Advertisement

Annals of Global Analysis and Geometry

, Volume 33, Issue 2, pp 161–205 | Cite as

A prime geodesic theorem for SL4

  • Anton DeitmarEmail author
  • Mark Pavey
Original Paper

Abstract

A prime geodesic theorem is derived for rank-one geodesics in quotients of SL4. This has applications in class number asymptotics for quartic fields. For these applications it is necessary to prove a more general statement than in the literature: several regularity conditions have to be abandoned. As a consequence, the analytical difficulties multiply. The final result is obtained by a sandwiching argument from infinitely many independent asymptotics.

Keywords

Prime geodesic theorem Trace formula Ruelle zeta function 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Borel A. (1969). Introduction aux groupes arithmétiques. Hermann, Paris zbMATHGoogle Scholar
  2. 2.
    Borel, A., Wallach, N.: Continuous Cohomology, Discrete Groups, and Representations of Reductive Groups. Ann. Math. Stud. Vol. 94, Princeton (1980)Google Scholar
  3. 3.
    Cartan H. and Eilenberg S. (1956). Homological Algebra.. Princeton University Press, Princeton, NJ Google Scholar
  4. 4.
    Chandrasekharan, K.: Introduction to Analytic Number Theory. Springer-Verlag (1968)Google Scholar
  5. 5.
    Conway J. (1978). Functions of One Complex Variable. Springer-Verlag, New York Google Scholar
  6. 6.
    Deitmar A. (1995). Higher torsion zeta functions. Adv. Math. 110: 109–128 zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Deitmar A. (2000). Geometric zeta-functions of locally symmetric spaces. Am. J. Math. 122(5): 887–926 zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Deitmar A. (2002). Class numbers of orders in cubic fields. J. Number Theory 95: 150–166 zbMATHMathSciNetGoogle Scholar
  9. 9.
    Deitmar A. (2004). A prime geodesic theorem for higher rank spaces. Geom Funct Anal 14: 1238–1266 CrossRefMathSciNetGoogle Scholar
  10. 10.
    Deitmar A. and Hoffman J. (2006). The Ihara-Selberg zeta function for PGL3 and Hecke operators. Int. J. Math. 17: 143–156 zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Deitmar, A.: A prime geodesic theorem for higher rank spaces II: singular geodesics. Rocky Mountain J. Math. (2006)Google Scholar
  12. 12.
    Deitmar, A., Pavey, M.: Class numbers of orders in complex quartic fields. Mathematische Annalen. (to appear)Google Scholar
  13. 13.
    Dieudonné, J.: Treatise on Analysis. Academic Press (1976)Google Scholar
  14. 14.
    Gangolli R. (1977). The length spectrum of some compact manifolds of negative curvature. J. Diff. Geom. 12: 403–426 zbMATHMathSciNetGoogle Scholar
  15. 15.
    Gelfand, I.M., Graev, M.I., Pyatetskii-Shapiro, I.I.: Representation Theory and Automorphic Functions. Saunders (1969)Google Scholar
  16. 16.
    Harish–Chandra J. (1975). Harmonic analysis on real reductive groups I. The theory of the constant term. J. Func. Anal. 19: 104–204 zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Hecht H. and Schmid W. (1983). Characters, asymptotics and \({\mathfrak{n}}\) -homology of Harish–Chandra modules. Acta Math. 151: 49–151 zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Hejhal, D.: The Selberg Trace Formula for \(PSL_2({\mathbb{R}})\) I. Springer Lecture Notes 548 (1976)Google Scholar
  19. 19.
    Hochschild G. and Serre J.-P. (1953). Cohomology of Lie algebras. Ann. Math. 57: 591–603 CrossRefMathSciNetGoogle Scholar
  20. 20.
