Characteristic ideal of the fine Selmer group and results on $μ$-invariance under isogeny in the function field case
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916356989911040 |
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| author | Ghosh, Sohan Ray, Jishnu |
| author_facet | Ghosh, Sohan Ray, Jishnu |
| contents | Consider a function field $K$ with characteristic $p>0$. We investigate the $Λ$-module structure of the Mordell-Weil group of an abelian variety over $\mathbb{Z}_p$-extensions of $K$, generalizing results due to Lee. Next, we study the algebraic structure and prove a control theorem for the S-fine Mordell-Weil groups, the function field analogue for Wuthrich's fine Mordell-Weil groups, over a $\mathbb{Z}_p$-extension of $K$. In case of unramified $\mathbb{Z}_p$-extension, $K_\infty$, we compute the characteristic ideal of the Pontryagin dual of the S-fine Mordell group. This provides an answer to an analogue of Greenberg's question for the characteristic ideal of the dual fine Selmer group in the function field setup. In the $\ell\neq p$ case, we prove the triviality of the $μ$-invariant for the Selmer group (same as the fine Selmer group in this case) of an elliptic curve over a non-commutative $GL_2(\mathbb{Z}_\ell)$-extension of $K$ and thus extending Conjecture A. In the $\ell=p$ case, we compute the change of $μ$-invariants of the dual Selmer groups of elliptic curves under isogeny, giving a lower bound for the $μ$-invariant. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_03201 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Characteristic ideal of the fine Selmer group and results on $μ$-invariance under isogeny in the function field case Ghosh, Sohan Ray, Jishnu Number Theory Consider a function field $K$ with characteristic $p>0$. We investigate the $Λ$-module structure of the Mordell-Weil group of an abelian variety over $\mathbb{Z}_p$-extensions of $K$, generalizing results due to Lee. Next, we study the algebraic structure and prove a control theorem for the S-fine Mordell-Weil groups, the function field analogue for Wuthrich's fine Mordell-Weil groups, over a $\mathbb{Z}_p$-extension of $K$. In case of unramified $\mathbb{Z}_p$-extension, $K_\infty$, we compute the characteristic ideal of the Pontryagin dual of the S-fine Mordell group. This provides an answer to an analogue of Greenberg's question for the characteristic ideal of the dual fine Selmer group in the function field setup. In the $\ell\neq p$ case, we prove the triviality of the $μ$-invariant for the Selmer group (same as the fine Selmer group in this case) of an elliptic curve over a non-commutative $GL_2(\mathbb{Z}_\ell)$-extension of $K$ and thus extending Conjecture A. In the $\ell=p$ case, we compute the change of $μ$-invariants of the dual Selmer groups of elliptic curves under isogeny, giving a lower bound for the $μ$-invariant. |
| title | Characteristic ideal of the fine Selmer group and results on $μ$-invariance under isogeny in the function field case |
| topic | Number Theory |
| url | https://arxiv.org/abs/2406.03201 |