Coordinate Transformation in Faltings' Extension

TL;DR

This paper computes the structure of certain differentials in a p-adic extension, finds linear relations among specific differential forms, and introduces a computable map to express these relations with polynomial coefficients.

Contribution

It introduces a differential version of Fontaine's map for explicit computation of linear relations among differential forms in Faltings' extension.

Findings

Linear dependence of specific differential forms in the extension.

Explicit polynomial expressions for coefficients in the linear equations.

Development of a computable map to express these equations.

Abstract

Analogue to Fontaine's computation for $\Omega_{\bar{\mathbb{Z}}_p/\mathbb{Z}_p}$, we compute the structure of $\Omega_{\mathcal{O}_{\bar{K}_0}/\mathcal{O}_{K_0}}$ (here $K_0$ is the completion of $\mathbb{Q}_p(T)$ at place $p$) and prove that $p^{1-1/p^n}\mathrm{d}p^{1/p^n}$, $T^{1-1/p^n}\mathrm{d}T^{1/p^n}$ and $S^{1-1/p^n}\mathrm{d}S^{1/p^n}$ are linearly dependent (Here $S := 1-T$). The main aim of this article is to find the linear equations for these three differential forms. Then we define a map which is called "differential version" of Fontaine's map to express the equations in a computable way. Finally, we prove that the coefficients in the equation can be expressed in some polynomial forms and compute some examples.