Eichler cohomology in general weights using spectral theory

In this paper, we construct a pairing between modular forms of positive real weight and elements of certain Eichler cohomology groups that were introduced by Knopp in 1974. We use spectral theory of automorphic forms to show that this pairing is perfect for all positive weights except 1. The approach in this paper gives a new proof of a theorem by Knopp and Mawi from 2010 for all real weights excluding 1 and also a version of this theorem for vector-valued modular forms.


Introduction
Let ⊆ SL 2 (R) be a finitely generated Fuchsian group of the first kind that contains translations and −I , where I is the identity matrix. The interpretation of modular forms for as elements in certain cohomology groups was first discovered by Eichler [5]. The following theorem is due to him in the case of even weights and trivial multiplier system. The general case was proved later by Gunning [10]. Theorem 1 Let r be a non-positive integer, v a weight 2 − r multiplier system for and P r the vector space of polynomials with coefficients in C of degree ≤ −r . Then Here P r is viewed as a -module with the | r,v action and H 1 r,v is the first cohomology group. Theorem 1 has many applications in the theory of modular forms and the study of critical values of their L-functions, e.g. in algebraicity results like Manin's period theorem [16].
The subject of this paper is a variant of Theorem 1 in the case of arbitrary real weight. Knopp first formulated it in 1974 [12]. To a cusp form g of real weight 2 − r and multiplier system v, he associated a cocycle with values in a space of functions P by This induces an injective map from S 2−r ( , v) to H 1 r,v ( , P) (see Corollary 2) and he conjectured that this map is actually an isomorphism, but was only able to prove this for the cases r ≤ 0 and r ≥ 2. In the case r < 0, he relied heavily on the previous work by Niebur [17] on automorphic integrals. Later, in 2000, a partial result on the missing cases in Knopp's conjecture was obtained by Wang [23] and it was resolved in 2010 by Knopp and Mawi [14], using Petersson's principal part theorem and generalised Poincaré series.
Theorem 2 (Knopp-Mawi) For all r ∈ R, we have A recent preprint [2] by Bruggeman and coworkers gives a similar isomorphism for a much wider class of automorphic forms. They also provide several motivations to study cocycles of real weight. One of them is a formula of Goldfeld [8] that suggests a connection between special values of derivatives of L-functions and cocycles. To be precise, let f = n≥1 a n q n be a Hecke cusp form of weight 2 for the group 0 (N ), and assume that f is invariant under the Fricke involution W N = 0 −1 N 0 . The L-function of f , L f (s), is defined as the analytic continuation to C of the Dirichlet series a n n −s . In [2, §9.4], it is shown that Goldfeld's formula leads to the following expression: where f r (z) = f (z)(η(z)η(N z)) r is a cusp form of weight 2 + r and S = 0 1 −1 0 .
Another reason to study cocycles in the case of half-integral weight k is given in [1]. Bringmann and Rolen use the non-holomorphic Eichler integral g * (this is essentially the auxiliary integral G in Sect. 2 and closely connected to the cocycle φ g ) to show that the function defined on Q is a quantum modular form.
In this article, we present a new proof of Theorem 2 for positive weights 2 − r = 1 that views the isomorphism in Knopp and Mawi's theorem as a duality. The key construction is a pairing between S 2−r ( , v) and H 1 r,v ( , P) which we introduce in Sect. 2. In Sect. 3 we show that this pairing is perfect, which implies Theorem 2 for the weights we consider. The proof also implies Theorem 2 for the weights 2 − r ≤ 0 and hence for all real weights, except 2 − r = 1.
The proof proceeds as follows: Theorem 5 and Corollary 3 show that every cocycle φ in Z 1 r,v ( , P) is a coboundary in Z 1 r,v ( , Q), where Q is a larger space of functions than P. This means that there exists g ∈ Q such that φ(γ ) = g| r,v γ − g for all γ ∈ . If 2 − r > 0, we assume in the next step that φ is orthogonal to all cusp forms with respect to the pairing we construct in Definition 4. Using the description of φ as a coboundary in Z 1 r,v ( , Q), we use classic results from the spectral theory of automorphic forms to show that y r +2 2 ∂g ∂z (z) is in the image of the Maass weightraising operator K −r (see Proposition 6). This then implies that φ is a coboundary in Z 1 r,v ( , P). In the case 2 − r = 1 only the last step of the proof fails, since some technical complications arise in the proof of Proposition 6.
One of the advantages of the new proof is that once all the constructions are in place the problem can be solved with standard techniques from the spectral theory of automorphic forms. The main references we use for spectral theory are the excellent articles [21] by W. Roelcke. Another advantage is that the proof can easily be generalised to the case of vector-valued cusp forms. We sketch this generalisation in the last section of this article.

