Subconvexity for sup-norms of cusp forms on \(\mathrm{PGL}(n)\)
type: Item
instance of: scholarly article
title: Subconvexity for sup-norms of cusp forms on \(\mathrm{PGL}(n)\)
zbMATH Open document ID: 1352.11054
author: Valentin Blomer
author: Péter Maga
published in: Selecta Mathematica. New Series
publication date: 2016-08-22T00:00:00Z
review text: The main theorem of the paper may be summarized as follows. Let \(G = \mathrm{PGL}_n(\mathbb{R})\), \(K = \mathrm{PO}_n(\mathbb{R})\) and let \(F\) be a Hecke Maass cusp form for \(\Gamma_0(N) \subset G\), with spectral parameter \(\mu\) and \(\mu^* = \mathrm{Re}(\mu)\). For any \(\varepsilon > 0\) and any compact subset \(\Omega\) of the Riemannian symmetric space \(G/K\), there is an estimate \[ \| F \|_{L^\infty(\Omega)} \ll_{\Omega, \varepsilon} N^\varepsilon D(\mu^*)^{\frac{1}{2} - \delta}, \] for some effectively computable \(\delta = \delta_n > 0\). Here \(D(\lambda) := \prod_\alpha (1 + |\langle \alpha, \lambda \rangle|)^{m(\alpha)}\) for \(\alpha\) ranging over the positive roots and \(m(\alpha)\) standing for the dimension of \(\mathfrak{g}_\alpha\). In particular, we have \[ \| F \|_{L^\infty(\Omega)} \ll_{\Omega, \varepsilon} N^\varepsilon \lambda_F^{\frac{n(n-1)}{4} \cdot (\frac{1}{2} - \delta)}, \] where \(\lambda_F\) denotes the Laplacian eigenvalue of \(F\). This leads to a subconvexity bound for sup-norms of cusp forms, which is in clear analogy with the well-know problem for critical \(L\)-values. It seems to be the first subconvexity result of this kind on groups of arbitrary rank. The main tool is a comparison of spectral and geometric expansions of the automorphic kernel, i.e., an amplified trace formula. In the geometric side, the key point is a counting problem of integral matrices.
reviewed by: Wen-Wei Li
Mathematics Subject Classification ID: 11F55
Mathematics Subject Classification ID: 11F72
Mathematics Subject Classification ID: 11D75
zbMATH DE Number: 6618476
zbMATH Keywords: sup-norms
zbMATH Keywords: Hecke operators
zbMATH Keywords: trace formula
zbMATH Keywords: Diophantine approximation
zbMATH Keywords: amplification
zbMATH Keywords: \(\mathrm{GL}(n)\)
MaRDI profile type: Publication
full work available at URL: s00029-015-0219-5
OpenAlex ID: W2253117354
cites work: On the Linear Independence of Fractional Powers of Integers
cites work: Twisted \(L\)-functions over number fields and Hilbert's eleventh problem
cites work: Bounds for eigenforms on arithmetic hyperbolic 3-manifolds
cites work: The Sup-Norm Problem for PGL(4)
cites work: The sup-norm problem on the Siegel modular space of rank two
cites work: Large values of cusp forms on \(\mathrm{GL}_n\)
cites work: The subconvexity problem for Rankin-Selberg \(L\)-functions and equidistribution of Heegner points. II
cites work: Nodal domains of Maass forms. I
cites work: Q4092049
cites work: The amplification method in the \(\mathrm{GL}(3)\) Hecke algebra
cites work: Q4830109
cites work: \(L^ \infty\) norms of eigenfunctions of arithmetic surfaces
cites work: Subconvexity for twisted \(L\)-functions over number fields via shifted convolution sums
cites work: Q5717222
cites work: The behaviour of eigenstates of arithmetic hyperbolic manifolds
cites work: Recent progress on the quantum unique ergodicity conjecture
cites work: Recent developments in mathematical Quantum Chaos
DOI: 10.1007/S00029-015-0219-5
description: scientific article; zbMATH DE number 6618476