Submission date: 31 October 2023

Abstract:

Let X be a definable group definable over a small model M_0. Recall that a global type p on X is definable f-generic over M_0 if every left translate of p is definable over M_0. We call p strongly f-generic over M_0 if every left translate of p does not fork over M_0.
Let H be a group definable over the field ℚ_p of p-adic numbers admitting global definable f-generic types over ℚ_p. 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 ℚ_p and a ℚ_p-definable function θ such that θ(r) is finitely satisfiable in ℚ_p.
Recall that the μ-type μ(x) on H is the partial type consisting of the formulas over ℚ_p 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 ⊧ μ and a ⊧ r, we have ε.a ⊧ r. Let G be groups definable over ℚ_p 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.

Mathematics Subject Classification: 03C98 37B05 11E95 03C64 43A07

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Full text arXiv 2310.20110: pdf, ps.