2019/11/04 by Pillay, Anand, Yao, Ningyuan
#03C98 #20G25 #22E35 #FOS: Mathematics #Logic (math.LO)
paper · doi:10.48550/arxiv.1911.01833
The aim of this paper is to develop the theory of groups definable in the p-adic field \mathbb Qp, with ``definable f-generics" in the sense of an ambient saturated elementary extension of \mathbb Qp. We call such groups definable f-generic groups. So, by a ``definable f-generic'' or dfg group we mean a definable group in a saturated model with a global f-generic type which is definable over a small model. In the present context the group is definable over \mathbb Qp, and the small model will be \mathbb Qp itself. The notion of a dfg group is dual, or rather opposite to that of an fsg group (group with ``finitely satisfiable generics") and is a useful tool to describe the analogue of torsion free o-minimal groups in the p-adic context. In the current paper our group will be definable over \mathbb Qp in an ambient saturated elementary extension \mathbb K of \mathbb Qp, so as to make sense of the notions of f-generic etc. In this paper we will show that every definable f-generic group definable in \mathbb Qp is virtually isomorphic to a finite index subgroup of a trigonalizable algebraic group over \mathbb Qp. This is analogous to the o-minimal context, where every connected torsion free group definable in \mathbb R is isomorphic to a trigonalizable algebraic group (Lemma 3.4, \citeCOS). We will also show that every open definable f-generic subgroup of a definable f-generic group has finite index, and every f-generic type of a definable f-generic group is almost periodic, which gives a positive answer to the problem raised in \citeP-Y of whether f-generic types coincide with almost periodic types in the p-adic case.