{
  "id": "1609.02528",
  "version": "v1",
  "published": "2016-09-08T18:48:28.000Z",
  "updated": "2016-09-08T18:48:28.000Z",
  "title": "On the p-adic Birch and Swinnerton-Dyer conjecture for elliptic curves over number fields",
  "authors": [
    "Daniel Disegni"
  ],
  "comment": "28 pages, comments welcome",
  "categories": [
    "math.NT"
  ],
  "abstract": "We formulate a multi-variable p-adic Birch and Swinnerton-Dyer conjecture for p-ordinary elliptic curves A over number fields K. It generalises the one-variable conjecture of Mazur-Tate-Teitelbaum, who studied the case K=Q and the phenomenon of exceptional zeros. We discuss old and new theoretical evidence towards our conjecture and in particular we fully prove it, under mild conditions, in the following situation: K is imaginary quadratic, A=E_K is the base-change to K of an elliptic curve over the rationals, and the rank of A is either 0 or 1. The proof is naturally divided into a few cases. Some of them are deduced from the purely cyclotomic case of elliptic curves over Q, which we obtain from a refinement of recent work of Venerucci alongside the results of Greenberg-Stevens, Perrin-Riou, and the author. The only genuinely multi-variable case (rank 1, two exceptional zeros, three partial derivatives) is newly established here. Its proof generalises to show that the `almost-anticyclotomic' case of our conjecture is a consequence of conjectures of Bertolini-Darmon on families of Heegner points, and of (partly conjectural) p-adic Gross--Zagier and Waldspurger formulas in families.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-09-08T18:48:28.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G40"
    ],
    "keywords": [
      "swinnerton-dyer conjecture",
      "number fields",
      "exceptional zeros",
      "p-ordinary elliptic curves",
      "purely cyclotomic case"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}