A class of non-holomorphic modular forms I

Abstract: This introductory paper studies a class of real analytic functions on the upper half plane satisfying a certain modular transformation property. They are not eigenfunctions of the Laplacian and are quite distinct from Maass forms. These functions are modular equivariant versions of real and imaginary parts of iterated integrals of holomorphic modular forms, and are modular analogues of single-valued polylogarithms. The coefficients of these functions in a suitable power series expansion are periods. They are related both to mixed motives (iterated extensions of pure motives of classical modular forms), as well as the modular graph functions arising in genus one string perturbation theory. In an appendix, we use weakly holomorphic modular forms to write down modular primitives of cusp forms. Their coefficients involve the full period matrix (periods and quasi-periods) of cusp forms.