{
  "id": "1806.04240",
  "version": "v1",
  "published": "2018-06-11T20:52:31.000Z",
  "updated": "2018-06-11T20:52:31.000Z",
  "title": "Congruences with Eisenstein series and mu-invariants",
  "authors": [
    "Joël Bellaïche",
    "Robert Pollack"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "We study the variation of mu-invariants in Hida families with residually reducible Galois representations. We prove a lower bound for these invariants which is often expressible in terms of the p-adic zeta function. This lower bound forces these mu-invariants to be unbounded along the family, and moreover, we conjecture that this lower bound is an equality. When U_p-1 generates the cuspidal Eisenstein ideal, we establish this conjecture and further prove that the p-adic L-function is simply a power of p up to a unit (i.e. lambda=0). On the algebraic side, we prove analogous statements for the associated Selmer groups which, in particular, establishes the main conjecture for such forms.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-06-11T20:52:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11F33",
      "11R23"
    ],
    "keywords": [
      "eisenstein series",
      "mu-invariants",
      "congruences",
      "conjecture",
      "lower bound forces"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}