Abelian groups definable in $p$-adically closed fields
| Title: | Abelian groups definable in $p$-adically closed fields |
|---|---|
| Authors: | Johnson, Will, Yao, Ningyuan |
| Publication Year: | 2022 |
| Collection: | Mathematics |
| Subject Terms: | Mathematics - Logic, 03C60 |
| More Details: | Recall that a group $G$ has finitely satisfiable generics ($fsg$) or definable $f$-generics ($dfg$) if there is a global type $p$ on $G$ and a small model $M_0$ such that every left translate of $p$ is finitely satisfiable in $M_0$ or definable over $M_0$, respectively. We show that any abelian group definable in a $p$-adically closed field is an extension of a definably compact $fsg$ definable group by a $dfg$ definable group. We discuss an approach which might prove a similar statement for interpretable abelian groups. In the case where $G$ is an abelian group definable in the standard model $\mathbb{Q}_p$, we show that $G^0 = G^{00}$, and that $G$ is an open subgroup of an algebraic group, up to finite factors. This latter result can be seen as a rough classification of abelian definable groups in $\mathbb{Q}_p$. Comment: 20 pages; updated references and fixed subscript typo in abstract |
| Document Type: | Working Paper |
| Access URL: | http://arxiv.org/abs/2206.14364 |
| Accession Number: | edsarx.2206.14364 |
| Database: | arXiv |
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| Items | – Name: Title Label: Title Group: Ti Data: Abelian groups definable in $p$-adically closed fields – Name: Author Label: Authors Group: Au Data: <searchLink fieldCode="AR" term="%22Johnson%2C+Will%22">Johnson, Will</searchLink><br /><searchLink fieldCode="AR" term="%22Yao%2C+Ningyuan%22">Yao, Ningyuan</searchLink> – Name: DatePubCY Label: Publication Year Group: Date Data: 2022 – Name: Subset Label: Collection Group: HoldingsInfo Data: Mathematics – Name: Subject Label: Subject Terms Group: Su Data: <searchLink fieldCode="DE" term="%22Mathematics+-+Logic%22">Mathematics - Logic</searchLink><br /><searchLink fieldCode="DE" term="%2203C60%22">03C60</searchLink> – Name: Abstract Label: Description Group: Ab Data: Recall that a group $G$ has finitely satisfiable generics ($fsg$) or definable $f$-generics ($dfg$) if there is a global type $p$ on $G$ and a small model $M_0$ such that every left translate of $p$ is finitely satisfiable in $M_0$ or definable over $M_0$, respectively. We show that any abelian group definable in a $p$-adically closed field is an extension of a definably compact $fsg$ definable group by a $dfg$ definable group. We discuss an approach which might prove a similar statement for interpretable abelian groups. In the case where $G$ is an abelian group definable in the standard model $\mathbb{Q}_p$, we show that $G^0 = G^{00}$, and that $G$ is an open subgroup of an algebraic group, up to finite factors. This latter result can be seen as a rough classification of abelian definable groups in $\mathbb{Q}_p$.<br />Comment: 20 pages; updated references and fixed subscript typo in abstract – Name: TypeDocument Label: Document Type Group: TypDoc Data: Working Paper – Name: URL Label: Access URL Group: URL Data: <link linkTarget="URL" linkTerm="http://arxiv.org/abs/2206.14364" linkWindow="_blank">http://arxiv.org/abs/2206.14364</link> – Name: AN Label: Accession Number Group: ID Data: edsarx.2206.14364 |
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| RecordInfo | BibRecord: BibEntity: Subjects: – SubjectFull: Mathematics - Logic Type: general – SubjectFull: 03C60 Type: general Titles: – TitleFull: Abelian groups definable in $p$-adically closed fields Type: main BibRelationships: HasContributorRelationships: – PersonEntity: Name: NameFull: Johnson, Will – PersonEntity: Name: NameFull: Yao, Ningyuan IsPartOfRelationships: – BibEntity: Dates: – D: 28 M: 06 Type: published Y: 2022 |
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