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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 NN_{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}). In this paper, we prove the bound f_{∞} «_{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_{∞}.
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 Naspect; it had been shown by the author that f_{∞} «_{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_{∞} «_{ε} N_{1}^{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 padic 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.
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 Naspect; it had been shown by the author that f_{∞} «_{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_{∞} «_{ε} N_{1}^{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 padic 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.
Original language  English 

Pages (fromto)  10091045 
Number of pages  37 
Journal  Algebra and Number Theory 
Volume  11 
Issue number  5 
Early online date  12 Jul 2017 
DOIs  
Publication status  Published  12 Jul 2017 
Keywords
 Amplification
 Automorphic form
 Maass form
 Newform
 Supnorm
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Projects
 1 Finished

Arithmetic aspects of automorphic forms: Petersson norms and special values of Lfunctions.
Saha, A.
1/09/14 → 1/09/16
Project: Research