Preliminaries
The group SL 2 (R) acts on C ∪ {∞} by We also define the function j (γ , z) = cz + d. For any other element δ of SL 2 (R), one has the relation for all z in the upper half plane H = {z = x + iy ∈ C| y > 0}. Let r ∈ R. Two useful functions when dealing with real weights, introduced by Petersson [18], are and σ r (γ , δ) = e 2πirω(γ,δ) .
Here arg, the argument, is always chosen to lie in (−π, π]. The value of ω(γ, δ) is independent of z and in {−1, 0, 1}. From the definition, it follows that Here j (γ , z) r = exp(r · log( j (γ , z)) and log is the principal branch of the complex logarithm satisfying log(z) = log |z| + i arg(z) for all z = 0.
In this article, we will work with modular forms with respect to a Fuchsian group of the first kind. We sketch the definition of such groups here and refer the reader to [22, §1] for a more thorough introduction. Let be a discrete subgroup of SL 2 (R) or of SL 2 (R)/{±I }. A cusp of is any element of R ∪ {∞} that is fixed by a parabolic element of , i.e. an element of that has only one fixed point in R ∪ {∞}. Let H * be the union of H with the cusps of . The quotient space \H * can be given the structure of a Riemann surface such that the natural projection is an open map. is called a Fuchsian group of the first kind, if \H * is compact. For the rest of this article we assume that ⊆ SL 2 (R) is a Fuchsian group of the first kind that contains a translation. This condition is not very restrictive. Any Fuchsian group of the first kind that has cusps is conjugate to a Fuchsian group of the first kind that contains translations. For convenience, we will also assume that contains −I .
A multiplier system of weight r for is a function v : → C which satisfies the consistency condition Note that v is also a multiplier system of any weight r ∈ R with r ≡ r mod 2 and v is a multiplier system of weight −r . A multiplier system is called unitary if |v(γ )| = 1 for all γ ∈ . For the rest of this article, we fix a unitary multiplier system v of weight r .
For a function f on the upper half plane H and γ ∈ SL 2 (R), the slash operators | r,v and | r are defined by The consistency condition for v implies that Let q 0 = ∞ and q 1 , . . . , q m be a set of representatives of the cusps of . For every cusp q, the stabiliser subgroup q is generated by −I and one generator σ q ∈ . For q = ∞ we choose σ ∞ = 1 λ 0 1 , the minimal translation matrix in with λ > 0. Let implies that f has a Fourier expansion at ∞ of the form where κ i ∈ [0, 1) is defined for any cusp by v(σ q i ) = e 2πiκ i . To find the expansion at the other cusps, choose σ q i so that if The Fourier expansion of f at q i is then given by Definition 1 Let f be holomorphic in H and invariant under | r,v . Then f is called a modular form 1 of weight r and multiplier system v with respect to , if in the Fourier expansions in (2) and (3) all a n,i with n + κ i < 0 are zero. If in addition all a n,i with n + κ i = 0 vanish, then f is called a cusp form. The set of modular forms is denoted by M r ( , v), the set of cusp forms by S r ( , v).
Remark 1 By the main theorem of [11], the only modular form of negative weight is the zero function. By [15], the only non-zero modular forms of weight 0 are constant functions.

