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

TL;DR

This paper classifies all possible torsion subgroups of $Q$-curves over number fields of degrees 3, 5, and 7, completing the prime degree case and linking torsion structures to rational $j$-invariants.

Contribution

It provides a complete classification of torsion subgroups of $Q$-curves over prime degree number fields 3, 5, and 7, extending previous results and establishing new isogeny relations.

Findings

All torsion subgroups over degree 3, 5, and 7 fields are classified.

Every torsion subgroup over these fields occurs for a curve with rational $j$-invariant.

The torsion subgroup of an elliptic curve isogenous to a rational $j$-invariant curve matches that over a degree $p$ field.

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.