Title
Beyond the spherical sup-norm problem
Document Type
Article
Version
Final Published Version
Publication Title
Journal de Mathématiques Pures et Appliquées
Volume
168
Publication Date
2022
Abstract
We open a new perspective on the sup-norm problem and propose a version for non-spherical Maa ss forms when the maximal compact K is non-abelian and the dimension of the K -type gets large. We solve this problem for an arithmetic quotient of G = SL2(C) with K = SU2(C). Our results cover the case of vector-valued Maa ss forms as well as all the individual scalar-valued Maa ss forms of the Wigner basis, reaching sub-Weyl exponents in some cases. On the way, we develop analytic theory of independent interest, including uniform strong localization estimates for generalized spherical functions of high K -type and a Paley-Wiener theorem for the corresponding spherical transform acting on the space of rapidly decreasing functions. The new analytic properties of the generalized spherical functions lead to novel counting problems of matrices close to various manifolds that we solve optimally.
Citation
Blomer, V., Harcos, G., Maga, P. & D. Milićević. 2022 "Beyond the spherical sup-norm problem." Journal de Mathématiques Pures et Appliquées 168: 1-64.
DOI
https://doi.org/10.1016/j.matpur.2022.09.009
Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License.