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
f is a weight 2, level N Hecke cusp form) and cocycles. This is made more explicit in [1] where it is shown that where f r (z) = f (z)(η(z)η(Nz)) r is a cusp form of weight 2 + r and S = 0 1 −1 0 .
In this article we present a new proof of Theorem 1.2 for 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 (Γ, v) and H 1 r,v (Γ, P) which we introduce in Section 2. In Section 3 we show that this pairing is perfect, which implies Theorem 1.2 for the weights we consider. The proof also implies Theorem 1.2 for the weights 2 − r < 0.
The proof proceeds as follows: Theorem 3.2 and Corollary 3.5 show that every cocycle φ in Z 1 r,v (Γ, P) is a coboundary in Z 1 r,v (Γ, 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 − g for all γ ∈ Γ. In the next step we assume that φ is orthogonal to all cusp forms. Using the description of φ as a coboundary in Z 1 r,v (Γ, Q) Proposition 3.11 shows that this orthogonality is equivalent to y r+2 2 ∂g ∂z (z) being in the image of the Maass raising operator K −r . Finally we apply standard results from the spectral theory of automorphic forms to show that φ must be a coboundary in Z 1 r,v (Γ, P). One of the advantages of the new proof is that once all the constructions are in place the theorem 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 [14] by W. Roelcke. Another advantage of the new proof is that it can easily be generalised to the case of vector-valued cusp forms. We sketch this generalisation in the last section of this article.

Preliminaries
For any matrix γ = a b c d ∈ SL 2 (R) we define the function j(γ, z) = cz +d.
Here j(γ, z) r = exp(r · log(j(γ, z)) and log is the principal branch of the complex logarithm satisfying log(z) = |z| + i arg(z) for all z = 0.
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 ∈ 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 f | r γ(z) = j(γ, z) −r f (γz).
The consistency condition for v implies that Let q 0 = ∞ and q 1 , . . . , q m be a set of representatives of the cusps of Γ. For every cusp q the stabiliser subgroup Γ q is generated by −1 and one generator σ q ∈ Γ. For q = ∞ it takes the form σ ∞ = 1 λ 0 1 with where κ i ∈ [0, 1) is defined for any cusp by v(σ i ) = e 2πiκ i . At the other cusps the expansion is of the form Here λ i > 0 is given by  (2) and (3) all a n,i with n + κ i < 0 (resp. ≤ 0) are zero. The set of modular forms is denoted by M r (Γ, v), the set of cusp forms by S r (Γ, v).
Remark. By the main theorem of [9] all modular forms of negative weight are 0.

Cohomology
Definition 1.2. Let P be the space of holomorphic functions on H that satisfy |f (z)| < K(|z| A + y −B ), ∀z ∈ H, for positive constants K, A and B.
A cocycle of weight r and multiplier system v with values in P is a function φ : Γ → P that satisfies We denote the space of cocycles by Z 1 r,v (Γ, P). There is a natural map d from P to Z 1 r,v (Γ, 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 B 1 r,v (Γ, P). The (first) Eichler cohomology group We denote the space of parabolic cocycles byZ 1 r,v (Γ, P). Since coboundaries are clearly parabolic we can form the parabolic cohomology group H 1 r,v (Γ, P) =Z 1 r,v (Γ, P)/B 1 r,v (Γ, P). It turns out that all cocycles are parabolic. This follows from a result that Knopp attributes to B.A. Taylor in [10]. Proposition 1.3. Let ǫ ∈ C with |ǫ| = 1 and g ∈ P. Then there exists an f ∈ P with ǫf (z + 1) − f (z) = g(z), ∀z ∈ H.
Proof. This is Proposition 9 in [10] 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 ǫ n g 1 (z + n) and f 2 (z) = − ∞ n=0 ǫ n g 2 (z + n) converge and are in P.
Proof. Let φ ∈ Z 1 r,v (Γ, P). We will show that for every parabolic γ ∈ Γ there exists f ∈ P such that First suppose γ = 1 s 0 1 is a translation by s = 0. Then by Corollary 1.4 a function f ∈ P with the desired property exists.
For the general case let γ = a b c d ∈ Γ fix a cusp q. Then there exists an s such that Replacing z by A −1 z in equation (6) we see that it is sufficient to show the Equation (1) implies the two relations After multiplying equation (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 1.4.

