Curve-excluding fields

TL;DR

This work introduces $C\mathrm XF$, the model companion for fields of characteristic $0$ avoiding a curve $C$ of genus at least $2$, and provides a complete classification of completions via the relative algebraic closure Abs$(K_0)$ in models. It develops a computable axiomatization, proves existence of the model companion, and shows rich field-theoretic and model-theoretic behavior: $C\mathrm XF$ fields are Hilbertian with unbounded and $\\omega$-free Galois groups, algebraically bounded with acl coinciding with field-theoretic algebraic closure, and they enjoy quantifier elimination after solvability predicates; they are NSOP$_4$ with TP$_2$, and contain SOP$_3$ examples, yielding strictly NSOP$_4$ pure fields. The completions are controlled by Abs$(M)$, enabling decidable, non-large models, while also providing virtually large fields and a framework for constructing fields with intricate combinations of logical and algebraic properties. The paper also derives an explicit bound on the complexity of rational morphisms and outlines broad generalizations to other varieties and logical theories, opening several avenues for further exploration in algebraic geometry and model theory of 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.

Theorems & Definitions (93)

• Theorem 1.5: = Corollary \ref{['cor:non-large-model-complete']}, Theorem \ref{['thm:unbdd']}
• Theorem 1.6: = Theorem \ref{['thm:cxf-exist']}, Remark \ref{['rem:computable']}
• Theorem 1.7
• Theorem 1.8: = Theorem \ref{['large-characterize']}
• Corollary 1.9: = Corollary \ref{['cor:decidable-non-large']}
• Theorem 1.10
• Theorem 1.11
• Theorem 1.12: = Theorem \ref{['nsop4']}, Corollary \ref{['tp2']}
• Theorem 1.13: = Theorem \ref{['thm:some-sop3']}
• Corollary 1.14: = Corollary \ref{['strict-nsop4']}