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Torsion of $\mathbb Q$-curves over number fields of small odd prime degree

Ivan Novak

Abstract

We determine all groups which occur as torsion subgroups of $\mathbb Q$-curves defined over number fields of degrees $3$, $5$ and $7$. In particular, we prove that every torsion subgroup of a $\mathbb Q$-curve defined over a number field of degree $3,5$ or $7$ already occurs as a torsion subgroup of an elliptic curve with rational $j$-invariant. As the quadratic case has been solved by Le Fourn and Najman, and the case of extensions of prime degree greater than $7$ has been solved by Cremona and Najman, this paper completes the classification of torsion of $\mathbb Q$-curves over number fields of prime degree. We also establish that the torsion subgroup an elliptic curve over a number field $K$ of prime degree which is isogenous to an elliptic curve with rational $j$-invariant is equal to the torsion subgroup of some elliptic curve defined over a degree $p$ number field with rational $j$-invariant.

Torsion of $\mathbb Q$-curves over number fields of small odd prime degree

Abstract

We determine all groups which occur as torsion subgroups of -curves defined over number fields of degrees , and . In particular, we prove that every torsion subgroup of a -curve defined over a number field of degree or already occurs as a torsion subgroup of an elliptic curve with rational -invariant. As the quadratic case has been solved by Le Fourn and Najman, and the case of extensions of prime degree greater than has been solved by Cremona and Najman, this paper completes the classification of torsion of -curves over number fields of prime degree. We also establish that the torsion subgroup an elliptic curve over a number field of prime degree which is isogenous to an elliptic curve with rational -invariant is equal to the torsion subgroup of some elliptic curve defined over a degree number field with rational -invariant.

Paper Structure

This paper contains 14 sections, 41 theorems, 20 equations.

Table of Contents

  1. Introduction
  2. General results
  3. Isogeny graphs
  4. Torsion over cubic number fields
  5. Eliminating $\mathbb{Z}/11\mathbb{Z}$
  6. Eliminating $\mathbb{Z}/15\mathbb{Z}$
  7. Eliminating $\mathbb{Z}/16\mathbb{Z}$
  8. Eliminating $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/10\mathbb{Z}$ and $\mathbb{Z}/20\mathbb{Z}$
  9. Eliminating $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/12\mathbb{Z}$
  10. Torsion over number fields of degree $5$ or $7$
  11. Possible prime power orders of points
  12. Possible odd orders of points
  13. Combining the $2^k$-torsion with the odd part
  14. Quadratic torsion of elliptic curves isogenous to those with rational $j$-invariant

Key Result

Theorem 1.1

Mazur Let $E/\mathbb{Q}$ be an elliptic curve. Then $E(\mathbb{Q})_{tors}$ is isomorphic to one of the following $15$ groups: There exist infinitely many $\overline{\mathbb{Q}}$-isomorphism classes for each such torsion subgroup.

Theorems & Definitions (70)

  • Theorem 1.1
  • Definition 1.2
  • Proposition 1.3
  • Remark 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Remark 1.8
  • Theorem 1.9
  • Theorem 1.10
  • ...and 60 more