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

Abstract

Abstract Given cusp forms f and g of integral weight k⩾2, the depth two holomorphic iterated Eichler–Shimura integral If,g is defined by ∫τi∞f(z)(X−z)k−2Ig(z;Y)dz, where Ig 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 If,g. We show that this vector‐valued modular form gives rise to a scalar‐valued iterated Eichler integral of depth two, denoted by Ef,g, that can be seen as a higher depth generalization of the scalar‐valued Eichler integral Ef 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 Ef,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 Δ. This allows for effective computation of Ef,g.