On simple groups definable in some valued fields

TL;DR

This paper demonstrates that non-abelian definably simple groups in certain valued fields are essentially linear algebraic groups, extending understanding of their structure in model theory and algebraic geometry.

Contribution

It establishes that such groups are essentially linear algebraic, providing a classification in 1-h-minimal henselian and algebraically closed valued fields.

Findings

Non-abelian definably simple groups are linear algebraic in 1-h-minimal henselian fields.

Similar results hold in algebraically closed valued fields with subgroup assumptions.

The work bridges model theory and algebraic group classification.

Abstract

We prove that non-abelian definable, definably simple groups in 1-h-minimal henselian valued fields are essentially already linear algebraic groups. Here, the group is assumed to live in the home sort. We have a similar result in pure algebraically closed valued fields of positive characteristic, under the additional assumption that the definable group is a subgroup of a linear algebraic group.