Algebra & Number Theory
- Algebra Number Theory
- Volume 11, Number 5 (2017), 1009-1045.
Hybrid sup-norm bounds for Maass newforms of powerful level
Let be an -normalized Hecke–Maass cuspidal newform of level , character and Laplace eigenvalue . Let denote the smallest integer such that and denote the largest integer such that . Let denote the conductor of and define . We prove the bound , which generalizes and strengthens previously known upper bounds for .
This is the first time a hybrid bound (i.e., involving both and ) has been established for in the case of nonsquarefree . The only previously known bound in the nonsquarefree case was in the -aspect; it had been shown by the author that provided . The present result significantly improves the exponent of in the above case. If is a squarefree integer, our bound reduces to , which was previously proved by Templier.
The key new feature of the present work is a systematic use of -adic representation theoretic techniques and in particular a detailed study of Whittaker newforms and matrix coefficients for where is a local field.
Algebra Number Theory, Volume 11, Number 5 (2017), 1009-1045.
Received: 13 October 2015
Revised: 25 October 2016
Accepted: 16 December 2016
First available in Project Euclid: 12 December 2017
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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
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