Curve-excluding fields

Abstract

If $C$ is a curve over $\mathbb{Q}$ with genus at least $2$ and $C(\mathbb{Q})$ is empty, then the class of fields $K$ of characteristic 0 such that $C(K)=\varnothing$ has a model companion, which we call $C\mathrm{XF}$. The theory $C\mathrm{XF}$ is not complete, but we characterize the completions. Using $C\mathrm{XF}$, we produce examples of fields with interesting combinations of properties. For example, we produce (1) a model-complete field with unbounded Galois group, (2) an infinite field with a decidable first-order theory that is not “large” in the sense of Pop, (3) a field that is algebraically bounded but not “very slim” in the sense of Junker and Koenigsmann, and (4) a pure field that is strictly NSOP${}_4$, i.e., NSOP${}_4$ but not NSOP${}_3$. Lastly, we give a new construction of fields that are virtually large but not large.