Cohomology
Definition 2 Let P be the space of holomorphic functions on H such that there exist positive constants K , A and B with A cocycle of weight r and multiplier system v with values in P is a function φ : → P that satisfies We denote the space of cocycles by Z 1 r,v ( , P). There is a natural map d from P to Z 1 r,v ( , P) that associates to a function g ∈ P the cocycle dg : γ → g| r,v γ − g.
A cocycle of the form dg for g ∈ P is called a coboundary and the space of coboundaries is denoted by We denote the space of parabolic cocycles byZ 1 r,v ( , P). Since coboundaries are clearly parabolic, we can form the parabolic cohomology groupH 1 . It turns out that all cocycles are parabolic. This follows from a result that Knopp attributes to B.A. Taylor [12].
Then there exists an f ∈ P with Proof This is Proposition 9 in [12] and a full proof is given there. We will only present the main idea here. A formal solution of (4) is given by the one-sided average However, this sum does not always converge. Knopp uses the fact that P is closed under integration and differentiation to replace g with a function g = g 1 + g 2 such that the one-sided averages f 1 (z) = − ∞ n=0 n g 1 (z + n) and f 2 (z) = − ∞ n=0 n g 2 (z + n) converge and are in P.
Now we treat the case s < 0. By the first part of this proof, there exists anf ∈ P that satisfies The function f (z) = − f (z − s) solves (5).
. We will show that for every parabolic γ ∈ there exists f ∈ P such that First suppose γ = 1 s 0 1 is a translation by s ∈ R \ {0}. Then by Corollary 1, a function f ∈ P with the desired property exists.
For the general case, let γ = a b c d ∈ and fix a cusp q. Then there exists an Replacing z by A −1 z in Eq. (6) we see that it is sufficient to show the existence of Equation (1) implies the two relations After multiplying Eq. (8) by j (A, A −1 z) r and using the two relations (9) and (10), we get where we set The existence of such an F ∈ P again follows from Corollary 1.

Petersson inner product
In this section we define the pairing that is essential for our proof of Theorem 2. We make use of the auxiliary integral of a cusp form of positive real weight. For weights greater than 2 it was introduced in [17] and for any positive weight it first appeared in [20], where also the transformation formula (12) is mentioned. Corollary 2 can also be deduced from results in these papers and [19] but the proof presented here is new.

Definition 3
Let r ∈ R with 2 −r > 0 and g be a cusp form for the group of weight 2 − r and unitary multiplier system v. The auxiliary integral of g is defined as Since g decays exponentially towards ∞ the integral converges and G is a smooth function from H to C. We can define a cocycle by .
Proof Let γ ∈ . We have In the last equality, we used To prove this, let We know that α ≡ β mod 2π and want to show α = β. Both (γ τ − γ z) and τ − z are in H, so their arguments are in (0, π). Furthermore, exactly one of j (γ , τ ) and j (γ , z) will be in H and one in H, so −π < β < 2π and 0 < α < −π . Together with β ≡ α mod 2π , this implies α = β. Now we use the modularity of g to obtain An application of Cauchy's theorem now gives us To see that φ ∞ g,γ is in P first note that (τ −z) −r is antiholomorphic in H as a function of z (actually even in the slit plane C\{R ≥0 +τ }) and the integrals in the definition of G and φ ∞ g converge absolutely because g is a cusp form. Therefore, φ ∞ g,γ (z) is holomorphic in H. To prove that φ ∞ g,γ is in P one can use simple bounds for |τ − z| −r . We sketch the procedure for the case −r ≥ 0 and Im(z) > 1. In this case One can use this to bound φ ∞ g,γ (z) by a polynomial in |z|. The other cases are dealt with similarly.
Let f be another modular form of the weight 2 − r and multiplier system v. Then, since f is holomorphic This is just a scalar times the integrand occurring in the Petersson inner product of g and f defined as Choose a fundamental domain of , F. Then by Stokes' theorem, we have So τ is the permutation that swaps 1 with 4 and 2 with 3.

Remark 2
For general Fuchsian groups of the first kind an example of such a fundamental domain is the Ford fundamental domain (see [6]) where λ, the width of the cusp ∞, was defined in the last section. For the rest of this article, we will fix this fundamental domain for .
We can restate Proposition 3 as Thus, the Petersson inner product of f and g becomes Using the modularity of f , the second integral in the sum becomes Finally, we arrive at Motivated by the previous calculations, we define a pairing between cusp forms and cocycles: The integrals in the sum converge because φ(α i m ) is in P and therefore can increase only polynomially towards the cusps, while f decreases exponentially. So Since ( f (z)h(z)) decays exponentially at the cusps, we can approach ( ∂F f (z)h(z)dz) by integrals over closed paths contained in (H). These integrals all vanish, because ( f (z)h(z)) is holomorphic, and so (( f, φ) = 0).

