A note on groups definable in the p-adic field

TL;DR

This paper investigates the structure of groups definable in p-adic fields, showing that commutative groups are essentially finite extensions of commutative algebraic groups, with specific results for one-dimensional cases.

Contribution

It proves that definable groups in p-adic fields are commutative-by-finite when the associated algebraic group is commutative, extending to p-adically closed fields.

Findings

Definable groups in Q_p are commutative-by-finite if the algebraic group is commutative.

One-dimensional definable groups in Q_p are necessarily commutative-by-finite.

Results extend to groups definable in p-adically closed fields.

Abstract

It is known that a group G definable in the field of p-adic numbers is definably locally isomorphic to the group of Q_p-points of a connected algebraic group H defined over Q_p. We show that if H is commutative then G is commutative-by-finite. It follows in particular that any one-dimensional group definable in Q_p is commutative-by-finite. The results extend to groups definable in p-adically closed fields.