{
  "id": "1402.3850",
  "version": "v2",
  "published": "2014-02-16T22:26:49.000Z",
  "updated": "2015-01-05T23:09:35.000Z",
  "title": "Modular symbols in Iwasawa theory",
  "authors": [
    "Takako Fukaya",
    "Kazuya Kato",
    "Romyar Sharifi"
  ],
  "comment": "43 pages",
  "journal": "Iwasawa Theory 2012 - State of the Art and Recent Advances, Springer, 2014, 177-219",
  "categories": [
    "math.NT",
    "math.AG"
  ],
  "abstract": "This survey paper is focused on a connection between the geometry of $\\mathrm{GL}_d$ and the arithmetic of $\\mathrm{GL}_{d-1}$ over global fields, for integers $d \\ge 2$. For $d = 2$ over $\\mathbb{Q}$, there is an explicit conjecture of the third author relating the geometry of modular curves and the arithmetic of cyclotomic fields, and it is proven in many instances by the work of the first two authors. The paper is divided into three parts: in the first, we explain the conjecture of the third author and the main result of the first two authors on it. In the second, we explain an analogous conjecture and result for $d = 2$ over $\\mathbb{F}_q(t)$. In the third, we pose questions for general $d$ over the rationals, imaginary quadratic fields, and global function fields.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-02-16T22:26:49.000Z",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2015-01-05T23:09:35.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11F67",
      "11R23",
      "11R34",
      "11G16",
      "11R58",
      "11Fxx",
      "11Gxx",
      "11Rxx"
    ],
    "keywords": [
      "iwasawa theory",
      "modular symbols",
      "third author",
      "global function fields",
      "imaginary quadratic fields"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 43,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1402.3850F"
    }
  }
}