{
  "id": "math/0609459",
  "version": "v1",
  "published": "2006-09-16T05:55:32.000Z",
  "updated": "2006-09-16T05:55:32.000Z",
  "title": "Modular Curves, Modular Surfaces, and Modular Fourfolds",
  "authors": [
    "Dinakar Ramakrishnan"
  ],
  "comment": "15 pages, Proceedings of a conference on Algebraic Cycles and Motives (in honor of Jacob Murre) in Leiden, The Netherlands",
  "categories": [
    "math.NT",
    "math.AG"
  ],
  "abstract": "This article surveys some recent work of the author on Hilbert modular fourfolds X. After some preliminaries on the cohomology and special, codimension 2 cycles Z on X of Hirzebruch-Zagier type, a proof of the Tate conjecture for X over abelian fields is exposed. It is followed by a description of a method to construct classes in the Bloch's Chow group CH^3(X,1) by using Hecke translates of the cycles Z as above with suitable intersections in translates of modular curves. The article ends with the introduction of a modular Gersten complex for a general Shimura variety X and the corresponding groups CH^p_{mod}(X) and CH^p_{mod}(X,1).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2006-09-16T05:55:32.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11F41",
      "11G35",
      "14C25",
      "14C35",
      "14J35"
    ],
    "keywords": [
      "modular curves",
      "modular surfaces",
      "hilbert modular fourfolds",
      "blochs chow group",
      "modular gersten complex"
    ],
    "tags": [
      "conference paper"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math......9459R"
    }
  }
}