{
  "id": "1609.02515",
  "version": "v1",
  "published": "2016-09-08T17:55:44.000Z",
  "updated": "2016-09-08T17:55:44.000Z",
  "title": "Growth of torsion groups of elliptic curves upon base change",
  "authors": [
    "Enrique González-Jiménez",
    "Filip Najman"
  ],
  "comment": "26 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "We study how the torsion of elliptic curves over number fields grows upon base change, and in particular prove various necessary conditions for torsion growth. For a number field $F$, we show that for a large set of number fields $L$, whose Galois group of their normal closure over $F$ has certain properties, it will hold that $E(L)_{tors}=E(F)_{tors}$ for all elliptic curves $E$ defined over $F$. Our methods turn out to be particularly useful in studying the possible torsion groups $E(K)_{tors}$, where $K$ is a number field and $E$ is a base change of an elliptic curve defined over $\\mathbb Q$. Suppose that $E$ is a base change of an elliptic curve over $\\mathbb Q$ for the remainder of the abstract. In this setting, we prove that $E(K)_{tors}=E(\\mathbb Q)_{tors}$ for all elliptic curves $E$ defined over $\\mathbb Q$ and all number fields $K$ of degree $d$, where $d$ is not divisible by a prime $\\leq 7$. Using this fact, we determine all the possible torsion groups $E(K)_{tors}$ over number fields $K$ of prime degree $p\\geq 7$. We determine all the possible degrees of $[\\mathbb Q(P):\\mathbb Q]$, where $P$ is a point of prime order $p$ for all $p$ such that $p\\not\\equiv 8 \\pmod 9$ or $\\left( \\frac{-D}{p}\\right)=1$ for any $D\\in \\{1,2,7,11,19,43,67,163\\}$; this is true for a set of density $\\frac{1535}{1536}$ of all primes and in particular for all $p<3167$. Using this result, we determine all the possible prime orders of a point $P\\in E(K)_{tors}$, where $[K:\\mathbb Q]=d$, for all $d\\leq 3342296$. Finally, we solve the outstanding problem of determining all the possible groups $E(K)_{tors}$, where $K$ is a quartic number field and $E$ is an elliptic curve defined over $\\mathbb Q$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-09-08T17:55:44.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "elliptic curve",
      "base change",
      "torsion groups",
      "prime order",
      "quartic number field"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}