Forms of an affinoid disc and ramification

TL;DR

This paper classifies tamely ramified forms of an affinoid disc over a nonarchimedean field and constructs explicit examples of wildly ramified forms, also computing related algebraic invariants.

Contribution

It provides a classification of tamely ramified forms and explicit constructions of wildly ramified forms of affinoid discs, extending previous work on algebraic forms.

Findings

Classification of tamely ramified forms of the affinoid disc

Explicit families of wildly ramified forms constructed

Computed class group and Grothendieck group in specific cases

Abstract

Let $k$ be a complete nonarchimedean field and let $X$ be an affinoid closed disc over $k$. We classify the tamely ramified twisted forms of $X$. Generalizing work of P. Russell on inseparable forms of the affine line we construct explicit families of wildly ramified forms of $X$. We compute the class group and the Grothendieck group of forms of $X$ in certain cases.