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On the Behavior of Eisenstein Series Through Elliptic Degeneration

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Abstract

Let Γ be a Fuchsian group of the first kind acting on the hyperbolic upper half plane \({\mathbb{H}}\), and let \({M = \Gamma\backslash \mathbb{H}}\) be the associated finite volume hyperbolic Riemann surface. If γ is a primitive parabolic, hyperbolic, resp. elliptic element of Γ, there is an associated parabolic, hyperbolic, resp. elliptic Eisenstein series. In this article, we study the limiting behavior of these Eisenstein series on an elliptically degenerating family of finite volume hyperbolic Riemann surfaces. In particular, we prove the following result. The elliptic Eisenstein series associated to a degenerating elliptic element converges up to a factor to the parabolic Eisenstein series associated to the parabolic element which fixes the newly developed cusp on the limit surface.

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References

  1. Abikoff W.: Degenerating families of Riemann surfaces. Ann. of Math. (2) 105(1), 29–44 (1977)

    Article  MathSciNet  Google Scholar 

  2. Beardon, A.F.: The geometry of discrete groups. Graduate Texts in Mathematics 91, New York: Springer-Verlag, 1983

  3. D’Hoker E., Phong D.H.: On determinants of Laplacians on Riemann surfaces. Commun. Math. Phys. 104(4), 537–545 (1986)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  4. Dodziuk, J., Jorgenson, J.: Spectral asymptotics on degenerating hyperbolic 3-manifolds. Mem. Amer. Math. Soc. 135(643) (1998)

  5. Falliero T.: Dégénérescence de séries d’Eisenstein hyperboliques. Math. Ann. 339(2), 341–375 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  6. Fay, J.: Theta functions on Riemann surfaces. Lecture Notes in Mathematics 352, Berlin: Springer-Verlag, 1973

  7. Garbin, D., Jorgenson, J.: Spectral convergence of elliptically degenerating Riemann surfaces. In preparation

  8. Garbin D., Jorgenson J., Munn M.: On the appearance of Eisenstein series through degeneration. Comment. Math. Helv. 83(4), 701–721 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  9. Hejhal, D.: The Selberg trace formula for PSL(2, R). Vol. 2. Lecture Notes in Mathematics 1001, Berlin: Springer-Verlag, 1983

  10. Hejhal, D.: Regular b-groups, degenerating Riemann surfaces, and spectral theory. Mem. Amer. Math. Soc. 88, no. 437 (1990)

    Google Scholar 

  11. Huntley J., Jorgenson J., Lundelius R.: On the asymptotic behavior of counting functions associated to degenerating hyperbolic Riemann surfaces. J. Funct. Anal. 149(1), 58–82 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  12. Imamoglu, Ö., O’Sullivan, C.: Parabolic, hyperbolic, and elliptic Poincaré series. http://arxiv.org/abs/0806.4398v1[math.NT], 2008

  13. Iwaniec, H.: Spectral methods of automorphic forms. Graduate Studies in Mathematics 53, Providence, RI: Amer. Math. Soc., 2002

  14. Ji L., Zworski M.: The remainder estimate in spectral accumulation for degenerating hyperbolic surfaces. J. Funct. Anal. 114(2), 412–420 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  15. Jorgenson J., Kramer J.: Bounds for special values of Selberg zeta functions of Riemann surfaces. J. Reine Angew. Math. 541, 1–28 (2001)

    MATH  MathSciNet  Google Scholar 

  16. Jorgenson, J., Kramer, J.: Canonical metrics, hyperbolic metrics, and Eisenstein series for \({{\rm PSL}_{2}(\mathbb{R})}\). Preprint, 2003

  17. Jorgenson, J., Kramer, J., v. Pippich, A.-M.: On the spectral expansion of hyperbolic Eisenstein series. Submitted

  18. Jorgenson J., Lundelius R.: Convergence of the heat kernel and the resolvent kernel on degenerating hyperbolic Riemann surfaces of finite volume. Quaestiones Math. 18(4), 345–363 (1995)

