Iwasawa theory of fine Selmer groups associated to Drinfeld modules
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arXiv
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866917706549166080 |
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| author | Ray, Anwesh |
| author_facet | Ray, Anwesh |
| contents | Let $q$ be a prime power and $F=\mathbb{F}_q(T)$ be the rational function field over $\mathbb{F}_q$, the field with $q$ elements. Let $ϕ$ be a Drinfeld module over $F$ and $\mathfrak{p}$ be a non-zero prime ideal of $A:=\mathbb{F}_q[T]$. Over the constant $\mathbb{Z}_p$-extension of $F$, we introduce the fine Selmer group associated to the $\mathfrak{p}$-primary torsion of $ϕ$. We show that it is a cofinitely generated module over $A_{\mathfrak{p}}$. This proves an analogue of Iwasawa's $μ=0$ conjecture in this setting, and provides context for the further study of the objects that have been introduced in this article. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_06499 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Iwasawa theory of fine Selmer groups associated to Drinfeld modules Ray, Anwesh Number Theory Algebraic Geometry 11R23, 11G09 Let $q$ be a prime power and $F=\mathbb{F}_q(T)$ be the rational function field over $\mathbb{F}_q$, the field with $q$ elements. Let $ϕ$ be a Drinfeld module over $F$ and $\mathfrak{p}$ be a non-zero prime ideal of $A:=\mathbb{F}_q[T]$. Over the constant $\mathbb{Z}_p$-extension of $F$, we introduce the fine Selmer group associated to the $\mathfrak{p}$-primary torsion of $ϕ$. We show that it is a cofinitely generated module over $A_{\mathfrak{p}}$. This proves an analogue of Iwasawa's $μ=0$ conjecture in this setting, and provides context for the further study of the objects that have been introduced in this article. |
| title | Iwasawa theory of fine Selmer groups associated to Drinfeld modules |
| topic | Number Theory Algebraic Geometry 11R23, 11G09 |
| url | https://arxiv.org/abs/2311.06499 |