Geometric and Functional Analysis

, Volume 27, Issue 6, pp 1378–1449 | Cite as

Analytic torsion of arithmetic quotients of the symmetric space \({\rm SL} (n,\mathbb{R})/ {\rm SO}(n)\)

  • Jasmin Matz
  • Werner Müller


In this paper we define a regularized version of the analytic torsion for arithmetic quotients of the symmetric space \({{\rm SL}(n,\mathbb{R})/ {\rm SO}(n)}\). The definition is based on the study of the renormalized trace of the corresponding heat operators, which is defined as the geometric side of the Arthur trace formula applied to the heat operator.

Keywords and phrases

Analytic torsion Locally symmetric spaces 

Mathematics Subject Classification

Primary: 58J52 Secondary: 11M36 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Art78.
    Arthur J.: A trace formula for reductive groups. I. Terms associated to classes in \({G(\mathbb{Q})}\) . Duke Math. J. 45(4), 911–952 (1978)MathSciNetCrossRefGoogle Scholar
  2. Art81.
    Arthur J.: The trace formula in invariant form. Ann. of Math. (2) 114, 1–74 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  3. Art85.
    Arthur J.: A measure on the unipotent variety. Canad. J. Math. 37(6), 1237–1274 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  4. Art88.
    Arthur J.: The local behaviour of weighted orbital integrals. Duke Math. J. 56(2), 223–293 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  5. Art81.
    J. Arthur. Automorphic representations and number theory. In: 1980 Seminar on Harmonic Analysis, Canadian Math. Soc., Conference Proceedings, Vol. 1. AMS, Providence, RI (1981).Google Scholar
  6. Art86.
    Arthur J.: On a family of distributions obtained from orbits. Canad. J. Math., 38(1), 179–214 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  7. Art82a.
    Arthur J.: On a family of distributions obtained from Eisenstein series. I. Application of the Paley–Wiener theorem. Amer. J. Math., 104(6), 1243–1288 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  8. Art82b.
    Arthur J.: On a family of distributions obtained from Eisenstein series. II. Explicit formulas. Amer. J. Math., 104(6), 1289–1336 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  9. Art05.
    J. Arthur. An introduction to the trace formula. In: Harmonic Analysis, the Trace Formula, and Shimura Varieties, Clay Math. Proc., 4 (2005), pp. 1–263.Google Scholar
  10. BM83.
    Barbasch D., Moscovici H.: L 2-index and the trace formula. J. Funct. Analysis 53, 151–201 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  11. Bor63a.
    Borel A.: Some finiteness properties of adele groups over number fields. Inst. Hautes Études Sci. Publ. Math., 16, 5–30 (1963)MathSciNetCrossRefzbMATHGoogle Scholar
  12. BV13.
    Bergeron N., Venkatesh A.: The asymptotic growth of torsion homology for arithmetic groups. J. Inst. Math. Jussieu, 12(2), 391–447 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  13. BGV92.
    Berline N., Getzler E., Vergne M.: Heat Kernels and Dirac Operators. Springer, Berlin (1992)CrossRefzbMATHGoogle Scholar
  14. BZ92.
    J.-M. Bismut, W. Zhang. An extension of a theorem by Cheeger and Müller. With an appendix by Franois Laudenbach. Astérisque No. 205 (1992).Google Scholar
  15. Bor63b.
    Borel A.: Compact Clifford–Klein forms of symmetric spaces. Topology, 2, 111–122 (1963)MathSciNetCrossRefzbMATHGoogle Scholar
  16. BG83.
    Borel A., Garland H.: Laplacian and the discrete spectrum of an arithmetic group. Amer. J. Math., 105(2), 309–335 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  17. BW00.
    A. Borel, N. Wallach. Continuous Cohomology, Discrete Subgroups, and Representations of Reductive Groups, Second edition. Mathematical Surveys and Monographs, Vol. 67. Amer. Math. Soc., Providence, RI (2000).Google Scholar
  18. BH99.
