# Eichler cohomology in general weights using spectral theory

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## Abstract

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.

## Keywords

Modular forms Eichler cohomology Real weight Spectral theory## Mathematics Subject Classification

11F12## 1 Introduction

Let \(\Gamma \subseteq \text{ SL }_2(\mathbb {R})\) 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 \(\Gamma \) 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**

*r*be a non-positive integer,

*v*a weight \(2-r\) multiplier system for \(\Gamma \) and \(P_{r}\) the vector space of polynomials with coefficients in \(\mathbb {C}\) of degree \(\le -r\). Then

Here \(P_r\) is viewed as a \(\Gamma \)-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].

*g*of real weight \(2-r\) and multiplier system \(\overline{v}\), he associated a cocycle with values in a space of functions \(\mathcal {P}\) by

### **Theorem 2**

*L*-functions and cocycles. To be precise, let \(f=\sum _{n\ge 1}a_n q^n\) be a Hecke cusp form of weight 2 for the group \(\Gamma _0(N)\), and assume that

*f*is invariant under the Fricke involution \(W_N=\left( \begin{array}{ll} 0 &{} -1\\ N&{}0 \end{array}\right) \). The

*L*-function of

*f*, \(L_f(s)\), is defined as the analytic continuation to \(\mathbb {C}\) of the Dirichlet series \(\sum a_n n^{-s}\). In [2, §9.4], it is shown that Goldfeld’s formula leads to the following expression:

*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 \(\phi _g\)) to show that the function

In this article, we present a new proof of Theorem 2 for positive weights \(2-r\ne 1\) that views the isomorphism in Knopp and Mawi’s theorem as a duality. The key construction is a pairing between \(S_{2-r}(\Gamma ,\overline{v})\) and \(H^1_{r,v}(\Gamma ,\mathcal {P})\) 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 \(2-r\le 0\) and hence for all real weights, except \(2-r=1\).

The proof proceeds as follows: Theorem 5 and Corollary 3 show that every cocycle \(\phi \) in \(Z^1_{r,v}(\Gamma ,\mathcal {P})\) is a coboundary in \(Z^1_{r,v}(\Gamma ,\mathcal {Q})\), where \(\mathcal {Q}\) is a larger space of functions than \(\mathcal {P}\). This means that there exists \(g\in \mathcal {Q}\) such that \(\phi (\gamma )=g|_{r,v}\gamma -g\) for all \(\gamma \in \Gamma \). If \(2-r>0\), we assume in the next step that \(\phi \) is orthogonal to all cusp forms with respect to the pairing we construct in Definition 4. Using the description of \(\phi \) as a coboundary in \(Z^1_{r,v}(\Gamma ,\mathcal {Q})\), we use classic results from the spectral theory of automorphic forms to show that \(y^{\frac{r+2}{2}}\overline{\frac{\partial g}{\partial \overline{z}}(z)}\) is in the image of the Maass weight-raising operator \(K_{-r}\) (see Proposition 6). This then implies that \(\phi \) is a coboundary in \(Z^1_{r,v}(\Gamma ,\mathcal {P})\). 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.

### 1.1 Preliminaries

*z*in the upper half plane \(\mathcal {H}=\{z=x+iy\in \mathbb {C}|\, y>0\}\). Let \(r\in \mathbb {R}\). Two useful functions when dealing with real weights, introduced by Petersson [18], are

*z*and in \(\{-1,0,1\}\). From the definition, it follows that

*cusp*of \(\Gamma \) is any element of \(\mathbb {R}\cup \{\infty \}\) that is fixed by a parabolic element of \(\Gamma \), i.e. an element of \(\Gamma \) that has only one fixed point in \(\mathbb {R}\cup \{\infty \}\). Let \(\mathcal {H}^*\) be the union of \(\mathcal {H}\) with the cusps of \(\Gamma \). The quotient space \(\Gamma \backslash \mathcal {H}^*\) can be given the structure of a Riemann surface such that the natural projection

*Fuchsian group of the first kind*, if \(\Gamma \backslash \mathcal {H}^*\) is compact. For the rest of this article we assume that \(\Gamma \subseteq {{\mathrm{SL}}}_2(\mathbb {R})\) 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 \(\Gamma \) contains \(-I\).

*multiplier system*of weight

*r*for \(\Gamma \) is a function \(v:\Gamma \rightarrow \mathbb {C}\) which satisfies the consistency condition

*v*is also a multiplier system of any weight \(r'\in \mathbb {R}\) with \(r'\equiv r\) mod 2 and \(\overline{v}\) is a multiplier system of weight \(-r\). A multiplier system is called

*unitary*if \(\left| v(\gamma )\right| =1\) for all \(\gamma \in \Gamma \). For the rest of this article, we fix a unitary multiplier system

*v*of weight

*r*.

