposition 2.4 we have, Since k ≤ r, s, this can be defined for all z ∈ H ∪H. The function R n,μ k satisfies Lemma 4.2 Set n = s − r and μ k = −k + (r + s + 1)/2 with k ≤ r, s.

2.
For each γ ∈ Γ and ζ ∈ H, we have Proof This is essentially Prop. 36 of [16] but there it is proved with the restriction that ζ ∈ R and j(γ , ζ ) > 0 due to the more general μ and n to which the proposition applies.
We now associate to F k Maass-Selberg forms which will be the basis for our construction of the period function for all functions in MI ! 1 .
where M k (resp. S k ) is the space of classical holomorphic modular (resp. cusp) forms of weight k for Γ . The isomorphism φ is induced by the assignment of f ∈ M k to the map We will now associate to the F k 's of the last section a 1-cocycle in the Γ -module P r+s−2k (C). We define it as the coboundary of a 0-cochain in a larger module than P −2k+r+s (C). The construction follows the definition of the "integral at a tangential base point at infinity" of [6], (Sect. 4).
For convenience of notation, we set η r−s (g; ζ ) := η r−s (y where the line of integration in the last integral includes the origin is well-defined. The differential forms to be integrated in v k can be written more explicitly in the form for each smooth function f : H ∪H → C.
Proof We first show the second assertion. With the definition of η r−s and Lemma 4.2, we have: Substituting the value for μ k , we get (27). From this, we deduce that, if f and ∂f /∂z decay exponentially as y → ∞, the same holds for η r−s (y r+s 2 f, R s−r,μ k (·, ζ )). This condition holds for f =F k . It also holds for f =F k as y → −∞.
The term corresponding to ay k−r−s in the first integral is O(y −2 ) as y → ∞, which assures convergence. (Note that each of the two summands in (27) individually has a term of order y −1 but they cancel each other out on the upper imaginary axis).
Since it is clear that the second integral in the definition of v f is convergent too, we deduce that, for each ζ ∈ C, all integrals are convergent.
Further, by Proposition 4.3, η r−s (F k + ay k−r−s ; ζ ), η r−s (by 1−k ; ζ ) and η r−s (F k , ζ ) are closed in H. The last form is also closed inH. Indeed, for each fixed ζ ∈ C, by (27), we have that d(η r−s (F k , ζ )) = P[e −2π iz ]dz∧dz, where P is a polynomial in e −2π iz whose coefficients are polynomials in z,z. Since η r−s (F k , ζ ) is closed in H, each of those polynomials are identically zero in H, and therefore, they vanish inH too.
We can now define the 1-cocyle σ k on Γ as We will show that, although v k does not belong to P r+s−2k (C), its differential does and, in fact, it belongs to a cohomology class analogous to that of (26) in the classical Eichler cohomology. 2. Letσ k : Γ → P r+s−2k (C) be the map given bỹ This gives a 1-cocycle which belongs to the same cohomology class as σ k .
Proof We occasionally use again the abbreviation η r−s (g; ζ ) := η r−s (y r+s 2 g, R s−r,μ k (·, ζ )). Since η r−s (g; ζ ) are closed for g =F k , ay k−r−s , by 1 where the last integral is also taken to be over a path that includes the origin. By (27), the last three terms of (28) are clearly in P r+s−2k (C). (However, note that to reach this conclusion, we first fix a specific path of integration and only then expand the integrand in ζ . If we first expanded, there would be no guarantee a priori that the resulting differentials are closed). Thus, the image of those integrals under the action by || (1), we have, for each γ ∈ Γ , where the action | s−r refers to the implied variable z. By Proposition 2.5 and Lemma 4.2(2), the last term equals η r−s (y r+s 2 F k , R s−r,μ k (·, ζ ))j(γ , ζ ) 1−2μ k . Therefore, which is in P r+s−2k (C). Therefore, σ k (γ ) ∈ P r+s−2k (C) and since, further, σ k is given as the differential of a 0-cochain, we deduce 1.
To derive 2., we see with (28) and (29) that, for all γ ∈ Γ and ζ ∈ H, Since the last integrals belong to P r+s−2k (C), we deduce, on the one hand, thatσ k is a 1-cocycle with coefficients in P r+s−2k (C) and, on the other, that σ k andσ k differ by a 1-coboundary in P r+s−2k (C). Therefore, they belong to the same cohomology class.
As in the case of the classical period polynomial, the value of this cocycle at the involution S encapsulates the critical values of the L-functions of F k . However, in the general case, its leading and constant terms must be "truncated".

