Structure of (Fine) Mordell--Weil Groups
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866916968931524608 |
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| author | Gambheera, Rusiru Kundu, Debanjana |
| author_facet | Gambheera, Rusiru Kundu, Debanjana |
| contents | In this article we study the algebraic structure of fine Mordell--Weil groups, plus/minus Mordell--Weil groups, Selmer groups, and plus/minus Selmer groups in the cyclotomic $\mathbb{Z}_p$-extensions of abelian number fields. As a first, we prove theorems on the equivariant structure of fine Mordell--Weil groups and plus/minus Mordell--Weil groups. In other words, we study the explicit shape of the fine, plus/minus objects as a $Λ(\mathcal{G})$-module with $\mathcal{G} \simeq \mathbb{Z}_p \times G$ and $G$ a finite abelian group. We prove refinements of previously known results over $\mathbb{Q}$ for the classical Selmer group and the plus/minus Selmer group, and subsequently also the Shafarevich--Tate group, and the plus/minus Shafarevich--Tate group. This gives new evidence towards an affirmative answer for the Kurihara--Pollack problem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_20341 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Structure of (Fine) Mordell--Weil Groups Gambheera, Rusiru Kundu, Debanjana Number Theory 11R23, 11R18, 14H52 In this article we study the algebraic structure of fine Mordell--Weil groups, plus/minus Mordell--Weil groups, Selmer groups, and plus/minus Selmer groups in the cyclotomic $\mathbb{Z}_p$-extensions of abelian number fields. As a first, we prove theorems on the equivariant structure of fine Mordell--Weil groups and plus/minus Mordell--Weil groups. In other words, we study the explicit shape of the fine, plus/minus objects as a $Λ(\mathcal{G})$-module with $\mathcal{G} \simeq \mathbb{Z}_p \times G$ and $G$ a finite abelian group. We prove refinements of previously known results over $\mathbb{Q}$ for the classical Selmer group and the plus/minus Selmer group, and subsequently also the Shafarevich--Tate group, and the plus/minus Shafarevich--Tate group. This gives new evidence towards an affirmative answer for the Kurihara--Pollack problem. |
| title | Structure of (Fine) Mordell--Weil Groups |
| topic | Number Theory 11R23, 11R18, 14H52 |
| url | https://arxiv.org/abs/2507.20341 |