Skip to main content
SpringerLink
Log in

Large Values of Eigenfunctions on Arithmetic Hyperbolic 3-Manifolds

  • Published:
Geometric and Functional Analysis Aims and scope Submit manuscript

Abstract

We prove that, on a distinguished class of arithmetic hyperbolic 3-manifolds, there is a sequence of L 2-normalized high-energy Hecke–Maass eigenforms \({\phi_{j}}\) which achieve values as large as \({\lambda^{1/4+o(1)}_{j}}\), where \({( \Delta+\lambda_{j} ) \phi_{j} = 0}\). Arithmetic hyperbolic 3-manifolds on which this exceptional behavior is exhibited are, up to commensurability, precisely those containing immersed totally geodesic surfaces. We adapt the method of resonators and connect values of eigenfunctions to the global geometry of the manifold by employing the pre-trace formula and twists by Hecke correspondences. Automorphic representations corresponding to forms appearing with highest weights in the optimized spectral averages are characterized both in terms of base change lifts and in terms of theta lifts from GSp2.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Asai T.: On certain Dirichlet series associated with Hilbert modular forms and Rankin’s method. Math. Ann. 226(1), 81–94 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  2. Asai T.: On the Doi–Naganuma lifting associated with imaginary quadratic fields. Nagoya Math. J. 71, 149–167 (1978)

    MathSciNet  MATH  Google Scholar 

  3. Bérard P.H.: On the wave equation on a compact Riemannian manifold without conjugate points. Math. Z. 155(3), 249–276 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  4. Berry M.V.: Regular and irregular semiclassical wavefunctions. J. Phys. A:Math. Gen. 10(12), 2083–2091 (1977)

    Article  Google Scholar 

  5. Burq N., Gérard P., Tzvetkov N.: Restrictions of the Laplace–Beltrami eigenfunctions to submanifolds. Duke Math. J. 138(3), 445–486 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  6. Donnelly H.: Exceptional sequences of eigenfunctions for hyperbolic manifolds. Proc. Amer. Math. Soc. 135(5), 1551–1555 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  7. Duistermaat J.J., Kolk J.A.C., Varadarajan V.S.: Functions, flows and oscillatory integrals on flag manifolds and conjugacy classes in real semisimple Lie groups. Compositio Math. 49(3), 309–398 (1983)

    MathSciNet  MATH  Google Scholar 

  8. M. Eichler, Lectures on Modular Correspondences, Tata Inst. Fund. Res. Lectures Math. Phys. 9 (1955).

  9. J. Elstrodt, F. Grünewald, J. Mennicke, Groups Acting on Hyperbolic Space: Harmonic Analysis and Number Theory, Springer, 1997.

  10. Elstrodt J., Grünewald F., Mennicke J.: Zeta functions of binary Hermitian forms and special values of Eisenstein series on three-dimensional hyperbolic space. Math. Ann. 277(4), 655–708 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  11. Flicker Y.: Twisted tensors and Euler products. Bull. Soc. Math. France 116(3), 295–313 (1988)

    MathSciNet  MATH  Google Scholar 

  12. Friedberg S.: OnMaass wave forms and the imaginary quadratic Doi–Naganuma lifting. Math. Ann. 263, 483–508 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  13. Ginzburg D., Jiang D., Soudry D.: Poles of L-functions and theta liftings for orthogonal groups. J. Inst. Math. Jussieu 8(4), 693–741 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  14. Gross B.H., Prasad D.: On the decomposition of a representation of SO n when restricted to SO n-1. Canad. J. Math. 44(5), 974–1002 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  15. M. Harris, R. Taylor, The Geometry and Cohomology of Some Simple Shimura Varieties, Annals of Mathematics Studies, 151, Princeton University Press (2001).

  16. D. Heitkamp, Hecke-Theorie zur \({{\rm SL}(2; \mathfrak{o})}\), Schriftenreihe des Mathematischen Instituts der Universität Münster, 3 Serie, Universität Münster, Mathematisches Institut, Münster, 1992.

  17. Hejhal D.A., Rackner B.N.: On the topography of Maass waveforms for PSL(2, Z). Experiment. Math. 1(4), 275–305 (1992)

    MathSciNet  MATH  Google Scholar 

  18. S. Helgason, Differential Geometry and Symmetric Spaces, Pure Appl. Math. XII, Acad. Press, 1962.

  19. Ichino A.: Pullbacks of Saito–Kurokawa lifts. Invent. Math. 162(3), 551–647 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  20. H. Iwaniec, Introduction to the Spectral Theory of Automorphic Forms, Revista Matemática Iberoamericana, 1995.

  21. Iwaniec H., Sarnak P.: L ∞ norms of eigenfunctions on arithmetic surfaces. Ann. of Math. 2nd Ser. 141(2), 301–320 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  22. Katok S., Sarnak P.: Heegner points, cycles and Maass forms. Israel J. Math. 84(1-2), 193–227 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  23. Y. Kitaoka, Arithmetic of Quadratic Forms. Cambridge Univ. Press, 1993.

