Advertisement

Asymptotics of Automorphic Spectra and the Trace Formula

  • Werner Müller
Conference paper
Part of the Simons Symposia book series (SISY)

Abstract

This paper is a survey article on the limiting behavior of the discrete spectrum of the right regular representation in \(L^{2}(\Gamma \setminus G)\) for a lattice \(\Gamma \) in a semisimple Lie group G. We discuss various aspects of the Weyl law, the limit multiplicity problem, and the analytic torsion.

1991 Mathematics Subject Classification.

Primary: 11F70 Secondary: 58J52 11F75 

References

  1. [AB1]
    M. Abert, N. Bergeron, I. Biringer, T. Gelander, N. Nikolov, J. Raimbault, I. Samet, On the growth of Betti numbers of locally symmetric spaces. C. R. Math. Acad. Sci. Paris 349 (15–16), 831–835 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  2. [AB2]
    M. Abert, N. Bergeron, I. Biringer, T. Gelander, N. Nikolov, J. Raimbault, I. Samet, On the Growth of L 2 -Invariants for Sequences of Lattices in Lie Groups (2012). arXiv:1210.2961Google Scholar
  3. [AR]
    P. Albin, F. Rochon, D. Sher, Analytic Torsion and R-Torsion of Witt Representations on Manifolds with Cusps (2014). arXiv:1504.02418Google Scholar
  4. [Ar1]
    J. Arthur, A trace formula for reductive groups. I. Terms associated to classes in \(G(\mathbb{Q})\). Duke Math. J. 45 (4), 911–952 (1978)Google Scholar
  5. [Ar2]
    J. Arthur, On a family of distributions obtained from Eisenstein series. I. Application of the Paley-Wiener theorem. Am. J. Math., 104(6), 1243–1288 (1982)MathSciNetzbMATHGoogle Scholar
  6. [Ar3]
    J. Arthur, On a family of distributions obtained from Eisenstein series. II. Explicit formulas. Am. J. Math., 104(6), 1289–1336 (1982)MathSciNetzbMATHGoogle Scholar
  7. [Ar4]
    J. Arthur, The local behavior of weighted orbital integrals. Duke Math. J. 56 (1988), 223–293MathSciNetzbMATHCrossRefGoogle Scholar
  8. [Ar5]
    J. Arthur, The trace formula in invariant form. Ann. of Math. (2), 114 (1), 1–74 (1981)Google Scholar
  9. [Ar6]
    J. Arthur, On a family of distributions obtained from orbits. Can. J. Math. 38 (1), 179–214 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  10. [Ar7]
    J. Arthur, A measure on the unipotent variety. Can. J. Math. 37 (6), 1237–1274 (1985)MathSciNetzbMATHCrossRefGoogle Scholar
  11. [Ar8]
    J. Arthur, The Endoscopic Classification of Representations Orthogonal and Symplectic Groups. American Mathematical Society Colloquium Publications, vol. 61. (American Mathematical Society, Providence, RI, 2013)Google Scholar
  12. [Av]
    V.G. Avakumović, Über die Eigenfunktionen auf geschlossenen Riemannschen Mannigfaltigkeiten. Math. Z. 65, 327–344 (1956)MathSciNetzbMATHCrossRefGoogle Scholar
  13. [BGV]
    N. Berline, E. Getzler, M. Vergne, Heat kernels and Dirac operators, in Grundlehren Text Editions. (Springer, Berlin, 2004)Google Scholar
  14. [BV]
    N. Bergeron, A. Venkatesh, The asymptotic growth of torsion homology for arithmetic groups. J. Inst. Math. Jussieu 12 (2), 391–447Google Scholar
  15. [BSV]
    N. Bergeron, M.H. Sengün, A. Venkatesh, Torsion Homology and Cycle Complexity of Arithmetic Manifolds (2014). arXiv: 1401.6989v1Google Scholar
  16. [BE]
    P. Borwein, T. Erdélyi, Sharp extensions of Bernstein’s inequality to rational spaces. Mathematika 43 (2), 413–423 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  17. [BZ]
