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2017 Hybrid sup-norm bounds for Maass newforms of powerful level
Abhishek Saha
Algebra Number Theory 11(5): 1009-1045 (2017). DOI: 10.2140/ant.2017.11.1009


Let f be an L2-normalized Hecke–Maass cuspidal newform of level N, character χ and Laplace eigenvalue λ. Let N1 denote the smallest integer such that N|N12 and N0 denote the largest integer such that N02|N. Let M denote the conductor of χ and define M1 = Mgcd(M,N1). We prove the bound fεN016+εN113+εM112λ524+ε, which generalizes and strengthens previously known upper bounds for f.

This is the first time a hybrid bound (i.e., involving both N and λ) has been established for f in the case of nonsquarefree N. The only previously known bound in the nonsquarefree case was in the N-aspect; it had been shown by the author that fλ,εN512+ε provided M = 1. The present result significantly improves the exponent of N in the above case. If N is a squarefree integer, our bound reduces to fεN13+ελ524+ε, which was previously proved by Templier.

The key new feature of the present work is a systematic use of p-adic representation theoretic techniques and in particular a detailed study of Whittaker newforms and matrix coefficients for GL2(F) where F is a local field.


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Abhishek Saha. "Hybrid sup-norm bounds for Maass newforms of powerful level." Algebra Number Theory 11 (5) 1009 - 1045, 2017.


Received: 13 October 2015; Revised: 25 October 2016; Accepted: 16 December 2016; Published: 2017
First available in Project Euclid: 12 December 2017

zbMATH: 06748165
MathSciNet: MR3671430
Digital Object Identifier: 10.2140/ant.2017.11.1009

Primary: 11F03
Secondary: 11F41 , 11F60 , 11F72 , 11F85 , 35P20

Keywords: amplification , automorphic form , Maass form , newform , sup-norm

Rights: Copyright © 2017 Mathematical Sciences Publishers


Vol.11 • No. 5 • 2017
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