The $p$-adic analytic subgroup theorem revisited

TL;DR

This paper revisits the $p$-adic analytic subgroup theorem, providing a new proof based on recent methods, thereby enhancing understanding of $p$-adic transcendence and its parallels to complex transcendence results.

Contribution

It offers a new proof of the $p$-adic analytic subgroup theorem using modern techniques, strengthening the theoretical framework in $p$-adic transcendence theory.

Findings

New proof of the $p$-adic analytic subgroup theorem

Clarification of the $p$-adic analogue of transcendence results

Enhanced understanding of $p$-adic transcendence properties

Abstract

It is well-known that the W\"ustholz' analytic subgroup theorem is one of the most powerful theorems in transcendence theory. The theorem gives in a very systematic and conceptual way the transcendence of a large class of complex numbers, e.g. the transcendence of $\pi$ which is originally due to Lindemann. In this paper we revisit the $p$-adic analogue of the analytic subgroup theorem and present a proof based on the method described and developed by the authors in a recent related paper.