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

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.

arXiv comment31 pages

References in corpus (1)

• Effective model-completeness for p-adic analytic structures

Cited by in corpus (3)

• Dp-minimal expansions of $(\mathbb{Z},+)$ via dense pairs via Mordell-Lang
• Expansions of the $p$-adic numbers that interprets the ring of integers
• Nippy proofs of p-adic results of Delon and Yao