Skip to content
Licensed Unlicensed Requires Authentication Published by De Gruyter June 13, 2019

A prime geodesic theorem for SL3(ℤ)

  • Anton Deitmar EMAIL logo , Polyxeni Spilioti and Yasuro Gon
From the journal Forum Mathematicum


We show a prime geodesic theorem for the group SL3() counting those geodesics whose lifts lie in the split Cartan subgroup. This is the first arithmetic prime geodesic theorem of higher rank for a non-cocompact group.

Communicated by Jan Bruinier

Award Identifier / Grant number: 248213549

Funding statement: The first author was supported by DFG grant 248213549.


[1] J. Arthur, A Paley–Wiener theorem for real reductive groups, Acta Math. 150 (1983), no. 1–2, 1–89. 10.1007/BF02392967Search in Google Scholar

[2] R. Ayoub, An Introduction to the Analytic Theory of Numbers, Math. Surveys 10, American Mathematical Society, Providence, 1963. Search in Google Scholar

[3] A. Borel and N. Wallach, Continuous Cohomology, Discrete Subgroups, and Representations of Reductive Groups, 2nd ed., Math. Surveys Monogr. 67, American Mathematical Society, Providence, 2000. 10.1090/surv/067Search in Google Scholar

[4] J. Cheeger, M. Gromov and M. Taylor, Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds, J. Differential Geom. 17 (1982), no. 1, 15–53. 10.4310/jdg/1214436699Search in Google Scholar

[5] L. Clozel and P. Delorme, Le théorème de Paley–Wiener invariant pour les groupes de Lie réductifs, Invent. Math. 77 (1984), no. 3, 427–453. 10.1007/BF01388832Search in Google Scholar

[6] A. Deitmar, A prime geodesic theorem for higher rank spaces, Geom. Funct. Anal. 14 (2004), no. 6, 1238–1266. 10.1007/s00039-004-0490-7Search in Google Scholar

[7] A. Deitmar, A higher rank Lefschetz formula, J. Fixed Point Theory Appl. 2 (2007), no. 1, 1–40. 10.1007/s11784-007-0028-3Search in Google Scholar

[8] A. Deitmar, A prime geodesic theorem for higher rank. II. Singular geodesics, Rocky Mountain J. Math. 39 (2009), no. 2, 485–507. 10.1216/RMJ-2009-39-2-485Search in Google Scholar

[9] A. Deitmar and S. Echterhoff, Principles of Harmonic Analysis, 2nd ed., Universitext, Springer, Cham, 2014. 10.1007/978-3-319-05792-7Search in Google Scholar

[10] A. Deitmar and W. Hoffmann, Asymptotics of class numbers, Invent. Math. 160 (2005), no. 3, 647–675. 10.1007/s00222-004-0423-ySearch in Google Scholar

[11] A. Deitmar and R. McCallum, A prime geodesic theorem for higher rank buildings, Kodai Math. J. 41 (2018), no. 2, 440–455. 10.2996/kmj/1530496852Search in Google Scholar

[12] A. Deitmar and M. Pavey, Class numbers of orders in complex quartic fields, Math. Ann. 338 (2007), no. 3, 767–799. 10.1007/s00208-007-0096-0Search in Google Scholar

[13] A. Deitmar and M. Pavey, A prime geodesic theorem for SL4, Ann. Global Anal. Geom. 33 (2008), no. 2, 161–205. 10.1007/s10455-007-9078-4Search in Google Scholar

[14] I. Efrat, Determinants of Laplacians on surfaces of finite volume, Comm. Math. Phys. 119 (1988), no. 3, 443–451. 10.1007/BF01218082Search in Google Scholar

[15] T. Finis, E. Lapid and W. Müller, On the spectral side of Arthur’s trace formula—absolute convergence, Ann. of Math. (2) 174 (2011), no. 1, 173–195. 10.4007/annals.2011.174.1.5Search in Google Scholar

[16] Y. Gon, Differences of the Selberg trace formula and Selberg type zeta functions for Hilbert modular surfaces, J. Number Theory 147 (2015), 396–453. 10.1016/j.jnt.2014.07.019Search in Google Scholar

[17] Harish-Chandra, Discrete series for semisimple Lie groups. II. Explicit determination of the characters, Acta Math. 116 (1966), 1–111. 10.1007/BF02392813Search in Google Scholar

[18] Harish-Chandra, Harmonic analysis on real reductive groups. I. The theory of the constant term, J. Funct. Anal. 19 (1975), 104–204. 10.1016/0022-1236(75)90034-8Search in Google Scholar

[19] Harish-Chandra, Supertempered distributions on real reductive groups, Studies in Applied Mathematics, Adv. Math. Suppl. Stud. 8, Academic Press, New York (1983), 139–153. 10.1007/978-1-4899-7407-5_85Search in Google Scholar

[20] H. Hecht and W. A. Schmid, Characters, asymptotics and n-homology of Harish-Chandra modules, Acta Math. 151 (1983), no. 1–2, 49–151. 10.1007/BF02393204Search in Google Scholar

[21] D. A. Hejhal, The Selberg Trace Formula for PSL(2,𝐑). Vol. 2, Lecture Notes in Math. 1001, Springer, Berlin, 1983. 10.1007/BFb0061302Search in Google Scholar

[22] H. Iwaniec, Introduction to the Spectral Theory of Automorphic Forms, Revista Matemática Iberoamericana, Madrid, 1995. Search in Google Scholar

[23] A. W. Knapp, Representation Theory of Semisimple Groups. An Overview Based on Examples, Princeton Landmarks Math., Princeton University, Princeton, 2001. Search in Google Scholar

[24] E. P. van den Ban and H. Schlichtkrull, Paley–Wiener spaces for real reductive Lie groups, Indag. Math. (N. S.) 16 (2005), no. 3–4, 321–349. 10.1016/S0019-3577(05)80031-XSearch in Google Scholar

[25] E. P. van den Ban and S. Souaifi, A comparison of Paley–Wiener theorems for real reductive Lie groups, J. Reine Angew. Math. 695 (2014), 99–149. 10.1515/crelle-2012-0105Search in Google Scholar

[26] N. R. Wallach, Real Reductive Groups. I, Pure Appl. Math. 132, Academic Press, Boston, 1988. Search in Google Scholar

Received: 2019-01-07
Revised: 2019-04-11
Published Online: 2019-06-13
Published in Print: 2019-09-01

© 2019 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 29.9.2023 from
Scroll to top button