{
  "id": "2401.06100",
  "version": "v1",
  "published": "2024-01-11T18:28:19.000Z",
  "updated": "2024-01-11T18:28:19.000Z",
  "title": "Special values of $p$-adic $L$-functions and Iwasawa $λ$-invariants of Dirichlet characters",
  "authors": [
    "Heiko Knospe"
  ],
  "comment": "12 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "We study the Iwasawa $\\lambda$-invariant of Dirichlet characters $\\chi$ for odd primes $p$. From the behaviour of the $p$-adic $L$-function at $s=0$, $s=1$ and $s=1-p$, we derive easily computable novel criteria to distinguish between the cases $\\lambda = 0$, $\\lambda = 1$, $\\lambda = 2$ and $\\lambda \\geq 3$. In particular, we look at the case where the $p$-adic $L$-function and the Iwasawa power series have a trivial (exceptional) zero. We give a condition for $\\lambda_p(\\chi) >1 $ using the $p$-adic valuation of the Bernoulli number $B_{p,\\chi \\omega^{-1}}$ or alternatively $L_p(1,\\chi)$. We also provide a similar condition for $\\lambda_p(\\chi) > 2$. The formulas are used to obtain numerical data on $\\lambda$-invariants, where either a prime or a Dirichlet character is fixed. The data is consistent with our earlier conjecture on the distribution of $\\lambda$-invariants. Now the case of a trivial zero is included, for which the distribution of $\\lambda$-values is shifted by $1$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-01-11T18:28:19.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11R23",
      "11R42",
      "11S80",
      "11M41"
    ],
    "keywords": [
      "dirichlet character",
      "special values",
      "iwasawa power series",
      "earlier conjecture",
      "computable novel criteria"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}