The Duality theorem
In this section we prove that the pairing we defined in Lemma 1, between S 2−r ( , v) and H 1 r,v ( , P), is perfect for 0 < 2 − r = 1. For such weights r this implies Theorem 2.
We already know that for every non-zero f in To show that the pairing is perfect, we therefore need to prove the following theorem.
Together with Corollary 2 this implies that S 2−r ( , v) and H 1 r,v ( , P) are dual to each other. The proof of Theorem 4 will be given at the end of this section. Most constructions that follow will be valid for any real r and so, if not explicitly stated otherwise, we work in this generality. In particular, we will also show Theorem 2 for r ≥ 2. Let H = H ∪ R ∪ {∞} be the closure of H in P 1 (C). A basis of neighbourhoods of ∞ in H is given by the sets Let q be a cusp with τ q ∞ = q for τ q ∈ SL 2 (R) such that τ −1 q q τ q is generated by We define a variation of the space P that will be useful in our proof. LetQ be the space of C ∞ -functions f on H such that, for every cusp q of , there exists a neighbourhood U q ⊆ H and K q , A q , B q > 0 such that f is holomorphic in U q and For the purpose of proving Theorem 4 we will actually be interested in a subspace Q ⊆Q, that we introduce in Definition 5.

Theorem 5 Every element of Z
Proof Let φ ∈ Z 1 r,v ( , P). We need to show that there exists a function G ∈Q with φ(γ ) = G| r,v γ − G for all γ in . Choose Y large enough, so that all the H Y (q) are disjoint and contain no elliptic fixed points. Define U = q cusp of H Y (q) and V = q cusp of H 2Y (q). Then U and V are -invariant. Recall that the projections π(U ) and π(V ) are open in \H * . By the smooth Urysohn lemma (see for example [4, Corollary 3.5.5]), there exists a smooth functionη on \H * such thatη(π(z)) = 1 for all π(z) ∈ π(V ) andη(π(z)) = 0 for all π(z) outside π(U ). Define η(z) =η(π(z)) to be the pullback ofη. It is a -invariant C ∞ -function on H that satisfies η(z) = 1 on V and η(z) = 0 outside U .
We will first construct a function that has ηφ as a coboundary. By Theorem 3, φ is a parabolic cocycle, so for every cusp q there exists a function g q ∈ P such that Note that this is equivalent to defining G| r,v δ(w) = φ(δ)(w) + G(w), so once we show that the definition of G(z) does not depend on the choice of δ, the coboundary of ηG will be ηφ. Suppose z = δw = δ w , for δ, δ ∈ and w, w ∈ H Y (q i ). We need to check that Multiplying both sides by v(δ) −1 j (δ, w) −r and using the consistency condition of the multiplier system v, we see that this is equivalent to This follows from the cocycle condition on φ and the choice of g q i . Indeed, since w ∈ δ −1 δ H Y (q i ) ∩ H Y (q i ) = ∅ and since we assumed that all the H Y (q) are disjoint, δ −1 δ must fix q i . Hence δ −1 δ = ±σ n q i for some n ≥ 1. This implies and so So ηG is a well-defined function inQ. We have thus shown that ηφ is a coboundary in Z 1 r,v ( ,Q).
It remains to show that (1 − η)φ is a coboundary. We first construct a partition of unity on H that is -invariant. The construction we describe here is due to Gunning [9]. Since acts discontinuously on H, every z ∈ H has a neighbourhood O z such Let V be as in the construction of η, a -invariant open set that contains all cusps of with η| V = 1. Since \H * is compact, there exist z 1 , . . . , z n ∈ H such that the sets π(O z i ) together with π(V ) cover \H * . Letˆ 1 , . . . ,ˆ n ,ˆ V be a partition of unity corresponding to this cover, i.e. smooth functions supported in π(O z 1 ), . . . , π(O z n ) and π(V ), respectively, satisfying n i=1ˆ i (π(z)) +ˆ V (π(z)) = 1, ∀z ∈ H.
We define functions H 1 , . . . , H n on H as follows. If there exists g i (z) ∈ such that where ( i ) is the stabiliser of (z i ), (| i |) is its order and ( i (z) =ˆ i (π(z))). This does not depend on the choice of g i (z): if γ z ∈ O z i with γ ∈ , then we must have γ −1 g i (z) ∈ i . Thus, the set i g i (z) is equal to i γ and we see that a different choice of g i (z) just permutes the summands in the definition of H i (z). If no such g i (z) ∈ exists, we set H i (z) = 0. Clearly, H i is a function inQ and defining H = n i=1 H i , we will see that H | r,v γ (z) − H (z) = (1 − η(z))φ(γ )(z) for all γ ∈ and z ∈ H. First note that if z is in V , then H (z) and H (γ z) vanish and so does (1 − η(z))φ(γ )(z). If z is not in V , we have where the first sum is over all i such that there exists a H (z)).
In the definition ofQ, the constants K q , A q , B q may vary from cusp to cusp; in the following definition, we impose stricter growth conditions, requiring the constants to be fixed.

