{
  "id": "1603.01283",
  "version": "v1",
  "published": "2016-03-03T21:14:40.000Z",
  "updated": "2016-03-03T21:14:40.000Z",
  "title": "Abelian part of compatible system and geometry of roots",
  "authors": [
    "Chun Yin Hui"
  ],
  "comment": "21",
  "categories": [
    "math.NT",
    "math.AG",
    "math.RT"
  ],
  "abstract": "Let {V_l} be a strictly compatible rational system of Galois representations arising from geometry. Let V_l^ss be the semisimplification of V_l, V_l^ab the maximal abelian subrepresentation of V_l^ss, and G_l the algebraic monodromy group of V_l^ss. Conjecturally, the abelian part {V_l^ab} is again a strictly compatible rational system (Conj. 1.2). The conjecture has meaningful consequences to arithmetic geometry. The main theme of this paper is to investigate the conjecture via l-independence of G_l. More precisely, we prove that the conjecture follows if (i) G_l is connected for all l, (ii) the tautological representation of G_l^der is independent of l, and (iii) the roots of G_l meet a criteria (Thm. 1.3). These conditions are satisfied under some group theoretic assumptions of G_l (Thm. 1.5), for example, (ii) and (iii) hold if G_l^der(C) is connected and semisimple of type A_4+A_6+A_6 for some l; (iii) holds if G_l^der(C) is almost simple of type different form A_7,A_8,B_4,D_8.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-03-03T21:14:40.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "abelian part",
      "compatible system",
      "strictly compatible rational system",
      "group theoretic assumptions",
      "maximal abelian subrepresentation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160301283H"
    }
  }
}