An $O_K$-basis for the image of a Lubin-Tate logarithm on $\pi$-regular extensions of $K$

TL;DR

This paper investigates the image of Lubin-Tate logarithms on $ ext{pi}$-regular extensions of a $p$-adic field, providing explicit bases and valuation bounds for the associated $O_K$-modules.

Contribution

It explicitly constructs an $O_K$-basis for the Lubin-Tate logarithm image on $ ext{pi}$-regular extensions and extends results to arbitrary finite extensions of $K$.

Findings

Computed a basis for the additive group $ ext{log}_{[ ext{pi}]}( ext{F}( ext{m}_L))$ as an $O_K$-module.

Determined the minimal valuation of elements in the Lubin-Tate logarithm image.

Extended results to arbitrary finite extensions and specific Lubin-Tate extensions.

Abstract

Let $K$ be a finite $p$-adic field with uniformiser $\pi$. In this paper we study the image of the logarithm attached to a Lubin-Tate series $[\pi](X)$ on the maximal ideal of so-called $\pi$-regular extensions of $K$; for such an extension $L|K$ we compute a basis for the additive group $\log_{[\pi]}(\mathcal{F}(\mathfrak{m}_L))$ 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 $[\pi](X)$, and determine the minimal valuation of the elements in $\log_{[\pi]}(\mathcal{F}(\mathfrak{m}_L))$. 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 $\log_{[\pi]}(\mathcal{F}(\mathfrak{m}_{K_{\pi^n}}))$, where $K_{\pi^n}$ is the Lubin-Tate extension of level $n\geq 1$.