    Bridson M., Haefliger A.: Metric Space of Non-positive Curvature, Grundlehren der Mathematischen Wissenschaften, Vol. 319. Springer, Berlin (1999)CrossRefGoogle Scholar
  19. CV.
    F. Calegari, A. Venkatesh. A torsion Jacquet–Langlands correspondence. arXiv:1212.3847.
  20. Che79.
    Cheeger J.: Analytic torsion and the heat equation. Ann. of Math. (2) 109(2), 259–322 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  21. CD84.
    Clozel L., Delorme P.: Le théorème de Paley–Wiener invariant pour les groupes de Lie réductifs. Invent. Math., 77(3), 427–453 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  22. Del86.
    Delorme P.: Formules limites et formules asymptotiques pour les multiplicités dans \({L^2(G/\Gamma)}\) . Duke Math. J., 53(3), 691–731 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  23. Don79.
    Donnelly H.: Asymptotic expansions for the compact quotients of properly discontinuous group actions. Illinois J. Math., 23, 485–496 (1979)MathSciNetzbMATHGoogle Scholar
  24. FL1.
    T. Finis, E. Lapid. On the continuity of the geometric side of the trace formula. Preprint 2015. arXiv:1512.08753v1.
  25. FL2.
    T. Finis, E. Lapid. On the analytic properties of intertwining operators I: Global normalizing factors. arXiv:1603.05475.
  26. FLM11.
    Finis T., Lapid E., Müller W.: On the spectral side of Arthur’s trace formula—absolute convergence. Ann. of Math. (2) 174, 173–195 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  27. FLM15.
    Finis T., Lapid E., Müller W.: Limit multiplicities for principal congruence subgroups of GL(n) and SL(n). J. Inst. Math. Jussieu, 14(3), 589–638 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  28. Fli82.
    Flicker Y.Z.: The Trace Formula and Base Change for GL(3), Lecture Notes in Mathematics, Vol. 927. Springer, Berlin (1982)CrossRefGoogle Scholar
  29. Gel96.
    S.S. Gelbart. Lectures on the Arthur–Selberg Trace Formula, Vol. 9. American Math. Soc., Providence (1996).Google Scholar
  30. Gil95.
    P.B. Gilkey. Invariance Theory, the Heat Equation, and the Atiyah–Singer Index Theorem, Second Edition, Studies in Advanced Mathematics. CRC Press, Boca Raton, Fl (1995).Google Scholar
  31. HH77.
    E. Heintze, H.-Ch. im Hof. Geometry of horospheres. J. Differential Geoemtry, 12 (1977), 481–491.Google Scholar
  32. Hel78.
    S. Helgason. Differential Geometry, Lie Groups, and Symmetric Space. Academic Press, Boston (1978).Google Scholar
  33. Hof08.
    Hoffmann W.: Geometric estimates for the trace formula. Ann. Global Anal. Geom., 34(3), 233–261 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  34. How74.
    Howe R.: The Fourier transform and germs of characters (case of GL(n) over a p-adic field). Math. Ann., 208(4), 305–322 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  35. Hum95.
    J.E. Humphreys. Conjugacy Classes in Semisimple Algebraic Groups, Mathematical Surveys and Monographs, vol. 43. American Mathematical Society, Providence, RI (1995).Google Scholar
  36. Kna01.
    Knapp A.W.: Representation Theory of Semisimple Groups. Princeton University Press, Princeton (2001)zbMATHGoogle Scholar
  37. LM09.
    E. Lapid, W. Müller. Spectral asymptotics for arithmetic quotients of \({{\rm SL}(n,\mathbb{R})/ {\rm SO}(n)}\) . Duke Math. J., 149 (2009).Google Scholar
  38. MM13.
    Marshall S., Müller W.: On the torsion in the cohomology of arithmetic hyperbolic 3-manifolds. Duke Math. J., 162(5), 863–888 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  39. MM63.
    Matsushima Y., Murakami S.: On vector bundle valued harmonic forms and automorphic forms on symmetric Riemannian manifolds. Ann. of Math., 78, 365–416 (1963)MathSciNetCrossRefzbMATHGoogle Scholar
  40. Mel93.
    R.B. Melrose, The Atiyah–Patodi–Singer Index Theorem, Research Notes in Mathematics, Vol. 4. A K Peters, Ltd., Wellesley, MA (1993).Google Scholar
  41. Mia80.
