Abelian groups definable in $p$-adically closed fields

TL;DR

This paper classifies abelian groups definable in p-adically closed fields, showing they are extensions of compact groups by groups with definable generics, and provides a detailed analysis for groups in the p-adic numbers.

Contribution

It establishes a structural decomposition of abelian definable groups in p-adic fields and relates them to algebraic groups, advancing understanding of their model-theoretic properties.

Findings

Abelian definable groups are extensions of fsg groups by dfg groups.

In dically closed fields, G^0 = G^{00} for definable abelian groups.

In dically closed fields, such groups are open subgroups of algebraic groups, up to finite factors.

Abstract

Recall that a group $G$ has finitely satisfiable generics ($fsg$) or definable $f$-generics ($dfg$) if there is a global type $p$ on $G$ and a small model $M_0$ such that every left translate of $p$ is finitely satisfiable in $M_0$ or definable over $M_0$, respectively. We show that any abelian group definable in a $p$-adically closed field is an extension of a definably compact $fsg$ definable group by a $dfg$ definable group. We discuss an approach which might prove a similar statement for interpretable abelian groups. In the case where $G$ is an abelian group definable in the standard model $\mathbb{Q}_p$, we show that $G^0 = G^{00}$, and that $G$ is an open subgroup of an algebraic group, up to finite factors. This latter result can be seen as a rough classification of abelian definable groups in $\mathbb{Q}_p$.