{
  "id": "2209.00488",
  "version": "v1",
  "published": "2022-09-01T14:17:09.000Z",
  "updated": "2022-09-01T14:17:09.000Z",
  "title": "Scalar-valued depth two Eichler-Shimura Integrals of Cusp Forms",
  "authors": [
    "Tobias Magnusson",
    "Martin Raum"
  ],
  "categories": [
    "math.NT"
  ],
  "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 $\\mathcal{E}_{f,g}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-09-01T14:17:09.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11F11",
      "11F30",
      "11F75"
    ],
    "keywords": [
      "cusp forms",
      "scalar-valued depth",
      "holomorphic iterated eichler-shimura integral",
      "explicit vector-valued modular form",
      "discriminant modular form"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}