Advertisement

Sup-Norm and Nodal Domains of Dihedral Maass Forms

  • Bingrong HuangEmail author
Article
  • 2 Downloads

Abstract

In this paper, we improve the sup-norm bound and the lower bound of the number of nodal domains for dihedral Maass forms, which are a distinguished sequence of Laplacian eigenfunctions on an arithmetic hyperbolic surface. More specifically, let \({\phi}\) be a dihedral Maass form with spectral parameter \({t_\phi}\), then we prove that \({\|\phi\|_\infty \ll t_\phi^{3/8+\varepsilon} \|\phi\|_2}\), which is an improvement over the bound \({t_\phi^{5/12+\varepsilon} \|\phi\|_2}\) given by Iwaniec and Sarnak. As a consequence, we get a better lower bound for the number of nodal domains intersecting a fixed geodesic segment under the Lindelöf Hypothesis. Unconditionally, we prove that the number of nodal domains grows faster than \({t_\phi^{1/8-\varepsilon}}\) for any \({\varepsilon>0}\) for almost all dihedral Maass forms.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgements

The author would like to thank Prof. Zeév Rudnick for suggesting thinking about dihedral forms, and for his valuable discussions and constant encouragement. He also wants to thank Professors Gergely Harcos, Peter Humphries, Junehyuk Jung, Djordje Milićević, and Matthew Young for their interest, comments, and suggestions. The author gratefully thanks the referees for the constructive comments and recommendations which definitely helped to improve the readability and quality of the paper, and especially simplifying the proof of Lemma 14.

