Definable subgroups in $SL\_2$ over a p-adically closed field

TL;DR

This paper characterizes all definable subgroups of SL2(K) over p-adically closed fields, providing a detailed description of their structure and properties, with implications for model theory and algebraic groups.

Contribution

It offers a comprehensive classification of definable subgroups in SL2(K) over p-adically closed fields, including nilpotent, solvable, and generic subgroups, extending previous results.

Findings

Complete description of definable subgroups in SL2(K) for p-adically closed fields

Introduction of frame subgroups containing all nilpotent and solvable subgroups

Insights into genericity and generosity in SL2(K)

Abstract

The aim of this paper is to describe all definable subgroups of SL2(K), for K a p-adically closed field. We begin by giving some "frame subgroups" which contain all nilpotent or solvable subgroups of SL2(K). A complete description is givien for K a p-adically closed field, some results are generalizable to K a field elementarily equivalent to a finite extension of Qp and an almost complete description is given for Q an p . We give also some indications about genericity and generosity in SL2(K).