{
  "id": "2207.09345",
  "version": "v1",
  "published": "2022-07-19T15:54:04.000Z",
  "updated": "2022-07-19T15:54:04.000Z",
  "title": "Local to global principles for homomorphisms of abelian schemes",
  "authors": [
    "Wojciech Gajda",
    "Sebastian Petersen"
  ],
  "categories": [
    "math.AG"
  ],
  "abstract": "Let $A$ and $B$ be abelian varieties defined over the function field $k(S)$ of a smooth algebraic variety $S/k.$ We establish criteria, in terms of restriction maps to subvarieties of $S,$ for existence of various important classes of $k(S)$-homomorphisms from $A$ to $B,$ e.g., for existence of $k(S)$-isogenies. Our main tools consist of Hilbertianity methods, Tate conjecture as proven by Tate, Zarhin and Faltings, and of the minuscule weights conjecture of Zarhin in the case, when the base field is finite.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-07-19T15:54:04.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "abelian schemes",
      "global principles",
      "homomorphisms",
      "minuscule weights conjecture",
      "smooth algebraic variety"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}