Definable groups and fields in t-minimal theories

TL;DR

This paper explores the properties of definable groups and fields within t-minimal theories, establishing conditions for finiteness, largeness, and topological structures, including definable manifolds.

Contribution

It characterizes definable fields as either finite or large and introduces a canonical topology on abelian definable groups, extending to non-abelian groups in visceral t-minimal theories.

Findings

Definable fields are either finite or large in the sense of Pop.

A canonical topology can be assigned to any abelian definable group in t-minimal theories.

In visceral t-minimal theories, the topology on non-abelian groups forms a definable manifold.

Abstract

Let $T$ be a theory which is t-minimal, meaning that with respect to some definable topology, a unary definable set $D \subseteq M$ has non-empty interior iff it is infinite. If $K$ is a definable field in $T$, then $K$ is finite or "large" in the sense of Pop: any smooth algebraic curve $C$ over $K$ with at least one $K$-rational point has infinitely many $K$-rational points. We also assign a canonical topology to any abelian definable group $G$ in a t-minimal theory. In the case where the t-minimal theory is "visceral" in the sense of Dolich and Goodrick, meaning that the definable topology is induced by a definable uniformity, we can drop the assumption of abelianity of $G$, and the resulting topology on $G$ is a definable manifold in the style of Acosta L\'opez and Hasson.