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Kloosterman Sums and their Applications: A Review

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The article reviews recent work in quantum chaos related to quantum mechanical systems which arise in the study of analytic number theory. The central themes of the review are Kloosterman sums, the Selberg-Kloosterman zeta function, the Kuznecov trace formula, the Sato-Tate distribution, and the appearance of Hecke operators in the study of quantum chaos.

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References

  • R. Aurich, E. B. Bogomolny, and F. Steiner, Periodic orbits on the regular hyperbolic octagon, Physica D48 (1991) 91–101.

    MathSciNet  Google Scholar 

  • R. Aurich, J. Bolte, C. Matthies, M. Sieber and F. Steiner, Crossing the entropy barrier of dynamical zeta functions, Physica D63 (1993) 71–86.

    MathSciNet  Google Scholar 

  • R. Aurich, J. Bolte and F. Steiner, Universal signatures of quantum chaos, Phys. Rev. Lett. 73 (1994) 1356–1359.

    Article  Google Scholar 

  • R. Aurich and J. Marklof, Trace formulae for three-dimensional hyperbolic lattices and application to a strongly chaotic tetrahedral billiard, DESY 95-009 (1995).

  • R. Aurich, C. Matthies, M. Sieber, and F. Steiner, A new rule for quantizing chaos (preprint, 1991); Novel rule for quantizing chaos, Phys. Rev. Lett. 68 (1992) 1629–1632.

    Article  MathSciNet  MATH  Google Scholar 

  • R. Aurich and F. Steiner, Periodic orbits of a strongly chaotic system, Physica D32 (1988) 451–460.

    MathSciNet  Google Scholar 

  • R. Aurich and F. Steiner, Periodic-orbit sum rules for the Hadamard-Gutzwiller model, Physica D39 (1989) 169–193.

    MathSciNet  Google Scholar 

  • R. Aurich and F. Steiner, Energy-level statistics of the Hadamard-Gutzwiller ensemble, Physica D43 (1990) 155–180.

    MathSciNet  Google Scholar 

  • R. Aurich and F. Steiner, Staircase functions, spectral rigidity, and a rule for quantizing chaos, Phys. Rev. A45 (1992) 583–592.

    MathSciNet  Google Scholar 

  • N. L. Balazs and A. Voros, Chaos on the pseudosphere, Phys. Rep. 143 (1986) 109–240.

    Article  MathSciNet  Google Scholar 

  • E. B. Bogomolny, B. Georgeot, M. J. Giannoni and C. Schmit, Chaotic billiards generated by arithmetic groups, Phys. Rev. Lett. 69 (1992) 1477–1480.

    Article  MathSciNet  MATH  Google Scholar 

  • E. B. Bogomolny, B. Georgeot, M. J. Giannoni and C. Schmit, Trace formulas for arithmetical systems, Phys. Rev. E47 (1993) R2217–R2220.

    MathSciNet  Google Scholar 

  • E. Bogomolny, F. Leyvraz and C. Schmit, Distribution of eigenvalues for the modular group (preprint, 1995).

  • J. Bolte, Some studies on arithmetical chaos in classical and quantum mechanics, Inter. J. Mod. Phys. B7 (1993) 4451–4553.

    Article  MathSciNet  Google Scholar 

  • J. Bolte, Periodic orbits in arithmetical chaos on hyperbolic surfaces, Nonlin. 6 (1993) 935–951.

    Article  MathSciNet  MATH  Google Scholar 

  • J. Bolte and C. Grosche, Selberg trace formula for bordered Riemann surfaces: hyperbolic, elliptic and parabolic conjugacy classes and determinants of Maass-Laplacians, Comm. math. phys. 163 (1994) 217–244.

    Article  MathSciNet  MATH  Google Scholar 

  • J. Bolte and F. Steiner, The Selberg trace formula for bordered Riemann surfaces, Comm. math. phys. 156 (1993) 1–16.

    Article  MathSciNet  MATH  Google Scholar 

  • J. Bolte, G. Steil, and F. Steiner, Arithmetical chaos and violation of universality in energy level statistics, Phys. Rev. Lett. 69 (1992) 2188–2191.

    Article  MathSciNet  MATH  Google Scholar 

  • R. W. Bruggeman, Fourier coefficients of cups forms, Invent, math. 45 (1978) 1–18.

    Article  MathSciNet  MATH  Google Scholar 

  • D. Bump, W. Duke, J. Hoffstein, and H. Iwaniec, An estimate for the Hecke eigenvalues of Maass forms, Duke Math. J. Inter. Math. Res. Not. 4 (1992) 75–81.

    Article  MathSciNet  Google Scholar 

  • A. Comtet, B. Georgeot and S. Ouvry, Trace formula for Riemann surfaces with magnetic field M. Degli Esposti, Classical and quantum equidistribution: an (easy) example, (thesis, Pennsylvania State University, 1994).