|F(z)| < K (|z| A + y −B ), ∀z ∈ H.
Note that the functions in P are the holomorphic functions in Q.
Proof This proof is similar to the proof of the main theorem of [13]. Let M be the set of matrices γ in with λ/2 ≤ Re(γ i) < λ/2. M is a complete set of coset representatives of ∞ \ . We need a technical lemma from [12]: Lemma 2 (Lemma 8 in [12]) There exist positive constants K 1 , Since only finitely many cusps are in F and since the real part of z ∈ F is bounded, we can also find positive As in the proof of Theorem 5, we use the fact that ψ is parabolic and hence there exists a function g ∞ ∈ P such that ψ( F is in Q if and only if F − g ∞ is in P, so we can assume without loss of generality that F(z + λ) = v(σ ∞ )F(z). Let z ∈ H. There exists τ ∈ F and γ ∈ such that z = γ τ . Since M is a complete set of representatives of ∞ \ , there is an integer m and δ ∈ M such that z = σ m ∞ δτ . If δ = I then we can deduce from Eq. (17) and the fact that |F| is ∞ -invariant. Suppose δ = a b c d is not the identity. Then c = 0, because the only member of M that fixes ∞ is I . We have By our choice of fundamental domain we have | j (δ, τ )| ≥ 1, since δ / ∈ ∞ . So y = Im(z) = Im(τ) | j (δ,τ )| 2 ≤ Im(τ ). On the other hand, using τ = δ −1 σ −m ∞ z, we have where c 0 > 0 depends only on . Such a c 0 exists because is discrete. Therefore, . These inequalities inserted into (20) lead to the desired inequality of the form for positive constants K , A, B and all z ∈ H.

Corollary 3 Every cocycle in Z
Let φ ∈ Z 1 r,v ( , P). By Corollary 3 there exists a function g ∈ Q such that g| r,v γ − g = φ(γ ) for all γ ∈ . By the same calculation as in Eq. (16), for any Here we note again that the integrals above exist because g can only increase polynomially towards the cusps of , while f decreases exponentially.

Spectral theory of automorphic forms
To carry out the proof of Theorem 4, we will apply spectral theory. We only give a very brief introduction here; for more details and proofs, see the exposition [21] by Roelcke. In these articles Roelcke uses a variation of the slash operator which we denote by | R The connection to our slash operator is given by the following lemma: . H r,v = H r ( , v) be the Hilbert space of functions f that are invariant under | R r,v and have finite norm with respect to the scalar product

Definition 6 Let
The weight r hyperbolic Laplacian and the Maass weight-raising and weight-lowering operators are defined as Before we sum up the main properties of these operators in Proposition 5, we recall some definitions from operator theory.

Definition 7
Let H and H be Hilbert spaces and let T be a linear operator from a subspace D of H to H . T is called closed if, for every sequence x n in D that converges to x ∈ H such that T x n converges to y ∈ H , we have x ∈ D and T x = y.

Definition 8
If D is dense in H then for any operator T from D to H , we can define its adjoint T * on the domain Any y in this set defines a linear functional on D by φ y : x → T x, y . This functional can be extended to H and by the Riesz representation theorem there exists z ∈ H such that φ y (x) = x, z for all x in H . We define T * y = z.
An operator is called self-adjoint if it is equal to its adjoint. An operator is called essentially self-adjoint if T ⊆ T * = (T * ) * , where T ⊆ T * means that T * extends T . Let Proposition 5 (i) r : D 2 r → H r,v is essentially self-adjoint. It has a self-adjoint extension to a dense subset of H r,v that we denote byD r .
(ii) The eigenfunctions of r are smooth (in fact they are real analytic). (iii) K r : D 2 r → H r +2,v and r : D 2 r → H r −2,v can be extended to closed operators defined onD r . For f ∈D r and g ∈D 2+r , we have (iv) Proof For proofs of the statements (i), (iii) and (iv) see [21]. (i) is Satz 3.2, (iii) follows from the discussion after the proof of Lemma 6.2 on page 332 and (iv) is Eq. (3.4) on page 305. Statement (ii) follows from the fact that r is an elliptic operator and elliptic regularity applies. For an introduction to the theory of elliptic operators, see [7]. The result needed here is Corollary 8.11 in [7].

