{
  "id": "2207.06287",
  "version": "v1",
  "published": "2022-07-13T15:33:51.000Z",
  "updated": "2022-07-13T15:33:51.000Z",
  "title": "On Iwasawa $λ$-invariants for abelian number fields and random matrix heuristics",
  "authors": [
    "Daniel Delbourgo",
    "Heiko Knospe"
  ],
  "comment": "159 pages, 2 figures",
  "categories": [
    "math.NT"
  ],
  "abstract": "Following both Ernvall-Mets\\\"{a}nkyl\\\"{a} and Ellenberg-Jain-Venkatesh, we study the density of the number of zeroes (i.e. the cyclotomic $\\lambda$-invariant) for the $p$-adic zeta-function twisted by a Dirichlet character $\\chi$ of any order. We are interested in two cases: (i) the character $\\chi$ is fixed and the prime $p$ varies, and (ii) $\\text{ord}(\\chi)$ and the prime $p$ are both fixed but $\\chi$ is allowed to vary. We predict distributions for these $\\lambda$-invariants using $p$-adic random matrix theory and provide numerical evidence for these predictions. We also study the proportion of $\\chi$-regular primes, which depends on how $p$ splits inside $\\mathbb{Q}(\\chi)$. Finally in an extensive Appendix, we tabulate the values of the $\\lambda$-invariant for every character $\\chi$ of conductor $\\leq 1000$ and for odd primes $p$ of small size.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-07-13T15:33:51.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11R23",
      "11R42",
      "11S80",
      "11M41"
    ],
    "keywords": [
      "abelian number fields",
      "random matrix heuristics",
      "adic random matrix theory",
      "dirichlet character",
      "odd primes"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 159,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}