Abstract
Abstract We construct an explicit family of modular iterated integrals which involves cusp forms. This leads to a new method of producing modular invariant functions based on iterated integrals of modular forms. The construction will be based on an extension of higher-order modular forms which, in contrast to the standard higher-order forms, applies to general Fuchsian groups of the first kind and, as such, is of independent interest.
Highlights
This paper deals with two classes of functions that have not been previously studied together, namely, modular iterated integrals and higher order modular forms. We show that they are interrelated in a way that key features of one of them can be elucidated through constructions in the other
We will define an explicit family of elements of MI′2 that will give an answer to Question
The standard higher-order modular forms become trivial in Γ0(1) because, as shown in [5], they are parametrised by weight 2 cusp forms which are trivial in SL2(Z). ( note that, in contrast to general iterated invariants, Mc(n−1)(O) = M (n−1)(O) because of the identity (γ − 1)(π − 1) =γ − (π − 1).)
Summary
This paper deals with two classes of functions that have not been previously studied together, namely, modular iterated integrals and higher order modular forms. A more explicit characterisation of the space MIl can be given in terms of Γ-invariant linear combinations of real and imaginary parts of iterated integrals of modular forms This is proven in the case of elements of MIl originating in Eisenstein series ([2]) and is conjectured to hold in general. Constructing such “invariant versions” of iterated integrals of modular forms is one of the important themes of [1], especially in relation to the applications to the theory of modular graph functions. This suggests a deeper relation between the two objects that are the subject of this paper
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