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

TL;DR

This paper introduces a higher-depth scalar-valued Eichler-Shimura integral for cusp forms, linking it to vector-valued modular forms and Eisenstein series, and provides methods for explicit computation.

Contribution

It constructs a scalar-valued depth two Eichler-Shimura integral from vector-valued modular forms and Eisenstein series, extending classical Eichler integrals to higher depth.

Findings

Explicit vector-valued modular form with top components as $I_{f,g}$

Expression of $ ext{E}_{f,g}$ in terms of Eisenstein series and modular forms

Provides an effective computational approach for $ ext{E}_{f,g}$

Abstract

Given cusp forms $f$ and $g$ of integral weight $k \geq 2$, 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 Pa\c{s}ol-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 $\mathcal{E}_{f,g}$.