Multiple modular L-functions and modular iterated integrals

TL;DR

This paper generalizes the relationship between multiple modular L-functions and modular iterated integrals to include forms with nonzero constant terms, providing new proofs, functional equations, and explicit examples.

Contribution

It removes previous restrictions on modular forms, establishing a broader connection and extending prior results by Choie-Ihara and Brown.

Findings

Established the relationship for general modular forms with nonzero constant terms.

Proved a functional equation for modular iterated integrals.

Computed explicit examples validating the theoretical framework.

Abstract

The connection between multiple modular L-functions, as defined by Manin in [5], and modular iterated integrals was made explicit by Choie and Ihara [3] under the restrictive assumption that all modular forms involved have vanishing constant terms in their q-expansions. In this paper, we remove the assumption and establish the relationship between modular iterated integrals and multiple modular L-functions for general modular forms, including those with nonzero constant terms. We also provide a proof of a functional equation for modular iterated integrals, which is a specialization of a general result obtained by Brown [2]. This leads us to a generalization of the result of Choie-Ihara [3]. In the final part of the paper, we compute explicit examples of modular iterated integrals. These calculations essentially reproduce the explicit initial computations carried out by Brown [2], but they also serve to validate the broader framework developed in this work.