Geodesic
Rankin-Selberg without unfolding and bounds for spherical Fourier coefficients of Maass forms
Reznikov, Andre1
1 Department of Mathematics, Bar-Ilan University, 52900 Ramat-Gan, Israel
Journal of the American Mathematical Society, Tome 21 (2008) no. 2, pp. 439-477

Voir la notice de l'article provenant de la source American Mathematical Society

We use the uniqueness of various invariant functionals on irreducible unitary representations of $PGL_2(\mathbb {R})$ in order to deduce the classical Rankin-Selberg identity for the sum of Fourier coefficients of Maass cusp forms and its new anisotropic analog. We deduce from these formulas non-trivial bounds for the corresponding unipotent and spherical Fourier coefficients of Maass forms. As an application we obtain a subconvexity bound for certain $L$-functions. Our main tool is the notion of a Gelfand pair from representation theory.
DOI : 10.1090/S0894-0347-07-00581-4

Reznikov, Andre 1

1 Department of Mathematics, Bar-Ilan University, 52900 Ramat-Gan, Israel 
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Reznikov, Andre. Rankin-Selberg without unfolding and bounds for spherical Fourier coefficients of Maass forms. Journal of the American Mathematical Society, Tome 21 (2008) no. 2, pp. 439-477. doi: 10.1090/S0894-0347-07-00581-4

[1] Bernstein, Joseph, Reznikov, Andre Analytic continuation of representations and estimates of automorphic forms Ann. of Math. (2) 1999 329 352

[2] Bernstein, Joseph, Reznikov, Andre Sobolev norms of automorphic functionals Int. Math. Res. Not. 2002 2155 2174

[3] Bernstein, Joseph, Reznikov, Andre Estimates of automorphic functions Mosc. Math. J. 2004

[4] Borel, Armand Automorphic forms on 𝑆𝐿₂(𝑅) 1997

[5] Borovikov, V. A. Uniform stationary phase method 1994

[6] Bump, Daniel The Rankin-Selberg method: an introduction and survey 2005 41 73

[7] Bruggeman, Roelof W., Motohashi, Yoichi A new approach to the spectral theory of the fourth moment of the Riemann zeta-function J. Reine Angew. Math. 2005 75 114

[8] Analysis. I 1989

[9] Gel′Fand, I. M., Shilov, G. E. Generalized functions. Vol. 1 1964

[10] Gel′Fand, I. M., Graev, M. I., Pyatetskii-Shapiro, I. I. Representation theory and automorphic functions 1990

[11] Good, A. Cusp forms and eigenfunctions of the Laplacian Math. Ann. 1981 523 548

[12] Gross, Benedict H. Some applications of Gel′fand pairs to number theory Bull. Amer. Math. Soc. (N.S.) 1991 277 301

[13] Helgason, Sigurdur Groups and geometric analysis 2000

[14] Hã¶Rmander, Lars The analysis of linear partial differential operators. III 1985

[15] Iwaniec, Henryk Spectral methods of automorphic forms 2002

[16] Iwaniec, H., Sarnak, P. Perspectives on the analytic theory of 𝐿-functions Geom. Funct. Anal. 2000 705 741

[17] Jacquet, Hervã©, Chen, Nan Positivity of quadratic base change 𝐿-functions Bull. Soc. Math. France 2001 33 90

[18] Kim, Henry H. Functoriality for the exterior square of 𝐺𝐿₄ and the symmetric fourth of 𝐺𝐿₂ J. Amer. Math. Soc. 2003 139 183

[19] Krã¶Tz, Bernhard, Stanton, Robert J. Holomorphic extensions of representations. I. Automorphic functions Ann. of Math. (2) 2004 641 724

[20] Kubota, Tomio Elementary theory of Eisenstein series 1973

[21] Kuznetsov, N. V. Sums of Kloosterman sums and the eighth power moment of the Riemann zeta-function 1989 57 117

[22] Lewis, John B. Eigenfunctions on symmetric spaces with distribution-valued boundary forms J. Functional Analysis 1978 287 307

[23] Maass, Hans Über eine neue Art von nichtanalytischen automorphen Funktionen und die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen Math. Ann. 1949 141 183

[24] Motohashi, Yoichi Spectral theory of the Riemann zeta-function 1997

[25] Oksak, A. I. Trilinear Lorentz invariant forms Comm. Math. Phys. 1973 189 217

[26] Petridis, Yiannis N., Sarnak, Peter Quantum unique ergodicity for 𝑆𝐿₂(𝒪)\𝐇³ and estimates for 𝐋-functions J. Evol. Equ. 2001 277 290

[27] Prasad, Dipendra Trilinear forms for representations of 𝐺𝐿(2) and local 𝜀-factors Compositio Math. 1990 1 46

[28] Rankin, R. A. Contributions to the theory of Ramanujan’s function 𝜏(𝑛) and similar arithmetical functions. I. The zeros of the function ∑^{∞}_{𝑛 Proc. Cambridge Philos. Soc. 1939 351 372

[29] Sarnak, Peter Integrals of products of eigenfunctions Internat. Math. Res. Notices 1994

[30] Selberg, Atle On the estimation of Fourier coefficients of modular forms 1965 1 15

[31] Stein, Elias M. Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals 1993

[32] Waldspurger, J.-L. Quelques propriétés arithmétiques de certaines formes automorphes sur 𝐺𝐿(2) Compositio Math. 1985 121 171

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