{
  "id": "2009.09885",
  "version": "v1",
  "published": "2020-09-21T14:08:57.000Z",
  "updated": "2020-09-21T14:08:57.000Z",
  "title": "Multiple zeta values and iterated Eisenstein integrals",
  "authors": [
    "Alex Saad"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "Brown showed that the affine ring of the motivic path torsor $\\pi_1^{\\text{mot}}(\\mathbb{P}^1 \\backslash \\left\\{0,1,\\infty\\right\\}, \\vec{1}_0, -\\vec{1}_1)$, whose periods are multiple zeta values, generates the Tannakian category $\\mathsf{MT}(\\mathbb{Z})$ of mixed Tate motives over $\\mathbb{Z}$. Brown also introduced multiple modular values, which are periods of the relative completion of the fundamental group of the moduli stack $\\mathcal{M}_{1,1}$ of elliptic curves. We prove that all motivic multiple zeta values may be expressed as $\\mathbb{Q}[2 \\pi i]$-linear combinations of motivic iterated Eisenstein integrals along elements of $\\pi_1 (\\mathcal{M}_{1,1}) \\cong SL_2(\\mathbb{Z})$, which are examples of motivic multiple modular values. This provides a new modular generator for $\\mathsf{MT}(\\mathbb{Z})$. We also explain how the coefficients in this linear combination may be partially determined using the motivic coaction.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-09-21T14:08:57.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11M32",
      "11F57",
      "14F35",
      "18M25"
    ],
    "keywords": [
      "linear combination",
      "motivic multiple modular values",
      "motivic multiple zeta values",
      "motivic path torsor",
      "motivic iterated eisenstein integrals"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}