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The L2 restriction norm of a GL3 Maass form

Part of: Zeta and $L$-functions: analytic theory Spectral theory and eigenvalue problems Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  20 March 2012

Xiaoqing Li  and
Matthew P. Young
Xiaoqing Li
Affiliation:
Department of Mathematics, State University of New York at Buffalo, Buffalo, NY 14260, USA (email: [email protected])
Matthew P. Young
Affiliation:
Department of Mathematics, Texas A&M University, College Station, TX 77843-3368, USA (email: [email protected])
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Abstract

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We prove a sharp upper bound on the L2 norm of a GL3 Maass form restricted to GL2×ℝ+.

Keywords

Maass formrestriction normautomorphic formGL3

MSC classification

Secondary: 11F55: Other groups and their modular and automorphic forms (several variables) 11F72: Spectral theory; Selberg trace formula 11M41: Other Dirichlet series and zeta functions 35P20: Asymptotic distribution of eigenvalues and eigenfunctions
Type
Research Article
Information
Compositio Mathematica , Volume 148 , Issue 3 , May 2012 , pp. 675 - 717
Copyright
Copyright © Foundation Compositio Mathematica 2012

References

[BH10]Blomer, V. and Holowinsky, R., Bounding sup-norms of cusp forms of large level, Invent. Math. 179 (2010), 645–681.CrossRefGoogle Scholar
[BR09]Bourgain, J. and Rudnick, Z., Restriction of toral eigenfunctions to hypersurfaces, C. R. Acad. Sci. Paris Ser. I 347 (2009), 1249–1253.Google Scholar
[Bru06]Brumley, F., Second order average estimates on local data of cusp forms, Arch. Math. (Basel) 87 (2006), 19–32.CrossRefGoogle Scholar
[Bum88]Bump, D., Barnes’ second lemma and its application to Rankin–Selberg convolutions, Amer. J. Math. 110 (1988), 179–185.CrossRefGoogle Scholar
[BGT07]Burq, N., Gérard, P. and Tzvetkov, N., Restrictions of the Laplace–Beltrami eigenfunctions to submanifolds, Duke Math. J. 138 (2007), 445–486.CrossRefGoogle Scholar
[CP04]Cogdell, J. and Piatetski-Shapiro, I., Remarks on Rankin–Selberg convolutions, in Contributions to automorphic forms, geometry, and number theory (Johns Hopkins University Press, Baltimore, MD, 2004), 255–278.Google Scholar
[Gal70]Gallagher, P. X., A large sieve density estimate near σ=1, Invent. Math. 11 (1970), 329–339.CrossRefGoogle Scholar
[GJ78]Gelbart, S. and Jacquet, H., A relation between automorphic representations of GL(2) and GL(3), Ann. Sci. Éc. Norm. Supér. (4) 11 (1978), 471–542.CrossRefGoogle Scholar
[Gol06]Goldfeld, D., Automorphic forms and L-functions for the group GL(n,ℝ), Cambridge Studies in Advanced Mathematics, vol. 99 (Cambridge University Press, Cambridge, 2006), with an appendix by Kevin A. Broughan.Google Scholar
[GL06]Goldfeld, D. and Li, X., Voronoi formulas on GL(n), Int. Math. Res. Not. 2006 (2006), doi:10.1155/IMRN/2006/86295.Google Scholar
[GT06]Goldfeld, D. and Thillainatesan, M., Rank lowering linear maps and multiple Dirichlet series associated to GL(n,ℝ), Pure Appl. Math. Q. 2 (2006), 601–615, part 2.CrossRefGoogle Scholar
[GR00]Gradshteyn, I. S. and Ryzhik, I. M., Table of integrals, series, and products, sixth edition (Academic Press, San Diego, CA, 2000), translated from the Russian. Translation edited and with a preface by Alan Jeffrey and Daniel Zwillinger.Google Scholar
[GP92]Gross, B. and Prasad, D., On the decomposition of a representation of SO n when restricted to SO n−1, Canad. J. Math. 44 (1992), 974–1002.CrossRefGoogle Scholar
[HL94]Hoffstein, J. and Lockhart, P., Coefficients of Maass forms and the Siegel zero, Ann. of Math. (2) 140 (1994), 161–181, with an appendix by Dorian Goldfeld, Jeffrey Hoffstein and Daniel Lieman.CrossRefGoogle Scholar
[Hu09]Hu, R., L p norm estimates of eigenfunctions restricted to submanifolds, Forum Math. 21 (2009), 1021–1052.CrossRefGoogle Scholar
