The Ramanujan Journal

, Volume 27, Issue 2, pp 235–284

Mass equidistribution of Hilbert modular eigenforms

Article

Abstract

Let \(\mathbb{F}\) be a totally real number field, and let f traverse a sequence of non-dihedral holomorphic eigencuspforms on \(\operatorname{GL}_{2}/\mathbb{F}\) of weight \((k_{1},\ldots,k_{[\mathbb{F}:\mathbb{Q}]})\), trivial central character and full level. We show that the mass of f equidistributes on the Hilbert modular variety as \(\max(k_{1},\ldots,k_{[\mathbb{F}:\mathbb{Q}]}) \rightarrow \infty\).

Our result answers affirmatively a natural analog of a conjecture of Rudnick and Sarnak (Commun. Math. Phys. 161(1), 195–213, 1994). Our proof generalizes the argument of Holowinsky–Soundararajan (Ann. Math. 172(2), 1517–1528, 2010) who established the case \(\mathbb{F} = \mathbb{Q}\). The essential difficulty in doing so is to adapt Holowinsky’s bounds for the Weyl periods of the equidistribution problem in terms of manageable shifted convolution sums of Fourier coefficients to the case of a number field with nontrivial unit group.

Keywords

Modular forms Hilbert modular forms Number theory Quantum chaos Quantum unique ergodicity 

Mathematics Subject Classification (2000)

