ABELIAN GROUPS DEFINABLE IN p-ADICALLY CLOSED FIELDS

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

$\mathbb {Q}_p$

.