{
  "id": "0709.2908",
  "version": "v1",
  "published": "2007-09-18T19:53:15.000Z",
  "updated": "2007-09-18T19:53:15.000Z",
  "title": "Three lectures on elliptic surfaces and curves of high rank",
  "authors": [
    "Noam D. Elkies"
  ],
  "comment": "14 pages; extended abstract for an invited lecture series at the 7/2007 Oberwolfach workshop ``Explicit Methods in Number Theory'', expanded from my lecture notes for publication in the Oberwolfach Reports",
  "categories": [
    "math.NT",
    "math.AG"
  ],
  "abstract": "Over the past two years we have improved several of the (Mordell-Weil) rank records for elliptic curves over Q and nonconstant elliptic curves over Q(t). For example, we found the first example of a curve E/Q with 28 independent points P_i in E(Q) (the previous record was 24, by R.Martin and W.McMillen 2000), and the first example of a curve over Q with Mordell-Weil group isomorphic with (Z/2Z) x Z^18 (the previous rank record for a curve with a 2-torsion point was 15, by Dujella 2002). In these lectures we give some of the background, theory, and computational tools that led to these new records and related applications. I Context and overview: the theorems of Mordell(-Weil) and Mazur; the rank problem; the approaches of Neron--Shioda and Mestre; elliptic surfaces and Neron specialization; fields other than Q. II Elliptic surfaces and K3 surfaces: the Mordell-Weil and Neron-Severi groups; K3 surfaces of high Neron-Severi rank and their moduli; an elliptic K3 surface over Q of Mordell-Weil rank 17. Some other applications of K3 surfaces of high rank and their moduli. III Computational issues, techniques, and results: slices of Niemeier lattices; finding and transforming models of K3 surfaces of high rank; searching for good specializations. Summary of new rank records for elliptic curves.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2007-09-18T19:53:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14J27",
      "11G05"
    ],
    "keywords": [
      "high rank",
      "elliptic surfaces",
      "rank record",
      "first example",
      "elliptic k3 surface"
    ],
    "tags": [
      "lecture notes"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0709.2908E"
    }
  }
}