Local fields, iterated extensions, and Julia Sets
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866908955517648896 |
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| author | Lee, Pui Hang Manes, Michelle Truong, Nha Xuan |
| author_facet | Lee, Pui Hang Manes, Michelle Truong, Nha Xuan |
| contents | Let $K$ be a field complete with respect to a discrete valuation $v$ of residue characteristic $p$. For $α\in K$, let $K_\infty$ be the extension obtained by adjoining all iterated preimages of $α$ under a unicritical polynomial $f_c(z)=z^\ell - c \in K[z]$. We study the extension $K_\infty/K$ and show that its qualitative behavior depends only on the valuation of $c$. This removes the previous restrictions on $\ell$ in work of Anderson--Hamblen--Poonen--Walton and completes the classification for all $\ell \ge 2$. We also relate the ramification to the structure of the Berkovich Julia set of $f_c$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_17961 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Local fields, iterated extensions, and Julia Sets Lee, Pui Hang Manes, Michelle Truong, Nha Xuan Number Theory Dynamical Systems 11S82, 37P05, 37P20, 37P40 Let $K$ be a field complete with respect to a discrete valuation $v$ of residue characteristic $p$. For $α\in K$, let $K_\infty$ be the extension obtained by adjoining all iterated preimages of $α$ under a unicritical polynomial $f_c(z)=z^\ell - c \in K[z]$. We study the extension $K_\infty/K$ and show that its qualitative behavior depends only on the valuation of $c$. This removes the previous restrictions on $\ell$ in work of Anderson--Hamblen--Poonen--Walton and completes the classification for all $\ell \ge 2$. We also relate the ramification to the structure of the Berkovich Julia set of $f_c$. |
| title | Local fields, iterated extensions, and Julia Sets |
| topic | Number Theory Dynamical Systems 11S82, 37P05, 37P20, 37P40 |
| url | https://arxiv.org/abs/2501.17961 |