Prime order torsion on elliptic curves over number fields. Part I: Asymptotics
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912383782354944 |
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| author | Derickx, Maarten Stoll, Michael |
| author_facet | Derickx, Maarten Stoll, Michael |
| contents | We study the asymptotics of the set $S(d)$ of possible prime orders of $K$-rational points on elliptic curves over number fields $K$ of degree $d$ as $d$ tends to infinity. Assuming some conjectures on the sparsity of newforms of weight $2$ and prime level with unexpectedly high analytic rank, we show that $\max S(d) \le 3d + 1$ for sufficiently large even $d$ and $\max S(d) = o(d)$ for odd $d$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_14109 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Prime order torsion on elliptic curves over number fields. Part I: Asymptotics Derickx, Maarten Stoll, Michael Number Theory Algebraic Geometry 11G05, 11G18, 14G05, 14G25, 14G35 We study the asymptotics of the set $S(d)$ of possible prime orders of $K$-rational points on elliptic curves over number fields $K$ of degree $d$ as $d$ tends to infinity. Assuming some conjectures on the sparsity of newforms of weight $2$ and prime level with unexpectedly high analytic rank, we show that $\max S(d) \le 3d + 1$ for sufficiently large even $d$ and $\max S(d) = o(d)$ for odd $d$. |
| title | Prime order torsion on elliptic curves over number fields. Part I: Asymptotics |
| topic | Number Theory Algebraic Geometry 11G05, 11G18, 14G05, 14G25, 14G35 |
| url | https://arxiv.org/abs/2505.14109 |