[Iwa80]Iwaniec, H., Fourier coefficients of cusp forms and the Riemann zeta-function. Seminar on Number Theory, 1979–1980, Exp. No. 18, 36 pp. (Université Bordeaux I, Talence, 1980).Google Scholar
[Iwa90]Iwaniec, H., Small eigenvalues of Laplacian for Γ0(N), Acta Arith. 56 (1990), 65–82.CrossRefGoogle Scholar
[Iwa02]Iwaniec, H., Spectral methods of automorphic forms, Graduate Studies in Mathematics, vol. 53, second edition (American Mathematical Society, Providence, RI, 2002), Revista Matematica Iberoamericana, Madrid.CrossRefGoogle Scholar
[IK04]Iwaniec, H. and Kowalski, E., Analytic number theory, American Mathematical Society Colloquium Publications, vol. 53 (American Mathematical Society, Providence, RI, 2004).Google Scholar
[IS95]Iwaniec, H. and Sarnak, P., L ∞ norms of eigenfunctions of arithmetic surfaces, Ann. of Math. (2) 141 (1995), 301–320.CrossRefGoogle Scholar
[JPS79]Jacquet, H., Piatetski-Shapiro, I. and Shalika, J., Automorphic forms on GL(3). I, II, Ann. of Math. (2) 109 (1979), 169–258.CrossRefGoogle Scholar
[JPS83]Jacquet, H., Piatetski-Shapiro, I. and Shalika, J., Rankin–Selberg convolutions, Amer. J. Math. 105 (1983), 367–464.CrossRefGoogle Scholar
[Kac66]Kac, M., Can one hear the shape of a drum? Amer. Math. Monthly 73 (1966), 1–23, part II.CrossRefGoogle Scholar
[MV10]Michel, P. and Venkatesh, A., The subconvexity problem for GL 2, Publ. Math. Inst. Hautes Études Sci. 111 (2010), 171–271.CrossRefGoogle Scholar
[Mil10]Milićević, D., Large values of eigenfunctions on arithmetic hyperbolic surfaces, Duke Math. J. 155 (2010), 365–401.CrossRefGoogle Scholar
[Mil01]Miller, S. D., On the existence and temperedness of cusp forms for SL 3(ℤ), J. Reine Angew. Math. 533 (2001), 127–169.Google Scholar
[MS06]Miller, S. D. and Schmid, W., Automorphic distributions, L-functions, and Voronoi summation for GL(3), Ann. of Math. (2) 164 (2006), 423–488.CrossRefGoogle Scholar
[Mot97]Motohashi, Y., Spectral theory of the Riemann zeta-function, Cambridge Tracts in Mathematics, vol. 127 (Cambridge University Press, Cambridge, 1997).CrossRefGoogle Scholar
[RW03]Ramakrishnan, D. and Wang, S., On the exceptional zeros of Rankin–Selberg L-functions, Compositio Math. 135 (2003), 211–244.CrossRefGoogle Scholar
[Rez04]Reznikov, A., Norms of geodesic restrictions for eigenfunctions on hyperbolic surfaces and representation theory, Preprint (2004), http://arxiv.org/abs/math/0403437.Google Scholar
[RS94]Rudnick, Z. and Sarnak, P., The behaviour of eigenstates of arithmetic hyperbolic manifolds, Comm. Math. Phys. 161 (1994), 195–213.CrossRefGoogle Scholar
[RS96]Rudnick, Z. and Sarnak, P., Zeros of principal L-functions and random matrix theory, Duke Math. J. 81 (1996), 269–322, A celebration of John F. Nash, Jr.CrossRefGoogle Scholar
[Sar03]Sarnak, P., Spectra of hyperbolic surfaces, Bull. Amer. Math. Soc. (N.S.) 40 (2003), 441–478.CrossRefGoogle Scholar
[Sar04]Sarnak, P., Letter to Morawetz (2004); http://www.math.princeton.edu/sarnak/.Google Scholar
[Sar08]Sarnak, P., Letter to Reznikov (2008); http://www.math.princeton.edu/sarnak/.Google Scholar
[Sta90]Stade, E., On explicit integral formulas for GL(n,R)-Whittaker functions, Duke Math. J. 60 (1990), 313–362, with an appendix by Daniel Bump, Solomon Friedberg and Jeffrey Hoffstein.CrossRefGoogle Scholar
[Sta93]Stade, E., Hypergeometric series and Euler factors at infinity for L-functions on GL(3,R)×GL(3,R), Amer. J. Math. 115 (1993), 371–387.CrossRefGoogle Scholar
[Tem10]Templier, N., On the sup-norm of Maass cusp forms of large level, Selecta Math. 16 (2010), 501–531.CrossRefGoogle Scholar
[Tit86]Titchmarsh, E. C., The theory of the Riemann zeta-function, second edition (The Clarendon Press, Oxford University Press, New York, 1986), edited and with a preface by D. R. Heath-Brown.Google Scholar
[Xia07]Xia, H., On L ∞ norms of holomorphic cusp forms, J. Number Theory 124 (2007), 325–327.CrossRefGoogle Scholar
[You11]Young, M. P., The second moment of GL(3)×GL(2) L-functions integrated, Adv. Math. 226 (2011), 3550–3578.CrossRefGoogle Scholar