{
  "id": "2312.03570",
  "version": "v1",
  "published": "2023-12-06T16:02:31.000Z",
  "updated": "2023-12-06T16:02:31.000Z",
  "title": "Theta-Induced Diffusion on Tate Elliptic Curves over Non-Archimedean Local Fields",
  "authors": [
    "Patrick Erik Bradley"
  ],
  "comment": "28 pages",
  "categories": [
    "math.NT",
    "math.AG"
  ],
  "abstract": "A diffusion operator on the $K$-rational points of a Tate elliptic curve $E_q$ is constructed, where $K$ is a non-archimedean local field, as well as an operator on the Berkovich-analytification $E_q^{an}$ of $E_q$. These are integral operators for measures coming from a regular $1$-form, and kernel functions constructed via theta functions. The second operator can be described via certain non-archimedan curvature forms on $E_q^{an}$. The spectra of these self-adjoint bounded operators on the Hilbert spaces of $L^2$-functions are identical and found to consist of finitely many eigenvalues. A study of the corresponding heat equations yields a positive answer to the Cauchy problem, and induced Markov processes on the curve. Finally, some geometric information about the $K$-rational points of $E_q$ is retrieved from the spectrum.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-12-06T16:02:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14H52",
      "58J35"
    ],
    "keywords": [
      "non-archimedean local field",
      "tate elliptic curve",
      "theta-induced diffusion",
      "rational points",
      "corresponding heat equations yields"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}