{
  "id": "2506.17393",
  "version": "v1",
  "published": "2025-06-20T18:00:09.000Z",
  "updated": "2025-06-20T18:00:09.000Z",
  "title": "Extensions of Abelian Schemes and the Additive Group",
  "authors": [
    "Gabriel Ribeiro",
    "Zev Rosengarten"
  ],
  "comment": "40 pages, comments appreciated",
  "categories": [
    "math.AG",
    "math.NT"
  ],
  "abstract": "We compute extension sheaves of abelian schemes and of the additive group by the multiplicative group in the fppf topology. Our main results include a generalized and streamlined proof of the Barsotti--Weil formula, the vanishing of $\\underline{\\operatorname{Ext}}^2(A,\\mathbb{G}_m)$ for an abelian scheme $A$ over a general base, and a description of $\\underline{\\operatorname{Ext}}^1(\\mathbb{G}_a,\\mathbb{G}_m)$ in characteristic zero.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2025-06-20T18:00:09.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "abelian scheme",
      "additive group",
      "extension sheaves",
      "barsotti-weil formula",
      "fppf topology"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 40,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}