On the algebraic structure of iterated integrals of quasimodular forms
Nils Matthes
TL;DR
The paper determines the algebraic structure of iterated integrals of quasimodular forms for SL_2(Z). It shows that the smallest integration-closed extension ${\mathcal I}^{QM}$ is a polynomial algebra over the QM_*-algebra, generated by Lyndon words on a basis of $\mathbb{C}\cdot E_2 \oplus M_{\ast}$, via an explicit isomorphism ${\mathcal I}^{QM} \cong QM_{\ast}[Lyn({\mathcal B}^{*})]$, with a parallel result for modular forms ${\mathcal I}^{M}$. The arguments combine a DDMS-type linear-independence criterion, independence results for iterated integrals of Eisenstein and modular forms, and the Milnor–Moore/Radford shuffle-algebra framework to produce a canonical, infinite Lyndon-basis presentation. This provides a concrete, computable basis and situates iterated quasimodular integrals in a broad shuffle-algebra context, connecting to classical Eichler–Shimura-type structures and non-abelian cocycles in the appendix. The work yields explicit polynomial generators and paves the way for further arithmetic and geometric applications of quasimodular iterated integrals.
Abstract
We study the algebra $\mathcal{I}^{QM}$ of iterated integrals of quasimodular forms for $\operatorname{SL}_2(\mathbb{Z})$, which is the smallest extension of the algebra $QM_{\ast}$ of quasimodular forms, which is closed under integration. We prove that $\mathcal{I}^{QM}$ is a polynomial algebra in infinitely many variables, given by Lyndon words on certain monomials in Eisenstein series. We also prove an analogous result for the $M_{\ast}$-subalgebra $\mathcal{I}^{M}$ of $\mathcal{I}^{QM}$ of iterated integrals of modular forms.