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Lower bounds for Maass forms on semisimple groups

Part of: Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  17 April 2020

Farrell Brumley  and
Simon Marshall
Farrell Brumley
Affiliation:
Université Sorbonne Paris Nord, Laboratoire de Géométrie, Analyse et Applications, LAGA, CNRS, UMR 7539, F-93430, Villetaneuse, France email brumley@math.univ-paris13.fr
Simon Marshall
Affiliation:
Department of Mathematics, University of Wisconsin – Madison, 480 Lincoln Drive,MadisonWI 53706, USA email marshall@math.wisc.edu
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Abstract

Let $G$ be an anisotropic semisimple group over a totally real number field $F$. Suppose that $G$ is compact at all but one infinite place $v_{0}$. In addition, suppose that $G_{v_{0}}$ is $\mathbb{R}$-almost simple, not split, and has a Cartan involution defined over $F$. If $Y$ is a congruence arithmetic manifold of non-positive curvature associated with $G$, we prove that there exists a sequence of Laplace eigenfunctions on $Y$ whose sup norms grow like a power of the eigenvalue.

Keywords

sup normsMaass formssymmetric varietiestrace formulaautomorphic periods

MSC classification

Primary: 11F03: Modular and automorphic functions
Secondary: 11F70: Representation-theoretic methods; automorphic representations over local and global fields 11F72: Spectral theory; Selberg trace formula

Information

Type
Research Article
Information
Compositio Mathematica , Volume 156 , Issue 5 , May 2020 , pp. 959 - 1003
Copyright
© The Authors 2020

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Footnotes

FB is supported by ANR grant 14-CE25 and SM is supported by NSF grant DMS-1902173.

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