*f*on the upper half plane \(\mathcal {H}\) and \(\gamma \in \text{ SL }_2(\mathbb {R})\), the slash operators \(|_{r,v}\) and \(|_r\) are defined by

*v*implies that

*q*, the stabiliser subgroup \(\Gamma _q\) is generated by \(-I\) and one generator \(\sigma _q\in \Gamma \). For \(q=\infty \) we choose \(\sigma _\infty =\left( \begin{array}{ll} 1&{}\lambda \\ 0&{}1\end{array}\right) \), the minimal translation matrix in \(\Gamma \) with \(\lambda >0\). Let

*f*be holomorphic on \(\mathcal {H}\) and invariant under \(|_{r,v}\). The equation \(f(z+\lambda )=v(\sigma _\infty )f(z)\) implies that

*f*has a Fourier expansion at \(\infty \) of the form

*f*at \(q_i\) is then given by

### **Definition 1**

Let *f* be holomorphic in \(\mathcal {H}\) and invariant under \(|_{r,v}\). Then *f* is called a *modular form* ^{1} of weight *r* and multiplier system *v* with respect to \(\Gamma \), if in the Fourier expansions in (2) and (3) all \(a_{n,i}\) with \(n+\kappa _i< 0\) are zero. If in addition all \(a_{n,i}\) with \(n+\kappa _i =0\) vanish, then *f* is called a cusp form. The set of modular forms is denoted by \(M_r(\Gamma ,v)\), the set of cusp forms by \(S_r(\Gamma ,v)\).

### 1.2 Cohomology

### **Definition 2**

*K*,

*A*and

*B*with

*cocycle*of weight

*r*and multiplier system

*v*with values in \(\mathcal {P}\) is a function \(\phi :\Gamma \rightarrow \mathcal {P}\) that satisfies

*d*from \(\mathcal {P}\) to \(Z^1_{r,v}(\Gamma ,\mathcal {P})\) that associates to a function \(g\in \mathcal {P}\) the cocycle

*coboundary*and the space of coboundaries is denoted by \(B^1_{r,v}(\Gamma ,\mathcal {P})\). The (first)

*Eichler cohomology group*\(H^1_{r,v}(\Gamma ,\mathcal {P})\) is the quotient space \(Z^1_{r,v}(\Gamma ,\mathcal {P})/B^1_{r,v}(\Gamma ,\mathcal {P})\).

*parabolic*if for all cusps \(q_i\), there exists a function \(g_{q_i}\in \mathcal {P}\) such that

### **Proposition 1**

### *Proof*

*g*with a function \(g'=g_1+g_2\) such that the one-sided averages \(f_1(z)= -\sum _{n=0}^{\infty }\overline{\epsilon }^ng_1(z+n)\) and \(f_2(z)=-\sum _{n=0}^{\infty }\overline{\epsilon }^ng_2(z+n)\) converge and are in \(\mathcal {P}\). \(\square \)

### **Corollary 1**

### *Proof*

### **Theorem 3**

### *Proof*

*q*. Then there exists an \(s\in \mathbb {R}\setminus \{0\}\) such that

*z*by \(A^{-1}z\) in Eq. (6) we see that it is sufficient to show the existence of \(f\in \mathcal {P}\) with

## 2 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**

*g*be a cusp form for the group \(\Gamma \) of weight \(2-r\) and unitary multiplier system \(\overline{v}\). The

*auxiliary integral*of

*g*is defined as

*t*ranges from 0 to \(\infty \).

*g*decays exponentially towards \(\infty \) the integral converges and

*G*is a smooth function from \(\mathcal {H}\) to \(\mathbb {C}\). We can define a cocycle by

### **Proposition 2**

### *Proof*

*g*to obtain

*z*(actually even in the slit plane \(\mathbb {C}\setminus \lbrace \mathbb {R}_{\ge 0}+\overline{\tau }\rbrace \)) and the integrals in the definition of

*G*and \(\phi _g^{\infty }\) converge absolutely because

*g*is a cusp form. Therefore, \(\phi ^{\infty }_{g,\gamma }(z)\) is holomorphic in \(\mathcal {H}\). To prove that \(\phi ^\infty _{g,\gamma }\) is in \(\mathcal {P}\) one can use simple bounds for \(\left| \tau -\overline{z}\right| ^{-r}\). We sketch the procedure for the case \(-r\ge 0\) and \(\text{ Im }(z)>1\). In this case

*f*be another modular form of the weight \(2-r\) and multiplier system \(\overline{v}\). Then, since

*f*is holomorphic

*g*and

*f*defined as

### **Proposition 3**

^{2}for \(i=1,\ldots ,2n\) (the indices are taken modulo 2

*n*) and \(\alpha _i\in \Gamma \) for \(i=1,\ldots ,2n\) such that there exists an involution of \(\{1,\ldots ,2n\}\), denoted by \(\tau \), such that

- 1.
\(\tau \) does not have any fixed points,

- 2.
\(\alpha _i A_i=A_{\tau (i)+1},~\alpha _i A_{i+1}=A_{\tau (i)}\),

- 3.
\(\alpha _{\tau (i)}=\alpha _i^{-1}\) and

- 4.
\(\alpha _i \text{ maps } [A_i,A_{i+1}[ \text{ to } [A_{\tau (i)+1},A_{\tau (i)}[\).

### *Example 1*

For \(\Gamma =\text{ SL }_2(\mathbb {Z})\), we choose the classic fundamental domain with \(A_1=\infty ,A_2=e^{2\pi i/3},A_3=i,A_4=A_2+1\). Then \(\alpha _1=T=\left( \begin{array}{ll} 1&{}1\\ 0&{}1 \end{array}\right) \) maps \([A_1,A_2[\) to \([A_1,A_4[\) and \(\alpha _2=S=\left( \begin{array}{ll} 0&{}1\\ -1&{}0 \end{array}\right) \) maps \([A_2,A_3[\) to \([A_4,A_3[\). So \(\tau \) is the permutation that swaps 1 with 4 and 2 with 3.

### *Remark 2*

*f*and

*g*becomes

*f*, the second integral in the sum becomes

### **Definition 4**

*f*decreases exponentially.

### **Lemma 1**

### *Proof*

*f*(

*z*)

*h*(

*z*)) decays exponentially at the cusps, we can approach \((\int _{\partial \mathcal {F}}f(z)h(z){\text{ d }}z)\) by integrals over closed paths contained in \((\mathcal {H})\). These integrals all vanish, because (

*f*(

*z*)

*h*(

*z*)) is holomorphic, and so \(((f,\phi )=0)\). \(\square \)

### **Corollary 2**

The map \(f\mapsto [\phi ^\infty _f]\) from \(S_{2-r}(\Gamma ,\overline{v})\) to \(H^1_{r,v}(\Gamma ,\mathcal {P})\) is injective.

### *Proof*

If \([\phi ^\infty _f]\) is a coboundary in \(Z^1_{r,v}(\Gamma ,\mathcal {P})\), then by the above calculations \(0=(f,\phi ^\infty _f)=(f,f)\) and hence \(f=0\). \(\square \)

## 3 The Duality theorem

In this section we prove that the pairing we defined in Lemma 1, between \(S_{2-r}(\Gamma ,\overline{v})\) and \(H^1_{r,v}(\Gamma ,\mathcal {P})\), is perfect for \(0<2-r\ne 1\). For such weights *r* this implies Theorem 2.

We already know that for every non-zero *f* in \(S_{2-r}(\Gamma ,\overline{v})\) there exists a cocycle \(\phi \) such that \((f,[\phi ])\ne 0\), since \((f,[\phi ^\infty _f])=(f,f)\ne 0\). To show that the pairing is perfect, we therefore need to prove the following theorem.

### **Theorem 4**

Let \(1\ne r<2\) and \([\phi ]\in H^1_{r,v}(\Gamma ,\mathcal {P})\). If \((f,[\phi ])=0\) for all \(f\in S_{2-r}(\Gamma ,\overline{v})\), then \([\phi ]=0\). Together with Corollary 2 this implies that \(S_{2-r}(\Gamma ,\overline{v})\) and \(H_{r,v}^1(\Gamma ,\mathcal {P})\) 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 \(r\ge 2\).

*q*be a cusp with \(\tau _q \infty =q\) for \(\tau _q\in \text{ SL }_2(\mathbb {R})\) such that \(\tau _q^{-1}\Gamma _q\tau _q\) is generated by \(T=\left( \begin{array}{ll} 1&{}1\\ 0&{}1 \end{array}\right) \). Then the open sets \(H_Y(q)=\tau _qH_Y(\infty )\) for \(Y>0\) form a basis of neighbourhoods of

*q*.

*f*on \(\mathcal {H}\) such that, for every cusp

*q*of \(\Gamma \), there exists a neighbourhood \(U_q\subseteq \mathcal {H}\) and \(K_q,A_q,B_q>0\) such that

*f*is holomorphic in \(U_q\) and

### **Theorem 5**

Every element of \(Z^1_{r,v}(\Gamma ,\mathcal {P})\) is a coboundary in \(Z^1_{r,v}(\Gamma ,\tilde{\mathcal {Q}})\).

### *Proof*

Let \(\phi \in Z^1_{r,v}(\Gamma ,\mathcal {P})\). We need to show that there exists a function \(G\in \tilde{\mathcal {Q}}\) with \(\phi (\gamma )=G|_{r,v}\gamma -G\) for all \(\gamma \) in \(\Gamma \). Choose *Y* large enough, so that all the \(H_Y(q)\) are disjoint and contain no elliptic fixed points. Define \(U=\bigcup _{q \text{ cusp } \text{ of } \Gamma }H_Y(q)\) and \(V=\bigcup _{q \text{ cusp } \text{ of } \Gamma }H_{2Y}(q)\). Then *U* and *V* are \(\Gamma \)-invariant. Recall that the projections \(\pi (U)\) and \(\pi (V)\) are open in \(\Gamma \backslash \mathcal {H}^*\). By the smooth Urysohn lemma (see for example [4, Corollary 3.5.5]), there exists a smooth function \(\hat{\eta }\) on \(\Gamma \backslash \mathcal {H}^*\) such that \(\hat{\eta }(\pi (z))=1\) for all \(\pi (z)\in \pi (V)\) and \(\hat{\eta }(\pi (z))=0\) for all \(\pi (z)\) outside \(\pi (U)\). Define \(\eta (z)=\hat{\eta }(\pi (z))\) to be the pullback of \(\hat{\eta }\). It is a \(\Gamma \)-invariant \(C^{\infty }\)-function on \(\mathcal {H}\) that satisfies \(\eta (z)=1\) on *V* and \(\eta (z)=0\) outside *U*.

*q*there exists a function \(g_q\in \mathcal {P}\) such that \(\phi (\sigma _q)=g_q|_{r,v}\sigma _q-g_q\), where \(\sigma _q\) is the generator of \(\Gamma _q/\{\pm I\}\). We define

*G*on

*U*as follows: if \(z\in H_Y(q_i)\) for some

*i*then \(G(z)=g_{q_i}(z)\). If \(z=\delta w\) for \(\delta \in \Gamma \) and \(w\in H_Y(q_i)\) we define

*G*(

*z*) does not depend on the choice of \(\delta \), the coboundary of \(\eta G\) will be \(\eta \phi \). Suppose \(z=\delta w=\delta ' w'\), for \(\delta ,\delta '\in \Gamma \) and \(w,w'\in H_Y(q_i)\). We need to check that

*v*, we see that this is equivalent to

*z*) and \(\gamma O_z \cap O_z = \emptyset \) if \(\gamma \in \Gamma \setminus \Gamma _z\). Let

*V*be as in the construction of \(\eta \), a \(\Gamma \)-invariant open set that contains all cusps of \(\Gamma \) with \(\eta |_V = 1\). Since \(\Gamma \backslash \mathcal {H}^*\) is compact, there exist \(z_1,\ldots ,z_n\in \mathcal {H}\) such that the sets \(\pi (O_{z_i})\) together with \(\pi (V)\) cover \(\Gamma \backslash \mathcal {H}^*\). Let \(\hat{\epsilon }_1,\ldots ,\hat{\epsilon }_n,\hat{\epsilon }_V\) be a partition of unity corresponding to this cover, i.e. smooth functions supported in \(\pi (O_{z_1}),\ldots , \pi (O_{z_n})\) and \(\pi (V)\), respectively, satisfying

*z*is in

*V*, then

*H*(

*z*) and \(H(\gamma z)\) vanish and so does \((1-\eta (z))\phi (\gamma )(z)\). If

*z*is not in

*V*, we have

*i*such that there exists a \(g_i(\gamma z)\in \Gamma \) with \(g_i(\gamma z)\gamma z\in O_{z_i}\). Now we choose \(g_i(\gamma z)=g_i(z)\gamma ^{-1}\), to get that \(H|_{r,v}\gamma (z)\) equals

In the definition of \(\tilde{\mathcal {Q}}\), 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.

### **Definition 5**

*F*in \(\tilde{\mathcal {Q}}\) such that there exist positive constants

*K*,

*A*,

*B*with

Note that the functions in \(\mathcal {P}\) are the holomorphic functions in \(\mathcal {Q}\).

### **Proposition 4**

Let *F* be in \(\tilde{\mathcal {Q}}\). If \(\gamma \mapsto F|_{r,v}\gamma -F=\psi (\gamma )\) is in \(Z^1_{r,v}(\Gamma ,\mathcal {P})\) then *F* is in \(\mathcal {Q}\).

### *Proof*

This proof is similar to the proof of the main theorem of [13]. Let *M* be the set of matrices \(\gamma \) in \(\Gamma \) with \(\lambda /2\le \text{ Re }(\gamma i)<\lambda /2\). *M* is a complete set of coset representatives of \(\Gamma _\infty \setminus \Gamma \). We need a technical lemma from [12]:

### **Lemma 2**

*F*is in \(\mathcal {Q}\) if and only if \(F-g_\infty \) is in \(\mathcal {P}\), so we can assume without loss of generality that \(F(z+\lambda )=v(\sigma _\infty )F(z)\). Let \(z\in \mathcal {H}\). There exists \(\tau \in \overline{\mathcal {F}}\) and \(\gamma \in \Gamma \) such that \(z=\gamma \tau \). Since

*M*is a complete set of representatives of \(\Gamma _\infty \setminus \Gamma \), there is an integer

*m*and \(\delta \in M\) such that \(z=\sigma _\infty ^m\delta \tau \). If \(\delta =I\) then we can deduce

*M*that fixes \(\infty \) is

*I*. We have

*K*,

*A*,

*B*and all \(z\in \mathcal {H}\). \(\square \)

### **Corollary 3**

Every cocycle in \(Z^1_{r,v}(\Gamma ,\mathcal {P})\) is a coboundary in \(Z^1_{r,v}(\Gamma ,\mathcal {Q})\).

*g*can only increase polynomially towards the cusps of \(\Gamma \), while

*f*decreases exponentially.

### 3.1 Spectral theory of automorphic forms

### **Lemma 3**

*f*is invariant under \(|_{r,v}\) if and only if \(F(z)=y^{\frac{r}{2}}f(z)\) is invariant under \(|^R_{r,v}\).

### **Definition 6**

*f*that are invariant under \(|^R_{r,v}\) and have finite norm with respect to the scalar product

*r*hyperbolic Laplacian and the Maass weight-raising and weight-lowering operators are defined as

### **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 \(x\in H\) such that \(Tx_n\) converges to \(y\in H'\), we have \(x\in D\) and \(Tx=y\).

### **Definition 8**

*D*is dense in

*H*then for any operator

*T*from

*D*to

*H*, we can define its

*adjoint*\(T^{*}\) on the domain

*y*in this set defines a linear functional on

*D*by \(\phi _y: x\mapsto \langle Tx,y \rangle \). This functional can be extended to

*H*and by the Riesz representation theorem there exists \(z\in H\) such that \(\phi _y(x)=\langle x,z\rangle \) 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 \(T\subseteq T^*=(T^*)^*\), where \(T\subseteq T^*\) means that \(T^*\) extends *T*.

### **Proposition 5**

- (i)
\(\Delta _r:\mathcal {D}^2_r\rightarrow H_{r,v}\) is essentially self-adjoint. It has a self-adjoint extension to a dense subset of \(H_{r,v}\) that we denote by \(\tilde{\mathcal {D}}_r\).

- (ii)
The eigenfunctions of \(\Delta _r\) are smooth (in fact they are real analytic).

- (iii)\(K_r: \mathcal {D}^2_r\rightarrow H_{r+2,v}\) and \(\Lambda _r:\mathcal {D}^2_r\rightarrow H_{r-2,v}\) can be extended to closed operators defined on \(\tilde{\mathcal {D}}_r\). For \(f\in \tilde{\mathcal {D}}_r\) and \(g\in \tilde{\mathcal {D}}_{2+r}\), we have$$\begin{aligned} (K_r f,g)^R=(f,\Lambda _{2+r}g)^R. \end{aligned}$$
- (iv)$$\begin{aligned} -\Delta _r=\Lambda _{r+2}K_r-\frac{r}{2}\left( 1+\frac{r}{2}\right) =K_{r-2}\Lambda _r+\frac{r}{2}\left( 1-\frac{r}{2}\right) . \end{aligned}$$

### *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 \(\Delta _r\) 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]. \(\square \)

### **Definition 9**

A *cuspidal Maass wave form* in \(H_{r,v}\) with eigenvalue \(\lambda \) is an eigenfunction of \(-\Delta _r\) with eigenvalue \(\lambda \) that decays exponentially at the cusps of \(\Gamma \).

### *Remark 3*

By [21, Satz 5.2], all eigenfunctions in \(H_{r,v}\) of \(-\Delta _{r}\) of eigenvalue \(\frac{r}{2}(1-\frac{r}{2})\) are of the form \(y^{\frac{r}{2}}f\), where *f* is a modular form in \(M_{r}(\Gamma ,v)\) that has finite Petersson norm, i.e. \((f,f)<\infty \). If *f* is a cusp form, then \(y^{\frac{r}{2}}f\) is a cuspidal Maass wave form.

*q*be a cusp of \(\Gamma \), \(\sigma _q\) the generator of \(\Gamma _q/\{\pm I\}\) and \(A_q\in \text{ SL }_2(\mathbb {R})\) chosen such that \(q=A_q^{-1}\infty \). The cusp

*q*is called

*singular for the multiplier system*

*v*, if \(v(\sigma _q)=1\) and

*regular for*

*v*otherwise. Let \(q_1,\ldots ,q_{m^*}\) be a set of representatives of the cusps of \(\Gamma \) that are singular for

*v*. For each of these cusps, we define the Eisenstein series

*z*,

*s*) in sets of the form \(K\times \{s|\text{ Re } s\ge 1+\epsilon \}\), where

*K*is a compact subset of \(\mathbb {C}\) and \(\epsilon >0\). For a fixed

*s*with real part \(\ge 1+\epsilon \), one can use the absolute and uniform convergence of the series to see that \(E_{r,v}^q(\cdot ,s)\) is invariant under \(|_{r,v}^R\) and that

### **Theorem 6**

- (i)
For fixed \(z\in \mathcal {H}\), the Eisenstein series \(E_{r,v}^q(z,\cdot )\) can be meromorphically continued to the whole complex plane.

- (ii)If, for one fixed
*z*, \(E_{r,v}^q(z,\cdot )\) has a pole of order*n*at \(s_0\), then the function \(f(z):=\lim _{s\rightarrow s_0}(s-s_0)^nE_{r,v}^q(z,s)\) is real analytic, invariant under \(|_{r,v}^R\) and satisfiesIf$$\begin{aligned} -\Delta _r f=s_0(1-s_0)f. \end{aligned}$$*n*is chosen so that*f*(*z*) has no poles in \(\mathcal {H}\), then*f*grows at most polynomially at each cusp of \(\Gamma \), i.e. if*q*is a cusp of \(\Gamma \) and \(\tau _q\infty =q\) for \(\tau _q\in {{\mathrm{SL}}}_2(\mathbb {R})\), then there exists \(A\in \mathbb {R}\) such that \(f|_{r}\tau _q(z) = \mathcal {O}(y^A)\) as \(y\rightarrow \infty \). In particular, if \(E_{r,v}^q(z,s)\) is holomorphic at \(s=s_0\), thenFurthermore, we have the following equalities:$$\begin{aligned} -\Delta _rE_{r,v}^q(\cdot ,s_0)=s_0(1-s_0)E_{r,v}^q(\cdot ,s_0). \end{aligned}$$$$\begin{aligned} K_rE_{r,v}^q(\cdot ,s_0)&=\left( \frac{r}{2}+s_0\right) E_{r+2,v}^q(\cdot ,s_0),\end{aligned}$$(21)$$\begin{aligned} \Lambda _r E_{r,v}^q(\cdot ,s_0)&=\left( \frac{r}{2}-s_0\right) E_{r-2,v}^q(\cdot ,s_0). \end{aligned}$$(22)

### **Theorem 7**

^{3}of \(\Delta _r\). Then

*f*has a spectral expansion

*f*has compact support mod \(\Gamma \), i.e. \(\pi (\text{ supp }(f))\) is compact in \(\Gamma \backslash \mathcal {H}^{*}\), then both parts of the spectral expansion, \(\sum (e_n,f)^R e_n\) and \(\sum _{i=1}^{m^*}\frac{1}{4\pi }\int _{-\infty }^{\infty }(E_{r,v}^{q_i}(\cdot ,\frac{1}{2}+i\rho ),f)^RE_{r,v}^{q_i}\left( z,\frac{1}{2}+i\rho \right) {\text {d}}\rho \), converge absolutely and uniformly on compact subsets of \(\mathcal {H}\).

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.

*G*vanishes in a neighbourhood of every cusp since

*g*is holomorphic there, so

*G*has compact support mod \(\Gamma \) and is in \(H_{2-r,\overline{v}}\).

To prove Theorem 4, we have to show that if \(\phi \) is orthogonal to \(S_{2-r}(\Gamma ,\overline{v})\), then \(g\in \mathcal {Q}\) can be chosen to be holomorphic. This implies that \(\phi \) is a coboundary in \(Z^1_{r,v}(\Gamma ,\mathcal {P})\).

### **Lemma 4**

Let \(2-r>0\) and \(\phi \), *g* and *G* be as above. Then \((f,\phi )=0\) for all \(f\in S_{2-r}(\Gamma ,\overline{v})\) if and only if \((\tilde{f},G)^R=0\) for all cuspidal Maass wave forms \(\tilde{f}\) with eigenvalue \(\frac{r}{2}(1-\frac{r}{2})\).

### *Proof*

We can now use spectral theory to characterise functions which are orthogonal to cuspidal Maass wave forms of eigenvalue \(\frac{r}{2}(1-\frac{r}{2})\).

### **Proposition 6**

*H*be a smooth function in \(H_{2-r,\overline{v}}\) with compact support mod \(\Gamma \). Then the following are equivalent:

- (i)
\((\tilde{f},H)^R=0\) for all cuspidal Maass wave forms \(\tilde{f}\) with eigenvalue \(\frac{r}{2}(1-\frac{r}{2})\).

- (ii)
\(H=K_{-r}F + K_{-r}E\), where

*F*is a smooth function in \(H_{-r,\overline{v}}\) and*E*is a linear combination of the functions \(E_{-r,\overline{v}}^{q_i}(z,\frac{r}{2})\). If \(2-r>1\) or \(2-r<0\) this implies \(E=0\).

### *Remark 4*

By [11, 15] we have \(S_{2-r}(\Gamma ,\overline{v})=\{0\}\), if \(2-r\le 0\). Since, by [21, Satz 5.2], all cuspidal Maass wave forms of eigenvalue \(\frac{r}{2}(1-\frac{r}{2})\) are of the form \(y^{\frac{r}{2}}f\), where \(f\in S_{2-r}(\Gamma ,\overline{v})\), the first condition is always satisfied in the case \(2-r\le 0\).

### *Proof*

- 1.
Images of eigenfunctions of \(\Delta _{-r}\) under the Maass raising operator \(K_{-r}=(z-\overline{z})\frac{\partial }{\partial z} - \frac{r}{2}\). We denote these by \(K_{-r}e_n\). By [21, Satz 6.3] these eigenfunctions cannot have eigenvalue \(\frac{r}{2}(1-\frac{r}{2})\).

- 2.
A (finite) orthonormal basis of the eigenfunctions of eigenvalue \(\frac{r}{2}(1-\frac{r}{2})\). By Remark 3 this set is of the form \(\{y^{\frac{2-r}{2}}f_1,\ldots ,y^{\frac{2-r}{2}}f_N\}\), where the \(f_i\) form an orthonormal basis of the subspace of \(M_{2-r}(\Gamma ,\overline{v})\) of modular forms with finite Petersson norm. If \(2-r\ge 1\) this subspace is equal to \(S_{2-r}(\Gamma ,\overline{v})\), while for \(2-r<1\) every modular form in \(M_{2-r}(\Gamma ,\overline{v})\) has finite Petersson norm.

*H*is of the form

*F*is smooth we apply \(\Lambda _{2-r}\) to (26) and obtain

*F*is a solution of an elliptic differential equation and so, by elliptic regularity,

*F*is smooth.

*H*is orthogonal to all cuspidal Maass wave forms with eigenvalue \(\frac{r}{2}(1-\frac{r}{2})\), we see that in the expansion

Theorem 4 now follows from Proposition 6.

### *Proof*

(of Theorem 4 and of Theorem 2 for \(2-r\ne 1\).)

*g*and

*G*be constructed as in (23) and (24). In the case \(2-r>0\) suppose additionally that \((f,\phi )=0\) for all \(f\in S_{2-r}(\Gamma ,\overline{v})\). By Lemma 4 in the case \(2-r>0\) and Remark 4 in the case \(2-r\le 0\),

*G*satisfies condition (i) of Proposition 6. Hence there is a smooth \(F\in H_{-r,\overline{v}}\) and a linear combination of Eisenstein series \(E(z)=\sum _{i=1}^{m^*} a_i E_{-r,\overline{v}}^{q_i}(z,\frac{r}{2})\), with

*E*is only non-zero if \(0\le 2-r<1\), and in this case the Eisenstein series \(E_{-r,\overline{v}}^{q_i}(\cdot ,\frac{r}{2})\) are smooth functions that grow at most polynomially at each cusp of \(\Gamma \). Since

*F*is smooth and in \(H_{-r,\overline{v}}\),

*F*also grows at most polynomially at each cusp and so the same is true for \(D = E+F\). We have

*D*is invariant under \(|^R_{-r,\overline{v}}\), \(\overline{D}\) is invariant under \(|^R_{r,v}\). By Lemma 3, the function \(\tilde{D}(z)=-2iy^{-\frac{r}{2}}\overline{D}\) is invariant under \(|_{r,v}\). This invariance implies that \(\tilde{g}=g-\tilde{D}\) satisfies \(\tilde{g}|_{r,v}\gamma -\tilde{g}=\phi (\gamma )\) for all \(\gamma \in \Gamma \). Since \(\tilde{D}\) grows at most polynomially at the cusps of \(\Gamma \), \(\tilde{g}\) satisfies the growth conditions for functions in \(\tilde{Q}\). Proposition 4 now tells us that \(\tilde{g}\in \mathcal {Q}\). Note also that equation (29) implies that \(\tilde{g}\) is holomorphic, so \(\tilde{g}\in \mathcal {P}\). We finally conclude that \(\phi \) is indeed a coboundary in \(Z^1_{r,v}(\Gamma ,\mathcal {P})\).

