{
  "id": "1210.3526",
  "version": "v1",
  "published": "2012-10-12T14:18:24.000Z",
  "updated": "2012-10-12T14:18:24.000Z",
  "title": "Standard models of abstract intersection theory for operators in Hilbert space",
  "authors": [
    "Grzegorz Banaszak",
    "Yoichi Uetake"
  ],
  "comment": "22 pages",
  "categories": [
    "math.NT",
    "math.AG",
    "math.SP"
  ],
  "abstract": "For an operator in a possibly infinite-dimensional Hilbert space of a certain class, we set down axioms of an abstract intersection theory, from which the Riemann hypothesis regarding the spectrum of that operator follows. In our previous paper [BU] we constructed a GNS (Gelfand-Naimark-Segal) model of abstract intersection theory. In this paper we propose another model, which we call a standard model of abstract intersection theory. We show that there is a standard model of abstract intersection theory for a given operator if and only if the Riemann hypothesis and semi-simplicity hold for that operator. (For the definition of semi-simplicity of an operator in Hilbert space, see the definition in Introduction.) We show this result under a condition for a given operator which is much weaker than the condition in the previous paper. The operator satisfying this condition can be constructed by the method of automorphic scattering in [U]. Combining this with a result from [U], we can show that an Dirichlet $L$-function, including the Riemann zeta-function, satisfies the Riemann hypothesis and its all nontrivial zeros are simple if and only if there is a corresponding standard model of abstract intersection theory. Similar results can be proven for GNS models since the same technique of proof for standard models can be applied.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-10-12T14:18:24.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11M26",
      "47A10"
    ],
    "keywords": [
      "abstract intersection theory",
      "standard model",
      "riemann hypothesis",
      "possibly infinite-dimensional hilbert space",
      "semi-simplicity hold"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1210.3526B"
    }
  }
}