    Hochschild G. and Serre J.-P. (1953). Cohomology of group extensions. Trans. Am. Math. Soc. 74: 110–134 zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Hopf H. and Samelson H. (1941). Ein Satz über die Wirkungsräume geschlossener Liescher Gruppen. Comment. Math. Helv. 13: 240–251 zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Howe R. and Moore C. (1979). Asymptotic properties of unitary representations. J. Funct. Anal. 32: 72–96 zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Juhl A. Cohomological Theory of Dynamical Zeta functions. Progress in Mathematics, Vol. 194. Birkhäuser Verlag, Basel (2001)Google Scholar
  24. 24.
    Knapp, A.: Representation Theory of Semisimple Lie Groups. Princeton University Press (1986)Google Scholar
  25. 25.
    Knieper G. (1997). On the asymptotic geometry of nonpositively curved manifolds. Geom. Funct. Anal. 7: 755–782 zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Koyama S. (1998). Prime geodesic theorem for arithmetic compact surfaces. Internat. Math. Res. Notices 8: 383–388 CrossRefMathSciNetGoogle Scholar
  27. 27.
    Labesse J.P. (1991). Pseudo-coefficients très cuspidaux et K-théorie. Math. Ann. 291: 607–616 zbMATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Luo W. and Sarnak P. (1995). Quantum ergodicity of eigenfunctions on \({PSL}_2(Z)\backslash H^2\) Inst. Hautes ÈTudes Sci. Publ. Math. 81: 207–237 zbMATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Margulis G.A. (1969). Certain applications of ergodic theory to the investigation of manifolds of negative curvature. Funkcional. Anal. i Priložen. 3(4): 89–90 MathSciNetGoogle Scholar
  30. 30.
    Pierce R.S. (1982). Associative Algebras. Springer-Verlag, New York zbMATHGoogle Scholar
  31. 31.
    Pollicott M. and Sharp R. (1998). Exponential error terms for growth functions on negatively curved surfaces. Am. J. Math. 120(5): 1019–1042 zbMATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    Randol B. (1977). On the asymptotic distribution of closed geodesics on compact Riemann surfaces. Trans. AMS 233: 241–247 CrossRefMathSciNetGoogle Scholar
  33. 33.
    Randol B. (1978). The Riemann hypothesis for Selberg’s zeta-function and the asymptotic behavior of eigenvalues of the Laplace operator. Trans. Am. Math. Soc. 236: 209–223 zbMATHCrossRefMathSciNetGoogle Scholar
  34. 34.
    Sarnak P. (1982). Class numbers of indefinite binary quadratic forms. J. Number Theory 15: 229–247 zbMATHCrossRefMathSciNetGoogle Scholar
  35. 35.
    Selberg, A.: On Discontinuous Groups in Higher-dimensional Symmetric Spaces. 1960 Contributions to function theory (internat. Colloq. Function Theory, Bombay, 1960), pp. 147–164. Tata Institute of Fundamental Research, Bombay.Google Scholar
  36. 36.
    Selberg A. (1956). Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series. J. Indian Math. Soc. 20: 47–87 MathSciNetGoogle Scholar
  37. 37.
    Speh B. (1981). The unitary dual of Gl(3,R) and Gl(4,R). Math. Ann. 258: 113–133 zbMATHCrossRefMathSciNetGoogle Scholar
  38. 38.
    Wallach N. (1976). On the Selberg Trace Formula in the case of compact quotient. Bull. AMS 82(2): 171–195 zbMATHMathSciNetGoogle Scholar
  39. 39.
    Witte, D.: Introduction to arithmetic groups. http://arxiv.org/abs/math/0106063.Google Scholar
  40. 40.
    Wolf J. (1962). Discrete groups, symmetric spaces and global holonomy. Am. J. Math. 84: 527–542 zbMATHCrossRefGoogle Scholar
  41. 41.
    Zelditch S. (1989). Trace formula for compact Γ \ PSL 2(R) and the equidistribution theory of closed geodesics. Duke Math. J. 59(1): 27–81 zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science + Business Media B.V. 2007

Authors and Affiliations

  1. 1.Mathematics DepartmentUniversity of TuebingenTuebingenGermany

Personalised recommendations