Bounding hyperbolic and spherical coefficients of Maass forms

Valentin Blomer[1]; Farrell Brumley[2]; Alex Kontorovich[3]; Nicolas Templier[4]

  • [1] Mathematisches Institut, Bunsenstr. 3-5, 37073 Göttingen, Germany
  • [2] Institut Galilée, Université Paris 13 99 avenue J.-B. Clément 93430 Villetaneuse, France
  • [3] Department of Mathematics Yale University New Haven, CT 06511 USA
  • [4] Department of Mathematics Fine Hall, Washington Road Princeton, NJ 08544 USA

Journal de Théorie des Nombres de Bordeaux (2014)

  • Volume: 26, Issue: 3, page 559-578
  • ISSN: 1246-7405

Abstract

top
We develop a new method to bound the hyperbolic and spherical Fourier coefficients of Maass forms defined with respect to arbitrary uniform lattices.

How to cite

top

Blomer, Valentin, et al. "Bounding hyperbolic and spherical coefficients of Maass forms." Journal de Théorie des Nombres de Bordeaux 26.3 (2014): 559-578. <http://eudml.org/doc/275819>.

@article{Blomer2014,
abstract = {We develop a new method to bound the hyperbolic and spherical Fourier coefficients of Maass forms defined with respect to arbitrary uniform lattices.},
affiliation = {Mathematisches Institut, Bunsenstr. 3-5, 37073 Göttingen, Germany; Institut Galilée, Université Paris 13 99 avenue J.-B. Clément 93430 Villetaneuse, France; Department of Mathematics Yale University New Haven, CT 06511 USA; Department of Mathematics Fine Hall, Washington Road Princeton, NJ 08544 USA},
author = {Blomer, Valentin, Brumley, Farrell, Kontorovich, Alex, Templier, Nicolas},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {Maass forms; Fourier coefficients; geodesics; periods; equidistribution; Sobolev norms; wave front lemma},
language = {eng},
month = {12},
number = {3},
pages = {559-578},
publisher = {Société Arithmétique de Bordeaux},
title = {Bounding hyperbolic and spherical coefficients of Maass forms},
url = {http://eudml.org/doc/275819},
volume = {26},
year = {2014},
}

TY - JOUR
AU - Blomer, Valentin
AU - Brumley, Farrell
AU - Kontorovich, Alex
AU - Templier, Nicolas
TI - Bounding hyperbolic and spherical coefficients of Maass forms
JO - Journal de Théorie des Nombres de Bordeaux
DA - 2014/12//
PB - Société Arithmétique de Bordeaux
VL - 26
IS - 3
SP - 559
EP - 578
AB - We develop a new method to bound the hyperbolic and spherical Fourier coefficients of Maass forms defined with respect to arbitrary uniform lattices.
LA - eng
KW - Maass forms; Fourier coefficients; geodesics; periods; equidistribution; Sobolev norms; wave front lemma
UR - http://eudml.org/doc/275819
ER -

References

top
  1. M. B. Bekka and M. Mayer, Ergodic theory and topological dynamics of group action on homogeneous spaces, London Mathematical Society Lecture Note Series, 269, Cambridge University Press, Cambridge, (2000), x+200 pp. Zbl0961.37001MR1781937
  2. J. Bernstein and A. Reznikov, Sobolev norms of automorphic functionals, IMRN (2002), 2155–2174. Zbl1022.11025MR1930758
  3. J. Bernstein and A. Reznikov, Subconvexity bounds for triple L -functions and representation theory. Ann. of Math. (2) 172, (2010), no. 3, 1679–1718. Zbl1225.11068MR2726097
  4. D. Bump, Automorphic forms and representations. Cambridge Studies in Advanced Mathematics 55, Cambridge University Press (1996). Zbl0868.11022MR1431508
  5. W. Duke, Z. Rudnick, and P. Sarnak, Density of integer points on affine homogeneous varieties. Duke Math. J. 71, (1993), no. 1, 143–179. Zbl0798.11024MR1230289
  6. A. Eskin and C. McMullen, Mixing, counting, and equidistribution in Lie groups. Duke Math. J. 71, (1993), no. 1, 181–209. Zbl0798.11025MR1230290
  7. I. Gelfand, M. Graev, and I. Piatetski-Shapiro, Representation Theory and Automorphic Functions. W.B. Saunders Co., Philadelphia, (1969). Zbl0177.18003MR233772
  8. A. Good, Cusp forms and eigenfunctions of the Laplacian. Math. Ann., 255, (1981), 523–548. Zbl0439.30031MR618183
  9. H. Oh and N. Shah, Limits of translates of divergent geodesics and integral points on one-sheeted hyperboloids, Israel J. Math, to appear. Zbl1298.22009MR3219562
  10. A. Reznikov, Rankin-Selberg without unfolding and bounds for spherical Fourier coefficients of Maass forms. J. Amer. Math. Soc. 21, (2008), no. 2, 439–477. Zbl1188.11024MR2373356
  11. A. Reznikov, Geodesic restrictions for the Casimir operator. J. Funct. Anal. 261, (2011), no. 9, 2437–2460. Zbl1229.53048MR2826400
  12. P. Sarnak, Fourth moments of Grössencharakteren zeta functions. Comm. Pure Appl. Math. 38, (1985), no. 2, 167–178. Zbl0577.10026MR780071
  13. P. Sarnak, Integrals of products of eigenfunctions. Internat. Math. Res. Notices (1994), no. 6, 251–260. Zbl0833.11020MR1277052
  14. A. Venkatesh, Sparse equidistribution problems, period bounds and subconvexity. Ann. of Math. (2) 172, (2010), no. 2, 989–1094. Zbl1214.11051MR2680486

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.