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Torsion groups of elliptic curves over quadratic fields

Sheldon Kamienny, Filip Najman

Abstract

We describe methods to determine all the possible torsion groups of an elliptic curve that actually appear over a fixed quadratic field. We use these methods to find, for each group that can appear over a quadratic field, the field with the smallest absolute value of it's discriminant such that there exists an elliptic curve with that torsion. We also examine the interplay of the torsion and rank over a fixed quadratic field and see that what happens is very different than over $\Q$. Finally we give some results concerning the number and density of fields with an elliptic curve with given torsion over them.

Torsion groups of elliptic curves over quadratic fields

Abstract

We describe methods to determine all the possible torsion groups of an elliptic curve that actually appear over a fixed quadratic field. We use these methods to find, for each group that can appear over a quadratic field, the field with the smallest absolute value of it's discriminant such that there exists an elliptic curve with that torsion. We also examine the interplay of the torsion and rank over a fixed quadratic field and see that what happens is very different than over . Finally we give some results concerning the number and density of fields with an elliptic curve with given torsion over them.

Paper Structure

This paper contains 6 sections, 19 theorems, 45 equations.

Table of Contents

  1. Introduction.
  2. Finding the possible torsions over a fixed quadratic field.
  3. Smallest field with a given torsion
  4. The Method of Mazur for $X_1(13)$ and $X_1(18)$
  5. Interplay of rank and torsion
  6. Number of fields having given torsion

Key Result

Theorem 1

Let $K$ be a quadratic field and $E$ an elliptic curve over $K$. Then the torsion subgroup $E(K)_{tors}$ of $E(K)$ is isomorphic to one of the following $26$ groups:

Theorems & Definitions (36)

  • Theorem 1
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Theorem 4
  • proof
  • Theorem 5
  • proof
  • Remark 6
  • ...and 26 more