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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2409.08214 |
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| _version_ | 1866917946104741888 |
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| author | Bourdon, Abbey Genao, Tyler |
| author_facet | Bourdon, Abbey Genao, Tyler |
| contents | In 1996, Merel showed there exists a function $B\colon \mathbb{Z}^+\rightarrow \mathbb{Z}^+$ such that for any elliptic curve $E/F$ defined over a number field of degree $d$, one has the torsion group bound $\# E(F)[\textrm{tors}]\leq B(d)$. Based on subsequent work, it is conjectured that one can choose $B$ to be polynomial in the degree $d$. In this paper, we show that such bounds exist for torsion from the family $\mathcal{I}_{\mathbb{Q}}$ of elliptic curves which are geometrically isogenous to at least one rational elliptic curve. More precisely, we show that for each $ε>0$, there exists $c_ε>0$ such that for any elliptic curve $E/F\in \mathcal{I}_{\mathbb{Q}}$, one has \[ E(F)[\textrm{tors}]\leq c_ε\cdot [F:\mathbb{Q}]^{3+ε}. \] This generalizes work of the second author for elliptic curves within a fixed rational geometric isogeny class. For the family of elliptic curves with rational $j$-invariant, we also obtain bounds which improve those of Clark and Pollack. In this case, our bounds on the exponent of $E(F)[\textrm{tors}]$ are optimal if one does not exclude elliptic curves with complex multiplication. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_08214 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Uniform polynomial bounds on torsion from rational geometric isogeny classes Bourdon, Abbey Genao, Tyler Number Theory 11G05 In 1996, Merel showed there exists a function $B\colon \mathbb{Z}^+\rightarrow \mathbb{Z}^+$ such that for any elliptic curve $E/F$ defined over a number field of degree $d$, one has the torsion group bound $\# E(F)[\textrm{tors}]\leq B(d)$. Based on subsequent work, it is conjectured that one can choose $B$ to be polynomial in the degree $d$. In this paper, we show that such bounds exist for torsion from the family $\mathcal{I}_{\mathbb{Q}}$ of elliptic curves which are geometrically isogenous to at least one rational elliptic curve. More precisely, we show that for each $ε>0$, there exists $c_ε>0$ such that for any elliptic curve $E/F\in \mathcal{I}_{\mathbb{Q}}$, one has \[ E(F)[\textrm{tors}]\leq c_ε\cdot [F:\mathbb{Q}]^{3+ε}. \] This generalizes work of the second author for elliptic curves within a fixed rational geometric isogeny class. For the family of elliptic curves with rational $j$-invariant, we also obtain bounds which improve those of Clark and Pollack. In this case, our bounds on the exponent of $E(F)[\textrm{tors}]$ are optimal if one does not exclude elliptic curves with complex multiplication. |
| title | Uniform polynomial bounds on torsion from rational geometric isogeny classes |
| topic | Number Theory 11G05 |
| url | https://arxiv.org/abs/2409.08214 |