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Twisted Poincaré series and zeta functions on finite quotients of buildings

  • Ming-Hsuan Kang
  • Rupert McCallum
Article
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Abstract

In the case where \(G=\hbox {SL}_{2}(F)\) for a non-archimedean local field F and \(\varGamma \) is a discrete torsion-free cocompact subgroup of G, there is a known relationship between the Ihara zeta function for the quotient of the Bruhat–Tits tree of G by the action of \(\varGamma \), and an alternating product of determinants of twisted Poincaré series for parabolic subgroups of the affine Weyl group of G. We show how this can be generalized to other split simple algebraic groups of rank two over F and formulate a conjecture about how this might be generalized to groups of higher rank.

Keywords

Building Ihara zeta function Coxeter group Poincaré series 

Notes

Acknowledgements

The authors would like to thank Professor Anton Deitmar and Professor Jiu-Kang Yu for their valuable discussions and also thank the referee for the constructive comments. The research was mainly performed while the first author was visiting Professor Deitmar in University of Tübingen. The authors would like to thank the university for its hospitality.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Applied MathematicsNational Chiao-Tung UniversityHsinchuTaiwan
  2. 2.Department of MathematicsUniversity of TübingenTübingenGermany

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