{
  "id": "1606.06103",
  "version": "v1",
  "published": "2016-06-20T13:07:27.000Z",
  "updated": "2016-06-20T13:07:27.000Z",
  "title": "On $\\ell$-torsion in class groups of number fields",
  "authors": [
    "Jordan Ellenberg",
    "Lillian B. Pierce",
    "Melanie Matchett Wood"
  ],
  "comment": "25 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "For each integer $\\ell \\geq 1$, we prove an unconditional upper bound on the size of the $\\ell$-torsion subgroup of the class group, which holds for all but a zero-density set of field extensions of $\\mathbb{Q}$ of degree $d$, for any fixed $d \\in \\{2,3,4,5\\}$ (with the additional restriction in the case $d=4$ that the field be non-$D_4$). For sufficiently large $\\ell$ (specified explicitly), these results are as strong as a previously known bound that is conditional on GRH. As part of our argument, we develop a probabilistic \"Chebyshev sieve,\" and give uniform, power-saving error terms for the asymptotics of quartic (non-$D_4$) and quintic fields with chosen splitting types at a finite set of primes.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-06-20T13:07:27.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "class group",
      "number fields",
      "unconditional upper bound",
      "finite set",
      "zero-density set"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}