EICHLER INTEGRALS FOR MAASS CUSP FORMS OF HALF-INTEGRAL WEIGHT

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1 EICHLER INTEGRALS FOR MAASS CUSP FORMS OF HALF-INTEGRAL WEIGHT T. MÜHLENBRUCH AND W. RAJI Abstract. In this paper, we define and discuss Eichler integrals for Maass cusp forms of half-integral weight on the full modular group. We discuss nearly periodic functions associated to the Eichler integrals, introduce period functions for such Maass cusp forms, and show that the nearly periodic functions and the period functions are closely related. Those functions are an extension of the periodic functions and period functions for Maass cusp forms of weight on the full modular group introduced by Lewis and Zagier. 1. Introduction Recall that modular cusp forms of weight k N for the group SL, Z)) are holomorphic functions u h from the upper half-plane H {z x + iy; x, y R, y > } to C, satisfying the u h z + 1) u h z) and u h 1/z) z k u h z), and vanish as y. More details can be found in e.g. [1] and [15]. In the context of the Eichler-Shimura theorem, we attach to each modular cusp form a polynomial p of degree k. One way to define it is by the following integral transformation: 1.1) p) : z) k u h z) dz C). This integral transformation goes back to Eichler in [6]. One important property is that each period polynomial satisfies the identities ) 1 p) + k p and 1.) ) ) 1 1 p) + + 1) k p + k p + 1 for each C. Some more details on the Eichler-Shimura theorem can be found in e.g. [1, Chapter V and VI], [9, 1.1] and [14] Mathematics Subject Classification. Primary 11F37; Secondary 11F5, 11F7. 1

2 T. MÜHLENBRUCH AND W. RAJI There exists another way to define the period polynomial, involving a variant of the above integral. Consider the integral transformation 1.3) f h ) : z) k u h z) dz H), defined only on the upper half-plane H. f h is obviously a holomorphic function; and it easily seen that f h is periodic. The period polynomial p now appears in the calculation ) 1 i f h ) k f h z) k u h z) dz z) k u h z) dz z) k u h z) dz p) H). One important extension of the Eichler-Shimura isomorphism was done by Lewis and Zagier in [11]. They found a one-to-one correspondence between Maass cusp forms of weight for SL, Z)) and functions called period functions. Part of their main result is the following Theorem [11]). Let s be a complex number with Re s) 1. There is an isomorphism between the following two function spaces: 1) The space of Maass cusp forms of weight with eigenvalue s1 s) for SL, Z): Maass cusp forms of weight for SL, Z) are real-analytic functions u : H C, satisfying the transformation properties uz + 1) uz) and u 1/z) uz), are eigenfunctions of the hyperbolic Laplacian y x + y) with eigenvalue s1 s), and vanish as y. ) The space of holomorphic functions f on C R, satisfying fz + 1) fz) and bounded by Im z) A for some A >, such that the function fz) z s f 1/z) extends holomorphically across the positive real axis and is bounded by a multiple of min{1, z 1 } in the right halfplane. 3) The space of holomorphic solutions ψ : C C of the three-term functional equation ) ψ) ψ + 1) + + 1) s ψ + 1 on C C R which satisfy the growth condition { ) O 1 as, Re > ) and ψ) O 1) as, Re > ).

3 EICHLER INTEGRALS FOR MAASS CUSP FORMS OF HALF-INTEGRAL WEIGHT 3 In analogy to the case of modular cusp forms, Lewis and Zagier also introduce two integral transformations in [11, Chapter II] from Maass cusp forms to period functions which are the functions ψ above) and the periodic holomorphic functions f on C R. In this paper, we extend the integral transformations of [11] to the context of Maass forms of half-integral weights with a multiplier system. Our main result is the following: Theorem 1.1. Let ν be a purely imaginary number complex number, i.e. ν ir, k 1 Z a weight and v a compatible multiplier as defined in.. For each Maass cusp form u of weight k, eigenvalue 1 4 ν and multiplier v for SL, Z), see Definition 3.1 for details, there exist the following functions: 1) A holomorphic function function f : C R C, which is nearly periodic, i.e. fz + )) 1) afz) for some a C with a 1, such that 1 fz) v z 1 ν 1 f 1/z) extends holomorphically across the positive real axis and is bounded by a multiple of min{1, z 1 } in the right half-plane. ) A holomorphic solution P : C C of the three-term functional equation )) 1 )) 1 ) P ) v P + 1) + v + 1) ν 1 P on C which satisfies the growth condition { ) O 1 as, Re > ) and P ) O 1) as, Re > ). We call such a function P a period function. The proof of this theorem follows ideas presented above for modular cusp forms for positive even weight: We use a Maass-Selberg differential form to define the kernel of integral transformations similarly as z) k u h z) dz is used above. We then describe properties of the determined nearly periodic functions and period functions, i.e., the images of our integral transformations. Summarizing, we introduce the arrows of the following diagram and show that the diagram commutes: Maass cusp forms of half-integer weight 3 Nearly periodic functions Period functions

4 4 T. MÜHLENBRUCH AND W. RAJI It is our hope that the results of this paper form a first step towards a working Eichler-Shimura theory for Maass cusp forms of half-integral weight for SL, Z)) since we establish one direction of a possible bijection between Maass cusp forms and period functions. We will discuss this and some other related questions briefly in 8. The paper is organized as follows: Section contains the preliminaries, like defining properly the group SL, Z) and its linear fractional transformations, the multiplier systems and the slash and double-slash notations. In 3 we define the Maass cusp forms for half-integral weight. The next section introduces the R-function and the Maass-Selberg form. The sections 5 and 6 contain the definitions of the integral transformations from Maass cusp forms to nearly periodic functions on one hand and to period functions on the other hand. These sections contain also our main result in a more detailed version. We use 7 to compare and relate our integral transforms to the one appearing in the setting of the classical modular cusp forms. The remaining Section 8 contains a short discussion and outlook.. Preliminaries.1. The matrix group SL, Z) and its linear fractional transformations. Let SL, R) denote the group of matrices with real entries and determinant 1. The subgroup SL, Z) SL, R) denotes the full modular group, that is the subgroup of matrices with integer entries. It is generated by.1) S These satisfy ) 1 1 and T.) S ST ) 3 1), ) where 1) SL, Z) is the matrix with 1 on the diagonal and on the off-diagonal entries. 1 denotes the identity matrix. We denote ).3) T 1 : T ST. 1 1 Note that.4) ST ) and ST ST T 1 S ) The group SL, R) acts on the upper half-plane H {z C; Im z) > } and its boundary P R R { } and the lower half-plane H {z

5 EICHLER INTEGRALS FOR MAASS CUSP FORMS OF HALF-INTEGRAL WEIGHT 5 C; Im z) < } by fractional linear transformations a ) a b c if z,.5) z : if z c d d c with c, and otherwise. az+b cz+d We also need the following µ-function: ) ) a b.6) µ : SL, R) C C; µ, z : cz + d. c d Obviously µ satisfies the cocycle-relation for every γ, δ SL, Z). Moreover, we have µγδ, z) µγ, δ z) µδ, z).7) Im γz) Im z) µγ, z), d dz γz 1 µγ, z)), d d z γ z 1 z µγ, z)) and γ γz µγ, ) µγ, z) for every γ SL, R)... Multiplier systems. We call a function v : SL, Z) C multiplier or multiplier system compatible with the half-integral weight k if v satisfies.8) vγδ) e ikargµγδ,z)) vγ)vδ) e ikargµγ,δ z)) e ikargµδ,z)) for every γ, δ SL, Z) and z H. Remark.1. 1) The range of arg ) is π < arg z) π for all z C. ) Condition.8) implies that the system of equations.9) fγz) vγ) e ikargµγ,z)) fz) z H, γ SL, Z)), allows non-zero solutions f : H C. 3) We have in particular.1) v 1) ) e ikπ, since.9) with γ 1) implies v 1) ) e ikarg 1) 1, if f does not vanish everywhere.

6 6 T. MÜHLENBRUCH AND W. RAJI.3. The slash and double-slash notations. We define arbitrary powers w s with possibly complex) s by using the standard branch of the logarithm: z s z s e isargz) with arg z) π, π] for every z C. We introduce the slash and the double-slash notation. Let k 1 Z, ν C and v be a multiplier. For f : H C and γ SL, Z) we define f v k γ) z) : e ikargµγ,z)) vγ) 1 fγ z) and.11) f v ν γ) z) : vγ) 1 µγ, z) ) ν 1 fγ z) for every z H. For example.9) reads as f v k γ f. We define the slash and double-slash notation also for functions f : H C on the lower half-plane, since γ z H in.5) for every γ SL, Z) and z H. As slight abuse of notation, we also use the slash notation.1) f ) 1k γ z) e ikargµγ,z)) fγ z) for matrices γ SL, R). Consider the subset SL, Z) + SL, Z), containing ) all matrices γ a b SL, Z) with only nonnegative entries, i.e., all SL, Z) satisfying c d a, b, c, d. These matrices have the property that they map the cut-plane C C, ] into itself: for every z C and γ SL, Z) + we have γz C. The slash and double-slash notation in.11) is also well defined for functions f : C H and all γ SL, Z) +. For given real z we may even extend the slash and double-slash notations to certain matrices γ SL, Z) which satisfy µγ, z) > on occasion. Remark.. The slash notation in.11) can be viewed as a group operation of SL, Z) on the space of functions on the upper half-plane. Indeed, relation.8) implies that f v ) k γδ f v γ) v δ holds. k k The double-slash notation in.11) is, as the name indicates, an abbreviation for the given expression. It is not a group operation in general. The same is true for the slash-notation except in the case mentioned above. 3. Maass Cusp Forms of Half-Integral Weight and Maass Operators In this section, we define Maass cusp form and useful Maass operators that will be used later. As usual, we write z x + iy for complex z with real part x and imaginary part y. Definition 3.1. Let k be a half-integral weight and v : SL, Z) C a compatible multiplier system. A Maass cusp form of weight k and multiplier v for SL, Z) is a real-analytic function u : H C satisfying

7 EICHLER INTEGRALS FOR MAASS CUSP FORMS OF HALF-INTEGRAL WEIGHT 7 1) u v γ u for every γ SL, Z), k ) u is an eigenfunction of the Laplace operator k with eigenvalue λ, i.e., k u λ u where 3.1) k y x + y) + iky x with z x + iy H. 3) u satisfies the growth condition uz) O y c ) as y for every c R. It is known, see for example [], that the eigenvalue λ is real. It is convenient to write λ 1 4 ν with suitable spectral parameter ν R ir. In the following lemma, we extend the definition of the Maass cusp form to the lower half-plane by considering the conjugate of the form defined on the upper half-plane. Lemma 3.. Let u be a Maass cusp form of weight k, multiplier system v and eigenvalue λ. Defining ũ : H C; z ũz) : u z) for a Maass cusp form u defines a real-analytic function on the lower half-plane which satisfies 1) ũ v γ ũ for every γ SL, Z), k ) ũ is an eigenfunction of the Laplace operator k with eigenvalue λ, and 3) ũ satisfies the growth condition ũz) O y c ) as y for every c R. Proof. Using the identity arg ) arg ), C, ], the transformation property follows immediately: ũ γ z) u γ z) e ikargµγ, z)) vγ) u z) e i k)argµγ,z)) vγ) ũz) for every z H and γ SL, Z). The substitution y y i.e. z z) gives k ũz) [ y x + y) + i k)y x ] u z) [ y) x + 1) y) + i k) y) x ] uz) using y y) k uz) λ uz) λ u z) using y y) for z H. This shows the second property. The growth condition follows directly from the definition ũz) u z). The raising and lowering Maass operators acting on the space of cusp forms of given weight k, multiplier v, and eigenvalue λ are given by 3.) E ± k ±iy x + y y ± k.

8 8 T. MÜHLENBRUCH AND W. RAJI Equivalently, it is sometimes convenient to write 3.3) E + k 4iy z + k and E k 4iy z k. They satisfy the identity 3.4) E ± k E k 4 k kk ). This implies: If u : H C is an eigenfunction of k with spectral value ν, i.e. k u 1 4 ν) u, then u satisfies 3.5) E ± k E k u 1 + ν k ) 1 + ν ± k ) u. The slash notation defined in.11) commutes with the Laplace operator, 3.6) k f v k γ) k f ) v k γ, and interacts as follows with the Maass-operators 3.7) E ± k f v k γ) E ± k f) v k± γ for every k R and u real-analytic. γ SL, Z)), 4. The Maass-Selberg Differential Form and the R-function We need to define the Maass-Selberg differential form. This form is used later to define the kernel of the associated integrals of the Maass cusp forms. First we define what is known as the R-function. It will play an important role in the construction of the kernel The R-function. We define hz) : Im z) for z H. For k 1 Z and ν C, it is easy to see that hz) is real-analytic and positive for z H, and that h satisfies the differential equations ) 4.1) k h 1 1 ν 4 ν h 1 ν and E ± k h 1 ν 1 ν ± k) h 1 ν. Define ) k ) 1 z Im z) ν 4.) R k,ν z, ) :. z z) z) for, z C such that 4.3) z, z R holds. Remark 4.1. The square roots z and z on the right hand side of 4.) are well defined since we require that z and z are in C R. The square roots in this situation, interpreted as principal square roots, are holomorphic. The R-function has the following properties:

9 EICHLER INTEGRALS FOR MAASS CUSP FORMS OF HALF-INTEGRAL WEIGHT 9 Proposition ) 4.5) 1) The function z R k,ν z, ) is smooth in the real and imaginary part of z if 4.3) holds. ) The map R k,ν z, ) is holomorphic on C {z r, z r; r }. 3) R k,ν has the form R k,ν z, ) e ikarg z) Im z) z) z) h 1 ν )) 1 1 z) k 1 ) 1 ν for real and z H. 4) Assume the usual restriction 4.3) for z and. The function H C; z R k,ν z, ) satisfies the differential equations ) 1 k R k,ν, ) 4 ν R k,ν, ) The function satisfies and E ± k R k,ν, ) 1 ν ± k) R k±,ν, ). H C; z R k,ν z, ) k R k,ν, ) ) 1 4 ν R k,ν, ) and E ± k R k,ν, ) 1 ν ± k) R k±,ν, ). Proof. We prove the proposition step by step. 1) For fixed C assume that z C satisfies 4.3). It is then obvious from 4.) that the function z R k,ν z, ) is smooth in real and imaginary part of z. Observe that the values under the square-roots are never negative by condition 4.3).) ) Fix z C this time. Again, it is obvious from 4.) that the function R k,ν z, ) is holomorphic for all C satisfying condition 4.3). Noticing that satisfies 4.3) is equivalent with is element of the two-cut plane C z + R z + R ) shows the second part of the proposition.

10 1 T. MÜHLENBRUCH AND W. RAJI 3) The first equality follows by rewriting the right hand side of 4.) using the identity z z 4.6) e iarg z), z z z which is correct under the given assumptions R and z H. The second equality follows by the slash notation in.1). 4) We assume real and z H for the moment. Combining the last expression of R k,ν z, ) in 4.4) with 3.6) and then with 4.1) gives k R k,ν, ) k h 1 ν )) 1 1 k k h 1 ν) 1 k 1 4 ν) h 1 ν 1 k 1 ) 1 1 ) ν) R k,ν, ). Analogously, just using 3.7) instead of 3.6) shows E ± k R k,ν, ) 1 ν ± k)r k±,ν, ). Next, using the substitution arguments in the proof of Lemma 3. shows that R k,ν z, ) satisfies the stated properties for z H. The last step is to extend from real values to complex values. This can be done since is just a constant for the differential operators. This shows that the stated differential equations hold also for complex as long as z and satisfy the condition 4.3). Remark 4.3. The R-function appeared first in [11], where Lewis and Zagier introduced R, 1 z, ) y x ) + y i 1 z 1 ) z in [11, p.11, above.6)]. Their notation for the above expression was R z). Before we describe the transformation law of R k,ν z, ), we need one trivial auxiliary lemma which will allow us to perform a certain factorization in the proof of the forthcoming Lemma 4.6. Lemma 4.4. Let z, w C and α C be complex numbers satisfying either 1) z R > or ) the product zw R >. Then the identity z w ) α z α w α holds.

11 EICHLER INTEGRALS FOR MAASS CUSP FORMS OF HALF-INTEGRAL WEIGHT11 Proof. 1) Assume that z is real and positive and w C. This ensures arg w) arg zw) π, π). Writing w in its polar coordinates gives ) ) α z w ) α ) α z w e iargw) z α w e iargw) z α w α. ) Assume that zw is real and positive and both z, w C. This ensures arg z) arg w) π, π). Writing z and w in its polar coordinates gives ) α z w z α zw ) ) α ) α α z α z e iargz) w e z iargw) 1 e iargz) ) α ) α z e iargz) w e iargw) z 1 e iargz) using case 1)) ) α z e iargz) w e iargw) z 1 e iargz) ) α w e iargw) w α, where we used that zw R >, z 1 C implies zw z 1 ) α zw) α z 1) α as shown above. The identity z w) α z α w α holds in both situations. Remark 4.5. It is important that z, w R < in the second case of the above auxiliary lemma. If z, w are both real and negative, then zw itself is positive. Due to the choice involved in the argument function, see Remark.1, the arguments arg z) π and arg w) π are not anymore of opposite sign: arg z) arg w). Hence the factorization in the proof of the auxiliary lemma does not work anymore. In what follows, we show the transformation law of R k,ν, z). Lemma 4.6. Let γ SL, Z),, z C satisfying 4.3) and µγ, ), µγ, z) C with Re µγ, )) >. Additionally, assume that and z satisfy one of the following three conditions: 1) µγ, ) R >, ) H and γz γ + ir > or 3) H and γ z γ + ir >. Then, the function, z) R k,ν z, ) satisfies the transformation formula 4.7) R k,ν γz, γ) e ikargµγ,z)) µγ, ) ) 1 ν Rk,ν z, ).

12 1 T. MÜHLENBRUCH AND W. RAJI Proof. Take γ SL, Z) and, z C satisfying 4.3) and µγ, z) C. One key part is the observation that the factorization ) 1 Im γz) ν 4.8) µγ, ) ) 1 ν Im z) γ γz)γ γ z) z) z) ) 1 ν holds if and z satisfy one of the additional assumptions. First, we assume µγ, ) R >. Using identities in.7) and Lemma 4.4 gives 1 ) 1 Im γz) ν Imz) ν µγ,z) µγ, z) γ γz)γ γ z) z z µγ,) µγ,z) µγ,) µγ, z) ) 1 ν Im z) z z µγ,) µγ,) µγ, ) ) 1 ν Im z) z) z) ) 1 ν. Next, we assume the second case H and γz γ + ir >. In particular, we have that γ γz and also γ γ z are non-vanishing purely imaginary: γ γz it and γ γz it for some t, t R. We have in fact t t + Re γ). Hence the expression Im γz) Im γz) Im γz) γ γz)γ γ z) it)it tt is positive. On the other hand, we have Im γz) γ γz)γ γ z) Imz) µγ,z) µγ, z) z z µγ,) µγ,z) µγ,) µγ, z) We may apply Lemma 4.4 to µγ, ) and Imz) z) z) Im z) ), µγ, ) z) z) since the assumption Re µγ, )) > implies arg µγ, )) π and hence arg µγ, ) ) ) < π: ) 1 Im γz) ν γ γz)γ γ z) Im z) z z µγ,) µγ,) µγ, ) ) 1 ν ) 1 ν Im z) z) z) ) 1 ν. Finally, we assume the third case H and γz γ + ir >. We show that 4.8) holds by interchanging the role of z and z in the calculation above. This proves the identity 4.8) for all three cases.

13 EICHLER INTEGRALS FOR MAASS CUSP FORMS OF HALF-INTEGRAL WEIGHT13 We need also the identity 4.9) γ γz γ γ z ) k e ikargµγ,z)) z z ) k. Indeed, it follows also by applying.7) and the assumption µγ, z) C as the following calculation shows: k ) k z γ γz µγ,) µγ,z) γ γ z z µγ,) µγ, z) ) k ) k µγ, z) z µγ, z) z e ikargµγ,z)) z z ) k. We also used 4.6) and that k 1 Z is real. To finally prove the lemma we combine the identities 4.8) and 4.9). We have R k,ν γz, γ) ) k ) 1 γ γz Im γz) ν γ γ z γ γz)γ γ z) e ikargµγ,z)) µγ, ) ) ) k 1 ν z Im z) z z) z) e ikargµγ,z)) µγ, ) ) 1 ν Rk,ν z, ). ) 1 ν Remark 4.7. The second assumption on and z in Lemma 4.6 is: H and γz γ + ir >. This is equivalent with saying that z lies in the open geodesic ray connection H to the cusp γ 1 i ). The third assumption can be rephrased analogously: The third condition is equivalent with saying that z lies in the geodesic ray connection H to the cusp γ 1 i ). 4.. The Maass-Selberg Differential Form. We recall differential forms presented in [1] and observe the action of Maass raising and lowering operators applied to those differential forms. Let f, g be real-analytic functions and write again z x + iy. We define 4.1) { f, g } +z) fz)gz) dz y and { f, g } z) fz)gz) d z y.

14 14 T. MÜHLENBRUCH AND W. RAJI We extend the slash-notation to linear combinations of the differential forms {f, g} ± : We define 4.11) { f, g } + v k γz) e ikargµγ,z)) vγ) 1 fγz)gγz) dγz) Im γz) { f, g } v k γz) e ikargµγ,z)) vγ) 1 fγz)gγz) dγz) Im γz), and extend in the obvious way to linear combinations. Lemma 4.8. Let v be a multiplier and k, q 1 Z. 1) We have for any matrix γ SL, Z) that {{ { } ± 4.1) f, g v f v k+q γ k γ, g 1 q γ} ± {f 1 k γ, g v q γ}±. ) { f, g } ± { g, f } ±. Proof. The last property follows directly from the definition in 4.1). We prove 4.1) by direct calculation, using the identities in.7). example, we have and { } + f, g v z) k+q vγ) 1 e k q+)iargµγ,z)) fγz) gγz) dγz) Im γz) vγ) 1 e k q)iargµγ,z)) dz fγz) gγz) Im z) { f v k γ, g 1 q γ} + z) For for all z H. The other identities follow analogously. Combining the property 4.1) of the 1-forms in 4.1) with Maass operators, we see that { 4.13) E ± k f v γ), g 1 γ} ± { k q E ± k f 1 γ), g v γ} ± { k q E ± k f, g} ± v γ k+q for every γ SL, R). We also have the relations 4.14) and { E + k f, g} + { f, E + k g} + + 4i z fg)dz { E k f, g} { f, E k g} 4i z fg)d z. { E k f, g} + { f, E k g} + 4i z fg)dz { E + k f, g} { f, E + k g} + 4i z fg)d z. We are now able to define the Maass-Selberg form. and and

15 EICHLER INTEGRALS FOR MAASS CUSP FORMS OF HALF-INTEGRAL WEIGHT15 Definition 4.9. Let f, g be real-analytic and k 1 Z. We define the Maass- Selberg form η k by 4.15) η k f, g) { E + k f, g} + { f, E k g}. Lemma 4.1. Let f, g be real-analytic and k 1 Z. The Maass-Selberg form has the following properties: 1) We have the equations 4.16) η k f, g) + g, f) 4i dfg) and η k f, g) g, f) 4 [gf y fg y ]dx + [fg x gf x + ik y fg]dy) where f x denotes x f, f y y f, g x x g and g y y g, respectively. ) If there exists a λ R such that f and g satisfy k f λf and k g λg, then the Maass-Selberg form is closed. 3) We have that 4.17) η k f, g) v γ η k f v γ, g 1 γ) η k k k f 1 γ, g v γ) k k 4.18) for any multiplier system v. 4) Let ν C and assume that f and g are eigenfunctions of the operators k and k, respectively, both with eigenvalue 1 4 ν. Then, we have that η k+ E + k f, E k g) 1 + ν + k)1 ν + k) η kf, g) + 4i d E + k f)e k g)). Proof. Recall that z 1 x i y, z 1 x + i y, dz dx + idy, d z dx idy and df z fdz + z fd z for any function f smooth in x and y. 1) We prove the first item via direct computation: η k f, g) + g, f) 4i [ fg z + gf z ) dz + fg z + gf z ) d z ] 4idfg) and [ gfz η k f, g) g, f) 4i fg z ik y fg) dz gf z fg z ik ] y fg) d z [ gfy ) 4 fg y dx + fgx gf x + i k ] y fg) dy, where we used 3.3) for the Maass operators.

16 16 T. MÜHLENBRUCH AND W. RAJI ) We want to show that d η k f, g) under the given conditions. Since η k f, g) η k f, g) + g, f) ) + η k f, g) g, f) ), and since we already proved the first part of the lemma, it is enough to show that η k f, g) g, f) is closed. We find, after some computation, that d η k f, g) g, f) ) [ f k g g k f ] dx dy y. 3) This follows directly from 4.13). 4) It follows directly from the equations in 4.14) that η k+ E + k f, E k g) 4id [ E + k f)e k g)] {E + k f, E+ k E k g}+ + {E k+ E+ k f, E k g}+. We may apply 3.5), since we assume that f and g are eigenfunctions of the Laplace operators k and k,respectively, with the same eigenvalue. The statement in the lemma follows. Remark The first three items of Lemma 4.1 are generalizations of the Lemma given in [11, II.]. We have that η f, g) [f, g] where [, ] is defined in [11, Chapter II, ]. Our form {, } ± in 4.1) differs from the form {, } in [11, II.,.5)], contrary to the notational resemblance. Lemma 4.1. Let f and g be smooth functions in x and y) on H H satisfying fz) f z) and gz) g z). 1) We can extend the Maass-Selberg form to smooth in x and y) functions f, g defined on the lower half-plane. ) The Maass-Selberg form satisfies 4.19) η k f, g) z) η k g, f) z). Proof. 1) All differentials and other components used in the definition of the Maass-Selberg-form are well defined for smooth in x and y) functions on the lower half-plane H. Hence the extension makes sense. ) This follows by direct calculations. First use the relations 4.15), 4.1) and 3.) followed by 3.3) to rewrite everything depending on the pair z, z) respectively x, y). Then, use the substitution z, z) z, z) respectively x, y) x, y). As final step, use the above relations in reverse order.

