Definably semisimple groups interpretable in $p$-adically closed fields

TL;DR

This paper proves that definably semisimple groups interpretable in p-adically closed fields are essentially finite extensions of K-linear groups, extending the result to models of the theory of the p-adic field with analytic structure.

Contribution

It establishes a structural classification of definably semisimple interpretable groups in p-adic fields, showing they are closely related to linear algebraic groups over the field.

Findings

Definably semisimple groups have a finite normal subgroup with a linear quotient.

The classification extends to models of the theory of p-adic fields with analytic structure.

Provides a structural understanding of interpretable groups in p-adic contexts.

Abstract

Let $K$ be a $p$-adically closed field and $G$ a group interpretable in $K$. We show that if $G$ is definably semisimple (i.e. $G$ has no definable infinite normal abelian subgroups) then there exists a finite normal subgroup $H$ such that $G/H$ is definably isomorphic to a $K$-linear group. The result remains true in models of $\mathrm{Th}(\mathbb{Q}_p^{an})$.