Remote Access Bulletin of the American Mathematical Society

Bulletin of the American Mathematical Society

ISSN 1088-9485(online) ISSN 0273-0979(print)

Journals Home Search My Subscriptions Subscribe
Previous issue | This issue | Most recent issue | All issues (1891–Present) | Next issue | Previous article | Articles in press | Recently published articles | Next article
Request Permissions
 
 

 

Spectra of hyperbolic surfaces


Author: Peter Sarnak
Journal: Bull. Amer. Math. Soc. 40 (2003), 441-478
MSC (2000): Primary 11F03, 11N75, 11R42, 35P30
DOI: https://doi.org/10.1090/S0273-0979-03-00991-1
Published electronically: July 17, 2003
MathSciNet review: 1997348
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: These notes attempt to describe some aspects of the spectral theory of modular surfaces. They are by no means a complete survey.

References [Enhancements On Off] (What's this?)

  • [A] J. Arthur, The trace formula and Hecke operators, Number Theory, Trace Formulas and Discrete Groups, Academic Press (1989), 11-27. MR 90e:11072
  • [Ar] E. Artin, Abh. Math. Sem., Hamburg (1924), 170-175.
  • [Ber] M. Berry, Regular and irregular semiclassical wavefunctions, J. Phys. A10 (1977), 2083-2091. MR 58:8961
  • [Be] L. Bers, Spaces of Riemann surfaces, ICM Proceedings (Edin., 1958), Cambridge Univ. Press., New York (1960), 349-361. MR 23:A1796
  • [B-S] E. Bogomolny and C. Schmit, ``Nonlinearity'' 6 (1993), 523-547. MR 94i:81027
  • [Bo] E. Bombieri, Lectures on the zeta function (Italy, 2002).
  • [Boo] A. Booker, ``Poles of Artin $L$-functions and the strong Artin Conjecture,'' preprint (2002).
  • [Bor] A. Borel, Automorphic forms on $SL_2(\mathbb{R} )$, Camb. Tracts in Math. 130 (1977). MR 98j:11028
  • [B-L] J. Bourgain and E. Lindenstrauss, ``Entropy of quantum limits,'' Comm. Math. Phys. 233 (2003), no. 1, 153-171.
  • [Br] R. Brauer, On Artin's $L$-series with general group characters, Annals of Math., Vol. 48, 2 (1947), 502-551. MR 8:503g
  • [B-D-S-T] K. Buzzard, M. Dickinson, N. Shepherd-Barron and R. Taylor, On icosahedral Artin representations, D.M.J., Vol. 109 (2001), 283-317. MR 2002k:11078
  • [Ca] P. Cartier, Computers in Number Theory, ed. Atkin-Birch, Academic Press (1971), 37-48, MR 47:3285
  • [C-F] J. Cassels and A. Frohlich, eds., ``Algebraic Number Theory,'' Academic Press, London; Thompson Book Co., Inc., Washington, D.C. (1967). MR 35:6500
  • [Cl] L. Clozel, ``Spectral theory of automorphic forms'', Lectures, Park City (2002).
  • [Co-PS-S] J. Cogdell, I. Piatetsky-Shapiro and P. Sarnak, ``Estimates for Hilbert modular $L$-functions and applications'' (in preparation).
  • [Cog] J. Cogdell, ``On sums of 3 squares'' (to appear), J. Théor. Nombres Bordeaux (2003).
  • [Co] Y. Colin de Verdiere, Ergodiciti et fonctions propres du laplacien, Comm. Math. Phys. 102 (1985), 497-502. MR 87d:58145
  • [Co2] Y. Colin de Verdiere, Pseudo-laplaciens. II. Ann Inst. Fourier 32 (1983), 275-286; and 33 (1983), 87-113. MR 84k:58222
