Vol. 11, No. 5, 2017

Download this article
Download this article For screen
For printing
Recent Issues

Volume 11
Issue 6, 1243–1488
Issue 5, 1009–1241
Issue 4, 767–1007
Issue 3, 505–765
Issue 2, 253–503
Issue 1, 1–252

Volume 10, 10 issues

Volume 9, 10 issues

Volume 8, 10 issues

Volume 7, 10 issues

Volume 6, 8 issues

Volume 5, 8 issues

Volume 4, 8 issues

Volume 3, 8 issues

Volume 2, 8 issues

Volume 1, 4 issues

The Journal
Cover
Editorial Board
Editors' Addresses
Editors' Interests
About the Journal
Scientific Advantages
Submission Guidelines
Submission Form
Subscriptions
Editorial Login
Contacts
Author Index
To Appear
 
ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
Hybrid sup-norm bounds for Maass newforms of powerful level

Abhishek Saha

Vol. 11 (2017), No. 5, 1009–1045
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.

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