Communications in Mathematical Physics

, Volume 338, Issue 3, pp 919–951 | Cite as

Arithmetic, Zeros, and Nodal Domains on the Sphere

  • Michael MageeEmail author


We obtain lower bounds for the number of nodal domains of Hecke eigenfunctions on the sphere. Assuming the generalized Lindelöf hypothesis we prove that the number of nodal domains of any Hecke eigenfunction grows with the eigenvalue of the Laplacian. By a very different method, we show unconditionally that the average number of nodal domains of degree l Hecke eigenfunctions grows significantly faster than the uniform growth obtained under Lindelöf.


Automorphic Form Automorphic Representation Quantum Chaos Nodal Domain Maass Form 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Berry M.V.: Regular and irregular semiclassical wavefunctions. J. Phys. A 10(12), 2083–2091 (1977)zbMATHADSCrossRefMathSciNetGoogle Scholar
  2. 2.
    Blum G., Gnutzmann S., Smilanksy U.: Nodal domains statistics: a criterion for quantum chaos. Phys. Rev. Lett. 88, 114101 (2002)ADSCrossRefGoogle Scholar
  3. 3.
    Bogomolny E., Schmit C.: Percolation model for nodal domains of chaotic wave functions. Phys. Rev. Lett. 88, 114102 (2002)ADSCrossRefGoogle Scholar
  4. 4.
    Böcherer S., Sarnak P., Schulze-Pillot R.: Arithmetic and equidistribution of measures on the sphere. Commun. Math. Phys. 242(1–2), 67–80 (2003)zbMATHADSCrossRefGoogle Scholar
  5. 5.
    Bourgain J., Lindenstrauss E.: Entropy of quantum limits. Commun. Math. Phys. 233(1), 153–171 (2003)zbMATHADSCrossRefMathSciNetGoogle Scholar
  6. 6.
    Cheng S.Y.: Eigenfunctions and nodal sets. Comment. Math. Helv. 51(1), 43–55 (1976)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Conrey J.B., Farmer D.W.: Mean values of L-functions and symmetry. Int. Math. Res. Notices 17, 883–908 (2000)CrossRefMathSciNetGoogle Scholar
  8. 8.
    Courant, R., Hilbert, D.: Methods of mathematical physics. Interscience Publishers, Inc., New York, NY (1953)Google Scholar
  9. 9.
    Deligne P.: La conjecture de Weil. I. Inst. Hautes Études Sci. Publ. Math. 43, 273–307 (1974)CrossRefMathSciNetGoogle Scholar
  10. 10.
    Edelman A., Kostlan E.: How many zeros of a random polynomial are real?. Bull. Am. Math. Soc. (N.S.) 32(1), 1–37 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Eichler, M.: The basis problem for modular forms and the traces of the Hecke operators. In: Modular functions of one variable, I (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972). Springer, Berlin, pp. 75–151. Lecture Notes in Math., vol. 320 (1973)Google Scholar
  12. 12.
    Ghosh A., Reznikov A., Sarnak P.: Nodal domains of Maass forms I. Geom. Funct. Anal. 23(5), 1515–1568 (2013)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Iwaniec H., Sarnak P.: L norms of eigenfunctions of arithmetic surfaces. Ann. Math. (2) 141(2), 301–320 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Jacquet H., Chen N.: Positivity of quadratic base change L-functions. Bull. Soc. Math. France 129(1), 33–90 (2001)zbMATHMathSciNetGoogle Scholar
  15. 15.
    Jung, J.: Quantitative quantum ergodicity and the nodal domains of Maass-Hecke cusp forms. Preprint, arXiv:1301.6211v2 [math.NT] (2014)
  16. 16.
    Jung, J., Zelditch, S.: Number of nodal domains and singular points of eigenfunctions of negatively curved surfaces with an isometric involution. Preprint, arXiv:1310.2919 [math.SP] To appear in J. Differ. Geom. (2013)
  17. 17.
    Kac M.: On the average number of real roots of a random algebraic equation. Bull. Am. Math. Soc. 49, 314–320 (1943)zbMATHCrossRefGoogle Scholar
  18. 18.
    Katz N.M., Sarnak P.: Zeroes of zeta functions and symmetry. Bull. Am. Math. Soc. (N.S.) 36(1), 1–26 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Lewy H.: On the minimum number of domains in which the nodal lines of spherical harmonics divide the sphere. Commun. Partial Differ. Equ. 2(12), 1233–1244 (1977)zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Martin, K., Whitehouse, D.: Central L-values and toric periods for GL(2). Int. Math. Res. Notices IMRN, 1, Art. ID rnn127, 141–191 (2009)Google Scholar
  21. 21.
    Michel P., Venkatesh A.: The subconvexity problem for GL2. Publ. Math. Inst. Hautes Études Sci. 111, 171–271 (2010)CrossRefMathSciNetGoogle Scholar
  22. 22.
    Milićević D.: Large values of eigenfunctions on arithmetic hyperbolic surfaces. Duke Math. J. 155(2), 365–401 (2010)zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Nastasescu, M.: The number of ovals of a random real plane curve. B.A. Thesis, Princeton University (2011)Google Scholar
  24. 24.
    Nazarov F., Sodin M.: On the number of nodal domains of random spherical harmonics. Am. J. Math. 131(5), 1337–1357 (2009)zbMATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Sarnak, P.: Arithmetic Quantum Chaos Lecture Notes. (1993)
  26. 26.
    Sarnak, P.: Letter to Andrei Reznikov. (2008)
  27. 27.
    Seeger A., Sogge C.: Bounds for eigenfunctions of differential operators. Indiana Univ. Math. J. 38, 669–682 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Soundararajan K.: Extreme values of zeta and L-functions. Math. Ann. 342(2), 467–486 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Szegö, G.: Orthogonal Polynomials. Amer. Math. Soc. Colloq. Publ. 23, Amer. Math. Soc., New York (1939)Google Scholar
  30. 30.
    Tate, J.: Number theoretic background. In: Automorphic forms, representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 2, Proc. Sympos. Pure Math., XXXIII. Amer. Math. Soc., Providence, RI, pp. 3–26 (1979)Google Scholar
  31. 31.
    VanderKam J.M.: L norms and quantum ergodicity on the sphere. Int. Math. Res. Notices 7, 329–347 (1997)CrossRefMathSciNetGoogle Scholar
  32. 32.
    Waldspurger J.-L.: Sur les valeurs de certaines fonctions L automorphes en leur centre de symétrie. Compositio Math. 54(2), 173–242 (1985)zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of California at Santa CruzSanta CruzUSA
  2. 2.School of MathematicsInstitute for Advanced StudyPrincetonUSA

Personalised recommendations