{
  "id": "1804.06840",
  "version": "v1",
  "published": "2018-04-18T17:59:11.000Z",
  "updated": "2018-04-18T17:59:11.000Z",
  "title": "The Mumford--Tate conjecture for products of abelian varieties",
  "authors": [
    "Johan Commelin"
  ],
  "comment": "34 pages. All comments are welcome!",
  "categories": [
    "math.AG"
  ],
  "abstract": "Let $X$ be a smooth projective variety over a finitely generated field $K$ of characteristic~$0$ and fix an embedding $K \\subset \\mathbb{C}$. The Mumford--Tate conjecture is a precise way of saying that certain extra structure on the $\\ell$-adic \\'etale cohomology groups of~$X$ (namely, a Galois representation) and certain extra structure on the singular cohomology groups of~$X$ (namely, a Hodge structure) convey the same information. The main result of this paper says that if $A_1$ and~$A_2$ are abelian varieties (or abelian motives) over~$K$, and the Mumford--Tate conjecture holds for both~$A_1$ and~$A_2$, then it holds for $A_1 \\times A_2$. These results do not depend on the embedding $K \\subset \\CC$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-04-18T17:59:11.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "abelian varieties",
      "extra structure",
      "adic etale cohomology groups",
      "mumford-tate conjecture holds",
      "singular cohomology groups"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 34,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}