Definable equivalence relations and zeta functions of groups

Ehud Hrushovski; Ben Martin; Silvain Rideau; Raf Cluckers

doi:10.4171/JEMS/817

Definable equivalence relations and zeta functions of groups

Ehud Hrushovski, Ben Martin, Silvain Rideau, Raf Cluckers

Mathematical Science

Research output: Contribution to journal › Article › peer-review

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Abstract

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.

Original language	English
Pages (from-to)	2467-2537
Number of pages	71
Journal	Journal of the European Mathematical Society
Volume	20
Issue number	10
DOIs	https://doi.org/10.4171/JEMS/817
Publication status	Published - 18 Jul 2018

Bibliographical note

The authors wish to thank Thomas Rohwer, Deirdre Haskell, Dugald Macpherson and Elisabeth Bouscaren for their comments on earlier drafts of this work, Martin Hils for suggesting that the proof could be adapted to finite extensions and
Zo´e Chatzidakis for pointing out an error in how constants were handled in earlier versions. The second author is grateful to Jamshid Derakhshan, Marcus du Sautoy, Andrei Jaikin-Zapirain, Angus Macintyre, Dugald Macpherson, Mark Ryten, Christopher Voll and Michele Zordan for helpful conversations. We are grateful to Alex Lubotzky for suggesting studying representation growth; several of the ideas in Section 8 are due to him. The first author was supported by the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013) / ERC Grant agreement no. 291111/ MODAG, the second author was supported by a Golda Meir Postdoctoral Fellowship at the Hebrew University of Jerusalem and the third author was partly supported by ANR MODIG (ANR-09-BLAN-0047) Model Theory and Interactions with Geometry. The author of the appendix would like to thank M. du Sautoy, C. Voll, and Kien Huu Nguyen for interesting discussions on this and related subjects. He was partially supported by the European Research Council under the European Community’s Seventh Framework Programme (FP7/2007-2013) with ERC Grant Agreement nr. 615722 MOTMELSUM and he thanks the Labex CEMPI (ANR-11-LABX-0007-01). We are grateful to the referee for their careful reading of the paper and for their many comments, corrections and suggestions for improving the exposition.

In memory of Fritz Grunewald.

Keywords

Elimination of imaginaries
invariant extensions of types
cell decompositions
rational zeta functions
subgroup zeta functions
representation zeta functions

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10.4171/JEMS/817Licence: Unspecified

Accepted Manuscript
The definitive version is published in Journal of the European Mathematical Society.
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https://arxiv.org/abs/math/0701011Licence: Unspecified

Ben Martin
- School of Natural & Computing Sciences, Mathematical Science - Personal Chair
Person: Academic

Cite this

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abstract = "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 fromdefinable 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.",

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author = "Ehud Hrushovski and Ben Martin and Silvain Rideau and Raf Cluckers",

note = "The authors wish to thank Thomas Rohwer, Deirdre Haskell, Dugald Macpherson and Elisabeth Bouscaren for their comments on earlier drafts of this work, Martin Hils for suggesting that the proof could be adapted to finite extensions and Zo´e Chatzidakis for pointing out an error in how constants were handled in earlier versions. The second author is grateful to Jamshid Derakhshan, Marcus du Sautoy, Andrei Jaikin-Zapirain, Angus Macintyre, Dugald Macpherson, Mark Ryten, Christopher Voll and Michele Zordan for helpful conversations. We are grateful to Alex Lubotzky for suggesting studying representation growth; several of the ideas in Section 8 are due to him. The first author was supported by the European Research Council under the European Union{\textquoteright}s Seventh Framework Programme (FP7/2007-2013) / ERC Grant agreement no. 291111/ MODAG, the second author was supported by a Golda Meir Postdoctoral Fellowship at the Hebrew University of Jerusalem and the third author was partly supported by ANR MODIG (ANR-09-BLAN-0047) Model Theory and Interactions with Geometry. The author of the appendix would like to thank M. du Sautoy, C. Voll, and Kien Huu Nguyen for interesting discussions on this and related subjects. He was partially supported by the European Research Council under the European Community{\textquoteright}s Seventh Framework Programme (FP7/2007-2013) with ERC Grant Agreement nr. 615722 MOTMELSUM and he thanks the Labex CEMPI (ANR-11-LABX-0007-01). We are grateful to the referee for their careful reading of the paper and for their many comments, corrections and suggestions for improving the exposition. In memory of Fritz Grunewald. ",

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AU - Hrushovski, Ehud

AU - Martin, Ben

AU - Rideau, Silvain

AU - Cluckers, Raf

N1 - The authors wish to thank Thomas Rohwer, Deirdre Haskell, Dugald Macpherson and Elisabeth Bouscaren for their comments on earlier drafts of this work, Martin Hils for suggesting that the proof could be adapted to finite extensions and Zo´e Chatzidakis for pointing out an error in how constants were handled in earlier versions. The second author is grateful to Jamshid Derakhshan, Marcus du Sautoy, Andrei Jaikin-Zapirain, Angus Macintyre, Dugald Macpherson, Mark Ryten, Christopher Voll and Michele Zordan for helpful conversations. We are grateful to Alex Lubotzky for suggesting studying representation growth; several of the ideas in Section 8 are due to him. The first author was supported by the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013) / ERC Grant agreement no. 291111/ MODAG, the second author was supported by a Golda Meir Postdoctoral Fellowship at the Hebrew University of Jerusalem and the third author was partly supported by ANR MODIG (ANR-09-BLAN-0047) Model Theory and Interactions with Geometry. The author of the appendix would like to thank M. du Sautoy, C. Voll, and Kien Huu Nguyen for interesting discussions on this and related subjects. He was partially supported by the European Research Council under the European Community’s Seventh Framework Programme (FP7/2007-2013) with ERC Grant Agreement nr. 615722 MOTMELSUM and he thanks the Labex CEMPI (ANR-11-LABX-0007-01). We are grateful to the referee for their careful reading of the paper and for their many comments, corrections and suggestions for improving the exposition. In memory of Fritz Grunewald.

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N2 - 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 fromdefinable 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.

AB - 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 fromdefinable 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.

KW - Elimination of imaginaries

KW - invariant extensions of types

KW - cell decompositions

KW - rational zeta functions

KW - subgroup zeta functions

KW - representation zeta functions

U2 - 10.4171/JEMS/817

DO - 10.4171/JEMS/817

M3 - Article

SN - 1435-9855

VL - 20

SP - 2467

EP - 2537

JO - Journal of the European Mathematical Society

JF - Journal of the European Mathematical Society

IS - 10

ER -

Definable equivalence relations and zeta functions of groups

Abstract

Bibliographical note

Keywords

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Ben Martin

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