17 EICHLER INTEGRALS FOR MAASS CUSP FORMS OF HALF-INTEGRAL WEIGHT Everything Combined. We will now insert the function R k,ν into the Maass-Selberg form and use the form to define nearly periodic functions. Lemma Let v be a multiplier which is compatible with the half-integral weight k, ν C, u a Maass cusp form with weight k multiplier v and eigenvalue 1 4 ν. Moreover let γ SL, Z) and, z C satisfying the assumptions of Lemma 4.6. It follows that ) η k u, R k,ν, γ) v γz) µγ, ) ) 1 ν ηk u, R k,ν, ) ) z) and 4.) ) R k,ν, γ), u v γz) µγ, ) ) 1 ν η k R k,ν, ), u ) z). Proof. We show the second identity: R k,ν, γ), u ) v γz) R k,ν, γ) 1 k γ, u v k γ) z) using 4.17) R k,ν, γ) 1 k γ, u) z) µγ, ) ) 1 ν ηk R k,ν, ), u ) z) using 4.7). The use of the transformation formula 4.7) in the calculation above is allowed since z and satisfy the assumptions of Lemma 4.6 and since E + k R k,ν z, ) appearing in the construction of η ±k R k,ν, ), u ) z) satisfies 4.5). The first identity follows by the same arguments. 5. Nearly Periodic Functions Let u be a Maass cusp form of weight k, multiplier v, and spectral value ν as defined in and below Definition 3.1. We define on C R C; 5.1) f) : R k,ν, ), u ) z) i if Im ) > and η k R k,ν, ), ũ ) z) if Im ) <, where ũz) u z) as defined in Lemma 3.. The path of integration is the geodesic ray connecting and the cusp i respectively i in the upper respectively lower half-plane. Remark ) We have made a choice in Definition 5.1 between integrating over the forms 1a) η k u, R k,ν, ) ) or 1b) R k,ν, ), u ) on H and a) η k R k,ν, ), ũ ) or b) ũ, R k,ν, ) )

18 18 T. MÜHLENBRUCH AND W. RAJI on H. We will see later in Remark 6. that each choice leads to the same period function on H prospectively H. Remark 5.4 compares our choice to the situation discussed in [11]. ) The reason, why we extended Maass cusp forms and the R-function to the lower half-plane H and also extended the Maass-Selberg-form η k to functions on the lower half-plane, see e.g. Lemma 3., Proposition 4. and Lemma 4.1 respectively, is the second case of 5.1). We want to be able to integrate along the geodesic ray ir > connecting H and i in the lower half-plane. In our opinion, this representation illustrates better how the function f is defined on H compared to the on H equivalent) integral representation in Lemma 5.3. Lemma 5.. For Re ν) < 1 the integration in 5.1) is well-defined along the geodesic paths connecting to i in H respectively to i in H. Proof. The singularity of R k,ν z, ) for z H respectively z H is of the form z) ν 1 respectively z) ν 1. The whole integrand has at most the same singularity since R k,ν is an eigenfunction of the Maass operators, see 4.5). The weight argument arg z) has also a well defined limit. Hence the integration is well defined for ν values satisfying Re ν) < 1. Lemma 5.3. For H we have 5.) f) Proof. For H we have f) i i η k u, R k,ν, ) ) z). η k R k,ν, ), ũ ) z) η k u, R k,ν, ) ) z) using 4.19) η k u, R k,ν, ) ) z), where we used uz) ũ z) for z H in Lemma 3.. Remark 5.4. For weight k we can compare the definition of f in 5.1) to the one in [11, page 1] since we have η f, g) [f, g], see Remark We find that [11] uses exactly the opposite choice: They use η u, R,ν, ) ) for H compared to plane they use η u, R,ν, ) ) in 5.)). η R,ν, ), u ) in 5.1)). On the lower half η R,ν, ), u ) for H compared to our integral

19 EICHLER INTEGRALS FOR MAASS CUSP FORMS OF HALF-INTEGRAL WEIGHT19 In the following lemma, we describe the transformation property of the function f). Lemma 5.5. Let Re ν) < 1. The function f defined in 5.1) satisfies 5.3) f v ν γz) vγ) 1 µγ, ) ) ν 1 fγ ) γ 1 i γ 1 i R k,ν, ), u ) z) η k u, R k,ν, ) ) z) if H and if H for every H H and γ SL, Z) satisfying Re µγ, )) >. The path of integration on the right hand side is the geodesic ray connecting respectively and γ 1 i ). Proof. Let H and γ SL, Z) such that Re µγ, )) > holds. We get fγ ) γ vγ) R k,ν, γ), u ) z) γ 1 i γ 1 i R k,ν, γ), u ) v γz). R k,ν, γ), u ) γz) Now, we would like to apply Lemma Therefore, we must check, if all z of the integration path satisfy the conditions on and z as given in Lemma 4.6. Remark 4.7 implies that we have to verify if the integration path is the geodesic ray connecting and γ 1 i ). This is indeed the case. Hence the second condition of Lemma 4.6 is satisfied since we assume Re µγ, )) >. Using the transformation formula 4.) in Lemma 4.13 gives fγ ) vγ) γ 1 i vγ) µγ, ) ) 1 ν R k,ν, γ), u ) v γz) γ 1 i R k,ν, ), u ) z). The same calculation for H, using the integral for f in 5.), gives vγ) 1 µγ, ) ) γ 1 i ν 1 fγ ) η k u, R k,ν, ) ) z). Definition 5.6. We call a function g nearly periodic if there exists an a C with a 1 such that gz + 1) a gz) holds for all z.

20 T. MÜHLENBRUCH AND W. RAJI We check that f) is nearly periodic as application of Lemma 5.5: For H we find vt ) 1 ft ) T 1 i f). R k,ν, ), u ) z) R k,ν, ), u ) z) where we use the invariance of the cusp i under translation and the trivial fact that Re µt, )) 1. The same arguments hold for H. Hence, we just proved the following Lemma 5.7. The function f in defined in 5.1) satisfies 5.4) vt ) 1 f + 1) f) for every C R. Written in the double-slash notation.11), we have 5.5) f v T f on C R. ν Similar to [11, Proposition ] we continue to prove an algebraic correspondence between f and a solution of a suitable three-term equation on C R. Lemma 5.8. Assume that k and ν satisfy e πiν 1) e πik. Put 5.6) c ± 1 e πik e ±πiν 1). Then, there exists a bijection between nearly periodic functions f satisfying 5.4) and solutions P of the three-term equation ) 5.7) P ) vt ) 1 P + 1) + vt ) 1 + 1) ν 1 P + 1 i.e. P v ν 1 T T ) ) for C R. The bijection is given by the formulas: 5.8) and 5.9) c ± f) P ) + vs) 1 ν 1 P S ) Im ) ) P v ν 1 + S ) ) P ) f) vs) 1 ν 1 f S ) C R) f v ν 1 S ) ). Remark 5.9. Let s see at a formal level that P, as defined in 5.9), satisfies the three-term functional equation 5.7): P v ν 1 T T ) f v ν 1 S) v ν 1 T T ST ) T ST T by.3))

