{
  "id": "2402.12781",
  "version": "v2",
  "published": "2024-02-20T07:46:23.000Z",
  "updated": "2024-10-30T17:52:08.000Z",
  "title": "On the anticyclotomic Iwasawa theory of newforms at Eisenstein primes of semistable reduction",
  "authors": [
    "Timo Keller",
    "Mulun Yin"
  ],
  "comment": "61 Pages. Comments welcome",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $f$ be a newform of weight $k=2r$ and level $N$ with trivial nebentypus. Let $\\mathfrak{p}\\nmid 2N$ be a maximal ideal of the ring of integers of the coefficient field of $f$ such that the self-dual twist of the mod-$\\mathfrak{p}$ Galois representation of $f$ is reducible with constituents $\\phi,\\psi$. Denote a decomposition group over the rational prime $p$ below $\\mathfrak{p}$ by $G_p$. We remove the condition $\\phi|_{G_p} \\neq \\mathbf{1}, \\omega$ from [CGLS22], and generalize their results to newforms of higher weights $2r$ with $r$ being odd. As a consequence, we prove some Iwasawa Main Conjectures and get the $p$-part of the strong BSD Conjecture for elliptic curves of analytic rank $0$ or $1$ over $\\mathbf{Q}$ in this setting. In particular, non-trivial $p$-torsion is allowed in the Mordell--Weil group. Using Hida families, we also prove an Iwasawa Main Conjecture for newforms of weight $2$ of multiplicative reduction at Eisenstein primes. In the above situations, we also get $p$-converse to the theorems of Gross--Zagier--Kolyvagin. The $p$-converse theorems have applications to Goldfeld's conjecture in certain quadratic twist families of elliptic curves having a $3$-isogeny.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2024-10-30T17:52:08.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G40",
      "11G05",
      "11G10",
      "14G10"
    ],
    "keywords": [
      "anticyclotomic iwasawa theory",
      "eisenstein primes",
      "semistable reduction",
      "iwasawa main conjecture",
      "elliptic curves"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 61,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}