  • M. Degli Esposti, S. Graffi and S. Isola, Classical limit of the quantized hyperbolic toral automorphism, Comm. Math. Phys. 167 (1995) 471–507.

    Article  MathSciNet  MATH  Google Scholar 

  • M. Degli Esposti, S. Graffi and S. Isola, Equidistribution of periodic orbits: an overview of classical vs quantum results, Lect. Notes Math. 1589 (1994) 65–91.

    Article  MathSciNet  Google Scholar 

  • M. Degli Esposti and S. Isola, Distribution of closed orbits for linear automorphisms of tori(preprint 1994).

  • P. Deligne, La conjecture de Weil 1, Publ. Math. 1HES 43 (1974) 273–307.

    MathSciNet  Google Scholar 

  • P. Deligne, Application de la formule de traces aux sommes trigonometriques, LNM 569 (1977) 168–232.

    MathSciNet  Google Scholar 

  • P. Deligne and J.-P. Serre, Formes modulaires de poids 1, Ann. Sci. Ec. Norm. Sup. 7 (1974) 507–530.

    MathSciNet  MATH  Google Scholar 

  • J.-M. Deshouillers and H. Iwaniec, Kloosterman sums and Fourier coefficients of cusp forms, Invent, math. 70 (1982) 219–288.

    Article  MathSciNet  MATH  Google Scholar 

  • M. Eisele and D. Mayer, Dynamical zeta functions for Artin’s billiard and the Venkov-Zograf factorization formula, (preprint, 1995).

  • J. Elstrodt, F. Grunewald, and J. Mennicke, Kloosterman sums for Clifford algebras and a lower bound for the positive eigenvalues of the Laplacian for congruence subgroups acting on hyperbolic spaces, Invent, math. 101 (1990) 641–685.

    Article  MathSciNet  MATH  Google Scholar 

  • J. D. Fay, Fourier coefficients of the resolvent for a Fuschsian group, J. f. reine ang. Math. 293/294 (1977) 143–203.

    MathSciNet  Google Scholar 

  • S. Gelbart, Automorphic Forms on Adele Groups (Princeton University Press, Princeton, 1975).

    MATH  Google Scholar 

  • I. M. Gel’fand, M. I. Graev, and I. I. Pyatetskii-Shapiro, Representation Theory and Automorphic Functions (W. B. Saunders, Philadelphia, 1969).

    Google Scholar 

  • D. Goldfeld and P. Sarnak, Sums of Kloosterman sums, Invent, math 71 (1983) 243–250.

    Article  MathSciNet  MATH  Google Scholar 

  • D. Hejhal, The Selberg trace formula and the Riemann zeta function, Duke Math. J. 43 (1976) 441–482.

    Article  MathSciNet  MATH  Google Scholar 

  • D. Hejhal, The Selberg Trace Formula for PSL(2, R), Vol. I, LNM 548; Vol. II, LNM 1001 (Springer, 1976, 1983).

  • D. A. Hejhal, Eigenvalues of the Laplacian for PSL(2, Z): some new results and computational techniques, in International Symposium in Memory of Hua Loo-Keng”, Vol. 1 (Springer, 1991) 59–102.

    MathSciNet  Google Scholar 

  • D. A. Hejhal, Eigenvalues of the Laplacian for Hecke triangle groups, Mem. AMS 97 (1992).

  • D. A. Hejhal, On the distribution of zeros of a certain class of Dirichlet series, Inter. Math. Res. Notes 4 (1992) 83–91.

    Article  MathSciNet  Google Scholar 

  • D. A. Hejhal and S. Arno, On Fourier coefficients of Maass waveforms for PSL(2, Z), Math. Comp. 61 (1993) 245–267.

    MathSciNet  MATH  Google Scholar 

  • M. Huxley, Introduction to Kloostermania, in Banach Center Publications, 17 (ed., H. Iwaniec, Warsaw, 1985).

  • H. Iwaniec, Prime geodesic theorem, J. Reine Angew. Math. 349 (1984) 136–159.

    MathSciNet  MATH  Google Scholar 

  • H. Iwaniec, Promenade along modular forms and analytic number theory, in Topics in Analytic Number Theory, S. W. Graham and J. D. Vaaler, eds. (University of Texas Press, Austin, 1985) 221–303.

    Google Scholar 

  • H. Iwaniec, Selberg’s lower bound of the first eigenvalue for congruence groups, in Number theory, trace formulas, discrete groups, (ed., E. J. Aver, Academic Press, 1989) 371-375.

  • D. Joyner, Distribution Theorems of L-functions (Pitnam-Wiley, New York, 1985).

