{
  "id": "2105.08125",
  "version": "v2",
  "published": "2021-05-17T19:30:10.000Z",
  "updated": "2021-08-25T14:35:19.000Z",
  "title": "Every finite abelian group is the group of rational points of an ordinary abelian variety over $\\mathbb{F}_2$, $\\mathbb{F}_3$ and $\\mathbb{F}_5$",
  "authors": [
    "Stefano Marseglia",
    "Caleb Springer"
  ],
  "comment": "7 pages. The title has been slightly changed to reflect the additional results now proved in the paper",
  "categories": [
    "math.NT"
  ],
  "abstract": "We show that every finite abelian group $G$ occurs as the group of rational points of an ordinary abelian variety over $\\mathbb{F}_2$, $\\mathbb{F}_3$ and $\\mathbb{F}_5$. We produce partial results for abelian varieties over a general finite field $\\mathbb{F}_q$. In particular, we show that certain abelian groups cannot occur as groups of rational points of abelian varieties over $\\mathbb{F}_q$ when $q$ is large. Finally, we show that every finite cyclic group arises as the group of rational points of infinitely many simple abelian varieties over $\\mathbb{F}_2$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2021-08-25T14:35:19.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14K15",
      "14G15",
      "11G10"
    ],
    "keywords": [
      "finite abelian group",
      "ordinary abelian variety",
      "rational points",
      "finite cyclic group arises",
      "simple abelian varieties"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 7,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}