{
  "id": "1802.08661",
  "version": "v1",
  "published": "2018-02-23T18:04:54.000Z",
  "updated": "2018-02-23T18:04:54.000Z",
  "title": "The modular Cauchy kernel for the Hilbert modular surface",
  "authors": [
    "Nina Sakharova"
  ],
  "categories": [
    "math.AG"
  ],
  "abstract": "In this paper we construct the modular Cauchy kernel on the Hilbert modular surface $\\Xi_{\\mathrm{Hil},m}(z)(z_2-\\bar{z_2})$, i.e. the function of two variables, $(z_1, z_2) \\in \\mathbb{H} \\times \\mathbb{H}$, which is invariant under the action of the Hilbert modular group, with the first order pole on the Hirzebruch-Zagier divisors. The derivative of this function with respect to $\\bar{z_2}$ is the function $\\omega_m (z_1, z_2)$ introduced by Don Zagier in \\cite{Za1}. We consider the question of the convergence and the Fourier expansion of the kernel function. The paper generalizes the first part of the results obtained in the preprint \\cite{Sa}",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-02-23T18:04:54.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "hilbert modular surface",
      "modular cauchy kernel",
      "hilbert modular group",
      "first order pole",
      "hirzebruch-zagier divisors"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}