Definable equivalence relations and zeta functions of groups

Journal article

We prove that the theory of the $p$-adics ${\mathbb Q}_p$ admits elimination of imaginaries provided we add a sort for ${\rm GL}_n({\mathbb Q}_p)/{\rm GL}_n({\mathbb Z}_p)$ for each $n$. We also prove that the elimination of imaginaries is uniform in $p$. Using $p$-adic and motivic integration, we deduce the uniform rationality of certain formal zeta functions arising from definable equivalence relations. This also yields analogous results for definable equivalence relations over local fields...

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Authors: Hrushovski, E; Martin, B; Rideau, S; Cluckers, R

• Publication status: Not published
• Peer review status: Not peer reviewed
• Journal: arXiv
• Publication date: 2015-08-21
• Keywords: cell decompositions; elimination of imaginaries; subgroup zeta functions; 03C60, 03C10, 11M41, 20E07, 20C15; rational zeta functions; representation zeta functions; math.LO; math.GR; invariant extensions of types