2023/10/31 by Ningyuan Yao, Zhentao Zhang, Yao, Ningyuan +1
Mathematics · #Advanced Topology and Set Theory #Mathematical and Theoretical Analysis #Algebraic Geometry and Number Theory
paper · pdf · doi:10.48550/arxiv.2310.20110
Let X be a definable group definable over a small model M0. Recall that a global type p on X is definable f-generic over M0 if every left translate of p is definable over M0. We call p strongly f-generic over M0 if every left translate of p does not fork over M0. Let H be a group definable over the field \mathbb Qp of p-adic numbers admitting global definable f-generic types over \mathbb Qp. We show that H has unboundedly many global weakly generic types iff there is a global type r on H which is strongly f-generic over \mathbb Qp and a \mathbb Qp-definable function θ such that θ(r) is finitely satisfiable in \mathbb Qp. Recall that the μ-type μ(x) on H is the partial type consisting of the formulas over \mathbb Qp which define open neighborhoods of the identity of H. We show that every global weakly generic type r on H is μ-invariant: For any ε\models μ and a\models r, we have ε⋅ a\models r. Let G be groups definable over \mathbb Qp such that H is a normal subgroup of G and G/H is a definably compact group. Then we show that the weakly generic types on G coincide with almost periodic types G iff G has boundedly many global weakly generic types.