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# Forum Mathematicum

Managing Editor: Bruinier, Jan Hendrik

Ed. by Blomer, Valentin / Cohen, Frederick R. / Droste, Manfred / Duzaar, Frank / Echterhoff, Siegfried / Frahm, Jan / Gordina, Maria / Shahidi, Freydoon / Sogge, Christopher D. / Takayama, Shigeharu / Wienhard, Anna

IMPACT FACTOR 2018: 0.867

CiteScore 2018: 0.71

SCImago Journal Rank (SJR) 2018: 0.898
Source Normalized Impact per Paper (SNIP) 2018: 0.964

Mathematical Citation Quotient (MCQ) 2018: 0.71

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1435-5337
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Volume 29, Issue 3

# Representation zeta functions of some nilpotent groups associated to prehomogeneous vector spaces

Alexander Stasinski
/ Christopher Voll
Published Online: 2016-06-21 | DOI: https://doi.org/10.1515/forum-2015-0099

## Abstract

We compute the representation zeta functions of some finitely generated nilpotent groups associated to unipotent group schemes over rings of integers in number fields. These group schemes are defined by Lie lattices whose presentations are modelled on certain prehomogeneous vector spaces. Our method is based on evaluating $𝔭$-adic integrals associated to certain rank varieties of matrices of linear forms.

MSC 2010: 22E55; 20F69; 05A15; 11S90; 11M41

## References

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Avni N., Klopsch B., Onn U. and Voll C., Representation zeta functions of compact p-adic analytic groups and arithmetic groups, Duke Math. J. 162 (2013), no. 1, 111–197. Google Scholar

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Björner A. and Brenti F., Combinatorics of Coxeter groups, Grad. Texts in Math. 231, Springer, New York, 2005. Google Scholar

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Denef J., Report on Igusa’s local zeta function, Séminaire Bourbaki, Vol. 1990/91, Exposés 730–744, Astérisque 201–203, Société Mathématique de France, Paris (1991), 359–386. Google Scholar

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du Sautoy M. P. F. and Woodward L., Zeta Functions of Groups and Rings, Lecture Notes in Math. 1925, Springer, Berlin, 2008. Google Scholar

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Dung D. H. and Voll C., Uniform analytic properties of representation zeta functions of finitely generated nilpotent groups, preprint 2015, http://arxiv.org/abs/1503.06947; to appear in Trans. Amer. Math. Soc.

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Igusa J.-I., An Introduction to the Theory of Local Zeta Functions, AMS/IP Stud. Adv. Math. 14, American Mathematical Society, Providence, 2000. Google Scholar

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Rossmann T., Stability results for local zeta functions of groups and related structures, preprint 2015, http://arxiv.org/abs/1504.04164.

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Rossmann T., Topological representation zeta functions of unipotent groups, J. Algebra 448 (2016), 210–237. Google Scholar

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Stasinski A. and Voll C., Representation zeta functions of nilpotent groups and generating functions for Weyl groups of type B, Amer. J. Math. 136 (2014), no. 2, 501–550. Google Scholar

Revised: 2016-05-23

Published Online: 2016-06-21

Published in Print: 2017-05-01

Funding Source: Deutsche Forschungsgemeinschaft

Award identifier / Grant number: Sonderforschungsbereich 701 at Bielefeld University

Funding Source: Engineering and Physical Sciences Research Council

Award identifier / Grant number: EP/F044194/1

Voll acknowledges support by the DFG through Sonderforschungsbereich 701 at Bielefeld University and helpful conversations with Jan Schepers. This research was supported by Engineering and Physical Sciences Research Council grant EP/F044194/1.

Citation Information: Forum Mathematicum, Volume 29, Issue 3, Pages 717–734, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741,

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