{
  "id": "2411.08269",
  "version": "v2",
  "published": "2024-11-13T00:46:12.000Z",
  "updated": "2024-11-29T02:03:42.000Z",
  "title": "Rings of Hilbert modular forms, computations on Hilbert modular surfaces, and the Oda-Hamahata conjecture",
  "authors": [
    "Adam Logan"
  ],
  "comment": "38 pages; version submitted to journal; only minor changes from previous version",
  "categories": [
    "math.NT",
    "math.AG"
  ],
  "abstract": "The modularity of an elliptic curve $E/\\mathbb Q$ can be expressed either as an analytic statement that the $L$-function is the Mellin transform of a modular form, or as a geometric statement that $E$ is a quotient of a modular curve $X_0(N)$. For elliptic curves over number fields these notions diverge; a conjecture of Hamahata asserts that for every elliptic curve $E$ over a totally real number field there is a correspondence between a Hilbert modular variety and the product of the conjugates of $E$. In this paper we prove the conjecture by explicit computation for many cases where $E$ is defined over a real quadratic field and the geometric genus of the Hilbert modular variety is $1$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2024-11-29T02:03:42.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14G35",
      "14Q10",
      "11G05",
      "11F41",
      "14Q25"
    ],
    "keywords": [
      "hilbert modular surfaces",
      "hilbert modular forms",
      "oda-hamahata conjecture",
      "hilbert modular variety",
      "elliptic curve"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 38,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}