A class of non-holomorphic modular forms II : equivariant iterated Eisenstein integrals

TL;DR

This paper introduces a new family of real analytic modular forms constructed from iterated integrals of Eisenstein series, expanding the understanding of mixed modular forms with algebraic and analytic properties similar to classical forms.

Contribution

It presents a novel class of non-holomorphic modular forms built from iterated Eisenstein integrals, with explicit algebraic and expansion properties, advancing the theory of mixed modular forms.

Findings

The new forms form an algebra with properties analogous to classical modular forms.

They admit expansions involving rational numbers and single-valued multiple zeta values.

The first non-trivial examples are real analytic Eisenstein series.

Abstract

We introduce a new family of real analytic modular forms on the upper half plane. They are arguably the simplest class of `mixed' versions of modular forms of level one and are constructed out of real and imaginary parts of iterated integrals of holomorphic Eisenstein series. They form an algebra of functions satisfying many properties analogous to classical holomorphic modular forms. In particular, they admit expansions in $q, \overline{q}$ and $\log |q|$ involving only rational numbers and single-valued multiple zeta values. The first non-trivial functions in this class are real analytic Eisenstein series.