Petersson inner product
In this section we define the pairing that is essential for our proof of Theorem 1.2.
Definition 2.1. Let r ∈ R with 2 − r > 0 and g be a cusp form for the group Γ of weight 2 − r and unitary multiplier system v. Let Since g decreases exponentially towards the cusps the integral converges and G is a smooth function from H → C. We can define a cocycle by Proof. Let γ ∈ Γ: In the last equality we used To prove this let α = arg(γτ − γz) and β = arg(τ − z) − arg(j(γ, τ )) − arg(j(γ, z)).
We know that α ≡ β mod 2π and want to show α = β. Both (γτ − γz) 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 holomorphic 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 Γ, F . Then by Stoke's theorem we have So τ is the permutation that swaps 1 with 4 and 2 with 3.
Remark. For general Fuchsian groups Γ of the first kind an example of such a fundamental domain is the Ford fundamental domain (see [5]) 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 Γ. We can restate Proposition 2.2 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 im is in P and therefore can increase only polynomially towards the cusps, while f decreases exponentially.
This pairing factors through H 1 r,v (Γ, P) as the following argument shows. Let φ be a coboundary. This means that there exists a function h ∈ P with The integral over the boundary is 0 because, since f (z)h(z) decreases exponentially towards the cusps we can approach ∂F f (z)h(z)dz by integrals over closed paths contained in H, which are all equal to zero, since f (z)h(z) is holomorphic.

Duality theorem
In this section we prove that the pairing we defined in Definition 2.2, viewed as a pairing between S 2−r (Γ, v) and H 1 r,v (Γ, P) is perfect for 2 − r > 1. This is equivalent to Theorem 1.2 (when 2 − r > 1).
We already know that for every non-zero f there exists a cocycle φ (e.g.
To show that the pairing is perfect we therefore need to prove the following theorem.
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 1.2 for 2 − r < 0.
A basis of neighbourhoods of ∞ is given by the sets We define a variation of the space P that will be useful in our proof. Let Q be the space of C ∞ -functions f on H such that for every cusp q of Γ there exists a neighbourhood U ⊆ H and K, A, on V and η(z) = 0 outside U (by the smooth Urysohn lemma such a function exists). We will first try to construct a function that has ηφ as a coboundary. By Proposition 1.5 φ is a parabolic cocycle so for every cusp q there exists a function g q ∈ P such that φ(σ q ) = g q |(σ q − 1), where σ q is the generator of Γ q . We define G on U as follows: if z ∈ H Y (q i ) for some i then G(z) = g q i (z). If z = δw for δ ∈ Γ and w ∈ H Y (q i ) we define 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.
Multiplying both sides by v(δ) −1 j(δ, w) −r and using the compatibility of the multiplier system v this is equivalent to This follows from the cocycle condition on φ and the choice of g q i . Indeed, since w ′ ∈ δ ′−1 δH 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 (Γ,Q). It remains to show that (1 − η)φ is a coboundary. For this purpose let U i , i = 1, . . . , n be a finite cover of H such that every U i is the Γ-orbit of an open set V i that contains at most one elliptic point of Γ. We denote the (finite) stabiliser of this fixed point by Γ i and require furthermore that for be a Γ-invariant C ∞ -partition of unity corresponding to this cover. We define where g i (z) is any element of Γ with z ∈ g i (z)V i . This does not depend on the choice of g i (z): If z ∈ γV i with γ ∈ Γ then we must have γ −1 g i (z) ∈ Γ i . Thus the set Γ i g i (z) is equal to Γ i γ. So a different choice of g i (z) just permutes the summands in the definition of H i (z).
Clearly H i is a function inQ and defining H = i H i we have In the definition ofQ the constants K, a, b may vary from cusp to cusp. Let Q be the space of functions F inQ such that there exist positive constants K, a, b with |F (z)| < K(|z| A + y −B ), ∀z ∈ H.
Note that the functions of P are the holomorphic functions in Q.
Proof. This proof is similar to the proof of the main theorem of [11]. Let M be the set of matrices γ in Γ with λ/2 ≤ Re(γi) < λ/2. M is a complete set of coset representatives of Γ ∞ \ Γ. We need a technical lemma from [10]: Since only finitely many cusps are in F and since the real part of z ∈ F is bounded we can also find positive K 2 , A 2 , B 2 with |F (τ )| < K 2 (Im(τ ) A 2 + Im(τ ) −B 2 ), ∀τ ∈ F ∩ H.
As in the proof of Theorem 3.2 we use the fact that ψ is parabolic and hence there exists a function g ∞ ∈ P such that ψ( F is in P if and only if F − g ∞ is in P so we can assume without loss of generality that F is invariant under σ ∞ . Let z ∈ H. There exists τ ∈ F and γ ∈ Γ such that z = γτ . Since M is a complete set of representatives of Γ ∞ \ Γ there is an integer m and δ ∈ M such that z = σ m ∞ δτ . If δ = Id then we can deduce |F (z)| < K 2 (Im(τ ) A 2 + Im(z) −B 2 ), from equation (16) and the fact that F is Γ ∞ -invariant. Suppose δ = a b c d is not the identity. Then c = 0, because the only member of M that fixes ∞ is Id. We have Since δ ∈ M we have |j(δ, τ )| ≥ 1 so y = Im(z) = Im(τ ) |j(δ,τ )| 2 ≤ Im(τ ). On the other hand, using where c 0 > 0 depends only on Γ. Such a c 0 exists because Γ is discrete.  Let φ ∈ Z 1 r,v (Γ, P). By Corollary 3.5 there exists a function g ∈ Q such that g| r,v γ − g = φ(γ) for all γ ∈ Γ. By the same calculation as in equation (15) we have

