Illinois Journal of Mathematics

Geometric zeta functions for higher rank $p$-adic groups

Anton Deitmar and Ming-Hsuan Kang

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The higher rank Lefschetz formula for $p$-adic groups is used to prove rationality of a several-variable zeta function attached to the action of a $p$-adic group on its Bruhat–Tits building. By specializing to certain lines one gets one-variable zeta functions, which then can be related to geometrically defined zeta functions.

Article information

Illinois J. Math., Volume 58, Number 3 (2014), 719-738.

Received: 31 March 2014
Revised: 10 March 2015
First available in Project Euclid: 9 September 2015

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11M41: Other Dirichlet series and zeta functions {For local and global ground fields, see 11R42, 11R52, 11S40, 11S45; for algebro-geometric methods, see 14G10; see also 11E45, 11F66, 11F70, 11F72}
Secondary: 11F70: Representation-theoretic methods; automorphic representations over local and global fields 11F72: Spectral theory; Selberg trace formula 11F75: Cohomology of arithmetic groups 20E42: Groups with a $BN$-pair; buildings [See also 51E24] 51E24: Buildings and the geometry of diagrams


Deitmar, Anton; Kang, Ming-Hsuan. Geometric zeta functions for higher rank $p$-adic groups. Illinois J. Math. 58 (2014), no. 3, 719--738. doi:10.1215/ijm/1441790387.

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  • A. Borel, Linear algebraic groups, 2nd ed., Graduate Texts in Mathematics, vol. 126, Springer, New York, 1991.
  • A. Borel, Introduction aux groupes arithmétiques, Publications de l'Institut de Mathématique de l'Université de Strasbourg, XV, Actualités Scientifiques et Industrielles, vol. 1341, Hermann, Paris, 1969.
  • A. Borel and N. Wallach, Continuous cohomology, discrete subgroups, and representations of reductive groups, 2nd ed., Mathematical Surveys and Monographs, vol. 67, American Mathematical Society, Providence, RI, 2000.
  • P. Cartier, Representations of $p$-adic groups: A survey, Automorphic forms, representations and $L$-functions \bmisc(Proc. Sympos. Pure Math., Oregon State Univ. Corvallis, OR, 1977), Proc. Sympos. Pure Math., vol. XXXIII, Amer. Math. Soc., Providence, RI, 1979, pp. 111–155.
  • A. Deitmar, Geometric zeta functions of locally symmetric spaces, Amer. J. Math. 122 (2000), no. 5, 887–926.
  • A. Deitmar, A prime geodesic theorem for higher rank spaces, Geom. Funct. Anal. 14 (2004), no. 6, 1238–1266.
  • A. Deitmar, Lefschetz formulae for $p$-adic groups, Chin. Ann. Math. Ser. B 28 (2007), no. 4, 463–474.
  • Y. Ihara, On discrete subgroups of the two by two projective linear group over $\mathfrak{p}$-adic fields, J. Math. Soc. Japan 18 (1966), 219–235.
  • M.-H. Kang and W.-C. W. Li, The zeta functions of complexes from $\mathrm{PGL}(3)$, Adv. Math. 256 (2014), 46–103.
  • M.-H. Kang, W.-C. W. Li and C.-J. Wang, The zeta functions of complexes from $\mathrm{PGL}(3)$: A representation-theoretic approach, Israel J. Math. 177 (2010), 335–348.
  • R. E. Kottwitz, Tamagawa numbers, Ann. of Math. (2) 127 (1988), no. 3, 629–646.
  • W.-C. W. Li, Ramanujan hypergraphs, Geom. Funct. Anal. 14 (2004), no. 2, 380–399.
  • A. Lubotzky, B. Samuels and U. Vishne, Ramanujan complexes of type $A_d$, probability in mathematics, Israel J. Math. 149 (2005), 267–299.
  • T. Sunada, $L$-Functions in geometry and some applications, Curvature and topology of Riemannian manifolds (Katata, 1985), Lecture Notes in Math., vol. 1201, Springer, Berlin, 1986, pp. 266–284.
  • J. Tits, Reductive groups over local fields, Automorphic forms, representations and $L$-functions \bmisc(Proc. Sympos. Pure Math., Oregon State Univ. Corvallis, OR, 1977), Proc. Sympos. Pure Math., vol. XXXIII, Amer. Math. Soc., Providence, RI, 1979, pp. 29–69.
  • J. A. Wolf, Discrete groups, symmetric spaces, and global holonomy, Amer. J. Math. 84 (1962), 527–542.