21 EICHLER INTEGRALS FOR MAASS CUSP FORMS OF HALF-INTEGRAL WEIGHT1 f v 1 S T + ST T ST + ST ST ) ν f v ν 1 S T + ST T ST + T 1 S) ST ST T 1 S by.4)) f v ν 1 T ) + T 1 S S) + ST T ST ) ) f v ν 1 T ) v ν 1 + T 1 S + ST ). However, the calculation is only formal, since the double-slash notation just hides the weight factors and the multipliers. In general, we do not know whether they match since the double-slash notation is not a group action. We have to check them on each occasion. Proof of Lemma 5.8. Let z C R. First, we compute vs) 1 vs) 1 ν 1 1) ν 1. We have ν 1 1 since arg ) + arg ) ν 1 e ν 1)i arg)+arg 1 ) ) e ±πiν 1) Im ) ) ) 1 ±π for Im ). The choices + and >, respectively and < correspond. The consistency relation.8) for multipliers implies vs)vs) e ikπ. Hence 5.1) vs) 1 vs) 1 ν 1 1) ν 1 e πik e ±πiν 1) holds. Next, we show that 5.8) and 5.9) are inverse to each other. On one hand, we have c ±f) P ) + vs) 1 ν 1 P 1) f) vs) 1 ν 1 f 1) + vs) 1 ν 1[ f 1) vs) 1 1) ] ν 1f) [ f) 1 vs) 1 vs) 1 ν 1 1 ) ] ν 1 f) [ 1 e πik e ±πiν 1)] for Im ) ). On the other hand, we have c ±P ) c [ ± f) vs) 1 ν 1 f 1) ]

22 T. MÜHLENBRUCH AND W. RAJI P ) + vs) 1 ν 1 P 1) vs) 1 ν 1[ P 1) + vs) 1 1) ] ν 1P ) P ) [ 1 e πik e ±πiν 1)] for Im ) ), using the same argument calculations as above. We now show, that f being nearly periodic corresponds to P satisfying the three-term equation. Let f be a nearly periodic function satisfying 5.4) and P function given by 5.9). Then, we find for H H ) P v ν 1 T T ) ) P ) vt ) 1 P T ) vt ) 1 + 1) ν 1 P T ) ) f) vs) 1 ν 1 fs) ) vt ) ft 1 ) vs) 1 T ) ν 1 fst ) vt ) 1 + 1) ν 1 ) ft ) vs) 1 T ) ν 1 fst ) ) f) vt ) 1 ft ) + vt ) 1 vs) 1 + 1) ν 1 T ) ν 1 fst ST ) ) vs) 1 ν 1 fs) + vt ) 1 vs) 1 T ) ν 1 fst ) ) vt ) 1 + 1) ν 1 ft ST ) f is nearly periodic and ST ST T 1 S) + vs) 1 ν 1 ) vt ) 1 1 ν 1 + 1)ν 1 T ) ν 1 ft 1 S) fs) + vt )vt ) 1 + 1) ν 1 vt )vt ) vs) 1 fst )

23 EICHLER INTEGRALS FOR MAASS CUSP FORMS OF HALF-INTEGRAL WEIGHT3 ) vt ) 1 ft ST ) ) vs) 1 vt ν 1 ) ft 1 S) fs) ) + vt )vt ) 1 + 1) fst ν 1 ) vt ) 1 ft ST ). We used several times multiplier identities based on the consistency relation.8). Hence, P satisfies the three-term equation 5.7). Conversely, let us assume that the function P satisfies the three-term equation 5.7) on C R. We have to show that f attached by 5.8) is indeed nearly periodic. Applying the three-term equation to P in and ST 1 +1 we obtain: P [ v 1 + T + T ]) v ] ν ν[ 1 ST ) [ P ) + P v T ) + P ] v T ) vst ) 1 + 1) ν 1 ν ν [ P 1 ) + P v + 1 T 1 ) ν + P v + 1 T 1 ) ] ν + 1 [ P z) + vt ) 1 P + 1) + vt ) 1 + 1) ν 1 P ) ] + 1 [ vst ) 1 + 1) ν 1 P 1 ) + vt ) 1 P ) + [ + 1 ) ν vt ) P )] ) ν 1 P ) + vst ) 1 vt ) 1 + 1) ν 1 P 1) ] + 1 [ + vt ) 1 P + 1) + vst ) 1 + 1) ν 1 P 1 ) ] + 1 [ + vt ) 1 + 1) ν 1 P T ) vst ) 1 vt ) 1 + 1) ν 1 P ) ] + 1 [ P ) + vs) 1 z ν 1 P 1) ] [ + vt ) 1 P T ) + vs) 1 T ) ν 1 P 1) ] + T

24 4 T. MÜHLENBRUCH AND W. RAJI c ±f) + vt ) 1 c ±f + 1) for Im ) ), using again multiplier identities derived from the consistency relation.8). This shows that if P satisfies the three-term equation then is f nearly periodic. 6. Period Functions 6.1. Period Functions by Integral Transforms. We follow [1,.3], which is an extension of [11, Chapter II, ] to real weights, and define the following integral transformation of a Maass cusp form. Definition 6.1. Let, ) and ν C, and let v and k 1 Z be compatible multiplier and weight. Let u be a Maass cusp form of weight k, multiplier v and eigenvalue 1 4 ν. We associate a function P k,ν :, ) C; P k,ν ) to the cusp form u by the integral transform 6.1) P k,ν ) R k,ν, ), u ) z) where the path of integration is the upper imaginary axis, i.e., the geodesic connecting and i. The integral transform above is well defined, as the following arguments show. Let, ) and consider the function R k,ν z, ). The construction of R k, ) implies polynomial growth for Im z) and Im z). The Maass cusp form u decays quicker than any polynomial at cusps, see Definition 3.1. Hence, the integral R k,ν, ), u ) z) is well defined. Remark 6.. The definition of P k,ν in [1, Definition 41] is P k,ν ) η k u, R k,ν, ) ) z) which seems to differ from the one we use in 6.1). However, u and R k,ν are eigenfunctions of k and k respectively. This implies that the Maass- Selberg form is closed, see Lemma 4.1, and we have R k,ν, ), u ) z) η k u, R k,ν, ) ) z) + d u ) R k,ν, ) ). Due to u being cuspidal, and hence vanishing in and i, we have d u ) R k,ν, ) ).