The proof above shows in particular that for \(2-r\le 0\) every cocycle in \(Z^1_{r,v}(\Gamma ,\mathcal {P})\) is a coboundary and hence \(H^1_{r,v}(\Gamma ,v)=\{0\}\). This proves Theorem 2 for \(2-r\le 0\), since \(S_{2-r}(\Gamma ,\overline{v})\) is also \(\{0\}\) in this case. \(\square \)

### *Remark 5*

*G*is in the image of \(K_{-r}\). In the case \(2-r=1\), we only obtain

## 4 Vector-valued modular forms

*v*a unitary multiplier system of weight

*r*. Let

*F*be a function from \(\mathcal {H}\) to \(\mathbb {C}^n\). The slash operator \(|_{\rho ,v,r}\) is defined by

### **Definition 10**

*modular form*for \(\Gamma \) of weight

*r*, representation \(\rho \) and multiplier system

*v*if the following conditions are satisfied:

- (i)
*f*is holomorphic on \(\mathcal {H}\). - (ii)
\(f(z)=f|_{r,v,\rho }\gamma (z)\) for all \(\gamma \in \Gamma \) and \(z\in \mathcal {H}\).

- (iii)If
*q*is a cusp of \(\Gamma \) and \(A\infty =q\), then for any \(\epsilon >0\)$$\begin{aligned} j(A,z)^{-r}f(Az) \text{ is } \text{ bounded } \text{ for } y\ge \epsilon . \end{aligned}$$

*f*satisfies the additional condition

- (iii’)If
*q*is a cusp of \(\Gamma \) and \(A\infty =q\), then there exists an \(\epsilon >0\) such that$$\begin{aligned} j(A,z)^{-r}f(Az)=\mathcal {O}_{y\rightarrow \infty }(e^{-\epsilon y}), \end{aligned}$$

*cusp form*. The set of modular forms or cusp forms of this kind is denoted by \(M_r(\Gamma ,v,\rho )\) and \(S_r(\Gamma ,v,\rho )\), respectively.

Let \(\mathcal {P}^n\) be the set of vector-valued functions \(f(z)=(f_1(z),\ldots ,f_n(z))\) such that all \(f_i\) are in \(\mathcal {P}\). The slash operator \(|_{r,v,\rho }\) defines a \(\Gamma \)-action on \(\mathcal {P}^n\) and so we can define the cohomology groups \(H^1_{r,v,\rho }(\Gamma ,\mathcal {P}^n)\) and \(\tilde{H}^1_{r,v,\rho }(\Gamma ,\mathcal {P}^n)\). 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**

### *Proof*

*U*is diagonalisable, there exists a \(V\in \text{ U }(n)\) and a diagonal matrix \(D\in \text{ U }(n)\) with

*V*, we get

*D*and \(G=Vg=(G_1,\ldots ,G_n)\in \mathcal {P}^n\). We can use Corollary 1 to find solutions \(F_i\in \mathcal {P}\) for

This can be used to show the following.

### **Theorem 8**

Every cocycle in \(Z^1_{v,\rho }(\Gamma ,\mathcal {P}^n)\) is parabolic.

### 4.1 Petersson inner product

*f*,

*g*be in \(S_{2-r}(\Gamma ,\overline{v},\rho ^{-1})\). The Petersson inner product of

*f*and

*g*is defined by

### **Lemma 5**

*g*be in \(S_{2-r}(\Gamma ,\overline{v},\rho ^{-1})\), then

### **Theorem 9**

*v*and \(\rho \) be as above and \(0<2-r\ne 1\). The pairing defined above is perfect, so the map \(f\mapsto \phi _f^\infty \) induces an isomorphism

### *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. \(\square \)

## Footnotes

- 1.
Another common term for modular forms that is used e.g. in [12], is

*entire automorphic forms*. - 2.
\([A_i,A_{i+1}[\) denotes the geodesic in \(\mathcal {H}\) that connects \(A_i\) and \(A_{i+1}\) and includes \(A_i\) but not \(A_{i+1}\).

- 3.
An

*orthonormal system*of eigenfunctions of an operator*T*on a Hilbert space*H*is a set of eigenfunctions of*T*that are pairwise orthogonal and have norm 1.

## Notes

### Acknowledgments

I am thankful to N. Diamantis for suggesting this topic to me and Y. Petridis for a helpful discussion on the proof of Proposition 6. Further thanks are due to F. Strömberg, T. Vavasour and the anonymous referee for a careful reading of this article and many helpful suggestions. I am particularly grateful for the countless comments, corrections and improvements that R. Bruggeman provided during the completion of this article.

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