    Miatello R.J.: The Minakshisundaram–Pleijel coefficients for the vector-valued heat kernel on compact locally symmetric spaces of negative curvature. Trans. Amer. Math. Soc., 260, 1–33 (1980)MathSciNetzbMATHGoogle Scholar
  42. Mul07.
    Müller W.: Weyl’s law for the cuspidal spectrum of SL_n. Annals of Math., 165, 275–333 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  43. Mul02.
    Müller W.: On the spectral side of the Arthur trace formula. Geom. Funct. Anal., 12, 669–722 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  44. Mul89.
    Müller W.: The trace class conjecture in the theory of automorphic forms. Annals of Math., 130, 473–529 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  45. Mul78.
    Müller W.: Analytic torsion and R-torsion of Riemannian manifolds. Adv. in Math., 28(3), 233–305 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  46. Mul93.
    Müller W.: Analytic torsion and R-torsion for unimodular representations. J. Amer. Math. Soc., 6(3), 721–753 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  47. Mul.
    W. Müller. On the analytic torsion of hyperbolic manifolds of finite volume. arXiv:1501.07851.
  48. MP12.
    Müller W., Pfaff J.: Analytic torsion of complete hyperbolic manifolds of finite volume. J. Funct. Anal., 263(9), 2615–2675 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  49. MP13.
    Müller W., Pfaff J.: Analytic torsion and L 2-torsion of compact locally symmetric manifolds. J. Diff. Geometry, 95(1), 71–119 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  50. MP14a.
    Müller W., Pfaff J.: The analytic torsion and its asymptotic behaviour for sequences of hyperbolic manifolds of finite volume. J. Funct. Anal., 267(8), 2731–2786 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  51. MP14b.
    Müller W., Pfaff J.: On the growth of torsion in the cohomology of arithmetic groups. Math. Ann., 359(1–2), 537–555 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  52. MS04.
    Müller W., Speh B.: Absolute convergence of the spectral side of the Arthur trace formula for GL(n) With an appendix by E. M. Lapid. Geom. Funct. Anal., 14(1), 58–93 (2004)CrossRefzbMATHGoogle Scholar
  53. MW95.
    C. Moeglin, J.-L. Waldspurger, Spectral Decomposition and Eisenstein Series. Une paraphrase de l’criture. Cambridge Tracts in Mathematics, Vol. 113. Cambridge University Press, Cambridge (1995).Google Scholar
  54. PR.
    J. Pfaff, J. Raimbault, The torsion in symmetric powers on congruence subgroups of Bianchi groups. arXiv:1503.04785.
  55. RS71.
    Ray D.B., Singer I.M.: R-torsion and the Laplacian on Riemannian manifolds. Advances in Math., 7, 145–210 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  56. Rai1.
    J. Raimbault, Asymptotics of analytic torsion for hyperbolic three-manifolds. arXiv:1212.3161 .
  57. Rai2.
    J. Raimbault, Analytic, Reidemeister and homological torsion for congruence three-manifolds. arXiv:1307.2845.
  58. Ran72.
    Ranga Rao R.: Orbital integrals on reductive groups. Ann. of Mathematics, 96(2), 505–510 (1972)MathSciNetzbMATHGoogle Scholar
  59. Sha81.
    Shahidi F.: On certain L-functions. Amer. J. Math., 103(2), 297–355 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  60. Sha88.
    Shahidi F.: On the Ramanujan conjecture and finiteness of poles for certain L-functions. Ann. of Math. (2), 127(3), 547–584 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  61. War79.
    G. Warner, Selberg’s Trace Formula for Nonuniform Lattices: The R-Rank One Case. Studies in Algebra and Number Theory, Adv. in Math. Suppl. Stud., Vol. 6. Academic Press, New York (1979).Google Scholar
  62. Wal88.
    N.R. Wallach, Real Reductive Groups I. Academic Press, Boston (1988).Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Einstein Institute of MathematicsHebrew University of JerusalemJerusalemIsrael
  2. 2.Mathematisches InstitutUniversität BonnBonnGermany

Personalised recommendations