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ABELIAN GROUPS DEFINABLE IN p-ADICALLY CLOSED FIELDS

Part of: Model theory

Published online by Cambridge University Press:  18 July 2023

WILL JOHNSON Open the ORCID record for WILL JOHNSON [Opens in a new window]  and
NINGYUAN YAO
WILL JOHNSON*
Affiliation:
SCHOOL OF PHILOSOPHY FUDAN UNIVERSITY 220 HANDAN ROAD GUANGHUA WEST BUILDING ROOM 2503, SHANGHAI 200433, CHINA E-mail: yaony@fudan.edu.cn URL: http://philosophy.fudan.edu.cn/64/8b/c14253a222347/page.htm
NINGYUAN YAO
Affiliation:
SCHOOL OF PHILOSOPHY FUDAN UNIVERSITY 220 HANDAN ROAD GUANGHUA WEST BUILDING ROOM 2503, SHANGHAI 200433, CHINA E-mail: yaony@fudan.edu.cn URL: http://philosophy.fudan.edu.cn/64/8b/c14253a222347/page.htm
*
E-mail: willjohnson@fudan.edu.cn
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Abstract

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 $\mathbb {Q}_p$.

Keywords

definable groupsp-adically closed fieldsfinitely satisfiable generics

MSC classification

Primary: 03C60: Model-theoretic algebra

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Type
Article
Information
The Journal of Symbolic Logic , Volume 90 , Issue 1 , March 2025 , pp. 460 - 481
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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