The intrinsic subgroup of an elliptic curve and Mazur's torsion theorem
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909993967550464 |
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| author | Yamazaki, Takao Yang, Yifan Yoo, Hwajong Yu, Myungjun |
| author_facet | Yamazaki, Takao Yang, Yifan Yoo, Hwajong Yu, Myungjun |
| contents | We define and study a biadditive symmetric (not necessarily perfect) pairing on the torsion part $\mathrm{Pic}(X)_{\mathrm{tors}}$ of the Picard group of a smooth projective curve $X$ over a field $k$ with values in $k^\times \otimes \mathbb{Q}/\mathbb{Z}$. We call its kernel the intrinsic subgroup of $X$. It turns out that some information on the reduction type of $X$ can be read off from the intrinsic subgroup. Mazur's torsion theorem says that there are exactly 15 isomorphism classes of abelian groups that appear as the rational torsion points of an elliptic curve $X$ over $\mathbb{Q}$ (identified with $\mathrm{Pic}(X)_{\mathrm{tors}}$). We refine this result by determining which subgroups of those 15 groups appear as the intrinsic subgroups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_00787 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The intrinsic subgroup of an elliptic curve and Mazur's torsion theorem Yamazaki, Takao Yang, Yifan Yoo, Hwajong Yu, Myungjun Number Theory Algebraic Geometry 11G05, 11G18, 14G25, 14G35 We define and study a biadditive symmetric (not necessarily perfect) pairing on the torsion part $\mathrm{Pic}(X)_{\mathrm{tors}}$ of the Picard group of a smooth projective curve $X$ over a field $k$ with values in $k^\times \otimes \mathbb{Q}/\mathbb{Z}$. We call its kernel the intrinsic subgroup of $X$. It turns out that some information on the reduction type of $X$ can be read off from the intrinsic subgroup. Mazur's torsion theorem says that there are exactly 15 isomorphism classes of abelian groups that appear as the rational torsion points of an elliptic curve $X$ over $\mathbb{Q}$ (identified with $\mathrm{Pic}(X)_{\mathrm{tors}}$). We refine this result by determining which subgroups of those 15 groups appear as the intrinsic subgroups. |
| title | The intrinsic subgroup of an elliptic curve and Mazur's torsion theorem |
| topic | Number Theory Algebraic Geometry 11G05, 11G18, 14G25, 14G35 |
| url | https://arxiv.org/abs/2512.00787 |