References

  1. 1.
    Assing, E.: On sup-norm bounds part II: \({ GL(2)}\) Eisenstein series (2017). ArXiv preprint arXiv:1710.00363
  2. 2.
    Bérard P.H.: On the wave equation on a compact Riemannian manifold without conjugate points. Math. Z. 155(3), 249–276 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Berry M.V.: Regular and irregular semiclassical wavefunctions. J. Phys. A 10(12), 2083–2091 (1977)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Blomer, V.: Epstein zeta-functions, subconvexity, and the purity conjecture. J. Inst. Math. Jussieu (2016) (to appear)Google Scholar
  5. 5.
    Blomer, V., Harcos, G., Maga, P., Milićević, D.: The sup-norm problem for \({GL(2)}\) over number fields, 39 pp (2016). arXiv:1605.09360
  6. 6.
    Blomer V., Holowinsky R.: Bounding sup-norms of cusp forms of large level. Invent. Math. 179(3), 645–681 (2010)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bogomolny, E., Schmit, C.: Percolation model for nodal domains of chaotic wave functions. Phys. Rev. Lett. 88, 114–102 (2002)Google Scholar
  8. 8.
    Coleman M.D.: A zero-free region for the Hecke L-functions. Mathematika 37(2), 287–304 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Ghosh A., Reznikov A., Sarnak P.: Nodal domains of Maass forms I. Geom. Funct. Anal. 23(5), 1515–1568 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Ghosh A., Reznikov A., Sarnak P.: Nodal domains of Maass forms, II. Am. J. Math. 139(5), 1395–1447 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Harcos G., Templier N.: On the sup-norm of Maass cusp forms of large level: II. Int. Math. Res. Not. IMRN 2012(20), 4764–4774 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Harcos G., Templier N.: On the sup-norm of Maass cusp forms of large level. III. Math. Ann. 356(1), 209–216 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Hecke E.: Eine neue Art von Zetafunktionen und ihre Beziehungen zur Verteilung der Primzahlen (zweite mitteilung). Math. Z. 6, 11–51 (1920)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Hejhal D.A., Rackner B.N.: On the topography of Maass waveforms for \({{\rm PSL}(2,\mathbb{Z})}\). Exp. Math. 1(4), 275–305 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Hejhal D.A., Strömbergsson A.: On quantum chaos and Maass waveforms of CM-type. Found. Phys. 31(3), 519–533 (2001). Invited papers dedicated to Martin C. Gutzwiller, Part IVGoogle Scholar
  16. 16.
    Hoffstein J., Lockhart P.: Coefficients of Maass forms and the Siegel zero. With an appendix by Dorian Goldfeld, Hoffstein, and Daniel Lieman. Ann. Math. (2) 140(1), 161–181 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Huang B.R., Xu Z.: Sup-norm bounds for Eisenstein series. Forum Math. 29(6), 1355–1369 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Iwaniec, H.: Topics in Classical Automorphic Forms. Graduate Studies in Mathematics, vol. 17. American Mathematical Society, Providence (1997)Google Scholar
  19. 19.
    Iwaniec, H.: Spectral Methods of Automorphic Forms, Volume 53 of Graduate Studies in Mathematics. American Mathematical Society, Providence; Revista Matemática Iberoamericana, Madrid, 2nd edn (2002)Google Scholar
  20. 20.
    Iwaniec, H., Kowalski, E.: Analytic Number Theory, Volume 53 of American Mathematical Society Colloquium Publications. American Mathematical Society, Providence (2004)Google Scholar
  21. 21.
    Iwaniec H., Sarnak P.: \({L^\infty}\) norms of eigenfunctions of arithmetic surfaces. Ann. Math. (2) 141(2), 301–320 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Jang S., Jung J.: Quantum unique ergodicity and the number of nodal domains of eigenfunctions. J. Am. Math. Soc. 31(2), 303–318 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Jung J.: Quantitative quantum ergodicity and the nodal domains of Hecke–Maass cusp forms. Commun. Math. Phys. 348(2), 603–653 (2016)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Jung, J., Young, M.P.: Sign changes of the Eisenstein series on the critical line. Int. Math. Res. Not. IMRN. (2016) (to appear)Google Scholar
  25. 25.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Lau, Y.-K., Liu, J., Ye, Y.: A new bound \({k^{2/3+\epsilon}}\) for Rankin–Selberg L-functions for Hecke congruence subgroups. IMRP Int. Math. Res. Pap., pages Art. ID 35090, 78 (2006)Google Scholar
  27. 27.
    Lindenstrauss E.: Invariant measures and arithmetic quantum unique ergodicity. Ann. Math. (2) 163(1), 165–219 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Luo, W.: Spectral mean values of automorphic L-functions at special points. In: Analytic Number Theory: Proceedings of a Conference in Honor of Heini Halberstam, Birkhäuser, pp. 621–632 (1996)Google Scholar
  29. 29.
    Luo W.: L 4-norms of the dihedral Maass forms. Int. Math. Res. Not. IMRN. 2014, 2294–2304 (2014)CrossRefzbMATHGoogle Scholar
  30. 30.
    Liu J., Ye Y.: Subconvexity for Rankin–Selberg L-functions of Maass forms. Geom. Funct. Anal. 12(6), 1296–1323 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Maass, H.: Über eine neue Art von nichtanalytischen automorphen Funktionen und die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen. Math. Ann. 121(1), 141–83 (1949)Google Scholar
  32. 32.
    Montgomery H.L., Vaughan R.C.: The large sieve. Mathematika 20, 119–134 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Nazarov F., Sodin M.: On the number of nodal domains of random spherical harmonics. Am. J. Math. 131(5), 1337–1357 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Rudnick Z., Sarnak P.: The behaviour of eigenstates of arithmetic hyperbolic manifolds. Commun. Math. Phys. 161(1), 195–213 (1994)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Rudnick, Z., Waxman, E.: Angles of Gaussian primes. arXiv:1705.07498 [math.NT]
  36. 36.
    Saha A.: Hybrid sup-norm bounds for Maass newforms of powerful level. Algebra Number Theory 11, 1009–1045 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Salem R., Zygmund A.: Some properties of trigonometric series whose terms have random signs. Acta Math. 91, 245–301 (1954)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Sarnak P.: Arithmetical quantum chaos. In: The Schur Lectures (Tel Aviv 1992), Israel Math. Conf. Proc. 8, Bar-Ilan University, Ramat Ga, pp. 183–236 (1995)Google Scholar
  39. 39.
    Sarnak P.: Spectra of hyperbolic surfaces. Bull. Am. Math. Soc. (N.S.) 40(4), 441–478 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Seeger A., Sogge C.D.: Bounds for eigenfunctions of differential operators. Indiana Univ. Math. J. 38(3), 669–682 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Soundararajan K.: Quantum unique ergodicity for \({{\rm SL}_2(\mathbb{Z}) \backslash H}\). Ann. Math. (2) 172(2), 1529–1538 (2010)MathSciNetzbMATHGoogle Scholar
  42. 42.
    Stern, A.: Bemerkungen über asymptotisches Verhalten von Eigenwerten und Eigenfunktionen. Math.- naturwiss. Diss. Göttingen, 30 S (1925)Google Scholar
  43. 43.
    Templier N.: On the sup-norm of Maass cusp forms of large level. Sel. Math. (N.S.) 16(3), 501–531 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Templier N.: Hybrid sup-norm bounds for Hecke–Maass cusp forms. J. Eur. Math. Soc. (JEMS) 17(8), 2069–2082 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Titchmarsh, E.C.: The Theory of the Riemann Zeta-Function, 2nd edn. Edited and with a preface by D. R. Heath-Brown. The Clarendon Press, Oxford University Press, New York, x+412 pp (1986)Google Scholar
  46. 46.
    Young, M.: A note on the sup norm of Eisenstein series. Q. J. Math. 69, 1151–1161 (2018).  https://doi.org/10.1093/qmath/hay019 Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Mathematical SciencesTel Aviv UniversityTel AvivIsrael

Personalised recommendations