Bounding hyperbolic and spherical coefficients of Maass forms
Journal de théorie des nombres de Bordeaux, Tome 26 (2014) no. 3, pp. 559-578.

On développe une nouvelle méthode pour majorer les coefficients de Fourier hyperboliques et sphériques des formes de Maass définies par rapport à des réseaux uniformes généraux.

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

DOI : 10.5802/jtnb.879
Classification : 11F70, 22E45
Mots clés : Maass forms, Fourier coefficients, geodesics, periods, equidistribution, Sobolev norms, wave front lemma
Blomer, Valentin 1 ; Brumley, Farrell 2 ; Kontorovich, Alex 3 ; Templier, Nicolas 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
@article{JTNB_2014__26_3_559_0,
     author = {Blomer, Valentin and Brumley, Farrell and Kontorovich, Alex and Templier, Nicolas},
     title = {Bounding hyperbolic and spherical coefficients  of {Maass} forms},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {559--578},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {26},
     number = {3},
     year = {2014},
     doi = {10.5802/jtnb.879},
     mrnumber = {3320492},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/jtnb.879/}
}
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
PY  - 2014
SP  - 559
EP  - 578
VL  - 26
IS  - 3
PB  - Société Arithmétique de Bordeaux
UR  - http://www.numdam.org/articles/10.5802/jtnb.879/
DO  - 10.5802/jtnb.879
LA  - en
ID  - JTNB_2014__26_3_559_0
ER  - 
%0 Journal Article
%A Blomer, Valentin
%A Brumley, Farrell
%A Kontorovich, Alex
%A Templier, Nicolas
%T Bounding hyperbolic and spherical coefficients  of Maass forms
%J Journal de théorie des nombres de Bordeaux
%D 2014
%P 559-578
%V 26
%N 3
%I Société Arithmétique de Bordeaux
%U http://www.numdam.org/articles/10.5802/jtnb.879/
%R 10.5802/jtnb.879
%G en
%F JTNB_2014__26_3_559_0
Blomer, Valentin; Brumley, Farrell; Kontorovich, Alex; Templier, Nicolas. Bounding hyperbolic and spherical coefficients  of Maass forms. Journal de théorie des nombres de Bordeaux, Tome 26 (2014) no. 3, pp. 559-578. doi : 10.5802/jtnb.879. http://www.numdam.org/articles/10.5802/jtnb.879/

[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. | MR | Zbl

[2] J. Bernstein and A. Reznikov, Sobolev norms of automorphic functionals, IMRN (2002), 2155–2174. | MR | Zbl

[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. | MR | Zbl

[4] D. Bump, Automorphic forms and representations. Cambridge Studies in Advanced Mathematics 55, Cambridge University Press (1996). | MR | Zbl

[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. | MR | Zbl

[6] A. Eskin and C. McMullen, Mixing, counting, and equidistribution in Lie groups. Duke Math. J. 71, (1993), no. 1, 181–209. | MR | Zbl

[7] I. Gelfand, M. Graev, and I. Piatetski-Shapiro, Representation Theory and Automorphic Functions. W.B. Saunders Co., Philadelphia, (1969). | MR | Zbl

[8] A. Good, Cusp forms and eigenfunctions of the Laplacian. Math. Ann., 255, (1981), 523–548. | MR | Zbl

[9] H. Oh and N. Shah, Limits of translates of divergent geodesics and integral points on one-sheeted hyperboloids, Israel J. Math, to appear. | MR | Zbl

[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. | MR | Zbl

[11] A. Reznikov, Geodesic restrictions for the Casimir operator. J. Funct. Anal. 261, (2011), no. 9, 2437–2460. | MR | Zbl

[12] P. Sarnak, Fourth moments of Grössencharakteren zeta functions. Comm. Pure Appl. Math. 38, (1985), no. 2, 167–178. | MR | Zbl

[13] P. Sarnak, Integrals of products of eigenfunctions. Internat. Math. Res. Notices (1994), no. 6, 251–260. | MR | Zbl

[14] A. Venkatesh, Sparse equidistribution problems, period bounds and subconvexity. Ann. of Math. (2) 172, (2010), no. 2, 989–1094. | MR | Zbl

Cité par Sources :