{
  "annotation_id": "c6653868-ab42-4a54-abd1-5ed8c2163d71",
  "date_created": "2026-02-17T05:53:16.129000Z",
  "date_modified": "2026-02-17T05:53:16.129000Z",
  "file_hash": "25ffedd95c370f153bafd1a0cfe8d2f91d50b98c405c7621abcb10bcce881ce3",
  "private": false,
  "record": {
    "abstract": "In this paper, we consider two infinite parametric families of elliptic curves defined over $\\mathbb{Q}$ given by the equations $E_{a,b} : y^{2} = x^{3} - a^{2}x + b^{2}$ and $E^{\\prime}_{a,b} : y^{2} = x^{3} - a^{2}x + b^{6}$, where $a,b \\in \\mathbb{N}$ satisfy certain mild conditions. We prove that the torsion group of $E_{a,b}(\\mathbb{Q})$ is trivial and the Mordell-Weil ranks of both $E_{a,b}(\\mathbb{Q})$ and $E^{\\prime}_{a,b}(\\mathbb{Q})$ are at least $3$ for infinitely many choices of $a$ and $b$ by using the N\\\u0027{e}ron-Tate height of a rational point and by exploiting the unit group of the ring of integers of $\\mathbb{Q}(\\sqrt{3})$. This is an extension of the results of Brown-Myres and Fujita-Nara where lower bounds of the ranks were provided under the assumption that $a = 1$ or $b = 1$. Also, our families of elliptic curves vastly generalize the curves recently investigated by Hatley and Stack.",
    "arxiv_id": "2601.08570",
    "authors": [
      "Pankaj Patel",
      "Debopam Chakraborty",
      "Jaitra Chattopadhyay"
    ],
    "categories": [
      "math.NT"
    ],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "title": "Two infinite families of elliptic curves with Mordell-Weil rank at least $3$",
    "url": "https://arxiv.org/abs/2601.08570",
    "version": "v1"
  },
  "schema_id": "dorsal/arxiv",
  "source": {
    "execution_id": "0d6b404f-3702-4655-afee-3210aeb57dda",
    "id": "arXiv Dataset",
    "type": "Model",
    "variant": "snapshot-2026-01-17",
    "version": "0.1.0"
  },
  "user_id": 1000002
}