Zeta functions from definable equivalence relations

Details Zeta functions from definable equivalence relations Zeta functions from definable equivalence relations

2006-12-31

arxiv.org/abs/math/0701011v1

Mathematics

Logic

39 pages

Scientific paper

We prove that the theory of the $p$-adics $Q_p$, together with a set of explicitly given sorts, admits elimination of imaginaries. Using $p$-adic integration, we deduce the rationality of certain formal zeta functions arising from definable equivalence relations. As an application, we prove rationality results for zeta functions obtained by counting isomorphism classes of irreducible representations of finitely generated nilpotent groups; these are analogous to similar results of Grunewald, Segal and Smith for subgroup growth zeta functions of finitely generated nilpotent groups.

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