Theorem 5.3
Assume that r ≡ s mod 4. Then, where σ 0 k (S)(ζ ) denotes the sum of the leading and constant term of σ k (ζ ) and Now, each smooth function h : H ∪H → C, we deduce from (27) that We further notice that η r−s (y k−r−s ; 0) = 0. Therefore, η r−s (y k−r−s ; ζ ) = η r−s (y k−r−s ; ζ ) − η r−s (y k−r−s ; 0) equals Each of the polynomials in ζ have degree ≤ s + r − 2k − 1. Hence, the integral i∞ i η r−s (y k−r−s ; ζ ) converges and, by integrating along the positive imaginary axis, we deduce that it equals Therefore, with (31) and (34) we deduce that σ k (S)(ζ ) − σ 0 k (S)(ζ ) equals i ∞ 1 t k−1 F k (it) + at k−r−s (R(t, ζ ) − R 0 (t, ζ ))dt where and R 0 (t, ζ ) is the sum of the constant and the leading term of the expansion of R(t, ζ ) in ζ . (The terms of order between 1 and s + r − 2k − 1 coming from 2ih(i)(ζ + i) s−k (ζ − i) r−k and 2i((ζ + i) s−k (ζ − i) r−k − 1) cancel because s ≡ r mod 4.) Using the binomial expansion, we see that the integral in the RHS of (35) equals, in the notation of the statement of the proposition: With the definition of L * F k , we deduce the proposition.
From Proposition 5.2, we obtain a map from the space of modular iterated integrals of length one to a direct sum of copies of the space of classical modular (resp. cusp) forms. Here [σ k ] stands for the cohomology class of the 1-cocycle defined in that proposition. The last isomorphism of the theorem follows from the Eichler-Shimura isomorphism [see (25)].

Corollary 5.5
Let F be a modular iterated integral of length one and weights (r, s) and let F = F 0 + · · · + F min(r,s) be its decomposition into eigenfunctions of the Laplacian. Then, for each k ∈ {0, . . . , min(r, s)}, there is a P k (ζ ) ∈ P r+s−2k (C) and unique f k ∈ S r+s−2k+2 , g k ∈ M r+s−2k+2 such that, for all γ ∈ Γ ,

An application to algebraicity
In [7] an Eichler-Shimura isomorphism for weakly holomorphic modular forms is proved, which respects rational structures. As a proof of concept for the "correctness" of our definition of the L-function in Sect. 3, we will use the results of [7] to show an analogue of Manin's Periods Theorem [15] for weakly holomorphic forms. It should be mentioned that K. Bringmann has shown us an alternative way, based on results of [2], to establish a statement implying the same result. Before stating and proving our result, we first summarise the setup of [7] and then show that it is compatible with the explicit expressions for the cocycles of the last section.
Let M ! k,Q , resp. S ! k,Q denote the Q-vector space of weight k weakly holomorphic modular, resp. cusp, forms for Γ =SL 2 (Z), with rational Fourier coefficients. Consider the differential operator D = 1 2π i d dz . In [12], it is shown that, although there are generally no Hecke eigenforms in S ! k,Q , there are well-defined operators on M ! k,Q /D k−1 M ! 2−k,Q induced by the standard Hecke operators and, within that space, there are Hecke invariant classes. With this terminology and notation, we have