{
  "id": "1307.4261",
  "version": "v3",
  "published": "2013-07-16T12:49:19.000Z",
  "updated": "2014-03-05T01:28:51.000Z",
  "title": "Selmer groups and class groups",
  "authors": [
    "Kestutis Cesnavicius"
  ],
  "comment": "17 pages; final version, to appear in Compositio Mathematica",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $A$ be an abelian variety over a global field $K$ of characteristic $p \\ge 0$. If $A$ has nontrivial (resp. full) $K$-rational $l$-torsion for a prime $l \\neq p$, we exploit the fppf cohomological interpretation of the $l$-Selmer group $\\mathrm{Sel}_l A$ to bound $\\#\\mathrm{Sel}_l A$ from below (resp. above) in terms of the cardinality of the $l$-torsion subgroup of the ideal class group of $K$. Applied over families of finite extensions of $K$, the bounds relate the growth of Selmer groups and class groups. For function fields, this technique proves the unboundedness of $l$-ranks of class groups of quadratic extensions of every $K$ containing a fixed finite field $\\mathbb{F}_{p^n}$ (depending on $l$). For number fields, it suggests a new approach to the Iwasawa $\\mu = 0$ conjecture through inequalities, valid when $A(K)[l] \\neq 0$, between Iwasawa invariants governing the growth of Selmer groups and class groups in a $\\mathbb{Z}_l$-extension.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2014-03-05T01:28:51.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G10",
      "11R23",
      "11R29",
      "11R58"
    ],
    "keywords": [
      "selmer group",
      "ideal class group",
      "global field",
      "iwasawa invariants",
      "number fields"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1307.4261C"
    }
  }
}