These inequalities inserted into (19) lead to the desired inequality of the form
Here we note again that the integrals above exist because g can only increase polynomially towards the cusps of Γ, while f decreases exponentially.

Spectral theory of automorphic forms
To finish the proof of Theorem 3.1 we will apply spectral theory. We only give a very brief introduction here, for more details and proofs see the exposition [14] 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: .

So a function f is invariant under
The weight r Laplacian and the Maass raising and lowering operators are defined as Before we sum up the main properties of these operators in Proposition 3.7 we recall some definitions from operator theory. Let x n be a sequence in D that converges to x ∈ H and suppose that T x n converges to y ∈ H ′ . Then x ∈ D and T x = y. 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 .
(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 even real analytic).
(iii) K r : D 2 r → H r+2,v and Λ 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 Proof. For proofs of the statements (i), (iii) and (iv) see [14]. (i) is Satz 3.2, (iii) follows from the discussion after the proof of Lemma 6.2 on page 332 and (iv) is equation (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 [3]. The result needed here is Corollary 8.11 in [3]. The main result in [14] is a spectral decomposition of ∆ r . For this purpose we introduce the Eisenstein series. Let q be a cusp of Γ, σ q the generator of Γ q and A q ∈ SL 2 (R) chosen such that q = A −1 ∞. The cusp q is called singular if v(σ q ) = 1 and regular otherwise. Let q 1 , . . . , q m * be a set of representatives of the singular cusps of Γ. For each of these cusps we define the Eisenstein series The definition of E q r,v depends on the choice of A but a different choice of A will only multiply the Eisenstein series by a constant of absolute value 1. The series above converges absolutely and uniformly for (z, s) in sets of the form K × {s|Re s ≥ 1 + ǫ}, where K is a compact subset of C. For a fixed s with real part ≥ 1 + ǫ one can use the absolute and uniform convergence of the series to see that E q r,v (·, s) is invariant under | R r,v and that −∆ r E q r,v (·, s) = s(1 − s)E q r,v (·, s).
These series can be meromorphically continued and play an important role in the spectral decomposition of ∆ r .
(i) For fixed z ∈ H the Eisenstein series E q r,v (z, ·) can be meromorphically continued to the whole complex plane.
(ii) If, for one fixed z, E q r,v (z, ·) has a pole of order n at s 0 the function In particular, if E q r,v (z, ·) is holomorphic at s 0 , then Furthermore we have the following equalities: The poles of E q r,v (z, ·) in the half plane defined by Re s ≥ 1 2 are all simple and in the interval ( 1 2 , 1]. In particular there are no poles on the line Re s = 1 2 . Theorem 3.9 (Spectral expansion). Let f ∈D r and e n a maximal orthonormal system of eigenfunctions of ∆ r . Then f has a spectral expansion If f has compact support mod Γ then both parts of the spectral expansion, (e n , f )e n and , converge absolutely and uniformly on compact subsets of H.
Proof. Both the properties of Eisenstein series and the spectral expansion are proved in the second part of [14] We turn back to the proof of Theorem 3.1: Let φ ∈ Z 1 r,v (Γ, P) and g ∈ Q such that φ(γ) = g| r,v γ − g for all γ ∈ Γ. By applying ∂ ∂z to the equation g| r,v γ − g = φ(γ) we see that ∂g ∂z (z) = v(γ)j(γ, z) −r j(γ, z) 2 ∂g ∂z (γz).
A short calculation shows that the function G : z → y Moreover G vanishes in a neighbourhood of every cusp since g is holomorphic there, so G has compact support mod Γ and is in H 2−r,v .
To prove Theorem 3.1 we have to show that if φ is orthogonal to S 2−r (Γ, v), then g ∈ Q can be chosen to be holomorphic. This implies that φ is a coboundary in Z 1 r,v (Γ, P). Proof. We have the equality According to the remark after Definition 3.4 these functions are exactly the cusp forms of eigenvalue r 2 (1 − r 2 ). We can now use spectral theory to characterise functions which are orthogonal to cusp forms of eigenvalue r 2 (1 − r 2 ). Proposition 3.11. Let 2 − r = 1 and H be a smooth function in H 2−r,v with compact support mod Γ. Then the following are equivalent: Remark. By [9] there exist no non-zero classic modular forms of negative weight. Since, by [14,Satz 5.2], all eigenfunctions of eigenvalue r 2 (1 − r 2 ) are of the form y Here we used that n (K −r e n , H)K −r e n converges absolutely and uniformly on compacta to swap differentiation and summation and write it as Since H is orthogonal to all cusp forms with eigenvalue r 2 (1 − r 2 ) we immediately see that E ∈ N must be orthogonal to [9], so in this case we also have E = 0.
Next we deal withF 2 : Applying equation (20) twice we see that If r = 1 the integral 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 Λ 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

