On profinite subgroups of an algebraic group over a local field

TL;DR

This paper explores the relationship between anisotropy and compactness in algebraic groups over local fields, establishing conditions for the existence of maximal compact subgroups and analyzing maximal pro-p subgroups using Bruhat-Tits theory.

Contribution

It provides new criteria linking anisotropy to compactness in algebraic groups over local fields and thoroughly investigates maximal pro-p subgroups in the semisimple case.

Findings

Equivalent conditions for rational points to admit maximal compact subgroups.

Characterization of maximal pro-p subgroups in semisimple groups.

Application of Bruhat-Tits theory to subgroup structure analysis.

Abstract

The purpose of this paper is to link anisotropy properties of an algebraic group together with compactness issues in the topological group of its rational points. We nd equivalent conditions on a smooth ane algebraic group scheme over a non-Archimedean local eld for the associated rational points to admit maximal compact subgroups. We use the structure theory of pseudo-reductive groups provided, whatever the characteristic, by Conrad, Gabber and Prasad. We also investigate thoroughly maximal prop subgroups in the semisimple case, using Bruhat-Tits theory.