Definition 9
A cuspidal Maass wave form in H r,v with eigenvalue λ is an eigenfunction of − r with eigenvalue λ that decays exponentially at the cusps of . The main result in [21] is a spectral decomposition of r . For this purpose we introduce the Eisenstein series. Let q be a cusp of , σ q the generator of q /{±I } and A q ∈ SL 2 (R) chosen such that q = A −1 q ∞. The cusp q is called singular for the multiplier system v, if v(σ q ) = 1 and regular for v otherwise. Let q 1 , . . . , q m * be a set of representatives of the cusps of that are singular for v. For each of these cusps, we define the Eisenstein series The definition of E q r,v depends on the choice of A q , but a different choice of A q will only multiply the Eisenstein series by a constant of absolute value 1. The series above converge absolutely and uniformly for (z, s) in sets of the form K × {s|Re s ≥ 1 + }, where K is a compact subset of C and > 0. For a fixed s with real part ≥ 1 + , one can use the absolute and uniform convergence of the series to see that E If n is chosen so that f (z) has no poles in H, then f grows at most polynomially at each cusp of , i.e. if q is a cusp of and τ q ∞ = q for τ q ∈ SL 2 (R), then Furthermore, we have the following equalities: The poles of E If f has compact support mod , i.e. π(supp( f )) is compact in \H * , then both parts of the spectral expansion, (e n , f ) R e n and m * i=1 1 4π 1 2 + iρ dρ, converge absolutely and uniformly on compact subsets of H.
Both the properties of Eisenstein series and the spectral expansion are proved in the second part of [21]. The Theorem we state is a combination of Satz 7.2 and the second part in Satz 12.3.
We turn back to the proof of Theorem 4: Let [φ] ∈ H 1 r,v ( , P) be represented by φ ∈ Z 1 r,v ( , P). By Corollary 3, there exists a function g ∈ Q such that By applying ∂ ∂z to (23), we see that A short calculation shows that the function is invariant under | R 2−r,v . Moreover, G vanishes in a neighbourhood of every cusp since g is holomorphic there, so G has compact support mod and is in H 2−r,v .
To prove Theorem 4, we have to show that if φ is orthogonal to S 2−r ( , v), then g ∈ Q can be chosen to be holomorphic. This implies that φ is a coboundary in Z 1 r,v ( , P). Lemma 4 Let 2 − r > 0 and φ, g and G be as above. . We can now use spectral theory to characterise functions which are orthogonal to cuspidal Maass wave forms of eigenvalue r 2 (1 − r 2 ). Proposition 6 Let 2 − r = 1 and H be a smooth function in H 2−r,v with compact support mod . Then the following are equivalent: Here we used that n (K −r e n , H ) R K −r e n converges absolutely and uniformly on compacta to swap differentiation and summation and write it as K −r F 1 = K −r n (K −r e n , H ) R e n . We now show thatF 2 = K −r F 2 for a smooth function F 2 ∈ H −r,v : Applying Eq. (21) twice and using Proposition 5, we see converges absolutely and uniformly on compacta. To see this note the integrand can be bounded above by converges absolutely and uniformly on compacta as it occurs in the spectral expansion of 2−r H . So when we apply K −r to F 2 = m * i=1 1 4π F i 2 , we can swap it with the integral and obtain F 2 is clearly in H −r,v by the bound we used for the F i 2 . We have thus shown that To see that F is smooth we apply 2−r to (26) and obtain We see that F is a solution of an elliptic differential equation and so, by elliptic regularity, F is smooth. It remains to show that y 2−r 2Ẽ is in the image of K −r . Since H is orthogonal to all cuspidal Maass wave forms with eigenvalue r 2 (1 − r 2 ), we see that in the expansioñ that are orthogonal to S 2−r ( , v) can occur. HenceẼ must be orthogonal to S 2−r ( , v) and has finite Petersson norm. If 2 − r ≥ 1 this implies E = 0. If 2 − r < 0, we have M 2−r ( , v) = {0} by [11], so in this case we also haveẼ = 0. We are left with the case 0 ≤ 2 − r < 1. In this case all modular forms in M 2−r ( , v) have finite Petersson norm, soẼ can be any form in the orthogonal complement of S 2−r ( , v). We can appeal to [21,Satz 11.2], to see thatẼ is a linear combination of residues of Eisenstein series at s = r 2 . Therefore, there exist a i ∈ C with = lim ).
The proof above shows in particular that for 2 − r ≤ 0 every cocycle in Z 1 r,v ( , P) is a coboundary and hence H 1 r,v ( , v) = {0}. This proves Theorem 2 for 2 − r ≤ 0, since S 2−r ( , v) is also {0} in this case.