Summing up we have
To see that F is smooth we apply Λ 2−r to equation (22): We see that F is a solution of an elliptic differential equation and so, by elliptic regularity, F is smooth.
(ii)⇒(i): Let H = K −r F + y 2−r 2 E for a smooth function F and letf be a cusp form with eigenvalue r 2 (1 − r 2 ). Then, since y The function f = y − 2−r 2f is in S 2−r (Γ, v) and hence holomorphic, so we have Theorem 3.1 now follows from Proposition 3.11.
Proof of Theorem 3.1 and proof of Theorem 1.2 for 2 − r < 0 and 2 − r > 1: Let φ ∈ Z 1 r,v (Γ, P) and g and G be constructed as above. In the case 2−r > 1 suppose additionally that (f, φ) = 0 for all f ∈ S 2−r (Γ, v). By Proposition 3.11 and the remark after it there is a smooth Dividing by y r+2 2 and taking the complex conjugate of both sides we arrive at ∂g ∂z This invariance implies thatg = g −F satisfiesg| r,v γ −g = φ(γ) for all γ ∈ Γ. Furthermore as a function in H r,v , F clearly satisfies the growth conditions for functions in Q sog ∈ Q. Now equation 3.1 implies thatg is holomorphic, hence in P and so φ is indeed a coboundary in Z 1 r,v (Γ, P). This shows in particular that for 2 − r < 0 every cocycle in Z 1 r,v (Γ, P) is a coboundary and hence H 1 r,v (Γ, v) = 0. This is the statement of 1.2, since S 2−r (Γ, v) is also 0 in this case.
Case 2 − r = 1: One can follow the proof of Proposition 3.11 to obtain To prove Theorem 1.2 in this case one would need to show that the second summand above is in the image of K −r .

Vector-valued modular forms
In this section we generalise Theorem 1.2 to vector-valued cusp forms. Let ρ : Γ → U(n) be a unitary representation of Γ 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 Definition 4.1. A function f : H → C n is a modular form for Γ 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 Γ and A∞ = q then for any ǫ > 0 j(A, z) −r f (Az) is bounded for y ≥ ǫ.
If f satsifies additionally the condition it is cusp form. The set of modular forms (resp. cusp forms) of this kind is denoted by M r (Γ, v, ρ) (resp. S r (Γ, v, ρ)).
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 Γ-action on P n and so we can define the cohomology groups H 1 r,v,ρ (Γ, P n ) andH 1 r,v,ρ (Γ, P n ). Just as in the 1-dimensional case they turn out to be the same. The proof of this fact relies on a generalisation of Corollary 1.4: Proposition 4.1. Let U ∈ U(n), s = 0 and g ∈ P n . Then there exists f ∈ P n such that U * f (z + s) − f (z) = g(z), ∀z ∈ H. (23) Proof. Since U is diagonalisable there exists a V ∈ U(n) and a diagonal D ∈ U(n) with U = V * DV.

Multiplying equation (23) with
Let ǫ 1 , . . . , ǫ n be the diagonal entries of D and G = V g = (G 1 , . . . , G n ) ∈ P n . We can use Corollary 1.4 to find solutions F i ∈ P for Then f = V −1 (F 1 , . . . , F n ) is in P n and satisfies (24).
This can be used to show

Petersson inner product
Let 2 − r > 0 and f, g be in S 2−r (Γ, 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 Section 2.

Theorem 4.4.
Let v and ρ be as above and 2 − r > 1. The pairing defined above is perfect, so the map f → φ ∞ f induces an isomorphism S 2−r (Γ, v, ρ −1 ) ∼ = H 1 r,v,ρ (Γ, P n ). If 2 − r < 0 we have S 2−r (Γ, v, ρ −1 ) ∼ = H 1 r,v,ρ (Γ, P n ) ∼ = {0}. Proof. All the constructions of Section 3 work in the vector-valued case. In particular every statement we cited from [14] is already formulated in the vector-valued case. The fact that every vector-valued modular form of negative weight is 0 is also stated in [14] as a consequence of Satz 5.3 and generalises the main theorem of [9].