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) ,
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
vii
Previous Page Next Page