    J.-M. Bismut, W. Zhang, An extension of a theorem by Cheeger and M’́uller. With an appendix by Frano̧ois Laudenbach. Astérisque 205, 235 pp. (1992); C.J. Bushnell, G. Henniart, An upper bound on conductors for pairs. J. Number Theory 65 (2),183–196 (1997)Google Scholar
  18. [BG]
    A. Borel, H. Garland, Laplacian and the discrete spectrum of an arithmetic group. Am. J. Math. 105 (2), 309–335 (1983).MathSciNetzbMATHCrossRefGoogle Scholar
  19. [CaV]
    F. Calegari, A. Venkatesh, A Torsion Jacquet–Langlands Correspondence (2012). arXiv:1212.3847Google Scholar
  20. [Ch]
    J. Cheeger, Analytic torsion and the heat equation. Ann. Math. (2) 109 (2), 259–322 (1979)Google Scholar
  21. [CD]
    L. Clozel, P. Delorme, Le théorème de Paley-Wiener invariant pour les groupes de Lie réductifs. Invent. Math. 77 (3), 427–453 (1984)Google Scholar
  22. [CKP]
    J. Cogdell, H. Kim, I. Piatetski-Shapiro, F. Shahidi, Functoriality for the classical groups. Publ. Math. Inst. Hautes Études Sci 99, 163–233 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  23. [CV]
    Y. Colin de Verdière, Pseudo-laplaciens. II. Ann. Inst. Fourier 33 (2), 87–113 (1983)MathSciNetzbMATHCrossRefGoogle Scholar
  24. [Co]
    L. Corwin, The Plancherel measure in nilpotent Lie groups as a limit of point measures, Math. Zeitschrift 155, 151–162 (1977)MathSciNetzbMATHCrossRefGoogle Scholar
  25. [DW1]
    D.L. DeGeorge, N.R. Wallach, Limit formulas for multiplicities in \(L^{2}(\Gamma \setminus G)\). Ann. Math. (2) 107, 133–150 (1978)Google Scholar
  26. [DW2]
    D.L. DeGeorge, N.R. Wallach, Limit formulas for multiplicities in \(L^{2}(\Gamma \setminus G)\). II. The tempered spectrum. Ann. Math. (2) 109 (3), 477–495 (1979)Google Scholar
  27. [DH]
    A. Deitmar, W. Hoffmann, Spectral estimates for towers of noncompact quotients. Can. J. Math. 51 (2), 266–293 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  28. [De]
    P. Delorme, Formules limites et formules asymptotiques pour les multiplicités dans \(L^{2}(G/\Gamma )\). Duke Math. J. 53, 691–731 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  29. [Di]
    J. Dixmier, Les C -algèbres et leurs représentations. Deuxième édition. Cahiers Scientifiques, Fasc. XXIX. Gauthier-Villars Éditeur, Paris (1969)Google Scholar
  30. [Do]
    H. Donnelly, On the cuspidal spectrum for finite volume symmetric spaces. J. Differ. Geom. 17 (2), 239–253 (1982)MathSciNetzbMATHGoogle Scholar
  31. [DG]
    J.J. Duistermaat, V.W. Guillemin, The spectrum of positive elliptic operators and periodic bicharacteristics. Invent. Math. 29 (1), 39–79 (1975)MathSciNetzbMATHCrossRefGoogle Scholar
  32. [DKV]
    J.J. Duistermaat, J.A.C. Kolk, V.S. Varadarajan, Spectra of compact locally symmetric manifolds of negative curvature. Invent. Math. 52, 27–93 (1979)MathSciNetzbMATHCrossRefGoogle Scholar
  33. [FL1]
    T. Finis, E. Lapid, On the spectral side of Arthur’s trace formula—combinatorial setup. Ann. Math. (2) 174 (1), 197–223 (2011)Google Scholar
  34. [FL2]
    T. Finis, E. Lapid, An Approximation Principle for Congruence Subgroups II: Application to the Limit Multiplicity Problem (2015). arXiv:1504.04795Google Scholar
  35. [FLM1]
    T. Finis, E. Lapid, W. Müller, On the spectral side of Arthur’s trace formula— absolute convergence. Ann. Math. (2) 174, 173–195 (2011)Google Scholar
