{
  "id": "math/0506461",
  "version": "v2",
  "published": "2005-06-22T20:21:29.000Z",
  "updated": "2005-11-30T15:49:56.000Z",
  "title": "Variation in the number of points on elliptic curves and applications to excess rank",
  "authors": [
    "Steven J. Miller"
  ],
  "comment": "14 pages (6 pages are appendices of calculations for additional families). Updated version with some minor typos corrected. A shorter version (minus the appendices) will appear in Comptes Rendus Mathematiques",
  "journal": "C. R. Math. Rep. Acad. Sci. Canada 27 (2005), no. 4, 111--120",
  "categories": [
    "math.NT"
  ],
  "abstract": "Michel proved that for a one-parameter family of elliptic curves over Q(T) with non-constant j(T) that the second moment of the number of solutions modulo p is p^2 + O(p^{3/2}). We show this bound is sharp by studying y^2 = x^3 + Tx^2 + 1. Lower order terms for such moments in a family are related to lower order terms in the n-level densities of Katz and Sarnak, which describe the behavior of the zeros near the central point of the associated L-functions. We conclude by investigating similar families and show how the lower order terms in the second moment may affect the expected bounds for the average rank of families in numerical investigations.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2005-11-30T15:49:56.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G05",
      "11G40"
    ],
    "keywords": [
      "elliptic curves",
      "lower order terms",
      "excess rank",
      "applications",
      "second moment"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005math......6461M"
    }
  }
}