Hybrid sup-norm bounds for Maass newforms of powerful level

Saha, Abhishek

doi:10.2140/ant.2017.11.1009

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Hybrid sup-norm bounds for Maass newforms of powerful level

Abhishek Saha

Vol. 11 (2017), No. 5, 1009–1045

DOI: 10.2140/ant.2017.11.1009

Abstract

Let $f$ be an $L^{2}$ -normalized Hecke–Maass cuspidal newform of level $N$ , character $χ$ and Laplace eigenvalue $λ$ . Let $N_{1}$ denote the smallest integer such that $N | N_{1}^{2}$ and $N_{0}$ denote the largest integer such that $N_{0}^{2} | N$ . Let $M$ denote the conductor of $χ$ and define $M_{1} = M ∕ gcd (M, N_{1})$ . We prove the bound $∥ f ∥_{\infty} ≪_{ε} N_{0}^{1 ∕ 6 + ε} N_{1}^{1 ∕ 3 + ε} M_{1}^{1 ∕ 2} λ^{5 ∕ 24 + ε}$ , which generalizes and strengthens previously known upper bounds for $∥ f ∥_{\infty}$ .

This is the first time a hybrid bound (i.e., involving both $N$ and $λ$ ) has been established for $∥ f ∥_{\infty}$ 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 $∥ f ∥_{\infty} ≪_{λ, ε} N^{5 ∕ 12 + ε}$ 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 $∥ f ∥_{\infty} ≪_{ε} N^{1 ∕ 3 + ε} λ^{5 ∕ 24 + ε}$ , 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 ${GL}_{2} (F)$ where $F$ is a local field.

Keywords

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

Mathematical Subject Classification 2010

Primary: 11F03

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

Milestones

Received: 13 October 2015

Revised: 25 October 2016

Accepted: 16 December 2016

Published: 12 July 2017

Authors

Abhishek Saha
Department of Mathematics University of Bristol Bristol BS81SN United Kingdom

Vol. 11, No. 5, 2017