{
  "id": "2110.02569",
  "version": "v3",
  "published": "2021-10-06T08:23:15.000Z",
  "updated": "2024-07-30T12:12:22.000Z",
  "title": "On the transcendence of special values of Goss $L$-functions attached to Drinfeld modules",
  "authors": [
    "Oğuz Gezmiş",
    "Changningphaabi Namoijam"
  ],
  "comment": "22 pages. The work in v2 of arXiv:2110.02569 has been divided into two papers: arXiv:2407.18916 and v3 of arXiv:2110.02569. The paper arXiv:2407.18916 includes generalizations of the results in section 4, section 5, and the appendix of v2 of arXiv:2110.02569",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $\\mathbb{F}_q$ be the finite field with $q$ elements and consider the rational function field $K:=\\mathbb{F}_q(\\theta)$. For a Drinfeld module $\\phi$ defined over $K$, we study the transcendence of special values of the Goss $L$-function attached to the abelian $t$-motive $M_{\\phi}$ of $\\phi$. Moreover, when $\\phi$ is a Drinfeld module of rank $r\\geq 2$ defined over $K$ which has everywhere good reduction, we prove that the value of the Goss $L$-function attached to the $(r-1)$-st exterior power of $M_{\\phi}$ at any positive integer is transcendental over $K$.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2024-07-30T12:12:22.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G09",
      "11M38",
      "11J93"
    ],
    "keywords": [
      "drinfeld module",
      "special values",
      "transcendence",
      "rational function field",
      "st exterior power"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}