  36. [FLM2]
    T. Finis, E. Lapid, W. Müller, Limit multiplicities for principal congruence subgroups of GL(n) and SL(n). J. Inst. Math. Jussieu 14 (3), 589–638 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  37. [FLM3]
    T. Finis, E. Lapid, W. Müller, On the degrees of matrix coefficients of intertwining operators. Pac. J. Math. 260 (2), 433–456 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  38. [Ga]
    L. Garding, Dirichlet’s problem for linear elliptic partial differential equations. Math. Scand. 1, 55–72 (1953)MathSciNetCrossRefGoogle Scholar
  39. [GGP]
    I.M. Gel’fand, M.I. Graev, I.I. Pyatetskii-Shapiro, Representation theory and automorphic functions. Reprint of the 1969 edition, in Generalized Functions, vol. 6 (Academic, Boston, MA, 1990)Google Scholar
  40. [Gi]
    P.B. Gilkey, Invariance theory, the heat equation, and the Atiyah-Singer index theorem, in Studies in Advanced Mathematics, 2nd edn. (CRC Press, Boca Raton, FL, 1995)Google Scholar
  41. [Ha]
    G. Harder, Eisenstein cohomology of arithmetic groups. The case GL2. Invent. Math. 89 (1), 37–118 (1987)Google Scholar
  42. [He]
    S. Helgason, Differential Geometry and Symmetric Spaces (Academic, New York, 1962)zbMATHGoogle Scholar
  43. [Ho]
    L. Hörmander, The spectral function of an elliptic operator. Acta Math. 121, 193–218 (1968)MathSciNetzbMATHCrossRefGoogle Scholar
  44. [Hu]
    M.N. Huxley, Scattering matrices for congruence subgroups, in Modular forms (Durham, 1983). Ellis Horwood Series in Mathematics and its Applications: Statistics, Operational Research, 141–156 (Horwood, Chichester, 1984 )Google Scholar
  45. [Ka]
    V. Kala, Density of Self-Dual Automorphic Representations of \(\mathop{\mathrm{GL}}\nolimits _{N}(\mathbb{A}_{\mathbb{Q}})\) (2014). arXiv:1406.0385Google Scholar
  46. [La1]
    R.P. Langlands, Euler Products (Yale University Press, New Haven, CT, 1971); A. James, K. Whittemore, Lecture in Mathematics given at Yale University, 1967, Yale Mathematical Monographs, 1Google Scholar
  47. [La2]
    R.P. Langlands, On the Functional Equations Satisfied by Eisenstein Series. Lecture Notes in Mathematics, vol. 544 (Springer, Berlin/New York, 1976)Google Scholar
  48. [LM]
    E. Lapid, W. Müller, Spectral asymptotics for arithmetic quotients of SL(n,R)/SO(n), Duke Math. J. 149 (1), 117–155 (2009)Google Scholar
  49. [LV]
    E. Lindenstrauss, A. Venkatesh, Existence and Weyl’s law for spherical cusps forms, Geom. Funct. Anal. 17 (1), 220–251 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  50. [Lo]
    J. Lott, Heat kernels on covering spaces and topological invariants. J. Differ. Geom. 35 (2), 471–510 (1992)MathSciNetzbMATHGoogle Scholar
  51. [Lub]
    A. Lubotzky, Subgroup growth and congruence subgroups. Invent. Math. 119, 267–295 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  52. [Lu1]
    W. Lück, Approximating L 2-invariants by their finite-dimensional analogues, Geom. Funct. Anal. 4 (4), 455–481 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  53. [Lu2]
    W. Lück, Survey on Approximating L 2 -Invariants by Their Classical Counterparts: Betti Numbers, Torsion Invariants and Homological Growth (2015). arXiv:1501.07446Google Scholar
  54. [MV]
    V. Mathai, L 2-analytic torsion. J. Funct. Anal. 107 (2), 369–386 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  55. [MM]
