{
  "id": "1806.07214",
  "version": "v1",
  "published": "2018-06-19T13:33:41.000Z",
  "updated": "2018-06-19T13:33:41.000Z",
  "title": "Codimension two cycles in Iwasawa theory and elliptic curves with supersingular reduction",
  "authors": [
    "Antonio Lei",
    "Bharathwaj Palvannan"
  ],
  "comment": "57 Pages, comments welcome",
  "categories": [
    "math.NT"
  ],
  "abstract": "A recent result of Bleher, Chinburg, Greenberg, Kakde, Pappas, Sharifi and Taylor has initiated the topic of higher codimension Iwasawa theory. As a generalization of the classical Iwasawa main conjecture, they prove a relationship between analytic objects (a pair of Katz's $2$-variable $p$-adic $L$-functions) and algebraic objects (two \"everywhere unramified\" Iwasawa modules) involving codimension two cycles in a $2$-variable Iwasawa algebra. We prove an analogous result by considering the restriction to an imaginary quadratic field $K$ (where an odd prime $p$ splits) of an elliptic curve $E$, defined over $\\mathbb{Q}$, with good supersingular reduction at $p$. On the analytic side, we consider eight out of the ten pairs of $2$-variable $p$-adic $L$-functions in this setup (four of the five $2$-variable $p$-adic $L$-functions have been constructed by Loeffler and the fifth $2$-variable $p$-adic $L$-function is due to Hida). On the algebraic side, we consider modifications of fine Selmer groups over the $\\mathbb{Z}_p^2$-extension of $K$. We also provide numerical evidence, using algorithms of Pollack, towards a pseudo-nullity conjecture of Coates-Sujatha.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-06-19T13:33:41.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11R23",
      "11G05",
      "11G07",
      "11R34",
      "11S25"
    ],
    "keywords": [
      "supersingular reduction",
      "elliptic curve",
      "higher codimension iwasawa theory",
      "fine selmer groups",
      "imaginary quadratic field"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 57,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}