Duke Mathematical Journal

Distribution of mass of holomorphic cusp forms

Valentin Blomer, Rizwanur Khan, and Matthew Young

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We prove an upper bound for the L 4 -norm and for the L 2 -norm restricted to the vertical geodesic of a holomorphic Hecke cusp form f of large weight k . The method is based on Watson’s formula and estimating a mean value of certain L -functions of degree 6. Further applications to restriction problems of Siegel modular forms and subconvexity bounds of degree 8 L -functions are given.

Article information

Source
Duke Math. J., Volume 162, Number 14 (2013), 2609-2644.

Dates
Received: 12 March 2012
Revised: 22 February 2013
First available in Project Euclid: 6 November 2013

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1383760700

Digital Object Identifier
doi:10.1215/00127094-2380967

Mathematical Reviews number (MathSciNet)
MR3127809

Zentralblatt MATH identifier
1312.11028

Subjects
Primary: 11F11: Holomorphic modular forms of integral weight 11F66: Langlands $L$-functions; one variable Dirichlet series and functional equations

Citation

Blomer, Valentin; Khan, Rizwanur; Young, Matthew. Distribution of mass of holomorphic cusp forms. Duke Math. J. 162 (2013), no. 14, 2609--2644. doi:10.1215/00127094-2380967. https://projecteuclid.org/euclid.dmj/1383760700


