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ABELIAN GROUPS DEFINABLE IN p-ADICALLY CLOSED FIELDS
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- Journal:
- The Journal of Symbolic Logic , First View
- Published online by Cambridge University Press:
- 18 July 2023, pp. 1-22
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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 
$\mathbb {Q}_p$.
