{
  "id": "2408.11473",
  "version": "v1",
  "published": "2024-08-21T09:40:08.000Z",
  "updated": "2024-08-21T09:40:08.000Z",
  "title": "Non-perfect pairings between Hecke algebra and modular forms over function fields",
  "authors": [
    "Cécile Armana"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "We study two analogs, for modular forms over $\\mathbb{F}_{q}(T)$, of the pairing between Hecke algebra and cusp forms given by the first coefficient in the expansion. For Drinfeld modular forms, the $\\mathbb{C}_{\\infty}$-pairing is provided by the first coefficient of their $t$-expansion at infinity. For $\\mathbb{Z}$-valued harmonic cochains, the $\\mathbb{Z}$-pairing is given by their Fourier coefficient with respect to the trivial ideal. We prove that, contrarily to classical cusp forms, both pairings in weight $2$ are not perfect in a quite general setting, namely for the congruence subgroup $\\Gamma_0(\\mathfrak{n})$ with any prime ideal $\\mathfrak{n}$ in $\\mathbb{F}_{q}[T]$ of degree $\\geq 5$. We show it by exhibiting a common element of the Hecke algebra in the kernels of both pairings and proving that it is non-zero using computations with modular symbols over $\\mathbb{F}_{q}(T)$. Finally we present computational data on other kernel elements of these pairings.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-08-21T09:40:08.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "hecke algebra",
      "function fields",
      "non-perfect pairings",
      "cusp forms",
      "first coefficient"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}