Enregistré dans:
| Auteur principal: | |
|---|---|
| Format: | Preprint |
| Publié: |
2026
|
| Sujets: | |
| Accès en ligne: | https://arxiv.org/abs/2604.27111 |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866918474855481344 |
|---|---|
| author | Harbor-Collins, Georgia |
| author_facet | Harbor-Collins, Georgia |
| contents | Let $K$ be a finite $p$-adic field with uniformiser $π$. In this paper we study the image of the logarithm attached to a Lubin-Tate series $[π](X)$ on the maximal ideal of so-called $π$-regular extensions of $K$; for such an extension $L|K$ we compute a basis for the additive group $\log_{[π]}(\mathcal{F}(\mathfrak{m}_L))$ as an $O_K$-module, where $\mathcal{F}(\mathfrak{m}_L)$ denotes the maximal ideal $\mathfrak{m}_L$ equipped with the $O_K$-module structure coming from the formal group associated to $[π](X)$, and determine the minimal valuation of the elements in $\log_{[π]}(\mathcal{F}(\mathfrak{m}_L))$. In the final section of this paper we discuss how some of these results extend to arbitrary finite extensions of $K$ and conclude by determining a basis of the $O_K$-module $\log_{[π]}(\mathcal{F}(\mathfrak{m}_{K_{π^n}}))$, where $K_{π^n}$ is the Lubin-Tate extension of level $n\geq 1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_27111 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | An $O_K$-basis for the image of a Lubin-Tate logarithm on $π$-regular extensions of $K$ Harbor-Collins, Georgia Number Theory Let $K$ be a finite $p$-adic field with uniformiser $π$. In this paper we study the image of the logarithm attached to a Lubin-Tate series $[π](X)$ on the maximal ideal of so-called $π$-regular extensions of $K$; for such an extension $L|K$ we compute a basis for the additive group $\log_{[π]}(\mathcal{F}(\mathfrak{m}_L))$ as an $O_K$-module, where $\mathcal{F}(\mathfrak{m}_L)$ denotes the maximal ideal $\mathfrak{m}_L$ equipped with the $O_K$-module structure coming from the formal group associated to $[π](X)$, and determine the minimal valuation of the elements in $\log_{[π]}(\mathcal{F}(\mathfrak{m}_L))$. In the final section of this paper we discuss how some of these results extend to arbitrary finite extensions of $K$ and conclude by determining a basis of the $O_K$-module $\log_{[π]}(\mathcal{F}(\mathfrak{m}_{K_{π^n}}))$, where $K_{π^n}$ is the Lubin-Tate extension of level $n\geq 1$. |
| title | An $O_K$-basis for the image of a Lubin-Tate logarithm on $π$-regular extensions of $K$ |
| topic | Number Theory |
| url | https://arxiv.org/abs/2604.27111 |