    Google Scholar 

  • N. M. Katz, Gauss Sums, Kloosterman Sums and Monodromy Groups (Princeton University Press, Princeton, 1988).

    MATH  Google Scholar 

  • H. D. Kloosterman, On the representation of numbers in the form ax 2 + by 2 + CZ2 + dt 2, Acta Math. 49 (1926) 407–464.

    Article  MathSciNet  Google Scholar 

  • A. Knapp, Elliptic Curves (Princeton University Press, Princeton, 1992).

    MATH  Google Scholar 

  • V. Kuznecov, Petersson conjecture for forms of weight zero and the Linnik conjecture, preprint (1977).

  • N.. Kuznecov, Petersson’s conjecture for cusp forms of weight zero and Linnik’s conjecture, Sums of Kloosterman sums, Math. USSR Sbornik 39 (1981) 299–342.

    Article  Google Scholar 

  • N. V. Kuznecov, The distribution of norms of primitive hyperbolic classes of the modular group and asymptotic formulas for the eigenvalues…, Sov. Math. Dokl. 19 (1978) 1053–1056.

    Google Scholar 

  • R. P. Langlands, Problems in the theory of automorphic forms, LNM 170 (1970) 18–61.

    MathSciNet  Google Scholar 

  • R. P. Langlands, On the functional equations satisfied by Eisenstein series, LNM 544 (1976).

  • D. H. Lehmer, Note on the distribution of Ramanujan’s tau function, Math. Comp. 24 (1970) 741–743.

    MathSciNet  MATH  Google Scholar 

  • Y. V. Linnik, Additive problems and eigenvalues of the modular operators, Proc. Inter. Congre. Math. (1962) 270-284.

  • G. Lion and M. Vergne, The Weil Representation, Maslov Index and Theta Series (Birkhauser, Boston, 1980).

    Google Scholar 

  • W. Luo, On the nonvanishing of Rankin-Selberg L-functions, Duke Math. J. 69 (1993) 411–425.

    Article  MathSciNet  MATH  Google Scholar 

  • W. Luo, Zeros of Hecke L-functions associated with cusp forms, Acta Arith. LXXI (1995) 139–158.

    Google Scholar 

  • W. Luo, Z. Rudnick and P. Sarnak, On Selberg’s eigenvalue conjecture, Geom. Func. Anal. 5 (1995) 387–401.

    Article  MathSciNet  MATH  Google Scholar 

  • W. Luo and P. Sarnak, Quantum ergodicity of eigenfunctions on PSL,2(Z)H2, Publ. IHES (to appear, 1994).

  • H. Maass, Über eine neue Art von nichtanalytischen automorphen Funktionen und die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen, Math. Ann. 121 (1949)141–183.

    Article  MathSciNet  MATH  Google Scholar 

  • J. Marklof, On multiplicities in length spectra of arithmetic hyperbolic three-orbifolds, DESY 95-055 (1995).

  • C. Matthies and F. Steiner, Selberg’s zeta function and the quantization of chaos, Phys. Rev. A44 (1991) R7877–R7880.

    MathSciNet  Google Scholar 

  • M. R. Murty, On the estimation of eigenvalues of Hecke operators, Rocky Mt. J. Math. 5 (1985) 521–533.

    Article  Google Scholar 

  • H. Ninnemann, Gutzwiller’s octagon and the triangular billiard T*(2, 3, 8) as models for the quantization of chaotic systems by Selberg’s formula (preprint, 1995).

  • H. Petersson, Über die Entwicklungkoeffizienten der automorphen Formen, Acta Math. 58 (1932) 169–215.

    Article  MathSciNet  Google Scholar 

  • T. Pignataro, Hausdorff dimension, spectral theory and applications to the quantization of geodesic flows on surfaces of constant negative curvature, (thesis, Princeton University, 1984).

  • N. V. Proskurin, The summation formulas lor general Kloosterman sums, Zap. Naucn. Sem Leningrad Otdel. Mat. Inst. Steklov 82 (1979) 103–135; J. Sov. Math.

    MathSciNet  MATH  Google Scholar 

  • B. Randol, Small eigenvalues of the Laplace operator on compact Riemann surfaces, Bull. Amer. Math. Soc. 80 (1974) 996–1000.

    Article  MathSciNet  MATH  Google Scholar 

  • W. Roelcke, Über die Wellengleichung bei Grenzkreisgruppen erster Art, Sitz. Ber. Heidi. Akad. der Wiss. (1956) 4.

  • Z. Rudnick and P. Sarnak, Zeros of principal L-functions and random matrix theory, (preprint, 1995).

  • H. Salié, Über die Kloostermanschen Summen S(u,v;q), Math. Z. 34 (1931) 91–109.

