The \(L^2\) restriction norm of a \(\mathrm{GL}_3\) Maass form
type: Item
instance of: scholarly article
zbMATH Open document ID: 1287.11066
DOI: 10.1112/S0010437X11007366
author: Xiaoqing Li
author: Matthew P. Young
published in: Compositio Mathematica
publication date: 2012-06-29T00:00:00Z
full work available at URL: https://arxiv.org/abs/1102.0195
full work available at URL: https://arxiv.org/abs/1212.4002
Mathematics Subject Classification ID: 11F55
Mathematics Subject Classification ID: 11F72
Mathematics Subject Classification ID: 11M41
Mathematics Subject Classification ID: 35P20
zbMATH DE Number: 6051022
zbMATH Keywords: \(\mathrm{GL}_3\) Maass form
zbMATH Keywords: restriction norm
zbMATH Keywords: automorphic form
MaRDI profile type: Publication
arXiv ID: 1212.4002
cites work: On \(L^{\infty }\) norms of holomorphic cusp forms
cites work: Hypergeometric Series and Euler Factors at Infinity for L-Functions on GL(3, R )×GL(3, R )
cites work: A relation between automorphic representations of ${\rm GL}(2)$ and ${\rm GL}(3)$
cites work: A large sieve density estimate near \(\sigma = 1\)
cites work: On explicit integral formulas for \(\text{GL}(n, \mathbb R)\)-Whittaker functions. With an appendix by Daniel Bump, Solomon Friedberg, and Jeffrey Hoffstein
cites work: Restrictions of the Laplace-Beltrami eigenfunctions to submanifolds
cites work: Barnes' Second Lemma and its Application to Rankin-Selberg Convolutions
cites work: Spectra of hyperbolic surfaces
cites work: Second order average estimates on local data of cusp forms
cites work: Zeros of principal \(L\)-functions and random matrix theory
cites work: Restriction of toral eigenfunctions to hypersurfaces
cites work: The behaviour of eigenstates of arithmetic hyperbolic manifolds
cites work: Bounding sup-norms of cusp forms of large level
cites work: Q4798000
cites work: Automorphic distributions, \(L\)-functions, and Voronoi summation for \(\text{GL}(3)\)
cites work: On the existence and temperedness of cusp forms for \(\text{SL}_3(\mathbb{Z})\)
cites work: Large values of eigenfunctions on arithmetic hyperbolic surfaces
cites work: Can One Hear the Shape of a Drum?
cites work: Rankin-Selberg Convolutions
cites work: Automorphic forms on \(GL(3)\). I, II
cites work: \(L^ \infty\) norms of eigenfunctions of arithmetic surfaces
cites work: The subconvexity problem for \(\mathrm{GL}_2\)
cites work: Lp norm estimates of eigenfunctions restricted to submanifolds
cites work: Coefficients of Maass forms and the Siegel zero. Appendix: An effective zero-free region, by Dorian Goldfeld, Jeffrey Hoffstein and Daniel Lieman
cites work: On the Decomposition of a Representation of SOn When Restricted to SOn-1
cites work: Rank lowering linear maps and multiple Dirichlet series associated to \(\text{GL}(n,\mathbb R)\)
cites work: The second moment of \(\text{GL}(3)\times\text{GL}(2)\) \(L\)-functions, integrated
cites work: On the sup-norm of Maass cusp forms of large level
title: The \(L^2\) restriction norm of a \(\mathrm{GL}_3\) Maass form
review text: From the introduction: ``In this paper, we study a novel restriction problem for a \(\mathrm{GL}_3\) Maass form restricted to a codimension two submanifold (essentially \(\mathrm{GL}_2 \times \mathbb{R}^+\)). Such a restricted function has nice invariance properties; it is invariant by \(\mathrm{SL}_2(\mathbb{Z})\) on the left and by \(\mathrm{O}_2(\mathbb{R})\) on the right, and it is natural to understand how it fits into the \(\mathrm{GL}_2\) picture. For instance, one can ask what is the inner product of this restricted function with a given \(\mathrm{SL}_2(\mathbb{Z})\) Maass form (or, more generally, we ask for the spectral decomposition). In fact, the Rankin-Selberg \(L\)-function for \(\mathrm{GL}_3 \times \mathrm{GL}_2\) is constructed along these lines. There are many examples of such period integrals giving values of \(L\)-functions''.NEWLINENEWLINE``Our main result is the following.NEWLINENEWLINETheorem. Let \(F\) be a Hecke-Maass form of type \((\nu_1, \nu_2)\) for \(\mathrm{SL}_3(\mathbb{Z})\) that is in the tempered spectrum of \(\Delta\) (meaning \(\mathrm{Re}(\nu_1) = \mathrm{Re}(\nu_2) = 1/3\) or alternately the Langlands parameters \(i\alpha\), \(i\beta\), \(i\gamma\)'' \(\ldots\) ``are purely imaginary), with Laplace eigenvalue \(\lambda_F (\Delta) = 1 + \frac{1}{2} ( \alpha^2 +\beta^2 + \gamma^2)\), and with \(L^2\) norm equal to 1. Then we have NEWLINE\[NEWLINE N(F) := \int_0^\infty \int_{\mathrm{SL}_2(\mathbb{Z}) \backslash \mathcal{H}^2} \left| F \begin{pmatrix} z_2y_1 & \\ & 1 \end{pmatrix} \right|^2 \frac{dx_2 \, dy_2}{y_2^2} \frac{dy_1}{y_1} \ll_\varepsilon \lambda_F(\Delta)^\varepsilon \, \left| A_F(1,1)\right|^2, NEWLINE\]NEWLINE where NEWLINE\[NEWLINE z_2 = \begin{pmatrix} 1& x_2 \\ & 1 \end{pmatrix} \begin{pmatrix} y_2 & \\ & 1 \end{pmatrix} y_2^{-\frac{1}{2}}, NEWLINE\]NEWLINE \(A_F (1,1)\) is the first Fourier coefficient of \(F\), and the implied constant depends only on \(\varepsilon > 0\).'' NEWLINENEWLINE\vdotsNEWLINENEWLINE `` C…
description: scientific article; zbMATH DE number 6051022