Manin Res Math Sci (2018) 5:2
Modular forms of real weights and
generalised Dedekind symbols
Yuri I. Manin
Mathematik, Bonn, Germany
In a previous paper, I have deﬁned non-commutative generalised Dedekind symbols
for classical PSL(2,
)-cusp forms using iterated period polynomials. Here I generalise
this construction to forms of real weights using their iterated period functions
introduced and studied in a recent article by R. Bruggeman and Y. Choie.
1 Introduction: generalised Dedekind symbols
The classical Dedekind symbol encodes an essential part of modular properties of the
Dedekind eta function and appears in many contexts seemingly unrelated to modular
forms (cf. [10,19]). Fukuhara in [6,7], and others ([1,5], gave an abstract deﬁnition of gen-
eralised Dedekind symbols with values in an arbitrary commutative group and produced
such symbols from period polynomials of PSL(2, Z)-modular forms of any even weight.
In the note , I have given an abstract deﬁnition of generalised Dedekind symbols for
the full modular group PSL(2, Z) taking values in arbitrary non-necessarily commutative
group and constructed such symbols from iterated versions of period integrals of modular
forms of integral weights considered earlier in [15,16].
In this article, I extend these constructions to cusp forms of real weights, studied in
particular in [3,11,12]. The essential ingredient here is furnished by the introduction of
iterated versions of their period integrals following .
1.1 Dedekind symbols
The Dedekind eta function is a holomorphic function of the complex variable z with
positive imaginary part given by
η(z) = e
(1 − e
It is a cusp form of weight 1/2, and from PSL(2, Z)-invariance of η(z)
it follows that
for any fractional linear transformation γ ∈ PSL(2, Z),
γ z :=
az + b
cz + d
with c > 0 we can obtain a rational number
The Author(s) 2018. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License
(http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium,
provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and
indicate if changes were made.