Bounding hyperbolic and spherical coefficients of Maass forms
Valentin Blomer1; Farrell Brumley2; Alex Kontorovich3; Nicolas Templier4
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, Volume 26 (2014) no. 3, pp. 559-578.

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

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.

DOI: 10.5802/jtnb.879
Classification: 11F70, 22E45
Keywords: Maass forms, Fourier coefficients, geodesics, periods, equidistribution, Sobolev norms, wave front lemma

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 
@article{JTNB_2014__26_3_559_0,
     author = {Valentin Blomer and Farrell Brumley and Alex Kontorovich and Nicolas Templier},
     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 = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.879/}
}
TY  - JOUR
AU  - Valentin Blomer
AU  - Farrell Brumley
AU  - Alex Kontorovich
AU  - Nicolas Templier
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  - https://jtnb.centre-mersenne.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 Valentin Blomer
%A Farrell Brumley
%A Alex Kontorovich
%A Nicolas Templier
%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 https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.879/
%R 10.5802/jtnb.879
%G en
%F JTNB_2014__26_3_559_0
Valentin Blomer; Farrell Brumley; Alex Kontorovich; Nicolas Templier. Bounding hyperbolic and spherical coefficients  of Maass forms. Journal de théorie des nombres de Bordeaux, Volume 26 (2014) no. 3, pp. 559-578. doi : 10.5802/jtnb.879. https://jtnb.centre-mersenne.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

Cited by Sources: