{
  "id": "1707.01485",
  "version": "v1",
  "published": "2017-07-05T17:34:47.000Z",
  "updated": "2017-07-05T17:34:47.000Z",
  "title": "On free resolutions of Iwasawa modules",
  "authors": [
    "Alexandra Nichifor",
    "Bharathwaj Palvannan"
  ],
  "comment": "39 pages. Comments welcome",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $\\Lambda$ (isomorphic to $\\mathbb{Z}_p[[T]]$) denote the usual Iwasawa algebra and $G$ denote the Galois group of a finite Galois extension $L/K$ of totally real fields. The main theorems in this article describe the precise conditions under which non-primitive Iwasawa modules, over the cyclotomic $\\mathbb{Z}_p$-extension, have a free resolution of length one over the group ring $\\Lambda[G]$. As one application, under these conditions of the main theorems, the validity of the non-commutative Iwasawa main conjecture allows us to find a representative for the non-primitive $p$-adic $L$-function (which is an element of a $K_1$-group) in a maximal $\\Lambda$-order. As another application, we consider an elliptic curve over $\\mathbb{Q}$ with a cyclic isogeny of degree $p^2$. We relate the characteristic ideal, in the ring $\\Lambda$, of the Pontryagin dual of its non-primitive Selmer group to two characteristic ideals, viewed as elements of group rings over $\\Lambda$, associated to two non-primitive classical Iwasawa modules.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-07-05T17:34:47.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11R23",
      "11R34",
      "11S25"
    ],
    "keywords": [
      "iwasawa modules",
      "free resolution",
      "characteristic ideal",
      "main theorems",
      "usual iwasawa algebra"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 39,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}