On signed Mordell-Weil groups for abelian varieties
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866917057160806400 |
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| author | Ray, Jishnu |
| author_facet | Ray, Jishnu |
| contents | In this short note, we work in the general framework of supersingular abelian varieties defined over $\mathbb{Q}$. Using Coleman maps constructed by Büyükboduk--Lei, we define some objects called ``the multi-signed Mordell-Weil groups" for supersingular abelian varieties, make comments on the structure of the dual of these groups as an Iwasawa module and show a (weak) control theorem. This recovers the case of elliptic curves over $\mathbb{Q}$ non-ordinary at the prime $p$ with $a_p=0$ studied by Antonio Lei. Using the multi-signed Mordell-Weil groups we define what we call ``the multi-signed Tate-Shafarevich groups" along the cyclotomic tower of $\mathbb{Q}$. Finally we pose some open questions related to our newly defined objects and make a remark on the asymptotic growth of these multi-signed Tate-Shafarevich groups along the cyclotomic tower using an idea of Meng Fai Lim. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2304_13452 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On signed Mordell-Weil groups for abelian varieties Ray, Jishnu Number Theory Primary: 11R23, Secondary: 11G10, 11G05, 11R18 In this short note, we work in the general framework of supersingular abelian varieties defined over $\mathbb{Q}$. Using Coleman maps constructed by Büyükboduk--Lei, we define some objects called ``the multi-signed Mordell-Weil groups" for supersingular abelian varieties, make comments on the structure of the dual of these groups as an Iwasawa module and show a (weak) control theorem. This recovers the case of elliptic curves over $\mathbb{Q}$ non-ordinary at the prime $p$ with $a_p=0$ studied by Antonio Lei. Using the multi-signed Mordell-Weil groups we define what we call ``the multi-signed Tate-Shafarevich groups" along the cyclotomic tower of $\mathbb{Q}$. Finally we pose some open questions related to our newly defined objects and make a remark on the asymptotic growth of these multi-signed Tate-Shafarevich groups along the cyclotomic tower using an idea of Meng Fai Lim. |
| title | On signed Mordell-Weil groups for abelian varieties |
| topic | Number Theory Primary: 11R23, Secondary: 11G10, 11G05, 11R18 |
| url | https://arxiv.org/abs/2304.13452 |