Model theory of the field of $p$-adic numbers expanded by a multiplicative subgroup

TL;DR

This paper develops a model-theoretic framework for the $p$-adic numbers expanded by a multiplicative subgroup, providing axioms, describing definable sets, and establishing NIP properties under certain conditions.

Contribution

It offers the first comprehensive axiomatization of the theory of $(Q_p, G)$ with a multiplicative subgroup satisfying Mann property, including definability and NIP results.

Findings

Axiomatization of the theory of $(Q_p, G)$ with such subgroups.

Description of definable sets when $G^{[n]}$ have finite index.

Proof that the theory is NIP under certain subgroup conditions.

Abstract

Let $G$ be a multiplicative subgroup of $\mathbb{Q}_p$. In this paper, we describe the theory of the pair $(\mathbb{Q}_p, G)$ under the condition that $G$ satisfies Mann property and is small as subset of a first-order structure. First, we give an axiomatisation of the first-order theory of this structure. This includes an axiomatisation of the theory of the group $G$ as valued group (with the valuation induced on $G$ by the $p$-adic valuation). If the subgroups $G^{[n]}$ of $G$ have finite index for all $n$, we describe the definable sets in this theory and prove that it is NIP. Finally, we extend some of our results to the subanalytic setting.