Algebra & Number Theory

Hybrid sup-norm bounds for Maass newforms of powerful level

Abhishek Saha

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Abstract

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.

Article information

Source
Algebra Number Theory, Volume 11, Number 5 (2017), 1009-1045.

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

Permanent link to this document
https://projecteuclid.org/euclid.ant/1513090721

Digital Object Identifier
doi:10.2140/ant.2017.11.1009

Mathematical Reviews number (MathSciNet)
MR3671430

Zentralblatt MATH identifier
06748165

Subjects
Primary: 11F03: Modular and automorphic functions
Secondary: 11F41: Automorphic forms on GL(2); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces [See also 14J20] 11F60: Hecke-Petersson operators, differential operators (several variables) 11F72: Spectral theory; Selberg trace formula 11F85: $p$-adic theory, local fields [See also 14G20, 22E50] 35P20: Asymptotic distribution of eigenvalues and eigenfunctions

Keywords
Maass form sup-norm automorphic form newform amplification

Citation

Saha, Abhishek. Hybrid sup-norm bounds for Maass newforms of powerful level. Algebra Number Theory 11 (2017), no. 5, 1009--1045. doi:10.2140/ant.2017.11.1009. https://projecteuclid.org/euclid.ant/1513090721


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