On anticyclotomic Selmer groups of elliptic curves
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866917014358982656 |
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| author | Longo, Matteo Ray, Jishnu Vigni, Stefano |
| author_facet | Longo, Matteo Ray, Jishnu Vigni, Stefano |
| contents | Let $p\geq5$ be a prime number and let $K$ be an imaginary quadratic field where $p$ is unramified. Under mild technical assumptions, in this paper we prove the non-existence of non-trivial finite $Λ$-submodules of Pontryagin duals of signed Selmer groups of a $p$-supersingular rational elliptic curve over the anticyclotomic $\mathbb Z_p$-extension of $K$, where $Λ$ is the corresponding Iwasawa algebra. In particular, we work under the assumption that our plus/minus Selmer groups have $Λ$-corank $1$, so they are not $Λ$-cotorsion. Our main theorem extends to the supersingular case analogous non-existence results by Bertolini in the ordinary setting; furthermore, since we cover the case where $p$ is inert in $K$, we refine previous results of Hatley-Lei-Vigni, which deal with $p$-supersingular elliptic curves under the assumption that $p$ splits in $K$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_01696 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On anticyclotomic Selmer groups of elliptic curves Longo, Matteo Ray, Jishnu Vigni, Stefano Number Theory Algebraic Geometry 11R223, 11G05 Let $p\geq5$ be a prime number and let $K$ be an imaginary quadratic field where $p$ is unramified. Under mild technical assumptions, in this paper we prove the non-existence of non-trivial finite $Λ$-submodules of Pontryagin duals of signed Selmer groups of a $p$-supersingular rational elliptic curve over the anticyclotomic $\mathbb Z_p$-extension of $K$, where $Λ$ is the corresponding Iwasawa algebra. In particular, we work under the assumption that our plus/minus Selmer groups have $Λ$-corank $1$, so they are not $Λ$-cotorsion. Our main theorem extends to the supersingular case analogous non-existence results by Bertolini in the ordinary setting; furthermore, since we cover the case where $p$ is inert in $K$, we refine previous results of Hatley-Lei-Vigni, which deal with $p$-supersingular elliptic curves under the assumption that $p$ splits in $K$. |
| title | On anticyclotomic Selmer groups of elliptic curves |
| topic | Number Theory Algebraic Geometry 11R223, 11G05 |
| url | https://arxiv.org/abs/2504.01696 |