Introduction

These notes proceed from the ideas and results of [21], in which Maass forms

for the full modular group were studied, but now treating arbitrary Fuchsian groups

and stressing the cohomological interpretation. They can be read independently

of [21].

The classical theory linking holomorphic automorphic forms to cohomology

starts with Eichler [10], and Shimura [31]. To an automorphic form F on the upper

half-plane with even weight k ≥ 2 one associates a cocycle with values in the space

of polynomial functions of degree at most k−2 by ψγ (t) =

z0

γ−1z0

(t−τ)k−2

F (τ) dτ,

with a base point z0 in the upper half-plane. If F is a cusp form, one can put the

base point at ∞. The coeﬃcients of the resulting polynomials are then values of

the L-function of F . All this has important number theoretical consequences. (See,

e.g., Manin [23].)

With the base point at ∞, the cocycle is, in the case of the modular group

SL2(Z), determined by its value on S =

(

0

1

−1

0

)

,

ψS(t) =

i∞

0

(t −

τ)k−2

F (τ) dτ ,

called the period function (or period polynomial) of F , and the condition of being

a cocycle is equivalent to the two functional equations

(1) ψ(t) +

tk−2

ψ

(

−

1)

t

= 0 , ψ(t) +

tk−2

ψ

(

1 −

1)

t

+ (1 −

t)k−2

ψ

(

1

1 − t

)

= 0 .

In this case, it is also known that the map assigning to a cusp form F (τ) the odd

part of the polynomial ψS is an isomorphism between the space of cusp forms of

weight k and the vector space of odd polynomials ψ(t) satisfying (1). An elementary

argument shows that this latter space can be characterized by a single functional

equation

ψ(t) = ψ(t + 1) +

tk−2

ψ(1 + 1/t) .

The starting point of [21] (see also the survey paper [20] and §2 of [33]) is the

observation that this functional equation is identical in form to the relation

(2) ψ(t) = ψ(t + 1) +

t−2s

ψ(1 + 1/t)

that occurred in the work of the second author [19], which gave a bijection between

the space of even Maass wave forms with spectral parameter s on the full modular

group and a class of holomorphic functions satisfying (2). Since (1) is just the

cocycle condition for SL2(Z), this immediately suggests the possibility of describing

Maass forms for arbitrary Fuchsian groups by an appropriate generalization of the

functional equation (2) having an interpretation in terms of cohomology.

The principal goal of these notes is to carry out this generalization by con-

structing explicit isomorphisms between, on the one hand, spaces of Maass wave

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