{
  "id": "2502.03562",
  "version": "v1",
  "published": "2025-02-05T19:18:16.000Z",
  "updated": "2025-02-05T19:18:16.000Z",
  "title": "Multiplicativity of Fourier Coefficients of Maass Forms for SL($n,\\mathbb Z$)",
  "authors": [
    "Dorian Goldfeld",
    "Eric Stade",
    "Michael Woodbury"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "The Fourier coefficients of a Maass form $\\phi$ for SL$(n,\\mathbb Z)$ are complex numbers $A_\\phi(M)$, where $M=(m_1,m_2,\\ldots,m_{n-1})$ and $m_1,m_2,\\ldots ,m_{n-1}$ are nonzero integers. It is well known that coefficients of the form $A_\\phi(m_1,1,\\ldots,1)$ are eigenvalues of the Hecke algebra and are multiplicative. We prove that the more general Fourier coefficients $A_\\phi(m_1,\\ldots,m_{n-1})$ are also eigenvalues of the Hecke algebra and satisfy the multiplicativity relations $$A_\\phi\\big(m_1m_1',\\;m_2m_2', \\;\\ldots\\; m_{n-1}m_{n-1}'\\big) = A_\\phi\\big(m_1,m_2,\\ldots,m_{n-1})\\cdot A_\\phi(m_1',m_2',\\ldots,m_{n-1}'\\big)$$ provided the products $\\prod\\limits_{i=1}^{n-1} m_i$ and $\\prod\\limits_{i=1}^{n-1} m_i'$ are relatively prime to each other.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2025-02-05T19:18:16.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11F55",
      "11F72"
    ],
    "keywords": [
      "maass form",
      "hecke algebra",
      "general fourier coefficients",
      "eigenvalues",
      "multiplicativity relations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}