Submission date: 29 June 2022

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 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 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 ℚ_p.

Mathematics Subject Classification: 03C60

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Full text arXiv 2206.14364: pdf, ps.