{
  "id": "2410.23241",
  "version": "v1",
  "published": "2024-10-30T17:27:46.000Z",
  "updated": "2024-10-30T17:27:46.000Z",
  "title": "$p$-converse theorems for elliptic curves of potentially good ordinary reduction at Eisenstein primes",
  "authors": [
    "Timo Keller",
    "Mulun Yin"
  ],
  "comment": "34 pages. Comments welcome!",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $E/\\mathbf{Q}$ be an elliptic curve and $p\\geq 3$ be a prime. We prove the $p$-converse theorems for elliptic curves of potentially good ordinary reduction at Eisenstein primes (i.e., such that the residual representation $E[p]$ is reducible) when the $p$-Selmer rank is $0$ or $1$. The key step is to obtain the anticyclotomic Iwasawa Main Conjectures for an auxiliary imaginary quadratic field $K$ where $E$ does not have CM similar to those in [CGLS22] and descent to $\\mathbf{Q}$. As an application we get improved proportions for the number of elliptic curves in quadratic twist families having rank $0$ or $1$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-10-30T17:27:46.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G40",
      "11G05",
      "11G10",
      "14G10"
    ],
    "keywords": [
      "elliptic curve",
      "eisenstein primes",
      "ordinary reduction",
      "converse theorems",
      "anticyclotomic iwasawa main conjectures"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 34,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}