{
  "id": "1407.5707",
  "version": "v2",
  "published": "2014-07-22T01:52:20.000Z",
  "updated": "2016-06-08T05:32:07.000Z",
  "title": "The Geometry of Hida Families I: $Λ$-adic de Rham cohomology",
  "authors": [
    "Bryden Cais"
  ],
  "comment": "This article is a revised version of part of arXiv:1209.0046",
  "categories": [
    "math.NT"
  ],
  "abstract": "We construct the $\\Lambda$-adic de Rham analogue of Hida's ordinary $\\Lambda$-adic \\'etale cohomology and of Ohta's $\\Lambda$-adic Hodge cohomology, and by exploiting the geometry of integral models of modular curves over the cyclotomic extension of $\\mathbf{Q}_p$, we give a purely geometric proof of the expected finiteness, control, and $\\Lambda$-adic duality theorems. Following Ohta, we then prove that our $\\Lambda$-adic module of differentials is canonically isomorphic to the space of ordinary $\\Lambda$-adic cuspforms. In the sequel to this paper, we construct the crystalline counterpart to Hida's ordinary $\\Lambda$-adic \\'etale cohomology, and employ integral $p$-adic Hodge theory to prove $\\Lambda$-adic comparison isomorphisms between all of these cohomologies. As applications of our work in this paper and the sequel, we will be able to provide a \"cohomological\" construction of the family of $(\\varphi,\\Gamma)$-modules attached to Hida's ordinary $\\Lambda$-adic \\'etale cohomology by the work of Dee, as well as a new and purely geometric proof of Hida's finitenes and control theorems. We are also able to prove refinements of theorems of Mazur-Wiles and of Ohta.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-07-22T01:52:20.000Z",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2016-06-08T05:32:07.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11F33",
      "11F67",
      "11G18",
      "11R23"
    ],
    "keywords": [
      "adic etale cohomology",
      "rham cohomology",
      "hida families",
      "hidas ordinary",
      "purely geometric proof"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1407.5707C"
    }
  }
}