Vol. 11, No. 5, 2017

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Hybrid sup-norm bounds for Maass newforms of powerful level

Abhishek Saha

Vol. 11 (2017), No. 5, 1009–1045

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.

Maass form, sup-norm, automorphic form, newform, amplification
Mathematical Subject Classification 2010
Primary: 11F03
Secondary: 11F41, 11F60, 11F72, 11F85, 35P20
Received: 13 October 2015
Revised: 25 October 2016
Accepted: 16 December 2016
Published: 12 July 2017
Abhishek Saha
Department of Mathematics
University of Bristol
United Kingdom