Duke Mathematical Journal

A sum formula for a pair of closed geodesics on a hyperbolic surface

Nigel J. E. Pitt

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Abstract

We consider an arbitrary pair of closed geodesics and the corresponding period integrals for the eigenfunctions of the Laplacian on a compact hyperbolic surface. A summation formula that relates geometric information about the geodesics (namely, the angles of intersection and lengths of common perpendiculars between them) to the period integrals is proved. As a corollary, an asymptotic is obtained for the second moment of the period integrals for a fixed geodesic as an average over the eigenvalue with an error term that can be interpreted in terms of the geometric data

Article information

Source
Duke Math. J., Volume 143, Number 3 (2008), 407-435.

Dates
First available in Project Euclid: 3 June 2008

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1212500462

Digital Object Identifier
doi:10.1215/00127094-2008-024

Mathematical Reviews number (MathSciNet)
MR2423758

Zentralblatt MATH identifier
1195.11073

Subjects
Primary: 11F72: Spectral theory; Selberg trace formula 11F67: Special values of automorphic $L$-series, periods of modular forms, cohomology, modular symbols

Citation

Pitt, Nigel J. E. A sum formula for a pair of closed geodesics on a hyperbolic surface. Duke Math. J. 143 (2008), no. 3, 407--435. doi:10.1215/00127094-2008-024. https://projecteuclid.org/euclid.dmj/1212500462


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References

  • A. F. Beardon, The Geometry of Discrete Groups, Grad. Texts in Math. 91, Springer, New York, 1983.
  • N. Burq, P. GéRard, and N. Tzvetkov, Restrictions of the Laplace-Beltrami eigenfunctions to submanifolds, Duke Math. J. 138 (2007), 445--486.
  • J. D. Fay, Fourier coefficients of the resolvent for a Fuchsian group, J. Reine Angew. Math. 293/294 (1977), 143--203.
  • A. Good, Local Analysis of Selberg's Trace Formula, Lecture Notes in Math. 1040, Springer, Berlin, 1983.
  • E. W. Hobson, The Theory of Spherical and Ellipsoidal Harmonics, Chelsea, New York, 1955.
  • H. Iwaniec, Prime geodesic theorem, J. Reine Angew. Math. 349 (1984), 136--159.
  • —, Introduction to the Spectral Theory of Automorphic Forms, Bibl. Rev. Mat. Iberoamericana, Rev. Mat. Iberoamericana, Dep. Mat., Univ. Autónoma Madrid, Madrid, 1995.
  • S. Katok, Closed geodesics, periods and arithmetic of modular forms, Invent. Math. 80 (1985), 469--480.
  • J. Lehner, Discontinuous Groups and Automorphic Functions, Math. Surveys 8, Amer. Math. Soc., Providence, 1964.
  • W. Z. Luo and P. Sarnak, Quantum ergodicity of eigenfunctions on PSL$_2(\Bbb Z)\backslash \Bbb H$, Inst. Hautes Études Sci. Publ. Math. 81 (1995), 207--237.
  • N. J. E. Pitt, On pairs of closed geodesics on hyperbolic surfaces, Ann. Inst. Fourier (Grenoble) 49 (1999), 1--25.
  • P. Sarnak, ``Arithmetic quantum chaos'' in The Schur Lectures (Tel Aviv, 1992), Israel Math. Conf. Proc. 8, Bar-Ilan Univ., Ramat Gan, Israel, 1995, 183--236.
  • A. Seeger and C. D. Sogge, Bounds for eigenfunctions of differential operators, Indiana Univ. Math. J. 38 (1989), 669--682.
  • A. Selberg, Collected Papers, Vol. I, Springer, Berlin, 1989.
  • E. C. Titchmarsh, Introduction to the Theory of Fourier Integrals, 3rd ed., Chelsea, New York, 1986.
  • E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, 4th ed., Cambridge Univ. Press, New York, 1962.
  • D. V. Widder, The Laplace Transform, Princeton Math. Ser. 6, Princeton Univ. Press, Princeton, 1941.
  • S. Zelditch, Selberg trace formulae, pseudodifferential operators, and geodesic periods of automorphic forms, Duke Math. J. 56 (1988), 295--344.