{
  "id": "math/9407202",
  "version": "v1",
  "published": "1994-07-12T13:31:22.000Z",
  "updated": "1994-07-12T13:31:22.000Z",
  "title": "Automorphic forms and cubic twists of elliptic curves",
  "authors": [
    "Daniel Lieman"
  ],
  "comment": "17 pages",
  "categories": [
    "math.NT",
    "math.RT"
  ],
  "abstract": "This paper surveys the connection between the elliptic curve E_D: x^3 + y^3 = D and a certain metaplectic form on the cubic cover of GL(3) which has the property that its m,n^{th} Whittaker--Fourier coefficient is essentially the L--series of the curve E_{m^2n}. One may obtain information about the collective behavior the curves E_D by exploiting this connection; for example, one can prove: Theorem: Fix any prime p \\ne 3, and any congruence class c mod p. Then there are infinitely many D congruent to c mod p such that the curve E_D has no rational solutions. This paper is fairly self-contained; no prior knowledge of algebraic number theory, analytic number theory or metaplectic forms is assumed. On the other hand, this paper is a survey, no proofs are included.",
  "revisions": [
    {
      "version": "v1",
      "updated": "1994-07-12T13:31:22.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "elliptic curve",
      "cubic twists",
      "automorphic forms",
      "metaplectic form",
      "algebraic number theory"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "1994math......7202L"
    }
  }
}