Definable equivalence relations and zeta functions of groups (with an appendix by Raf Cluckers)
We prove that the theory of the p-adics Qp admits elimination of imaginaries provided we add a sort for GLn(Qp)/GLn(Zp) 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 functi...
| Principais autores: | , , |
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| Formato: | Journal article |
| Idioma: | English |
| Publicado em: |
European Mathematical Society Publishing House
2018
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| _version_ | 1826292520510816256 |
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| author | Hrushovski, E Martin, B Rideau, S |
| author_facet | Hrushovski, E Martin, B Rideau, S |
| author_sort | Hrushovski, E |
| collection | OXFORD |
| description | We prove that the theory of the p-adics Qp admits elimination of imaginaries provided we add a sort for GLn(Qp)/GLn(Zp) 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 of positive characteristic. The appendix contains an alternative proof, using cell decomposition, of the rationality (for fixed p) of these formal zeta functions that extends to the subanalytic context.
As an application, we prove rationality and uniformity results for zeta functions obtained by counting twist isomorphism classes of irreducible representations of finitely generated nilpotent groups; these are analogous to similar results of Grunewald, Segal and Smith and of du Sautoy and Grunewald for subgroup zeta functions of finitely generated nilpotent groups. |
| first_indexed | 2024-03-07T03:15:58Z |
| format | Journal article |
| id | oxford-uuid:b5c6d6ba-734f-4dd6-accb-dcfea64b94b4 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-07T03:15:58Z |
| publishDate | 2018 |
| publisher | European Mathematical Society Publishing House |
| record_format | dspace |
| spelling | oxford-uuid:b5c6d6ba-734f-4dd6-accb-dcfea64b94b42022-03-27T04:36:12ZDefinable equivalence relations and zeta functions of groups (with an appendix by Raf Cluckers)Journal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:b5c6d6ba-734f-4dd6-accb-dcfea64b94b4EnglishSymplectic ElementsEuropean Mathematical Society Publishing House2018Hrushovski, EMartin, BRideau, SWe prove that the theory of the p-adics Qp admits elimination of imaginaries provided we add a sort for GLn(Qp)/GLn(Zp) 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 of positive characteristic. The appendix contains an alternative proof, using cell decomposition, of the rationality (for fixed p) of these formal zeta functions that extends to the subanalytic context. As an application, we prove rationality and uniformity results for zeta functions obtained by counting twist isomorphism classes of irreducible representations of finitely generated nilpotent groups; these are analogous to similar results of Grunewald, Segal and Smith and of du Sautoy and Grunewald for subgroup zeta functions of finitely generated nilpotent groups. |
| spellingShingle | Hrushovski, E Martin, B Rideau, S Definable equivalence relations and zeta functions of groups (with an appendix by Raf Cluckers) |
| title | Definable equivalence relations and zeta functions of groups (with an appendix by Raf Cluckers) |
| title_full | Definable equivalence relations and zeta functions of groups (with an appendix by Raf Cluckers) |
| title_fullStr | Definable equivalence relations and zeta functions of groups (with an appendix by Raf Cluckers) |
| title_full_unstemmed | Definable equivalence relations and zeta functions of groups (with an appendix by Raf Cluckers) |
| title_short | Definable equivalence relations and zeta functions of groups (with an appendix by Raf Cluckers) |
| title_sort | definable equivalence relations and zeta functions of groups with an appendix by raf cluckers |
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