    MATH  MathSciNet  Google Scholar 

  19. Jorgenson J., Lundelius R.: A regularized heat trace for hyperbolic Riemann surfaces of finite volume. Comment. Math. Helv. 72(4), 636–659 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  20. Jorgenson J., Lundelius R.: Convergence of the normalized spectral counting function on degenerating hyperbolic Riemann surfaces of finite volume. J. Funct. Anal. 149(1), 25–57 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  21. Judge, C.: The Laplace spectrum of surfaces with cone points. Ph.D. thesis, Univ. of Maryland College Park, 1993

  22. Judge C.: On the existence of Maass cusp forms on hyperbolic surfaces with cone points. J. Amer. Math. Soc. 8(3), 715–759 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  23. Judge C.: Conformally converting cusps to cones. Conform. Geom. Dyn. 2, 107–113 (1998) (electronic)

    Article  MATH  MathSciNet  Google Scholar 

  24. Kramer, J., v. Pippich, A.-M.: Elliptic Eisenstein series for \({{\rm PSL}_{2}(\mathbb{Z})}\). Submitted

  25. Kubota T.: Elementary theory of Eisenstein series. Kodansha Ltd., Tokyo (1973)

    MATH  Google Scholar 

  26. Kudla S.S., Millson J.J.: Harmonic differentials and closed geodesics on a Riemann surface. Invent. Math. 54(3), 193–211 (1979)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  27. Lundelius R.: Asymptotics of the determinant of the Laplacian on hyperbolic surfaces of finite volume. Duke Math. J. 71(1), 211–242 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  28. Obitsu, K.: Asymptotics of degenerating Eisenstein series. http://arxiv.org/abs/0801.3691v3[math.cv], 2008

  29. v. Pippich, A.-M.: Elliptische Eisensteinreihen. Diplomarbeit, Humboldt-Universität zu Berlin, 2005

  30. v. Pippich, A.-M.: The arithmetic of elliptic Eisenstein series. Ph.D. thesis, Humboldt-Universität zu Berlin. In preparation

  31. Polyakov A.M.: Quantum geometry of bosonic strings. Phys. Lett. B 103(3), 207–210 (1981)

    Article  ADS  MathSciNet  Google Scholar 

  32. Randol B.: Cylinders in Riemann surfaces. Comment. Math. Helv. 54(1), 1–5 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  33. Risager M.S.: On the distribution of modular symbols for compact surfaces. Int. Math. Res. Not. 41(41), 2125–2146 (2004)

    Article  MathSciNet  Google Scholar 

  34. Sarnak P.: Determinants of Laplacians. Commun. Math. Phys. 110(1), 113–120 (1987)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  35. Wolpert S.A.: Asymptotics of the spectrum and the Selberg zeta function on the space of Riemann surfaces. Commun. Math. Phys. 112(2), 283–315 (1987)

    Article  MATH  ADS  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Mathematics Ph.D. Program, The Graduate Center of CUNY, 365 Fifth Avenue, New York, NY, U.S.A.

    D. Garbin

  2. Mathematics Ph.D. Program, Humboldt-Universität zu Berlin, Institut für Mathematik, Rudower Chaussee 25, 12489, Berlin, Germany

    A.-M. v. Pippich

Authors
  1. D. Garbin
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  2. A.-M. v. Pippich
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Correspondence to D. Garbin.

Additional information

Communicated by L. Takhtajan

The first author acknowledges support from the PSC–CUNY grant 69288-00-38.

The second author acknowledges support from the DFG Graduate School Berlin Mathematical School and the DFG Research Training Group Arithmetic and Geometry.

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Garbin, D., Pippich, AM.v. On the Behavior of Eisenstein Series Through Elliptic Degeneration. Commun. Math. Phys. 292, 511–528 (2009). https://doi.org/10.1007/s00220-009-0892-3

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