  24. Koyama S.: L ∞-norms of eigenfunctions for arithmetic hyperbolic 3-manifolds. Duke Math. J. 77(3), 799–817 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  25. Krishnamurthy M.: The Asai transfer to GL4 via the Langlands–Shahidi method. Int. Math. Res. Not. 41, 2221–2254 (2003)

    Article  MathSciNet  Google Scholar 

  26. Kudla S., Millson J.: Intersection numbers of cycles on locally symmetric spaces and Fourier coefficients of holomorphic modular forms in several complex variables. Inst. Hautes Études Sci. Publ. Math. 71, 121–172 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  27. Lapid E., Offen O.: Compact unitary periods. Compos. Math. 143(2), 323–338 (2007)

    MathSciNet  MATH  Google Scholar 

  28. C. Maclachlan, A.W. Reid, The Arithmetic of Hyperbolic 3-Manifolds. Springer-Verlag, 2003.

  29. S. Marshall, L p bounds for higher rank eigenfunctions and asymptotic of spherical functions, preprint; arXiv:1106.0534v1

  30. Milićević D.: Large values of eigenfunctions on arithmetic hyperbolic surfaces. Duke Math. J. 155(2), 365–401 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  31. D. Milićević, R. Takloo-Bighash, Base change, theta lifting, and arithmetic hyperbolic 3-manifolds of Maclachlan–Reid type, in preparation.

  32. Ramakrishnan D.: Modularity of solvable Artin representations of GO(4)-type. Int. Math. Res. Not. 1, 1–54 (2002)

    Article  MathSciNet  Google Scholar 

  33. Rudnick Z., Sarnak P.: The behaviour of eigenstates of arithmetic hyperbolic manifolds. Comm. Math. Phys. 161(1), 195–213 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  34. P. Sarnak, Arithmetic quantum chaos, The R. A. Blyth Lectures, Univ. Toronto, 1993.

  35. P. Sarnak, A letter to Cathleen Morawetz, 2004.

  36. R. Schulze-Pillot, Representation by integral quadratic forms—a survey, Algebraic and Arithmetic Theory of Quadratic Forms, Contemp. Math. 344, Amer. Math. Soc. (2004), 323–337.

  37. Seeger A., Sogge C.D.: Bounds for eigenfunctions of differential operators. Indiana Univ. Math. J. 38(3), 669–682 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  38. Shimizu H.: Theta series and automorphic forms on GL2. J. Math. Soc. Japan 24(4), 638–683 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  39. Shintani T.: On construction of holomorphic cusp forms of half integral weight. Nagoya Math J. 58, 83–126 (1975)

    MathSciNet  MATH  Google Scholar 

  40. C.L. Siegel, Über die analytische Theorie der quadratischen Formen (I–III), Ann. of Math. 36 (1935), 527–606; Ann. of Math. 37 (1936), 230–263; Ann. of Math. 38 (1937), 212–291.

  41. Soundararajan K.: Extreme values of zeta and L-functions. Math. Ann. 342(2), 467–486 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  42. Takase K.: On certain Dirichlet series associated with automorphic forms on SL(2, C). Man. Math. 56, 293–312 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  43. K. Takase, Wave forms on O(1, q+1) and associated Dirichlet series, Proc. Japan Acad. 62, Ser. A (1986), 112–115.

  44. Y. Tanigawa, Selberg trace formula for Picard groups, Algebraic Number Theory, Kyoto Int. Symp. 1976 (S. Iyanaga,Ed.), Japan Soc. Promotion Sci., Tokyo (1977), 229–242.

  45. J.A. Toth, S. Zelditch, Norms of modes and quasi-modes revisited, Harmonic Analysis at Mount Holyoke. Contemp. Math. 320, Amer. Math. Soc. (2003), 435–458.

  46. V.S. Varadarajan, The method of stationary phase and applications to geometry and analysis on Lie groups, Algebraic and Analytic Methods in Representation Theory, European School of Group Theory 1994 (B. Ørsted, H. Schlichtkrull, eds.), Persp. Math. 17, Acad. Press. (1997), 167–242.

  47. Waldspurger J.-L.: Sur les valeurs de certaines fonctions L automorphes en leur centre de symétrie. Compositio Math. 54, 173–242 (1985)

    MathSciNet  MATH  Google Scholar 

  48. Walling L.: A remark on differences of theta series. J. Number Theory 48(2), 243–251 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  49. T.C. Watson, Rankin triple products and quantum chaos, Ann. Math. to appear.

Download references

Author information

Author notes

  1. Djordje Milićević

    Present address: Max-Planck-Institut für Mathematik, Vivatsgasse 7, D–53111, Bonn, Germany

Authors and Affiliations

  1. Department of Mathematics, Amherst College, Seeley Mudd Building, A/C Box 2239, Amherst, Massachusetts, 01002–5000, USA

    Djordje Milićević

Authors
  1. Djordje Milićević
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Djordje Milićević.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Milićević, D. Large Values of Eigenfunctions on Arithmetic Hyperbolic 3-Manifolds. Geom. Funct. Anal. 21, 1375–1418 (2011). https://doi.org/10.1007/s00039-011-0144-5

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00039-011-0144-5

Keywords and phrases

2010 Mathematics Subject Classification

Search

Navigation