ABELIAN GROUPS DEFINABLE IN p-ADICALLY CLOSED FIELDS

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 M₀ such that every left translate of p is finitely satisfiable in M₀ or definable over M₀ , 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 Qₚ , we show that G⁰ = G⁰⁰ , 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 Qₚ .