  • [Da] H. Davenport, Multiplicative number theory, G.T.M., Vol. 74, Springer (1980). MR 82m:10001
  • [De] P. Deligne, Formes modulaires et represéntations $l$-adiques, Lecture Notes in Math., Vol. 179, Springer (1971), 139-172.
  • [D-I] J. Deshouillers and H. Iwaniec, The nonvanishing of Rankin-Selberg $L$-functions at special points, Contemp. Math. 53 (1986), 51-97. MR 88d:11047
  • [D-F-I] W. Duke, J. Friedlander and H. Iwaniec, Equidistribution of roots of a quadratic congruence to prime moduli, Annals of Math. 141 (1995), 423-441. MR 95k:11124
  • [Du-Iw] W. Duke and H. Iwaniec, Estimates for coefficients of $L$-functions. IV. Amer. Jnl. Math. 116 (1994), 207-217. MR 95k:11073
  • [D-SP] W. Duke and R. Schulze-Pillot, Representation of integers by positive ternary quadratic forms and equidistribution of lattice points on ellipsoids, Invent. Math. 99 (1990), 49-57. MR 90m:11051
  • [Du-Gu] H. Duistermaat and V. Guillemin, The spectrum of positive elliptic operators and periodic bicharacteristics, Invent. Math. 29 (1975), 39-79. MR 53:9307
  • [E-F-K-A-M-M] B. Eckhardt, S. Fishman, J. Keating, O. Agam, J. Main, K. Muller, Approach to ergodicity in quantum wave functions, Phys. Rev. E, Vol. 52, No. 6 (1995), 5893-5903.
  • [Eg] Y. Egorov, The canonical transformations of pseudodifferential operators, Uspehi. Mat. Nauk 24 (1969) 5 (149), 235-236. MR 42:657
  • [E-K] M. Einsiedler and A. Katok, ``Invariant measures on $G/\Gamma$ for split simple Lie groups $G$'', Comm. Pure Appl. Math., Vol. LVI, no. 8 (2003), 1184-1221.
  • [F-P] M. Feingold and A. Peres, Distribution of matrix elements of chaotic systems, Phys. Rev. A, Vol. 34, No. 1 (1986), 591-595. MR 87i:81055
  • [F] J. Friedlander, Bounds for $L$-functions, Proc. ICM (Zurich 1994), Birkhäuser (1995), 363-373. MR 97e:11108
  • [Ga] P. Garrett, Decomposition of Eisenstein series: Rankin triple products, Annals of Math. (2) 125 (1987), 209-235. MR 88m:11033
  • [Ge] S. Gelbart, Automorphic forms on adèle groups, Ann. Math. Studies 83 (1975). MR 52:280
  • [Ge-Ja] S. Gelbart and H. Jacquet, A relation between automorphic representations of ${\rm GL}(2)$ and ${\rm GL}(3)$, Ann. Sci. École Norm. Sup. (4) 11 (1978), 471-552. MR 81e:10025
  • [Go-Ja] R. Godement and H. Jacquet, Zeta functions of simple algebras, L.N.M. 260, Springer-Verlag, Berlin-New York (1972). MR 49:7241
  • [G-S] V. Golovshanski and N. Smotrov, preprint, Inst. App. Math. Khabarovsk (1982).
  • [Har] G. Harcos, Uniform approximate functional equation for principal $L$-functions, IMRN (2002) 18, 923-932. MR 2003d:11074
  • [H-K] M. Harris and S. Kudla, The central critical value of a triple product $L$-function, Ann. of Math. 133 (1991), 605-672. MR 93a:11043
  • [Ha] H. Hasse, J. reine angew Math. 153 (1924) 113-130.
  • [Hec1] E. Hecke, Math. Zeit, Vol. 1 (1918), 357-376.
  • [Hec2] E. Hecke, Math. Ann., Vol. 114 (1937), 1-28.
  • [He1] D. Hejhal, Eigenvalues of the Laplacian for Hecke triangle groups, Memoirs of the AMS 97 (1992), no. 469. MR 93f:11043
