{
  "id": "2206.08595",
  "version": "v1",
  "published": "2022-06-17T07:38:13.000Z",
  "updated": "2022-06-17T07:38:13.000Z",
  "title": "Quadratic Chabauty and $p$-adic Gross-Zagier",
  "authors": [
    "Sachi Hashimoto"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $X$ be a quotient of the modular curve $X_0(N)$ whose Jacobian $J_X$ is a simple factor of $J_0(N)^{new}$ over $\\mathbb{Q}$. Let $f$ be the newform of level $N$ and weight 2 associated with $J_X$; assume $f$ has analytic rank 1. We give analytic methods for determining the rational points of $X$ using quadratic Chabauty by computing two $p$-adic Gross--Zagier formulas for $f$. Quadratic Chabauty requires a supply of rational points on the curve or its Jacobian; this new method eliminates this requirement. To achieve this, we give an algorithm to compute the special value of the anticyclotomic $p$-adic $L$-function of $f$ constructed by Bertolini, Darmon, and Prasanna, which lies outside of the range of interpolation.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-06-17T07:38:13.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "quadratic chabauty",
      "rational points",
      "adic gross-zagier formulas",
      "simple factor",
      "analytic rank"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}