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.
Similar content being viewed by others
References
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)
Blomer, V., Harcos, G.: Twisted L-functions over number fields and Hilbert’s eleventh problem. Geom. Funct. Anal. 20(1), 1–52 (2010)
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
Davenport, H.: Multiplicative Number Theory, 2nd edn. Graduate Texts in Mathematics, vol. 74. Springer, New York (1980). Revised by Hugh L. Montgomery
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)
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)
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)
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)
Harris, M., Kudla, S.S.: The central critical value of a triple product L-function. Ann. Math. 133(3), 605–672 (1991)
Hildebrand, A., Tenenbaum, G.: Integers without large prime factors. J. Théor. Nr. Bordx. 5(2), 411–484 (1993)
Hinz, J.G.: Methoden des grossen Siebes in algebraischen Zahlkörpern. Manuscr. Math. 57(2), 181–194 (1987)
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
Holowinsky, R.: A sieve method for shifted convolution sums. Duke Math. J. 146(3), 401–448 (2009)
Holowinsky, R.: Sieving for mass equidistribution. Ann. Math. 172(2), 1499–1516 (2010)
Holowinsky, R., Soundararajan, K.: Mass equidistribution for Hecke eigenforms. Ann. Math. 172(2), 1517–1528 (2010)
Ichino, A.: Trilinear forms and the central values of triple product L-functions. Duke Math. J. 145(2), 281–307 (2008)
Iwaniec, H.: Spectral Methods of Automorphic Forms, 2nd edn. Graduate Studies in Mathematics, vol. 53. American Mathematical Society, Providence (2002)
Iwaniec, H.: Notes on the quantum unique ergodicity for holomorphic cusp forms (2010)
Iwaniec, H., Kowalski, E.: Analytic Number Theory. American Mathematical Society Colloquium Publications, vol. 53. American Mathematical Society, Providence (2004)
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)
Jacquet, H., Langlands, R.P.: Automorphic Forms on GL(2). Lecture Notes in Mathematics, vol. 114. Springer, Berlin (1970)
Jacquet, H.: Automorphic Forms on GL(2). Part II. Lecture Notes in Mathematics, vol. 278. Springer, Berlin (1972)
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
Krause, Uwe: Abschätzungen für die Funktion Ψ K (x,y) in algebraischen Zahlkörpern. Manuscr. Math. 69(3), 319–331 (1990)
Labesse, J.-P., Langlands, R.P.: L-indistinguishability for SL(2). Can. J. Math. 31(4), 726–785 (1979)
Lindenstrauss, E.: Invariant measures and arithmetic quantum unique ergodicity. Ann. Math. 163(1), 165–219 (2006)
Luo, W., Sarnak, P.: Quantum ergodicity of eigenfunctions on \(\mathrm {PSL}_{2}(\bold Z)\backslash H^{2}\). Publ. Math. IHÉS 81, 207–237 (1995)
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
Marshall, S.: Mass equidistribution for automorphic forms of cohomological type on GL_2. ArXiv e-prints (June 2010)
Montgomery, H.L.: A note on the large sieve. J. Lond. Math. Soc. 43, 93–98 (1968)
Nair, M.: Multiplicative functions of polynomial values in short intervals. Acta Arith. 62(3), 257–269 (1992)
Nair, M., Tenenbaum, G.: Short sums of certain arithmetic functions. Acta Math. 180(1), 119–144 (1998)
Nelson, P.: Equidistribution of cusp forms in the level aspect. arXiv:1011.1292 [math.NT] (2010)
Rudnick, Z., Sarnak, P.: The behaviour of eigenstates of arithmetic hyperbolic manifolds. Commun. Math. Phys. 161(1), 195–213 (1994)
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)
Sarnak, P.: Recent progress on QUE. http://www.math.princeton.edu/sarnak/SarnakQUE.pdf (2009)
Schaal, W.: On the large sieve method in algebraic number fields. J. Number Theory 2, 249–270 (1970)
Shimura, G.: On the holomorphy of certain Dirichlet series. Proc. Lond. Math. Soc. (3) 31(1), 79–98 (1975)
Shimura, G.: The special values of the zeta functions associated with Hilbert modular forms. Duke Math. J. 45(3), 637–679 (1978)
Silberman, L., Venkatesh, A.: On quantum unique ergodicity for locally symmetric spaces. Geom. Funct. Anal. 17(3), 960–998 (2007)
Šnirel’man, A.I.: Ergodic properties of eigenfunctions. Usp. Mat. Nauk 29(6(180)), 181–182 (1974)
Soundararajan, K.: Arizona winter school lecture notes on quantum unique ergodicity and number theory. http://math.arizona.edu/~swc/aws/10/2010SoundararajanNotes.pdf (2010)
Soundararajan, K.: Quantum unique ergodicity for SL2(ℤ)\ℍ. Ann. Math. 172(2), 1529–1538 (2010)
Soundararajan, K.: Weak subconvexity for central values of L-functions. Ann. Math. 172(2), 1469–1498 (2010)
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
Venkatesh, A.: Sparse equidistribution problems, period bounds and subconvexity. Ann. Math. 172(2), 989–1094 (2010)
Watson, T.C.: Rankin triple products and quantum chaos. arXiv:0810.0425 [math.NT] (2008)
Weil, A.: Séries de Dirichlet et fonctions automorphes. In: Séminaire Bourbaki, vol. 10, Exp. No. 346, pp. 547–552. Soc. Math. France, Paris (1995)
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
Zelditch, S.: Uniform distribution of eigenfunctions on compact hyperbolic surfaces. Duke Math. J. 55(4), 919–941 (1987)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Nelson, P.D. Mass equidistribution of Hilbert modular eigenforms. Ramanujan J 27, 235–284 (2012). https://doi.org/10.1007/s11139-011-9319-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11139-011-9319-9