Sporadic Cubic Torsion
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arXiv
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| Format: | Preprint |
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2020
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| author | Derickx, Maarten Etropolski, Anastassia van Hoeij, Mark Morrow, Jackson S. Zureick-Brown, David |
| author_facet | Derickx, Maarten Etropolski, Anastassia van Hoeij, Mark Morrow, Jackson S. Zureick-Brown, David |
| contents | Let $K$ be a number field, and let $E/K$ be an elliptic curve over $K$. The Mordell--Weil theorem asserts that the $K$-rational points $E(K)$ of $E$ form a finitely generated abelian group. In this work, we complete the classification of the finite groups which appear as the torsion subgroup of $E(K)$ for $K$ a cubic number field.
To do so, we determine the cubic points on the modular curves $X_1(N)$ for \[N = 21, 22, 24, 25, 26, 28, 30, 32, 33, 35, 36, 39, 45, 65, 121.\] As part of our analysis, we determine the complete list of $N$ for which $J_0(N)$ (resp., $J_1(N)$, resp., $J_1(2,2N)$) has rank 0. We also provide evidence to a generalized version of a conjecture of Conrad, Edixhoven, and Stein by proving that the torsion on $J_1(N)(\mathbb{Q})$ is generated by $\text{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$-orbits of cusps of $X_1(N)_{\bar{\mathbb{Q}}}$ for $N\leq 55$, $N \neq 54$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2007_13929 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Sporadic Cubic Torsion Derickx, Maarten Etropolski, Anastassia van Hoeij, Mark Morrow, Jackson S. Zureick-Brown, David Number Theory Algebraic Geometry 11G18, (11G05, 14H45, 11Y50) Let $K$ be a number field, and let $E/K$ be an elliptic curve over $K$. The Mordell--Weil theorem asserts that the $K$-rational points $E(K)$ of $E$ form a finitely generated abelian group. In this work, we complete the classification of the finite groups which appear as the torsion subgroup of $E(K)$ for $K$ a cubic number field. To do so, we determine the cubic points on the modular curves $X_1(N)$ for \[N = 21, 22, 24, 25, 26, 28, 30, 32, 33, 35, 36, 39, 45, 65, 121.\] As part of our analysis, we determine the complete list of $N$ for which $J_0(N)$ (resp., $J_1(N)$, resp., $J_1(2,2N)$) has rank 0. We also provide evidence to a generalized version of a conjecture of Conrad, Edixhoven, and Stein by proving that the torsion on $J_1(N)(\mathbb{Q})$ is generated by $\text{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$-orbits of cusps of $X_1(N)_{\bar{\mathbb{Q}}}$ for $N\leq 55$, $N \neq 54$. |
| title | Sporadic Cubic Torsion |
| topic | Number Theory Algebraic Geometry 11G18, (11G05, 14H45, 11Y50) |
| url | https://arxiv.org/abs/2007.13929 |