The growth of Tate-Shafarevich groups of $p$-supersingular elliptic curves over anticyclotomic $\mathbb{Z}_p$-extensions at inert primes
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866914016546258944 |
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| author | Isik, Erman Lei, Antonio |
| author_facet | Isik, Erman Lei, Antonio |
| contents | Let $E$ be an elliptic curve defined over $\mathbb{Q}$, and let $K$ be an imaginary quadratic field. Consider an odd prime $p$ at which $E$ has good supersingular reduction with $a_p(E)=0$ and which is inert in $K$. Under the assumption that the signed Selmer groups are cotorsion modules over the corresponding Iwasawa algebra, we prove that the Mordell-Weil ranks of $E$ are bounded over any subextensions of the anticyclotomic $\mathbb{Z}_p$-extension of $K$. Additionally, we provide an asymptotic formula for the growth of the $p$-parts of the Tate-Shafarevich groups of $E$ over these extensions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_02202 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The growth of Tate-Shafarevich groups of $p$-supersingular elliptic curves over anticyclotomic $\mathbb{Z}_p$-extensions at inert primes Isik, Erman Lei, Antonio Number Theory 11R23 (primary), 11G05, 11R20 (secondary) Let $E$ be an elliptic curve defined over $\mathbb{Q}$, and let $K$ be an imaginary quadratic field. Consider an odd prime $p$ at which $E$ has good supersingular reduction with $a_p(E)=0$ and which is inert in $K$. Under the assumption that the signed Selmer groups are cotorsion modules over the corresponding Iwasawa algebra, we prove that the Mordell-Weil ranks of $E$ are bounded over any subextensions of the anticyclotomic $\mathbb{Z}_p$-extension of $K$. Additionally, we provide an asymptotic formula for the growth of the $p$-parts of the Tate-Shafarevich groups of $E$ over these extensions. |
| title | The growth of Tate-Shafarevich groups of $p$-supersingular elliptic curves over anticyclotomic $\mathbb{Z}_p$-extensions at inert primes |
| topic | Number Theory 11R23 (primary), 11G05, 11R20 (secondary) |
| url | https://arxiv.org/abs/2409.02202 |