  • [He2] D. Hejhal, The Selberg trace formula for ${\rm PSL}(2,R)$. Vol. I. S. L. Notes, Vol. 548 (1976); and Vol. 1001, Springer-Verlag, Berlin-New York (1980). MR 55:12641
  • [He3] D. Hejhal, On eigenfunctions of the Laplacian for Hecke triangle groups, ``Emerging Applications of Number Theory,'' IMA Series 109, Springer, New York (1999), 291-315. MR 2000f:11063
  • [H-R] D. Hejhal and Rackner, On the topography of Maass waveforms for ${\rm PSL}(2, Z)$, Exp. Math., 1 (1992), 275-305. MR 95f:11037
  • [Hel] R. Heller, Bound-state eigenfunctions of classically chaotic Hamiltonian systems: scars of periodic orbits, Phys. Lett. Review 53 (1984), 1515-1518. MR 85k:81055
  • [Hu] M. Huxley, Introduction to Kloostermania, Banach Center Publ. 17, PWN, Warsaw (1982), 217-306. MR 87j:11046
  • [Iw1] H. Iwaniec, The spectral growth of automorphic $L$-functions, J. reine angew Math. 428 (1992), 139-159. MR 93g:11049
  • [Iw2] H. Iwaniec, ``Introduction to the spectral theory of automorphic forms,'' Revista Mat. Iber. (1995). MR 96f:11078
  • [I-S1] H. Iwaniec and P. Sarnak, $L\sp \infty$ norms of eigenfunctions of arithmetic surfaces, Annals of Math. (2) 141 (1995), 301-320. MR 96d:11060
  • [I-S2] H. Iwaniec and P. Sarnak, Perspectives on the analytic theory of $L$-functions, GAFA 2000 (Tel Aviv, 1999), Geom. Funct. Anal. 2000, Special Volume, Part II, 705-741. MR 2002b:11117
  • [Ja-Sh] H. Jacquet and J. Shalika, On Euler products and the classification of automorphic representations. I. Amer. Jnl. Math. 103 (1981), 499-558. MR 82m:10050a
  • [Ja] D. Jakobson, Quantum unique ergodicity for Eisenstein series on ${\rm PSL}\sb 2( Z)\backslash {\rm PSL}\sb 2( R)$, Ann de l'Inst. Fourier 44 (1994), 1477-1504. MR 96b:11068
  • [Ju] C. Judge, On the existence of Maass cusp forms on hyperbolic surfaces with cone points, JAMS 8 (1995), 715-759. MR 96b:11069
  • [Ka-Sp] A. Katok and R. Spatzier, Invariant measures for higher-rank hyperbolic abelian actions, Erg. and Dyn. Systems 16 (1996), 751-778. MR 97d:58116
  • [Ka-Sa] N. Katz and P. Sarnak, Zeroes of zeta functions and symmetry, BAMS 36 (1999), 1-26. MR 2000f:11114
  • [Ke-Sn] J. Keating and N. Snaith, Random matrix theory and $L$-functions at $s=1/2$, CMP 214, No. 1 (2000), 91-111. MR 2002c:11108
  • [K] H. Kim, ``Functoriality of the exterior square of $GL_4$and the symmetric fourth power of $GL_2$'', JAMS, Vol. 16, 1 (2003), 139-183.
  • [Ki-Sa] H. Kim and P. Sarnak, Appendix to [K], JAMS 16 (2003), no. 1, 139-183 (electronic).
  • [K-S1] H. Kim and F. Shahidi, Functorial products for ${\rm GL}\sb 2\times{\rm GL}\sb 3$ and the symmetric cube for ${\rm GL}\sb 2$, Annals of Math. 155 (2002), 837-893.
  • [K-S2] H. Kim and F. Shahidi, Cuspidality of symmetric powers with applications, Duke Math. J. 112 (2002), 177-197. MR 2003a:11057
  • [Kl] F. Klein, ``Lectures on the Icosahedron'', Paul French Frübner Co. (1913).
  • [Kne] M. Kneser, ``Quadratische formen'', Lecture Notes, Göttingen (1974).
