Modular iterated integrals associated with cusp forms

TL;DR

This paper introduces a new method for constructing modular iterated integrals involving cusp forms, extending higher-order modular forms to general Fuchsian groups, with potential broad applications.

Contribution

It develops an explicit family of modular iterated integrals involving cusp forms and extends higher-order modular forms to all Fuchsian groups of the first kind.

Findings

Provides a new construction of invariant iterated integrals

Extends higher-order modular forms to general Fuchsian groups

Offers a framework applicable beyond classical modular groups

Abstract

We construct an explicit family of modular iterated integrals which involves cusp forms. This leads to a new method of producing "invariant versions" of 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.