Existence and non-existence of rational elliptic curves with prescribed torsion subgroups over quadratic fields
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866912887961812992 |
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| author | Avci, Omer |
| author_facet | Avci, Omer |
| contents | Let $K=\mathbb{Q}(\sqrt{-p})$ be a quadratic field for an odd prime $p$. We show that there exist infinitely many primes $p$ for which no elliptic curve $E/\mathbb{Q}$ has torsion subgroup $\mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/2N\mathbb{Z}$ over $K$ for $N=5,6$. We also prove that there exist infinitely many primes $p$ for which there are infinitely many elliptic curves $E/\mathbb{Q}$ with this torsion structure, conditional on the parity conjecture. Using these results, we obtain new torsion classification results over Kummer extensions of cyclotomic fields and over composites of $\mathbb{Z}_p$-extensions of number fields, refining and extending our previous work. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2602_07723 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Existence and non-existence of rational elliptic curves with prescribed torsion subgroups over quadratic fields Avci, Omer Number Theory 11G05, 14G05, 11R20, 14H52 Let $K=\mathbb{Q}(\sqrt{-p})$ be a quadratic field for an odd prime $p$. We show that there exist infinitely many primes $p$ for which no elliptic curve $E/\mathbb{Q}$ has torsion subgroup $\mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/2N\mathbb{Z}$ over $K$ for $N=5,6$. We also prove that there exist infinitely many primes $p$ for which there are infinitely many elliptic curves $E/\mathbb{Q}$ with this torsion structure, conditional on the parity conjecture. Using these results, we obtain new torsion classification results over Kummer extensions of cyclotomic fields and over composites of $\mathbb{Z}_p$-extensions of number fields, refining and extending our previous work. |
| title | Existence and non-existence of rational elliptic curves with prescribed torsion subgroups over quadratic fields |
| topic | Number Theory 11G05, 14G05, 11R20, 14H52 |
| url | https://arxiv.org/abs/2602.07723 |