  • [Ku] N. Kuznecov, The Petersson conjecture for cusp forms of weight zero and the Linnik conjecture. Sums of Kloosterman sums. (Russian), Mat. Sb. (NS) 111(153) (1980), no. 3, 334-383, 479. MR 81m:10053
  • [Lan1] S. Lang, $SL_2 (\mathbb{R} )$, Addison-Wesley (1975). MR 55:3170
  • [Lan2] S. Lang, ``Algebraic Number Theory'', Addison Wesley (1970). MR 44:181
  • [La1] R. Langlands, Base change for ${\rm GL}(2)$, Annals of Math. Studies 96, Princeton University Press (1980). MR 82a:10032
  • [La2] R. Langlands, Problems in the theory of automorphic forms. Problems in the theory of automorphic forms, Lectures in modern analysis and applications, III, S. L. Notes, Vol. 170 (1970), 18-61. MR 46:1758
  • [La3] R. Langlands, On the functional equations satisfied by Eisenstein series, S.L. Notes, Vol. 544 (1976). MR 58:28319
  • [La4] R. Langlands, Eisenstein series, the trace formula, and the modern theory of automorphic forms, Number Theory, Trace Formulas and Discrete Groups, Academic Press (1988), 125-155. MR 90e:11077
  • [L-R] E. Lapid and S. Rallis, Annals of Math. 157 (2003), 1-26.
  • [Lax] P. Lax, The Radon transform and translation representation, J. Evol. Equ. 1 (2001), 311-323. MR 2003a:58047
  • [L-P] P. Lax and R. Phillips, Scattering theory for automorphic functions, Annals of Math. Studies, Vol. 87 (1976). MR 58:27768
  • [Laz] V. F. Lazutkin, KAM Theory and Semi Classical Approximation to Eigenfunctions, Ergeb. Math., Vol. 24, Springer (1993). MR 94m:58069
  • [Li1] E. Lindenstrauss, ``Invariant measures and arithmetic quantum unique ergodicity", preprint (2003).
  • [Li2] E. Lindenstrauss, On quantum unique ergodicity for $\Gamma\backslash\mathbb{H}\times\mathbb{H} $, IMRN 17 (2001), 913-933. MR 2002k:11076
  • [L-P-S] A. Lubotzky, R. Phillips and P. Sarnak, Ramanujan graphs, Combinatorica 8, no. 3 (1988), 261-277. MR 89m:05099
  • [Lu] W. Luo, Nonvanishing of $L$-values and the Weyl law, Annals of Math. 154 (2001), 477-502. MR 2002i:11084
  • [L-R-S] W. Luo, Z. Rudnick and P. Sarnak, On Selberg's eigenvalue conjecture, GAFA 5 (1995), 387-401. MR 96h:11045
  • [L-S1] W. Luo and P. Sarnak, Quantum ergodicity of eigenfunctions on ${\rm PSL}\sb 2(\mathbf{Z})\backslash \mathbf{H}\sp 2$, Proc. Publ. IHES 81 (1995), 207-237. MR 97f:11037
  • [L-S2] W. Luo and P. Sarnak, ``Quantum ergodicity of eigenfunctions on $SL_2 ( \mathbb{Z} ) \backslash \mathbb{H}\; {\rm II}$'' (in preparation).
  • [Mas] H. Maass, Math. Ann. 121 (1949), 141-183. MR 11:163c
  • [M] G. Margulis, Explicit group-theoretic constructions of combinatorial schemes and their applications in the construction of expanders and concentrators, J. Prob. Inf. Transmission 24 (1988), 39-46. MR 89f:68054
  • [Mc] H. McKean, An upper bound to the spectrum of $\Delta $ on a manifold of negative curvature, J.D.G. 4 (1970), 359-366. MR 42:1009
  • [Mc-Si] H. McKean and I. Singer, Curvature and the eigenvalues of the Laplacian, Jnl. of Diff. Geom. 1 (1967), 43-69. MR 36:828
  • [Me] M. Mehta, Random Matrices, Academic Press (1980). MR 92f:82002