    Article  Google Scholar 

  • P. Sarnak, The arithmetic and geometry of some hyperbolic manifolds, Acta Math. 151 (1983) 253–295.

    Article  MathSciNet  MATH  Google Scholar 

  • P. Sarnak, Arithmetic quantum chaos, Schur Lectures, Tel Aviv (1992).

  • P. Sarnak, Selberg’s eigenvalue conjecture, Notices AMS 42 (1995) 1272–1277.

    MathSciNet  MATH  Google Scholar 

  • I. Sataka, Spherical functions and Ramanujan conjecture, Proc. Symp. in Pure Math. IX (1966) 258–264.

    Article  Google Scholar 

  • C. Schmit, Quantum and classical properties of some billiards on the hyperbolic plane, in Chaos and Quantum Physics, Proc. of Les Houches Summer School, LI1, ed. M. J. Giannoni, A. Voros, and J. Zinn-Justin (North Holland, New York, 1991) 333–369.

    Google Scholar 

  • C. Schmit, Triangular billiards on the hyperbolic plane: spectral properties (preprint, IPNO/TH 91-68, Orsay).

  • C. Schmit and C. Jacquemin, Classical quantization of a compact billiard on the pseudo-sphere (preprint, IPNO/TH 91-65, Orsay).

  • A. Selberg, Harmonic analysis and discontinuous groups in symmetric Riemannian spaces with applications to Dirichiefs series, J. Indian Math. Soc. 20 (1956) 41–87.

    MathSciNet  Google Scholar 

  • A. Selberg, On the estimation of Fourier coefficients of modular forms, Proc. Symposia in Pure Math. VIII (1965) 1–15.

    Article  MathSciNet  Google Scholar 

  • J.-P. Serre, Abelian 1-Adic Representations and Elliptic Curves (Benjamin, New York, 1968).

    Google Scholar 

  • F. Shahidi, Langlands’ functoriality conjecture, in Proc. of the 20th Annual Iranian Math. Conf., March 1989 (University of Tehran Press, 1991) 647-665.

  • F. Shahidi, Symmetric power L-functions for GL(2), preprint, 1992.

  • H. Shimizu, On traces of Hecke operators, J. Fac. Sci. Univ. Tokyo, Sect. 1, 10 (1963) 1–19.

    MATH  Google Scholar 

  • G. Shimura, Introduction to the arithmetic theory of automorphic functions, (Princeton University Press, Princeton, 1971).

    MATH  Google Scholar 

  • G. Steil, Über das diskrete energiespektrum des artinschen billiards, Diploma Thesis, University of Hamburg, 1992.

  • F. Steiner, in Universität Hamburg 1994: Schlaglichter der Forschung zum 75. Jahrestag, (ed. R. Ansorge, Hamburg, 1994) 543.

  • K. Takeuchi, A characterization of arithmetic Fuchsian groups, J. Math. Soc. Japan 27 (1975) 600–612.

    Article  MathSciNet  MATH  Google Scholar 

  • K. Takeuchi, Arithmetic triangle groups, J. Math. Soc. Japan 29 (1977) 91–106.

    Article  MathSciNet  MATH  Google Scholar 

  • A. Terras, Harmonic Analysis on Symmetric Spaces and Applications (Springer-Verlag, Berlin, 1985).

    Book  MATH  Google Scholar 

  • A. B. Venkov, Selberg’s trace formula for the Hecke operator generated by an involution and the eigenvalues of the Laplace-Beltrami operator on the fundamental domain of the modular group PSL(2, Z), lzv. Akad. Nauk SSSR Ser. Mat. 42 (1978) 484–499; Math. USSR lzv. 42 (1978) 448.

    MATH  Google Scholar 

  • A. B. Venkov, Spectral theory of automorphic functions, Proc. Steklov Inst. of Math. 153 (1981).

  • A. B. Venkov, Spectral Theory of Automorphic Functions and its Applications (Kluwer, Dordrecht, 1990).

    Book  MATH  Google Scholar 

  • M. F. Vignéras, Quaternions, Lecture Notes in Math. 800 (1980).

  • M.-F. Vignéras, Quelques remarques sur la conjecture λ1 ≥1/4, (preprint, 1983).

  • A. Weil, On some exponential sums, Proc. Nat. Acad. Sci. USA 34 (1948) 204–207.

    Article  MathSciNet  MATH  Google Scholar 

  • T. Yamada, On the distribution of the norms of the hyperbolic transformations, Osaka J. Math. 3 (1966) 29–37.

    MathSciNet  MATH  Google Scholar 

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    Norman E. Hurt

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Hurt, N.E. Kloosterman Sums and their Applications: A Review. Results. Math. 29, 16–41 (1996). https://doi.org/10.1007/BF03322202

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