    Y. Matsuhsima, S. Murakami, On vector bundle valued harmonic forms and automorphic forms on symmetric Riemannian manifolds. Ann. Math. (2) 78, 365–416 (1963)Google Scholar
  56. [Ma1]
    J. Matz, Weyl’s Law for Hecke Operators on GL(n) Over Imaginary Quadratic Number Fields (2013). arXiv:1310.6525Google Scholar
  57. [MT]
    J. Matz, N. Templier, Sato-Tate Equidistribution for Families of Hecke-Maass Forms on SL(n,R)/SO(n) (2015). arXiv:1505.07285Google Scholar
  58. [Mia]
    R.J. Miatello, The Minakshisundaram-Pleijel coefficients for the vector-valued heat kernel on compact locally symmetric spaces of negative curvature. Trans. Am. Math. Soc. 260 (1), 1–33 (1980)MathSciNetzbMATHGoogle Scholar
  59. [Mi]
    S.D. Miller, On the existence and temperedness of cusp forms for \(\mathrm{SL}_{3}(\mathbb{Z})\). J. Reine Angew. Math. 533, 127–169 (2001)MathSciNetGoogle Scholar
  60. [MP]
    S. Minakshisundaram, A. Pleijel, Some properties of the eigenfunctions of the Laplace-operator on Riemannian manifolds. Can. J. Math. 1, 242–256 (1949)MathSciNetzbMATHCrossRefGoogle Scholar
  61. [MW]
    C. Mœglin, J.-L. Waldspurger, Le spectre résiduel de GL(n), Ann. Sci. l’École Norm. Sup. (4) 22 (4), 605–674 (1989)Google Scholar
  62. [Mu1]
    W. Müller, Analytic torsion and R-torsion of Riemannian manifolds. Adv. Math. 28 (3), 233–305 (1978)MathSciNetzbMATHCrossRefGoogle Scholar
  63. [Mu2]
    W. Müller, The trace class conjecture in the theory of automorphic forms. Ann. Math. (2), 130 (3), 473–529 (1989)Google Scholar
  64. [Mu3]
    W. Müller, Analytic torsion and R-torsion for unimodular representations. J. Am. Math. Soc. 6 (3), 721–753 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  65. [Mu4]
    W. Müller, B. Speh, Absolute convergence of the spectral side of the Arthur trace formula for GLn. Geom. Funct. Anal. 14, 58–93 (2004). With an appendix by E.M. LapidGoogle Scholar
  66. [Mu5]
    W. Müller, Weyl’s law for the cuspidal spectrum of SLn. Ann. Math. (2) 165 (1), 275–333 (2007)Google Scholar
  67. [Mu6]
    W. Müller, On the spectral side of the Arthur trace formula. Geom. Funct. Anal. 12, 669–722 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  68. [Mu7]
    W. Müller, Weyl’s law in the theory of automorphic forms, in Groups and Analysis. London Mathematical Society Lecture Note Series, vol. 354 (Cambridge University Press, Cambridge, 2008), pp. 133–163Google Scholar
  69. [MP1]
    W. Müller, J. Pfaff, Analytic torsion of complete hyperbolic manifolds of finite volume. J. Funct. Anal. 263 (9), 2615–2675 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  70. [MP2]
    W. Müller, J. Pfaff, The analytic torsion and its asymptotic behavior for sequences of hyperbolic manifolds of finite volume, J. Funct. Anal. 267 (8), 2731–2786 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  71. [MS]
    W. Müller, B. Speh, 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)Google Scholar
  72. [Ol]
    M. Olbrich, L 2-invariants of locally symmetric spaces. Doc. Math. 7, 219–237 (2002)MathSciNetzbMATHGoogle Scholar
  73. [OW]
    S. Osborne, G. Warner, The theory of Eisenstein systems, in Pure and Applied Mathematics, vol. 99 (Academic, [Harcourt Brace Jovanovich, Publishers], New York/London, 1981)Google Scholar
