Bruggeman Res Math Sci (2018) 5:5
Holomorphic modular forms and cocycles
Universiteit Utrecht, Postbus
80010, 3508 TA Utrecht, The
This is a slightly expanded version of my lecture at the conference Modular forms are
everywhere at Bonn, May 2017, taking into account remarks by Don Zagier and Shaul
Zemel, and suggestions of the referees.
The relation between modular forms and cohomology may be studied in several contexts:
Classical For modular cusp forms of even weight the Eichler integral determines
for a given modular cusp form of even positive weight a cohomology class
with values in a module of polynomial functions. The names to mention
are Eichler  and Shimura .
Real weight There are interesting holomorphic modular cusp forms of positive real
weight. For instance the Dedekind eta function
η(τ ) = e
1 − e
. One can assign cohomology classes to such cusp forms. The
values are in a much larger module. The name to mention is Knopp .
Modular Maass wave forms are functions on the upper half-plane that are
invariant under the transformation of the modular group. They are not
holomorphic, but satisfy a second-order linear diﬀerential equation: There
are inﬁnitely many linearly independent Maass cusp forms, although none
of them can be given explicitly for the full modular group.
There is a way to associate cohomology classes to Maass cusp form, with
values in a large module of functions on the real projective line P
can be generalized to cuspidal automorphic forms on larger Lie groups
(R). I mention Bunke and Ohlbrich [3,4].
Don Zagier, John Lewis and I looked at the relation between Maass forms and coho-
mology. We could associate cocycles to Maass forms with large growth at the cusps and
established a quite satisfactory theory . In the Eichler–Shimura theory the cohomol-
ogy groups have ﬁnite dimension. So a linear map from inﬁnite-dimensional spaces of
modular forms (with unrestricted growth at the cusp) has a huge kernel. In the context of
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