  • [Mi] S.D. Miller, On the existence and temperedness of cusp forms for ${\rm SL}\sb 3(\mathbb{Z} )$, J. reine agnew. Math. 533 (2001), 127-169. MR 2002b:11070
  • [M-S] S.D. Miller and W. Schmid, ``Automorphic Distributions, $L$-functions and Voronoi Summation for $GL(3)$'' (2002), preprint.
  • [Mu] W. Müller (2002) (in preparation).
  • [Pe] Y. Petridis, Spectral data for finite volume hyperbolic surfaces at the bottom of the continuous spectrum, J. Funct. Anal. 124 (1994), no. 1, 61-94. MR 95g:11045
  • [P-S1] R. Phillips and P. Sarnak, On cusp forms for co-finite subgroups of ${\rm PSL}(2, R)$, Invent. Math. 80 (1985), 339-364. MR 86m:11037
  • [P-S2] R. Phillips and P. Sarnak, The Weyl theorem and the deformation of discrete groups, Comm. Pure and Appl. Math 38 (1985), 853-866. MR 87f:11035
  • [P-S3] R. Phillips and P. Sarnak, Perturbation theory for the Laplacian on automorphic functions, JAMS 5 (2), No. 1 (1992), 1-32. MR 92g:11056
  • [PS-R] I. Piatetsky-Shapiro and S. Rallis, Rankin triple $L$functions, Compositio Math. 64 (1987), 31-115. MR 89k:11037
  • [Ra] M. Ratner, The central limit theorem for geodesic flows on $n$-dimensional manifolds of negative curvature, Israel Jnl. of Math., Vol. 16 (1973), 181-197. MR 48:11446
  • [R-S] Z. Rudnick and P. Sarnak, The behaviour of eigenstates of arithmetic hyperbolic manifolds, CMP 161 (1994), 195-213. MR 95m:11052
  • [Ru] D. Rudolph, $\times 2$ and $\times 3$ invariant measures and entropy, Erg. Theory and Dyn. Systems 10 (1990), 395-406. MR 91g:28026
  • [Sa1] P. Sarnak, ``A panorama of number theory'', ed. Wustholz, Cambridge (2002), 121-127.
  • [Sa2] P. Sarnak, On cusp forms, The Selberg trace formula and related topics (Brunswick, Maine, 1984), AMS Contemp. Math. 53 (1986). MR 87j:11047
  • [Sa3] P. Sarnak, Letter to Z. Rudnick, August, 2002.
  • [Sa4] P. Sarnak, ``Arithmetic Quantum Chaos,'' Israel Math. Conf. Proc., Vol. 8, Ramat Gan (1995), 183-236. MR 96d:11059
  • [Sa5] P. Sarnak, Estimates for Rankin-Selberg $L$-functions and quantum unique ergodicity, J. Funct. Anal. 184 (2001), 419-453. MR 2003c:11050
  • [Sa-Wa] P. Sarnak and T. Watson, ``$L^4$ norms of eigenfunctions on the modular surface" (in preparation).
  • [SP] R. Schulze-Pillot, Exceptional integers for genera of integral ternary positive definite quadratic forms, D.M.J. 102 (2000), 351-357. MR 2001a:11068
  • [Sel3] A. Selberg, Collected Works, Springer (1988), 341-355.
  • [Sel1] A. Selberg, Collected Works, Vol. I, Springer (1989), 626-674. MR 92h:01083
  • [Sel2] A. Selberg, On the estimation of Fourier coefficients of modular forms, Proc. Symp. Pure Math., VIII, Amer. Math. Soc. (1965), 1-15. MR 32:93
  • [Se] J-P. Serre, Modular forms of weight one and Galois representations, Algebraic Number Fields, ed. Fröhlich, Academic Press (1977), 193-268. MR 56:8497
  • [Sh] F. Shahidi, Automorphic $L$-functions: a survey, Perspectives in Math., Vol. 10, Academic Press (1990), 415-437. MR 91i:11170