  74. [Pf]
    J. Pfaff, A Gluing Formula for the Analytic Torsion on Hyperbolic Manifolds With Cusps (2013). arXiv:1312.6384Google Scholar
  75. [Pr]
    K. Prachar, Primzahlverteilung (Springer, Berlin/G’́ottingen/Heidelberg, 1957)Google Scholar
  76. [Ra1]
    J. Raimbault, Asymptotics of Analytic Torsion for Hyperbolic Three-Manifolds (2012). arXiv:1212.3161Google Scholar
  77. [Ra2]
    J. Raimbault, Analytic, Reidemeister and Homological Torsion for Congruence Three-Manifolds (2013). arXiv:1307.2845Google Scholar
  78. [RS]
    D.B. Ray, I.M. Singer, R-torsion and the Laplacian on Riemannian manifolds. Adv. Math. 7, 145–210 (1971)MathSciNetzbMATHCrossRefGoogle Scholar
  79. [Rez]
    A. Reznikov, Eisenstein matrix and existence of cusp forms in rank one symmetric spaces. Geom. Funct. Anal. 3 (1), 79–105 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  80. [RoS]
    J. Rohlfs, B. Speh, On limit multiplicities of representations with cohomology in the cuspidal spectrum. Duke Math. J. 55, 199–211 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  81. [Sa1]
    P. Sarnak, Spectra of hyperbolic surfaces. Bull. Am. Math. Soc. (N.S.) 40 (4), 441–478 (2003)Google Scholar
  82. [Sa2]
    P. Sarnak, On cusp forms, in The Selberg Trace Formula and Related Topics (Brunswick, Maine, 1984). Contemporary Mathematics, vol. 53 (American Mathematical Society, Providence, RI, 1986), pp. 393–407Google Scholar
  83. [Sa3]
    P. Sarnak, Notes on the generalized Ramanujan conjectures, in Harmonic Analysis, the Trace Formula, and Shimura Varieties. Clay Mathematics Proceedings, vol. 4 (American Mathematical Society, Providence, RI, 2005), pp. 659–685Google Scholar
  84. [SST]
    P. Sarnak, S.-W. Shin, N. Templier, Families of L-Functions and Their Symmetry (2014). arXiv:1401.5507Google Scholar
  85. [Sau]
    F. Sauvageot, Principe de densité pour les groupes réductifs. Compositio Math. 108, 151–184 (1997)MathSciNetCrossRefGoogle Scholar
  86. [Sav]
    G. Savin, Limit multiplicities of cusp forms. Invent. Math., 95, 149–159 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  87. [Se1]
    A. Selberg, Harmonic Analysis, Collected Papers, vol. I (Springer, Berlin, 1989), pp. 624–674Google Scholar
  88. [Sha1]
    F. Shahidi, On certain L-functions. Am. J. Math. 103 (2), 297–355 (1981)MathSciNetzbMATHCrossRefGoogle Scholar
  89. [Sha2]
    F. Shahidi, On the Ramanujan conjecture and finiteness of poles for certain L-functions. Ann. Math. (2) 127 (3), 547–584 (1988)Google Scholar
  90. [Shi]
    S.-W. Shin, Automorphic Plancherel density theorem. Israel J. Math. 192 (1), 83–120 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  91. [Shu]
    M.A. Shubin, Pseudodifferential operators and spectral theory, 2nd edn. (Springer, Berlin, 2001)zbMATHCrossRefGoogle Scholar
  92. [Wa1]
    G. Warner, Selberg’s trace formula for nonuniform lattices: the R-rank one case, in Studies in Algebra and Number Theory. Advances in Mathematics Supplementary Studies, vol. 6 (Academic, New York/London, 1979)Google Scholar
  93. [We]
    H. Weyl, Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen (mit einer Anwendung auf die Theorie der Hohlraumstrahlung). Math. Annalen 71, 441–479 (1912)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Universität BonnMathematisches InstitutBonnGermany

Personalised recommendations