Torsion of $\mathbb Q$-curves over number fields of small odd prime degree
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866911466369581056 |
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| author | Novak, Ivan |
| author_facet | Novak, Ivan |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_00753 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Torsion of $\mathbb Q$-curves over number fields of small odd prime degree Novak, Ivan Number Theory 11G05 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. |
| title | Torsion of $\mathbb Q$-curves over number fields of small odd prime degree |
| topic | Number Theory 11G05 |
| url | https://arxiv.org/abs/2506.00753 |