  • [Shi1] G. Shimura Introduction to the arithmetic theory of automorphic functions, Princeton University Press (1971). MR 47:3318
  • [Shi2] G. Shimura, On the holomorphy of certain Dirichlet series, Proc. London Math. Soc. (3) 31 (1975), 79-98. MR 52:3064
  • [Shn] A. Shnirelman, Uspekhi Mat. Nauk 29/6 (1974), 181-182.
  • [Si] B. Simon, Resonances in $n$-body quantum systems with dilatation analytic potentials and the foundations of time-dependent perturbation theory, Ann. of Math. 2 97 (1973), 247-274. MR 50:6378
  • [So] C. Sogge, Concerning the $L\sp p$ norm of spectral clusters for second-order elliptic operators on compact manifolds, J. Func. Anal. 77 (1988), 123-138. MR 89d:35131
  • [Sp] F. Spinu, Thesis, Princeton University (2003).
  • [Sta] H. Stark, Modular Forms, R. Rankin, ed., Ellis Howard (1984), 263-268. MR 86f:11003
  • [St] G. Steil, Diplom. Math. Univ. Hamburg (1992).
  • [Ste] E. Stein, Oscillatory integrals in Fourier analysis, Lectures in Harmonic Analysis, Princeton University Press (1986), 307-356. MR 88g:42022
  • [Str] A. Strombergsson, private communication (2003).
  • [Ta] K. Takeuchi, A characterization of arithmetic Fuchsian groups, J. Math. Soc. Japan 27, 4 (1975), 600-612. MR 53:2842
  • [To] A. Toth, Roots of quadratic congruences, IMRN (2000), No. 14, 719-739. MR 2001m:11159
  • [Tu] J. Tunnell, Artin's conjecture for representations of octahedral type, Bull. Amer. Math. Soc. 5 (1981), 173-175. MR 82j:12015
  • [Ve] A. Venkov, Spectral theory of automorphic functions, A translation of Trudy Mat. Inst. Steklov. 153 (1981), Proc. Steklov Inst. Math. 1982, no. 4 (153) (1983). MR 85j:11060b
  • [Wa] T. Watson, Thesis, Princeton University (2002).
  • [Wei] A. Weil, Automorphic forms and Dirichlet series, Springer Lect. Notes, Vol. 189 (1971).
  • [W] M. Wolf, Infinite energy harmonic maps and degeneration of hyperbolic surfaces in moduli space, J.D.G. 33 (1991), 487-539. MR 92b:58055
  • [Wol1] S. Wolpert, Invent. Math., Vol. 108 (1992), 67-89, 91-129. MR 93b:58160
  • [Wol2] S. Wolpert, Disappearance of cusp forms in special families, Annals of Math. (2) 139 (1994), 239-291. MR 95e:11062
  • [Ze] S. Zelditch, Uniform distribution of eigenfunctions on compact hyperbolic surfaces, Duke Math. Jnl. 55 (1987), 919-941. MR 89d:58129

Similar Articles

Retrieve articles in Bulletin of the American Mathematical Society with MSC (2000): 11F03, 11N75, 11R42, 35P30

Retrieve articles in all journals with MSC (2000): 11F03, 11N75, 11R42, 35P30

Additional Information

Peter Sarnak
Affiliation: Courant Institute of Math. Sciences; and Department of Mathematics, Princeton University, Princeton, NJ 05840
Email: sarnak@Math.Princeton.EDU

DOI: https://doi.org/10.1090/S0273-0979-03-00991-1
Received by editor(s): March 6, 2003
Published electronically: July 17, 2003
Additional Notes: This paper is based on notes for the Colloquium Lecture presented at the Joint Mathematics Meetings in Baltimore in January 2003
Article copyright: © Copyright 2003 American Mathematical Society