The non-$p$-part of the fine Selmer group in a $\mathbf{Z}_p$-extension
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arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866929210339098624 |
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| author | Chakravarthy, Adithya |
| author_facet | Chakravarthy, Adithya |
| contents | Fix two distinct primes $p$ and $\ell$. Let $A$ be an abelian variety over $\mathbf{Q}(ζ_{\ell})$, the cyclotomic field of $\ell$-th roots of unity. Suppose that $A(\mathbf{Q}(ζ_{\ell}))[\ell] \neq 0$. We show that there exists a number field $L$ and a $\mathbf{Z}_p$ extension $L_{\infty}/L$ where the $\ell$-primary fine Selmer group of $A$ grows arbitrarily quickly. This is a fine Selmer group analogue of a theorem of Washington which says that there are certain (non-cyclotomic) $\mathbf{Z}_p$-extensions where the $\ell$-part of the class group can grow arbitrarily quickly. We also prove this for a wide class of non-commutative $p$-adic Lie extensions. Finally, we include several examples to illustrate this theorem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_07775 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The non-$p$-part of the fine Selmer group in a $\mathbf{Z}_p$-extension Chakravarthy, Adithya Number Theory Fix two distinct primes $p$ and $\ell$. Let $A$ be an abelian variety over $\mathbf{Q}(ζ_{\ell})$, the cyclotomic field of $\ell$-th roots of unity. Suppose that $A(\mathbf{Q}(ζ_{\ell}))[\ell] \neq 0$. We show that there exists a number field $L$ and a $\mathbf{Z}_p$ extension $L_{\infty}/L$ where the $\ell$-primary fine Selmer group of $A$ grows arbitrarily quickly. This is a fine Selmer group analogue of a theorem of Washington which says that there are certain (non-cyclotomic) $\mathbf{Z}_p$-extensions where the $\ell$-part of the class group can grow arbitrarily quickly. We also prove this for a wide class of non-commutative $p$-adic Lie extensions. Finally, we include several examples to illustrate this theorem. |
| title | The non-$p$-part of the fine Selmer group in a $\mathbf{Z}_p$-extension |
| topic | Number Theory |
| url | https://arxiv.org/abs/2401.07775 |