{
  "id": "2403.03073",
  "version": "v2",
  "published": "2024-03-05T16:00:16.000Z",
  "updated": "2025-01-27T21:08:22.000Z",
  "title": "Entanglement of elliptic curves upon base extension",
  "authors": [
    "Tori Day",
    "Rylan Gajek-Leonard"
  ],
  "comment": "Corrected minor typos",
  "categories": [
    "math.NT"
  ],
  "abstract": "Fix distinct primes $p$ and $q$ and let $E$ be an elliptic curve defined over a number field $K$. The $(p,q)$-entanglement type of $E$ over $K$ is the isomorphism class of the group $\\operatorname{Gal}(K(E[p])\\cap K(E[q])/K)$. The size of this group measures the extent to which the image of the mod $pq$ Galois representation attached to $E$ fails to be a direct product of the mod $p$ and mod $q$ images. In this article, we study how the $(p,q)$-entanglement group varies over different base fields. We prove that for each prime $\\ell$ dividing the greatest common divisor of the size of the mod $p$ and $q$ images, there are infinitely many fields $L/K$ such that the entanglement over $L$ is cyclic of order $\\ell$. We also classify all possible $(2,q)$-entanglement types that can occur as the base field $L$ varies.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2025-01-27T21:08:22.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G05",
      "11F80"
    ],
    "keywords": [
      "elliptic curve",
      "base extension",
      "base field",
      "entanglement type",
      "fix distinct primes"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}