25 EICHLER INTEGRALS FOR MAASS CUSP FORMS OF HALF-INTEGRAL WEIGHT5 Hence, the definition of P k,ν in 6.1) and in [1, Definition 41] agree. This also shows that the choice mentioned in Remark 5.1 does not matter for the period functions. Lemma 6.3. Let k, v, ν and u as in Definition 6.1, and let, ) and γ SL, Z) such that µγ, ) > and γ, ), ). The function P k,ν defined in 6.1) satisfies 6.) Pk,ν v ν γ) ) γ 1 γ 1 R k,ν, ), u ) z) where the path of integration is the geodesic connecting γ 1 and γ 1. Proof. We have Pk,ν v ν γ) ) vγ) 1 µγ) ) ν 1 γ 1 γ 1 R k,ν, γ), u ) z) R k,ν, ), u ) γ 1 z) using Lemma 4.13 R k,ν, ), u ) z). The use of Lemma 4.13 is valid since and µγ, ) are both positive real values. The path of integration of the last integral is the geodesic connecting γ 1 and γ 1 and lies in the upper left quadrant {z C; Re z), Im z) } of C. We show next that P k,ν satisfies the three-term equation on R +. Lemma 6.4. Let ν, k and v as in Definition 6.1 and u a Maass cusp form with weight k compatible multiplier v and eigenvalue 1 4 ν. The function P k,ν satisfies the three-term equation 6.3) P k,ν v ν 1 T T ) on, ). Proof. Let >. Lemma 6.3 allows us to write 1 ) R k,ν, ), u ) z) 1 P k,ν ) vt ) 1 P k,ν T ) vt ) 1 + 1) ν 1 P k,ν T ) P k,ν v ν 1 T T ) ). The next step is to extend P k,ν ) to the right half plane { C; Re ) > }. Let be in this right half-plane and recall that R k,ν z, ) is holomorphic in if Re z). Hence, The function P k,ν ), given by the integral transform 6.1) extends holomorphically to { C; Re ) > }. It is easily

26 6 T. MÜHLENBRUCH AND W. RAJI ) checked that P k,ν + 1) and P k,ν +1 have also holomorphic extensions to this right half-plane. The last step is to extend P k,ν to the cut plane C C, ]. Assume Re ) > for the moment. Since the differential form η k u, R k,ν, ) ) is closed, see Lemma 4.1, we replace vertical path of integration in 6.1) by a path which connects and i in the upper left quadrant and which passes to the left of either or. We then may move to any point for which either or is still right of the new integration path. This procedure extends P k,ν to a holomorphic function on C. Summarizing we have Theorem 6.5. Under the assumptions of Definition 6.1, the function P k,ν associated to u by 6.1) extends to a holomorphic function on the cut plane C which satisfies the three-term equation 6.3) on R >. It also satisfies the growth conditions 6.4) P k,ν ) { O z max{,reν) 1) as Im z), and O z min{,reν) 1) as Im z),. Proof. The first part of the proposition follows from the discussion above. The cusp form u is bounded on H since a cusp form vanishes in all cusps Q i and u is real-analytic on H. Also, E + k u is bounded since the Maass operator maps cusp forms of weight k to cusp forms of weight k +. Applying successively 6.1), 4.15), 4.1), 4.5) and 4.4) we find P k,ν ) i [ E + k R k,ν, ) ) z) uz) dz y R k,νz, ) E k u) z) d z ] y [ 1 ν k) R k,ν z, ) uz) dz y R k,νz, ) E k u) z) d z y ) 1 e ikarg iy) y ν iy) + iy) [ 1 ν k) e iarg iy) uiy) E k u) iy) ] dy y for >. Using the notation fz) gz) for fz) O gz)), we find the estimate 1 P k,ν ) y Reν) 6.5) + y { max E k u) iy) } dy, 1 ν k) uiy) y. ]

27 EICHLER INTEGRALS FOR MAASS CUSP FORMS OF HALF-INTEGRAL WEIGHT7 for >. The integral converges since u and hence uiy) and E k u) iy) are of quick decay as y and as y. Using the estimate y + y y in 6.5) gives P k,ν ) O Reν) 1) for every >. We have P k,ν ) O 1) for every > if we use y + y y 1 in 6.). This proves the stated growth condition. 6.. Period Functions and Nearly Periodic Functions. Let s start with a Maass cusp form u of weight k, multiplier v and spectral value ν as in Definition 3.1. We associated in 5 a nearly periodic function f by the integral transform 5.1): 5.1) C R C; f) : R k,ν, ), u ) z) i Then, we attached a period-function P by 5.9): if H and η k R k,ν, ), ũ ) z) if H. 5.9) P f v ν 1 S) on H H ) which satisfies the three-term equation 5.7) P v ν 1 T T ) on H H ). On the other hand, we have the integral transformation 6.1) from the Maass cusp form u to the period function P k,ν : 6.1) P k,ν ) which satisfies the three-term equation R k,ν, ), u ) z) on R > ) 6.3) P v ν 1 T T ) on R > ) and extends to C Theorem 6.5). Are both directions compatible? In other words, do we get the same function P on H H, regardless of using the intermediate periodic function via 5.1) and 5.9) of taking the formula 6.1)?

28 8 T. MÜHLENBRUCH AND W. RAJI Lemma 6.6. Let k, v, ν and u as in Definition 6.1 with additionally Re ν) < 1. The maps u 5.1) f 5.9) P and u 6.1) P k,ν give rise to the same function P P k,ν on { C; Re ) >, Im ) }. Proof. For H with Re ) > in the upper half-plane, we find P k,ν ) R k,ν, ), u ) z) R k,ν, ), u ) z) + R k,ν, ), u ) z) R k,ν, ), u ) z) S 1 i f) vs) 1 ν 1 fs ) using Lemma 5.5 P ). A similar calculation holds for H with Re ) > : P k,ν ) R k,ν, ), u ) z) η k u, R k,ν, ) ) z) using Lemma 4.16 η k u, R k,ν, ) ) z) η k u, R k,ν, ) ) z) f) vs) 1 ν 1 fs ) using Lemma 5.5 P ). Theorem 6.7. The function P given by 5.9) on H H is holomorphic, extends holomorphically to the cut-plane C C \, ], satisfies the threeterm-equation P v 1 T T ) on C, and satisfies the growth condition 6.4). ν Proof. Lemma 6.6 shows that P agrees on { C; Re ) >, Im ) } with P k,ν given by 6.1). The latter extends holomorphically to C and satisfies the growth condition 6.4) by Proposition Period Functions and Period Polynomials In the following section we compare the integral transformation 6.1) and the classical Eichler integral in 1.1) for holomorphic cusp forms. Let u h be a modular cusp form of weight k N as defined in the introduction. We attach a Maass cusp form u : H C to u h by 7.1) uz) : Im z) k u h z).

29 EICHLER INTEGRALS FOR MAASS CUSP FORMS OF HALF-INTEGRAL WEIGHT9 As shown in [13, 3.], u is indeed a Maass ) cusp form of weight k, trivial multiplier v 1 and eigenvalue k 1 k. Hence, u has spectral values ν { k 1, } 1 k. The following proposition compares the period functions attached to u and the period polynomial attached to u h. It is based on [1, Proposition 49]. Proposition 7.1. Let u be the Maass cusp form in 7.1), which is derived from a modular cusp form u h of weight k N. 1) The function P k, 1 k associated to u by 6.1) vanishes everywhere. ) The function P k, k 1 associated to u by 6.1) restricted to the right half-plane { C; Re ) > } is a multiple of the period polynomial p associated to u h by 1.1): P k, k 1 k) p. Proof. Since ) ) k 1 1 k k 1 k ), we see that k 1 are spectral values of u. Moreover, E k u as shown in [13, 3.]. Let P k,ν be the period function associated to u via 6.1): and 1 k P k,ν ) R k,ν, ), u)z). Using 4.15) and 4.1) we then get [ E + P k,ν ) k R k,ν, ) ) z) uz) dz y R k,νz, ) E k u) z) d z ]. y Recalling E + k R k,ν 1 ν k)r k,ν in 4.5) and E k u above, we find 7.) P k,ν ) 1 ν k) R k,ν z, ) uz) dz y for Re ) >. To prove the first part of the Proposition, we assume ν 1 k. Then, the factor 1 ν k in 7.) vanishes, implying P k, 1 k. To prove the second part of the Proposition, we assume ν k 1. By 4.) we have ) k ) k z Im z) R k, k 1 z, ) 7.3) z z) z) z) k Im z) k

