{
  "id": "1901.06894",
  "version": "v1",
  "published": "2019-01-21T12:01:20.000Z",
  "updated": "2019-01-21T12:01:20.000Z",
  "title": "L-series and isogenies of abelian varieties",
  "authors": [
    "Harry Smit"
  ],
  "comment": "20 pages, the notation section of this article overlaps partially with arXiv:1901.06198",
  "categories": [
    "math.NT"
  ],
  "abstract": "Faltings's isogeny theorem states that two abelian varieties are isogenous over a number field precisely when the characteristic polynomials of the reductions at almost all prime ideals of the number field agree. This implies that two abelian varieties over $\\mathbb{Q}$ with the same $L$-series are necessarily isogenous, but this is false over a general number field. Let $A$ and $A'$ be two abelian varieties, defined over number fields $K$ and $K'$ respectively. Our main result is that $A$ and $A'$ are isogenous after a suitable isomorphism between $K$ and $K'$ if and only if the Dirichlet character groups of $K$ and $K'$ are isomorphic and the $L$-series of $A$ and $A'$ twisted by the Dirichlet characters match.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-01-21T12:01:20.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G40",
      "11R37",
      "11R42",
      "14K02"
    ],
    "keywords": [
      "abelian varieties",
      "faltingss isogeny theorem states",
      "number field agree",
      "general number field",
      "dirichlet character groups"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}