{
  "id": "2310.06813",
  "version": "v1",
  "published": "2023-10-10T17:33:45.000Z",
  "updated": "2023-10-10T17:33:45.000Z",
  "title": "Anticyclotomic Iwasawa theory of abelian varieties of $\\mathrm{GL}_2$-type at non-ordinary primes II",
  "authors": [
    "Ashay Burungale",
    "Kâzım Büyükboduk",
    "Antonio Lei"
  ],
  "comment": "This work complements an earlier paper arXiv:2211.03722 by the same authors",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $p\\ge 5$ be a prime number, $E/\\mathbb{Q}$ an elliptic curve with good supersingular reduction at $p$, and $K$ an imaginary quadratic field such that the root number of $E$ over $K$ is $-1$. When $p$ is split in $K$, Castella and Wan formulated the plus and minus Heegner point main conjecture for $E$ along the anticyclotomic $\\mathbb{Z}_p$-extension of $K$, and proved it for semistable curves. We generalize their results to two settings: 1. Under the assumption that $p$ is split in $K$ but without assuming $a_p(E)=0$, we study Sprung-type main conjectures for $\\mathrm{GL}_2$-type abelian varieties at non-ordinary primes and prove it under some conditions. 2. We formulate, relying on the recent work of the first-named author with Kobayashi and Ota, plus and minus Heegner point main conjectures for elliptic curves when $p$ is inert in $K$, and prove the minus main conjecture for semistable elliptic curves. These results yield a $p$-converse to the Gross--Zagier and Kolyvagin theorem for $E$. Our method relies on Howard's framework of bipartite Euler system, Zhang's resolution of Kolyvagin's conjecture and the recent proof of Kobayashi's main conjecture.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-10-10T17:33:45.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "anticyclotomic iwasawa theory",
      "abelian varieties",
      "non-ordinary primes",
      "minus heegner point main conjecture",
      "elliptic curve"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}