Twisted Poincaré series and zeta functions on finite quotients of buildings

  • Ming-Hsuan KangEmail author
  • Rupert McCallum


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.


Building Ihara zeta function Coxeter group Poincaré series 



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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Authors and Affiliations

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

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