Export citation

References

  • [BR] J. Bernstein and A. Reznikov, Subconvexity bounds for triple $L$-functions and representation theory, Ann. of Math. (2) 172 (2010), 1679–1718.
  • [Be] M. V. Berry, Regular and irregular semiclassical wavefunctions, J. Phys. A 10 (1977), 2083–2091.
  • [Bi] A. Biró, A relation between triple products of weight $0$ and weight $\frac{1}{2}$ cusp forms, Israel J. Math. 182 (2011), 61–101.
  • [Bl] V. Blomer, On the $4$-norm of an automorphic form, J. Eur. Math. Soc. (JEMS) 15 (2013), 1825–1852.
  • [B] J. Bourgain, “Geodesic restrictions and $L^{p}$-estimates for eigenfunctions of Riemannian surfaces” in Linear and Complex Analysis, Amer. Math. Soc. Transl. Ser. 2 226, Amer. Math. Soc., Providence, 2009, 27–35.
  • [CFK+] J. B. Conrey, D. W. Farmer, J. P. Keating, M. O. Rubinstein, and N. C. Snaith, Integral moments of $L$-functions, Proc. Lond. Math. Soc. (3) 91 (2005), 33–104.
  • [EZ] M. Eichler and D. Zagier, The Theory of Jacobi Forms, Progr. Math. 55, Birkhäuser, Boston, 1985.
  • [Go] D. Goldfeld, Automorphic Forms and $L$-functions for the Group $\operatorname{GL} (n,\mathbb{R})$, with an appendix by K. Broughan, Cambridge Stud. Adv. Math. 99, Cambridge Univ. Press, Cambridge, 2006.
  • [GR] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 6th ed., Academic Press, San Diego, 2000.
  • [HR] D. A. Hejhal and B. N. Rackner, On the topography of Maass waveforms for $\operatorname{PSL} (2,\mathbb{Z})$, Exp. Math. 1 (1992), 275–305.
  • [HSt] D. A. Hejhal and A. Strömbergsson, On quantum chaos and Maass waveforms of CM-type, Found. Phys. 31 (2001), 519–533.
  • [HL] J. Hoffstein and P. Lockhart, Coefficients of Maass forms and the Siegel zero, with an appendix by D. Goldfeld, J. Hoffstein, and D. Lieman, Ann. of Math. (2) 140 (1994), 161–181.
  • [HSo] R. Holowinsky and K. Soundararajan, Mass equidistribution for Hecke eigenforms, Ann. of Math. (2) 172 (2010), 1517–1528.
  • [Ic] A. Ichino, Pullbacks of Saito-Kurokawa lifts, Invent. Math. 162 (2005), 551–647.
  • [Iw1] H. Iwaniec, Spectral Methods of Automorphic Forms, 2nd ed., Grad. Stud. Math. 53, Amer. Math. Soc., Providence, 2002.
  • [Iw2] H. Iwaniec, Topics in Classical Automorphic Forms, Grad. Stud. Math. 17, Amer. Math. Soc., Providence, 1997.
  • [IK] H. Iwaniec and E. Kowalski, Analytic Number Theory, Amer. Math. Soc. Colloq. Publ. 53, Amer. Math. Soc., Providence, 2004.
  • [JS] H. Jacquet and J. Shalika, “Exterior square $L$-functions” in Automorphic Forms, Shimura Varieties, and $L$-functions, Vol. II (Ann Arbor, MI, 1988), Perspect. Math. 11, Academic Press, Boston, 1990, 143–226.
  • [JM] M. Jutila and Y. Motohashi, Uniform bound for Hecke $L$-functions, Acta Math. 195 (2005), 61–115.
  • [KZ] W. Kohnen and D. Zagier, Values of $L$-series of modular forms at the center of the critical strip, Invent. Math. 64 (1981), 175–198.
  • [KR] P. Kurlberg and Z. Rudnick, Value distribution for eigenfunctions of desymmetrized quantum maps, Int. Math. Res. Not. IMRN 18 (2001), 985–1002.
  • [La] E. M. Lapid, On the nonnegativity of Rankin-Selberg $L$-functions at the center of symmetry, Int. Math. Res. Not. IMRN 2 (2003), 65–75.
  • [LW] Y.-K. Lau and J. Wu, A density theorem on automorphic $L$-functions and some applications, Trans. Amer. Math. Soc. 358 (2005), no. 1, 441–472.
  • [Li] X. Li, Bounds for $\operatorname{GL} (3)\times \operatorname{GL} (2)$ $L$-functions and $\operatorname{GL} (3)$ $L$-functions, Ann. of Math. (2) 173 (2011), 301–336.
  • [LY] S.-C. Liu and M. Young, Growth and nonvanishing of restricted Siegel modular forms arising as Saito-Kurokawa lifts, to appear in Amer. J. Math.
  • [Lu1] W. Luo, Values of symmetric square $L$-functions at $1$, J. Reine Angew Math. 506 (1999), 215–235.
  • [Lu2] W. Luo, $L^{4}$-norms of the dihedral Maass forms, to appear in Int. Math. Res. Not. IMRN.
  • [MS] S. D. Miller and W. Schmid, Automorphic distributions, $L$-functions, and Voronoi summation for $\operatorname{GL} (3)$, Ann. of Math. (2) 164 (2006), 423–488.
  • [Or] T. Orloff, Special values and mixed weight triple products (with an appendix by Don Blasius), Invent. Math. 90 (1987), 169–188.
  • [Pe] Z. Peng, Zeros and central values of automorphic $L$-functions, Ph.D. dissertation, Princeton University, Princeton, N.J., 2001.
  • [R1] A. Reznikov, Norms of geodesic restrictions for eigenfunctions on hyperbolic surfaces and representation theory, preprint, arXiv:math/0403437.
  • [R2] A. Reznikov, Estimates of triple products of automorphic functions, II, preprint, arXiv:math/1202.4766.
  • [Sa1] P. Sarnak, Spectra of hyperbolic surfaces, Bull. Amer. Math. Soc. (N.S.) 40 (2003), 441–478.
  • [Sa2] P. Sarnak, Letter to Andrei Reznikov, June 2008.
  • [So] K. Soundararajan, Quantum unique ergodicity for $\operatorname{SL} _{2}(\mathbb{Z})\backslash\mathbb{H}$, Ann. of Math. (2) 172 (2010), 1529–1538.
  • [Sp] F. Spinu, The $L^{4}$-norm of the Eisenstein series, Ph.D. dissertation, Princeton University, Princeton, N.J., 2003.
  • [Wa] T. C. Watson, Rankin triple products and quantum chaos, Ph.D. dissertation, Princeton University, Princeton, N.J., 2002.
  • [Xi] H. Xia, On $L^{\infty}$-norms of holomorphic cusp forms, J. Number Theory 124 (2007), 325–327.