Admissibility of groups over function fields of p-adic curves

TL;DR

This paper investigates the conditions under which finite groups are admissible over function fields of p-adic curves, providing necessary criteria, examples of non-admissibility, and a Hasse principle for division algebras.

Contribution

It introduces necessary conditions for group admissibility over these fields, presents a specific non-admissible group example, and establishes a Hasse principle for division algebras.

Findings

Identified necessary conditions for admissibility.

Constructed an example of a non-admissible group over Qp(t).

Proved a Hasse principle for division algebras over these fields.

Abstract

Let K be a field and G a finite group. The question of 'admissibility' of G over K was originally posed by Schacher, who gave partial results in the case K = Q. In this paper, we give necessary conditions for admissibility of a finite group G over function fields of curves over complete discretely valued fields. Using this criterion, we give an example of a finite group which is not admissible over Qp(t). We also prove a certain Hasse principle for division algebras over such fields.