Scalar-valued depth two Eichler–Shimura integrals of cusp forms

Journal article, 2023

Given cusp forms (Formula presented.) and (Formula presented.) of integral weight (Formula presented.), the depth two holomorphic iterated Eichler–Shimura integral (Formula presented.) is defined by (Formula presented.), where (Formula presented.) is the Eichler integral of (Formula presented.) and (Formula presented.) are formal variables. We provide an explicit vector-valued modular form whose top components are given by (Formula presented.). We show that this vector-valued modular form gives rise to a scalar-valued iterated Eichler integral of depth two, denoted by (Formula presented.), that can be seen as a higher depth generalization of the scalar-valued Eichler integral (Formula presented.) 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 (Formula presented.) 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 (Formula presented.). This allows for effective computation of (Formula presented.).