30 3 T. MÜHLENBRUCH AND W. RAJI for every and z with z, z R. Combining this with 7.1) in 7.) gives P k, k 1 ) k) for at least all in the right half-plane. z) k u h z)dz k) p) Can we also recover the periodic function f h? The answer is given in the following proposition. Proposition 7.. Let u be the Maass cusp form in 7.1), which is derived from a modular cusp form u h of weight k N. 1) The integral transformation 5.1) defining f) for H is welldefined for both spectral values ν { 1 k, } k 1. ) The function f associated to u by 5.1) with weight k and spectral value ν 1 k vanishes everywhere. 3) The function f associated to u by 5.1) with weight k and spectral value ν k 1 and restricted to the upper half-plane H is a multiple of f h associated to u h by 1.3): f k) f h. Proof. We follow the arguments of the proof of Proposition 7.1. For H is the nearly periodic function f associated to u given by 5.1). Using 4.15) and 4.1) we then get f) [ E + k R k,ν, ) ) z) uz) dz y R k,νz, ) E k u) z) d z y Recalling E + k R k,ν 1 ν k)r k,ν in 4.5) and E k u, we find 7.4) f) 1 ν k) R k,ν z, ) uz) dz y for H. To prove the second part of the Proposition, we assume ν 1 k. Then, the factor 1 ν k in 7.4) vanishes, implying P k, 1 k. To prove the third part of the Proposition, we assume ν k 1. Using 7.3) and 7.1) in 7.4) gives f) k) z) k u h z)dz k) f h ) for every H. The first part follows also from above calculations. Even if the calculations above are a priori formal, the well-definiteness of the results show that the original integral transforms are also well defined. We are just adding cleverly zeros. ].

31 EICHLER INTEGRALS FOR MAASS CUSP FORMS OF HALF-INTEGRAL WEIGHT31 8. Discussion and Outlook In this paper, we introduced and discussed Eichler integrals attached to Maass cusp forms of half-integral weight. We also introduced the corresponding period functions. This generalizes on one hand the classical case of period polynomials and periodic functions associated to holomorphic modular cusp forms, as shown in 7. On the other hand, our results fit neatly with the also known case of Maass cusp forms of weight and associated periodic and period functions, discussed in [11]. Obvious remaining questions are 1) For half-integral weight, do the period functions i.e., the space of holomorphic solutions of the three-term equation 5.7) which satisfy the growth condition 6.4)) bijectively correspond to Maass cusp forms? We only show one direction. ) Can we use the introduced period functions to describe a Eichler- Shimura-cohomology for the half-integral or real weight case? 3) Does everything also hold for real or complex weights and/or noncuspidal forms? For example, can we extend the results to the general Maass wave forms introduced in [13]? 4) What can we say about Eichler-Shimura theory of harmonic Maass wave forms? 5) How are period functions and L-series related. The first question is positively answered for Maass cusp forms of weight in [11] and for real weights in [1]. Also, Bruggeman, Lewis and Zagier discuss recently the case of Maass forms of weight and of polynomial growth in the cusps) and is associated group cohomology in [3]. Deitmar and Hilgert discuss the situation for subgroups of finite index and weight in SL, Z) in [5]. A recent result by Deitmar discusses the situation for Maass wave forms of higher order in [4]. To our knowledge, the second and third question are still open for general Maass wave forms with complex weight. Our results extend trivially to the case of Maass cusp forms with real weight by just replacing half-integer with real everywhere). The third question is also positively answered in [8] for generalized modular forms introduced in [7]). The fourth question is answered in [1]. They show an Eichler-Shimura-type result for harmonic Maass wave forms, see e.g. [1, Theorem 1.]. The last question is also discussed in [1], generalizing the first part of [11]. Acknowledgements The authors would like to thank the referee for his excellent recommendations. In addition, the authors would like to thank the Center for Advanced Mathematical Sciences CAMS) at the American University of Beirut for the

32 3 T. MÜHLENBRUCH AND W. RAJI support. The first-named author would like to express his gratitude towards CAMS for supporting his visit. References [1] K. Bringmann, P. Guerzhoy, Z. Kent, and K. Ono, Eichler-Shimura theory for mock modular forms, Math. Ann ), [] R.W. Bruggeman, Families of automorphic forms, Monographs in Mathematics 88, Birkhäuser, [3] R.W. Bruggeman, J. Lewis and D. Zagier, Period functions for Maass wave forms II: Cohomology, Preprint. [4] A. Deitmar, Lewis-Zagier correspondence for higher-order forms, Pacific J. Math ), [5] A. Deitmar and J. Hilgert, A Lewis correspondence for submodular groups, Forum Math. 19 7), [6] M. Eichler, Eine Verallgemeinerung der Abelschen Integrale, Math. Zeitschrift ), [7] M. Knopp and G. Mason, Generalized modular forms, Journal of Number Theory 99 3), [8] M. Knopp and W. Raji, Eichler Cohomology and generalized modular forms II, International Journal of Number Theory 6 1), [9] W. Kohnen and D. Zagier, Modular forms with rational periods, in Modular forms Durham, 1983), , Ellis Horwood Ser. Math. Appl. Statist. Oper. Res., Horwood, Chichester, [1] S. Lang, Introduction to modular forms, Springer Verlag, Third printing, 1. [11] J. Lewis and D. Zagier, Period functions for Maass wave forms. I, Annals of Mathematics, Second Series 153 1), [1] T. Mühlenbruch, Systems of automorphic forms and period functions. PhD-Thesis, Utrecht University, 3. [13] T. Mühlenbruch and W. Raji, Generalized Maass Wave forms, Accepted in Proceedings of AMS. [14] D. Zagier, Periods of modular forms and Jacobi theta functions, Invent. math ), [15] D. Zagier, Introduction to modular forms, in From Number Theory to Physics, eds. M. Waldschmidt et al, Springer-Verlag, Heidelberg 199), Department of Mathematics and Computer Science, FernUniversität in Hagen, 5884 Hagen, Germany address: tobias.muehlenbruch@fernuni-hagen.de Department of Mathematics, American University of Beirut, Beirut and fellow at Center for Advanced Mathematical Sciences, Beirut, Lebanon. address: wr7@aub.edu.lb