{
  "id": "1603.09407",
  "version": "v1",
  "published": "2016-03-30T22:45:40.000Z",
  "updated": "2016-03-30T22:45:40.000Z",
  "title": "Ordinary reductions of abelian varieties",
  "authors": [
    "Kirti Joshi"
  ],
  "comment": "13 pages",
  "categories": [
    "math.AG",
    "math.NT"
  ],
  "abstract": "I show that a conjecture of Joshi-Rajan on primes of Hodge-Witt reduction and in particular a conjecture of Jean-Pierre Serre on primes of good, ordinary reduction for an abelian variety over a number field follows from a certain conjecture on Galois rep- resentations which may perhaps be easier to prove (and I prove this conjecture for abelian compatible systems of a suitable type). This reduction (to a conjecture about certain sys- tems of Galois representations) is based on a new slope estimate for non Hodge-Witt abelian varieties. In particular for any abelian variety over a number field with at least one prime of good ordinary or split toric reduction, I show that the conjecture of Joshi-Rajan and the conjecture of Serre on ordinary reductions can be reduced to proving that a certain rational trace of Frobenius is in fact an integer. The assertion that this trace is an integer is proved for abelian systems of Galois representations (of suitable type).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-03-30T22:45:40.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "abelian variety",
      "ordinary reduction",
      "conjecture",
      "galois representations",
      "number field"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}