On the Image of the $p$-adic Logarithm on Annuli of Principal Units
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866908788739538944 |
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| author | Sarkar, Mabud Ali |
| author_facet | Sarkar, Mabud Ali |
| contents | Let $K$ be a finite extension of $\mathbb{Q}_p$, and let $\mathfrak{m}_K$ be its maximal ideal. The image of the group of principal units $1+\mathfrak{m}_K$ under $p$-adic logarithm plays important role in several areas of number theory. In general, when the ramification index of $K/\mathbb{Q}_p$ is greater or equal to $p-1$, the precise description of this image is not known. For the cyclotomic extension $K=\mathbb{Q}_p(ζ_p)$ of degree $p-1$, it was previously proved in \cite{MAS} that the image of the annulus region $(1+\mathfrak{m}_K) \setminus (1+\mathfrak{m}_K^2)$ by $p$-adic logarithm is exactly $\mathfrak{m}_K^2$. In this paper, we give a self-contained analytic proof of this result based on explicit $p$-adic logarithmic expansions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_18187 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the Image of the $p$-adic Logarithm on Annuli of Principal Units Sarkar, Mabud Ali Number Theory 11F85, ~11S15, ~11R18, ~11Y40 Let $K$ be a finite extension of $\mathbb{Q}_p$, and let $\mathfrak{m}_K$ be its maximal ideal. The image of the group of principal units $1+\mathfrak{m}_K$ under $p$-adic logarithm plays important role in several areas of number theory. In general, when the ramification index of $K/\mathbb{Q}_p$ is greater or equal to $p-1$, the precise description of this image is not known. For the cyclotomic extension $K=\mathbb{Q}_p(ζ_p)$ of degree $p-1$, it was previously proved in \cite{MAS} that the image of the annulus region $(1+\mathfrak{m}_K) \setminus (1+\mathfrak{m}_K^2)$ by $p$-adic logarithm is exactly $\mathfrak{m}_K^2$. In this paper, we give a self-contained analytic proof of this result based on explicit $p$-adic logarithmic expansions. |
| title | On the Image of the $p$-adic Logarithm on Annuli of Principal Units |
| topic | Number Theory 11F85, ~11S15, ~11R18, ~11Y40 |
| url | https://arxiv.org/abs/2601.18187 |