On non-trivial $Λ$-submodules with finite index of the plus/minus Selmer group over anticyclotomic $\mathbb{Z}_{p}$-extension at inert primes
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866911749226102784 |
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| author | Shii, Ryota |
| author_facet | Shii, Ryota |
| contents | Let $K$ be an imaginary quadratic field where $p$ is inert. Let $E$ be an elliptic curve defined over $K$ and suppose that $E$ has good supersingular reduction at $p$. In this paper, we prove that the plus/minus Selmer group of $E$ over the anticyclotomic $\mathbb{Z}_{p}$-extension of $K$ has no non-trivial $Λ$-submodules of finite index under mild assumptions for $E$. This is an analogous result to R. Greenberg and B. D. Kim for the anticyclotomic $\mathbb{Z}_{p}$-extension essentially. By applying the results of A. Agboola--B. Howard or A. Burungale--K. Büyükboduk--A. Lei, we can also construct examples satisfying the assumptions of our theorem. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2308_16384 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On non-trivial $Λ$-submodules with finite index of the plus/minus Selmer group over anticyclotomic $\mathbb{Z}_{p}$-extension at inert primes Shii, Ryota Number Theory Let $K$ be an imaginary quadratic field where $p$ is inert. Let $E$ be an elliptic curve defined over $K$ and suppose that $E$ has good supersingular reduction at $p$. In this paper, we prove that the plus/minus Selmer group of $E$ over the anticyclotomic $\mathbb{Z}_{p}$-extension of $K$ has no non-trivial $Λ$-submodules of finite index under mild assumptions for $E$. This is an analogous result to R. Greenberg and B. D. Kim for the anticyclotomic $\mathbb{Z}_{p}$-extension essentially. By applying the results of A. Agboola--B. Howard or A. Burungale--K. Büyükboduk--A. Lei, we can also construct examples satisfying the assumptions of our theorem. |
| title | On non-trivial $Λ$-submodules with finite index of the plus/minus Selmer group over anticyclotomic $\mathbb{Z}_{p}$-extension at inert primes |
| topic | Number Theory |
| url | https://arxiv.org/abs/2308.16384 |