{
  "id": "2208.09622",
  "version": "v1",
  "published": "2022-08-20T07:19:07.000Z",
  "updated": "2022-08-20T07:19:07.000Z",
  "title": "A formal power series over a noncommutative Hecke ring and the rationality of the Hecke series for $GSp_4$",
  "authors": [
    "Fumitake Hyodo"
  ],
  "comment": "27 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "The present paper studies Hecke rings derived by the automorphism groups of certain algebras $L_p$ over the ring of $p$-adic integers. Our previous work considered the case where $L_p$ is the Heisenberg Lie algebra (of dimension 3) over the ring of $p$-adic integers. Although this Hecke ring is noncommutative, we showed that a formal power series with coefficients in this Hecke ring satisfies an identity similar to the rationality of the Hecke series for $GL_2$ due to E.~Hecke. In the present paper, we establish an analogous result in the case of the Heisenberg Lie algebra of dimension 5 over the ring of $p$-adic integers. In this case, our identity is similar to the rationality of the Hecke series for $GSp_4$, due to G.~Shimura.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-08-20T07:19:07.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20C08",
      "20G25",
      "20G30",
      "11F03"
    ],
    "keywords": [
      "formal power series",
      "hecke series",
      "noncommutative hecke ring",
      "rationality",
      "heisenberg lie algebra"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}