{
  "id": "2409.18021",
  "version": "v2",
  "published": "2024-09-26T16:29:35.000Z",
  "updated": "2024-12-19T18:29:05.000Z",
  "title": "A bound on the $μ$-invariants of supersingular elliptic curves",
  "authors": [
    "Rylan Gajek-Leonard"
  ],
  "comment": "8 pages. Removed previous hypothesis that $\\lambda_p^\\pm\\leq 1$ and improved overall bound on $\\mu$-invariants",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $E/\\mathbb{Q}$ be an elliptic curve and let $p$ be a prime of good supersingular reduction. Attached to $E$ are pairs of Iwasawa invariants $\\mu_p^\\pm$ and $\\lambda_p^\\pm$ which encode arithmetic properties of $E$ along the cyclotomic $\\mathbb{Z}_p$-extension of $\\mathbb{Q}$. A well-known conjecture of B. Perrin-Riou and R. Pollack asserts that $\\mu_p^\\pm=0$. We provide support for this conjecture by proving that for any $\\ell\\geq 0$, we have $\\mu_p^\\pm\\leq 1$ for all but finitely many primes $p$ with $\\lambda_p^\\pm=\\ell$. As a corollary, we show that if $E$ has Mordell-Weil rank 0 then $\\mu_p^\\pm\\leq 1$ holds on density 1 set of good supersingular primes.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2024-12-19T18:29:05.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "supersingular elliptic curves",
      "encode arithmetic properties",
      "mordell-weil rank",
      "pollack asserts",
      "iwasawa invariants"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}