Scalar-valued depth two Eichler-Shimura Integrals of Cusp Forms

Abstract

Given cusp forms $f$ and $g$ of integral weight $k\ge 2$, the depth two holomorphic iterated Eichler-Shimura integral $I_{f,g}$ is defined by ${\int }_{\tau }^{i\infty }f(z)(X-z{)}^{k-2}{I}_g(z;Y)\mathrm{d}z$, where $I_g$ is the Eichler integral of $g$ and $X,Y$ are formal variables. We provide an explicit vector-valued modular form whose top components are given by $I_{f,g}$. We show that this vector-valued modular form gives rise to a scalar-valued iterated Eichler integral of depth two, denoted by ${\mathcal{E}}_{f,g}$, that can be seen as a higher-depth generalization of the scalar-valued Eichler integral ${\mathcal{E}}_f$ of depth one. As an aside, our argument provides an alternative explanation of an orthogonality relation satisfied by period polynomials originally due to Paşol-Popa. We show that ${\mathcal{E}}_{f,g}$ can be expressed in terms of sums of products of components of vector-valued Eisenstein series with classical modular forms after multiplication with a suitable power of the discriminant modular form $\Delta$. This allows for effective computation of ${\mathcal{E}}_{f,g}$.