|
Spectra of hyperbolic surfaces
Author(s):
Peter
Sarnak
Journal:
Bull. Amer. Math. Soc.
40
(2003),
441-478.
MSC (2000):
Primary 11F03, 11N75, 11R42, 35P30
Posted:
July 17, 2003
Retrieve article in:
PDF DVI PostScript
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:
- [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
-functions and the strong Artin Conjecture,'' preprint (2002). - [Bor]
- A. Borel, Automorphic forms on
, 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
-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
-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
-adiques, Lecture Notes in Math., Vol. 179, Springer (1971), 139-172. - [D-I]
- J. Deshouillers and H. Iwaniec, The nonvanishing of Rankin-Selberg
-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
-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
for split simple Lie groups '', 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
-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
and , 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
-functions, IMRN (2002) 18, 923-932. MR 2003d:11074 - [H-K]
- M. Harris and S. Kudla, The central critical value of a triple product
-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
. 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
, 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
-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,
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
-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
, 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
-functions at , CMP 214, No. 1 (2000), 91-111. MR 2002c:11108 - [K]
- H. Kim, ``Functoriality of the exterior square of
and the symmetric fourth power of '', 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
and the symmetric cube for , 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,
, Addison-Wesley (1975). MR 55:3170 - [Lan2]
- S. Lang, ``Algebraic Number Theory'', Addison Wesley (1970). MR 44:181
- [La1]
- R. Langlands, Base change for
, 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
, 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
-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
, Proc. Publ. IHES 81 (1995), 207-237. MR 97f:11037 - [L-S2]
- W. Luo and P. Sarnak, ``Quantum ergodicity of eigenfunctions on
'' (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
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
, J. reine agnew. Math. 533 (2001), 127-169. MR 2002b:11070 - [M-S]
- S.D. Miller and W. Schmid, ``Automorphic Distributions,
-functions and Voronoi Summation for '' (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
, 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
functions, Compositio Math. 64 (1987), 31-115. MR 89k:11037 - [Ra]
- M. Ratner, The central limit theorem for geodesic flows on
-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,
and 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
-functions and quantum unique ergodicity, J. Funct. Anal. 184 (2001), 419-453. MR 2003c:11050 - [Sa-Wa]
- P. Sarnak and T. Watson, ``
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
-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
-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
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:
10.1090/S0273-0979-03-00991-1
PII:
S 0273-0979(03)00991-1
Received by editor(s):
March 6, 2003
Posted:
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
Copyright of article:
Copyright
2003,
American Mathematical Society
|