58J51 11M36 

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References

  1. 1.
    Blasius, D.: Hilbert modular forms and the Ramanujan conjecture. In: Noncommutative Geometry and Number Theory. Aspects Math., vol. E37, pp. 35–56. Vieweg, Wiesbaden (2006) CrossRefGoogle Scholar
  2. 2.
    Blomer, V., Harcos, G.: Twisted L-functions over number fields and Hilbert’s eleventh problem. Geom. Funct. Anal. 20(1), 1–52 (2010) MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Colin de Verdière, Y.: Ergodicité et fonctions propres du laplacien. In: Bony–Sjöstrand–Meyer Seminar, 1984–1985, p. 8. École Polytech., Palaiseau (1985). Exp. No. 13 Google Scholar
  4. 4.
    Davenport, H.: Multiplicative Number Theory, 2nd edn. Graduate Texts in Mathematics, vol. 74. Springer, New York (1980). Revised by Hugh L. Montgomery MATHGoogle Scholar
  5. 5.
    Gelbart, S., Jacquet, H.: A relation between automorphic representations of GL(2) and GL(3). Ann. Sci. École Norm. Sup. (4) 11(4), 471–542 (1978) MathSciNetMATHGoogle Scholar
  6. 6.
    Gelbart, S., Jacquet, H.: Forms of GL(2) from the analytic point of view. In: Automorphic Forms, Representations and L-Functions, Part 1, Oregon State Univ., Corvallis, Ore., 1977. Proc. Sympos. Pure Math., vol. XXXIII, pp. 213–251. Amer. Math. Soc., Providence (1979) Google Scholar
  7. 7.
    Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series, and Products, 7th edn. Elsevier/Academic Press, Amsterdam (2007). Translated from the Russian, Translation edited and with a preface by Alan Jeffrey and Daniel Zwillinger, With one CD-ROM (Windows, Macintosh and UNIX) MATHGoogle Scholar
  8. 8.
    Greaves, G.: Sieves in Number Theory. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 43. Springer, Berlin (2001) MATHGoogle Scholar
  9. 9.
    Harris, M., Kudla, S.S.: The central critical value of a triple product L-function. Ann. Math. 133(3), 605–672 (1991) MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Hildebrand, A., Tenenbaum, G.: Integers without large prime factors. J. Théor. Nr. Bordx. 5(2), 411–484 (1993) MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Hinz, J.G.: Methoden des grossen Siebes in algebraischen Zahlkörpern. Manuscr. Math. 57(2), 181–194 (1987) MathSciNetMATHGoogle Scholar
  12. 12.
    Hoffstein, J., Lockhart, P.: Coefficients of Maass forms and the Siegel zero. Ann. Math. 140(1), 161–181 (1994). With an appendix by Dorian Goldfeld, Hoffstein and Daniel Lieman MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Holowinsky, R.: A sieve method for shifted convolution sums. Duke Math. J. 146(3), 401–448 (2009) MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Holowinsky, R.: Sieving for mass equidistribution. Ann. Math. 172(2), 1499–1516 (2010) MathSciNetMATHGoogle Scholar
  15. 15.
    Holowinsky, R., Soundararajan, K.: Mass equidistribution for Hecke eigenforms. Ann. Math. 172(2), 1517–1528 (2010) MathSciNetMATHGoogle Scholar
  16. 16.
    Ichino, A.: Trilinear forms and the central values of triple product L-functions. Duke Math. J. 145(2), 281–307 (2008) MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Iwaniec, H.: Spectral Methods of Automorphic Forms, 2nd edn. Graduate Studies in Mathematics, vol. 53. American Mathematical Society, Providence (2002) MATHGoogle Scholar
  18. 18.
    Iwaniec, H.: Notes on the quantum unique ergodicity for holomorphic cusp forms (2010) Google Scholar
  19. 19.
    Iwaniec, H., Kowalski, E.: Analytic Number Theory. American Mathematical Society Colloquium Publications, vol. 53. American Mathematical Society, Providence (2004) MATHGoogle Scholar
  20. 20.
    Iwaniec, H., Sarnak, P.: Perspectives on the analytic theory of L-functions. Geom. Funct. Anal. (Special Volume, Part II), 705–741 (2000). GAFA 2000 (Tel Aviv, 1999) Google Scholar
  21. 21.
    Jacquet, H., Langlands, R.P.: Automorphic Forms on GL(2). Lecture Notes in Mathematics, vol. 114. Springer, Berlin (1970) MATHGoogle Scholar
  22. 22.
    Jacquet, H.: Automorphic Forms on GL(2). Part II. Lecture Notes in Mathematics, vol. 278. Springer, Berlin (1972) MATHGoogle Scholar
  23. 23.
    Kowalski, E.: The Large Sieve and Its Applications. Cambridge Tracts in Mathematics, vol. 175. Cambridge University Press, Cambridge (2008). Arithmetic geometry, random walks and discrete groups MATHCrossRefGoogle Scholar
  24. 24.
