On the algebraic structure of iterated integrals of quasimodular forms

Matthes, Nils

doi:10.2140/ant.2017.11.2113

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On the algebraic structure of iterated integrals of quasimodular forms

Nils Matthes

Vol. 11 (2017), No. 9, 2113–2130

DOI: 10.2140/ant.2017.11.2113

Abstract

We study the algebra $ℐ^{QM}$ of iterated integrals of quasimodular forms for ${S L}_{2} (ℤ)$ , which is the smallest extension of the algebra ${QM}_{*}$ of quasimodular forms which is closed under integration. We prove that $ℐ^{QM}$ 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 $M_{*}$ -subalgebra $ℐ^{M}$ of $ℐ^{QM}$ of iterated integrals of modular forms.

Keywords

quasimodular forms, iterated integrals

Mathematical Subject Classification 2010

Primary: 11F11

Secondary: 11F67

Milestones

Received: 8 November 2016

Revised: 15 June 2017

Accepted: 8 September 2017

Published: 2 December 2017

Authors

Nils Matthes
Max-Planck-Institut für Mathematik Bonn Germany

Vol. 11, No. 9, 2017