{
  "id": "1808.05526",
  "version": "v1",
  "published": "2018-08-16T14:54:40.000Z",
  "updated": "2018-08-16T14:54:40.000Z",
  "title": "Newforms of half-integral weight: the minus space of S_{k+1/2}(Γ_0(8M))",
  "authors": [
    "Ehud Moshe Baruch",
    "Soma Purkait"
  ],
  "comment": "To appear in Israel Journal of Mathematics",
  "categories": [
    "math.NT"
  ],
  "abstract": "We compute generators and relations for a certain $2$-adic Hecke algebra of level $8$ associated with the double cover of $\\mathrm{SL}_2$ and a $2$-adic Hecke algebra of level $4$ associated with $\\mathrm{PGL}_2$. We show that these two Hecke algebras are isomorphic as expected from the Shimura correspondence. We use the $2$-adic generators to define classical Hecke operators on the space of holomorphic modular forms of weight $k+1/2$ and level $8M$ where $M$ is odd and square-free. Using these operators and our previous results on half-integral weight forms of level $4M$ we define a subspace of the space of half-integral weight forms as a common $-1$ eigenspace of certain Hecke operators. Using the relations and a result of Ueda we show that this subspace which we call the minus space is isomorphic as a Hecke module under the Ueda correspondence to the space of new forms of weight $2k$ and level $4M$. We observe that the forms in the minus space satisfy a Fourier coefficient condition that gives the complement of the plus space but does not define the minus space.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-08-16T14:54:40.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11F37",
      "11F12",
      "11F70"
    ],
    "keywords": [
      "adic hecke algebra",
      "half-integral weight forms",
      "define classical hecke operators",
      "fourier coefficient condition",
      "holomorphic modular forms"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}