Remark 5
The proof fails if 2 − r = 1, because Proposition 6 is not available in that case. The only point where we need the assumption 2 − r = 1 in the proof of that proposition is when we show thatF 2 is in the image of K −r , in particular for the construction of the functions F i 2 ∈ H −r,v in (25). The crucial consequence of Proposition 6 is that G is in the image of K −r . In the case 2 − r = 1, we only obtain In the notation of the proof of Proposition 6 we have F = F 1 and E = 0 since r = 1.
To prove Theorem 2 in this case, one would need to show that the second summand above is in the image of K −1 .

Vector-valued modular forms
In this section, we generalise Theorem 2 to vector-valued cusp forms. Let ρ : → U (n) be a unitary representation of on C n and v a unitary multiplier system of weight r . Let F be a function from H to C n . The slash operator | ρ,v,r is defined by F| r,v,ρ γ (z) = j (γ , z) −r v(γ )ρ(γ ) −1 F(γ z).
Definition 10 A function f : H → C n is a modular form for of weight r , representation ρ and multiplier system v if the following conditions are satisfied: (i) f is holomorphic on H.
(iii) If q is a cusp of and A∞ = q, then for any > 0 j (A, z) −r f (Az) is bounded for y ≥ .
If f satisfies the additional condition (iii') If q is a cusp of and A∞ = q, then there exists an > 0 such that it is a cusp form. The set of modular forms or cusp forms of this kind is denoted by M r ( , v, ρ) and S r ( , v, ρ), respectively.
Let P n be the set of vector-valued functions f (z) = ( f 1 (z), . . . , f n (z)) such that all f i are in P. The slash operator | r,v,ρ defines a -action on P n and so we can define the cohomology groups H 1 r,v,ρ ( , P n ) andH 1 r,v,ρ ( , P n ). Just as in the one-dimensional case, they turn out to be the same. The proof of this fact relies on a generalisation of Corollary 1: Proposition 7 Let U ∈ U(n), s ∈ R \ {0} and g ∈ P n . Then there exists an f ∈ P n such that Proof Since U is diagonalisable, there exists a V ∈ U(n) and a diagonal matrix D ∈ U(n) with Multiplying Eq. (30) by V , we get Let 1 , . . . , n be the diagonal entries of D and G = V g = (G 1 , . . . , G n ) ∈ P n . We can use Corollary 1 to find solutions F i ∈ P for Then f = V −1 (F 1 , . . . , F n ) is in P n and satisfies (31).
This can be used to show the following.

Petersson inner product
Let 2 − r > 0 and f, g be in S 2−r ( , v, ρ −1 ). The Petersson inner product of f and g is defined by where (a i ), (b i ) = n i=1 a i b i is the usual scalar product on C n . We will repeat the constructions of Sect. 2.
Again we can use Stokes' theorem to show Using this we define a pairing between S 2−r ( , v, ρ −1 ) and H 1 r,v,ρ ( , P n ) as follows. Let f ∈ S 2−r ( , ρ −1 , v) and [φ] ∈ H 1 r,v ( , P n ) be represented by φ. Then is well defined (independent of the representative φ), and furthermore we have the following theorem, analogous to Theorem 2.
Proof All the constructions of Sect. 3 work in the vector-valued case. In particular every statement we cited from [21] is already formulated for vector-valued functions. The fact that every vector-valued modular form of negative weight is 0 is also stated in [21] as a consequence of Satz 5.3, and this generalises the main theorem of [11]. It is also shown that a vector-valued modular form of weight 0 is constant.