    Krause, Uwe: Abschätzungen für die Funktion Ψ K(x,y) in algebraischen Zahlkörpern. Manuscr. Math. 69(3), 319–331 (1990) MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Labesse, J.-P., Langlands, R.P.: L-indistinguishability for SL(2). Can. J. Math. 31(4), 726–785 (1979) MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    Lindenstrauss, E.: Invariant measures and arithmetic quantum unique ergodicity. Ann. Math. 163(1), 165–219 (2006) MathSciNetMATHCrossRefGoogle Scholar
  27. 27.
    Luo, W., Sarnak, P.: Quantum ergodicity of eigenfunctions on \(\mathrm {PSL}_{2}(\bold Z)\backslash H^{2}\). Publ. Math. IHÉS 81, 207–237 (1995) MathSciNetMATHCrossRefGoogle Scholar
  28. 28.
    Luo, W., Sarnak, P.: Mass equidistribution for Hecke eigenforms. Commun. Pure Appl. Math. 56(7), 874–891 (2003). Dedicated to the memory of Jürgen K. Moser MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    Marshall, S.: Mass equidistribution for automorphic forms of cohomological type on GL_2. ArXiv e-prints (June 2010) Google Scholar
  30. 30.
    Montgomery, H.L.: A note on the large sieve. J. Lond. Math. Soc. 43, 93–98 (1968) MATHCrossRefGoogle Scholar
  31. 31.
    Nair, M.: Multiplicative functions of polynomial values in short intervals. Acta Arith. 62(3), 257–269 (1992) MathSciNetMATHGoogle Scholar
  32. 32.
    Nair, M., Tenenbaum, G.: Short sums of certain arithmetic functions. Acta Math. 180(1), 119–144 (1998) MathSciNetMATHCrossRefGoogle Scholar
  33. 33.
    Nelson, P.: Equidistribution of cusp forms in the level aspect. arXiv:1011.1292 [math.NT] (2010)
  34. 34.
    Rudnick, Z., Sarnak, P.: The behaviour of eigenstates of arithmetic hyperbolic manifolds. Commun. Math. Phys. 161(1), 195–213 (1994) MathSciNetMATHCrossRefGoogle Scholar
  35. 35.
    Sarnak, P.: Arithmetic quantum chaos. In: The Schur Lectures (1992) (Tel Aviv). Israel Math. Conf. Proc., vol. 8, pp. 183–236. Bar-Ilan Univ., Ramat Gan (1995) Google Scholar
  36. 36.
    Sarnak, P.: Recent progress on QUE. http://www.math.princeton.edu/sarnak/SarnakQUE.pdf (2009)
  37. 37.
    Schaal, W.: On the large sieve method in algebraic number fields. J. Number Theory 2, 249–270 (1970) MathSciNetMATHCrossRefGoogle Scholar
  38. 38.
    Shimura, G.: On the holomorphy of certain Dirichlet series. Proc. Lond. Math. Soc. (3) 31(1), 79–98 (1975) MathSciNetMATHCrossRefGoogle Scholar
  39. 39.
    Shimura, G.: The special values of the zeta functions associated with Hilbert modular forms. Duke Math. J. 45(3), 637–679 (1978) MathSciNetMATHCrossRefGoogle Scholar
  40. 40.
    Silberman, L., Venkatesh, A.: On quantum unique ergodicity for locally symmetric spaces. Geom. Funct. Anal. 17(3), 960–998 (2007) MathSciNetMATHCrossRefGoogle Scholar
  41. 41.
    Šnirel’man, A.I.: Ergodic properties of eigenfunctions. Usp. Mat. Nauk 29(6(180)), 181–182 (1974) Google Scholar
  42. 42.
    Soundararajan, K.: Arizona winter school lecture notes on quantum unique ergodicity and number theory. http://math.arizona.edu/~swc/aws/10/2010SoundararajanNotes.pdf (2010)
  43. 43.
    Soundararajan, K.: Quantum unique ergodicity for SL2(ℤ)\ℍ. Ann. Math. 172(2), 1529–1538 (2010) MathSciNetMATHGoogle Scholar
  44. 44.
    Soundararajan, K.: Weak subconvexity for central values of L-functions. Ann. Math. 172(2), 1469–1498 (2010) MathSciNetMATHGoogle Scholar
  45. 45.
    Titchmarsh, E.C.: The Theory of the Riemann Zeta-Function, 2nd edn. The Clarendon Press Oxford University Press, New York (1986). Edited and with a preface by D.R. Heath-Brown MATHGoogle Scholar
  46. 46.
    Venkatesh, A.: Sparse equidistribution problems, period bounds and subconvexity. Ann. Math. 172(2), 989–1094 (2010) MathSciNetMATHCrossRefGoogle Scholar
  47. 47.
    Watson, T.C.: Rankin triple products and quantum chaos. arXiv:0810.0425 [math.NT] (2008)
  48. 48.
    Weil, A.: Séries de Dirichlet et fonctions automorphes. In: Séminaire Bourbaki, vol. 10, Exp. No. 346, pp. 547–552. Soc. Math. France, Paris (1995) Google Scholar
  49. 49.
    Whittaker, E.T., Watson, G.N.: A Course of Modern Analysis. An Introduction to the General Theory of Infinite Processes and of Analytic Functions: With an Account of the Principal Transcendental Functions, 4th edn. Cambridge University Press, New York (1962). Reprinted MATHGoogle Scholar
  50. 50.
    Zelditch, S.: Uniform distribution of eigenfunctions on compact hyperbolic surfaces. Duke Math. J. 55(4), 919–941 (1987) MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.PasadenaUSA

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