{
  "id": "1405.2643",
  "version": "v2",
  "published": "2014-05-12T06:54:46.000Z",
  "updated": "2015-01-07T08:08:19.000Z",
  "title": "On Nekovář's heights, exceptional zeros and a conjecture of Mazur-Tate-Teitelbaum",
  "authors": [
    "Kazim Büyükboduk"
  ],
  "comment": "31 pages, submitted. Major revision and reorganization. Most notably, we have added an appendix where we give a proof of a Rubin-style formula alluded to in the Abstract",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $E/\\mathbb{Q}$ be an elliptic curve which has split multiplicative reduction at a prime $p$ and whose analytic rank $r_{an}(E)$ equals one. The main goal of this article is to relate the second order derivative of the Mazur-Tate-Teitelbaum $p$-adic $L$-function $L_p(E,s)$ of $E$ to Nekov\\'{a}\\v{r}'s height pairing evaluated on natural elements arising from the Beilinson-Kato elements. Along the way, we extend a Rubin-style formula of Nekov\\'a\\v{r} (or in an alternative wording, correct another Rubin-style formula of his) to apply in the presence of exceptional zeros. Our height formula allows us, among other things, to compare the order of vanishing of $L_p(E,s)$ at $s=1$ to its (complex) analytic rank $r_{an}(E)$ assuming the non-triviality of the height pairing. This has consequences towards a conjecture of Mazur, Tate and Teitelbaum.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-05-12T06:54:46.000Z",
      "abstract": "Let $E/\\mathbb{Q}$ be an elliptic curve which has split multiplicative reduction at a prime $p$ and whose analytic rank $r$ equals one. The main goal of this article is to relate the second order derivative of the Mazur-Tate-Teitelbaum $p$-adic $L$-function $L_p(E,s)$ attached to $E$ to Nekov\\'{a}\\v{r}'s height pairing evaluated on natural elements arising from Beilinson-Kato elements. Our height formula allows us, among other things, to compare the order of vanishing of $L_p(E,s)$ at $s=1$ to its (complex) analytic rank $r$, assuming the non-triviality of the height pairing. This has strong consequences towards a conjecture of Mazur, Tate and Teitelbaum.",
      "comment": "20 pages, submitted",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2015-01-07T08:08:19.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G05",
      "11G07",
      "11G40",
      "11R23",
      "14G10"
    ],
    "keywords": [
      "exceptional zeros",
      "nekovářs heights",
      "conjecture",
      "mazur-tate-teitelbaum",
      "analytic rank"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 31,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1405.2643B"
    }
  }
}