Enregistré dans:
| Auteurs principaux: | , |
|---|---|
| Format: | Preprint |
| Publié: |
2023
|
| Sujets: | |
| Accès en ligne: | https://arxiv.org/abs/2303.06063 |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866918035371065344 |
|---|---|
| author | Johnson, Will Ye, Jinhe |
| author_facet | Johnson, Will Ye, Jinhe |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_06063 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Curve-excluding fields Johnson, Will Ye, Jinhe Logic Algebraic Geometry 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. |
| title | Curve-excluding fields |
| topic | Logic Algebraic Geometry |
| url | https://arxiv.org/abs/2303.06063 |