Hybrid sup-norm bounds for Maass newforms of powerful level

Abhishek Saha*

*Corresponding author for this work

Research output: Contribution to journalArticle (Academic Journal)peer-review

17 Citations (Scopus)
254 Downloads (Pure)


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= M/gcd(M,N1). In this paper, we prove the bound ||f|| «ε N01/6 + ε N11/3+ε M11/2λ5/24+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 non-squarefree N. The only previously known bound in the non-squarefree case was in the N-aspect; it had been shown by the author that ||f|| «λ,ε N5/12+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 N11/3+ε λ5/24 + 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 GL2(F) where F is a local field.
Original languageEnglish
Pages (from-to)1009-1045
Number of pages37
JournalAlgebra and Number Theory
Issue number5
Early online date12 Jul 2017
Publication statusPublished - 12 Jul 2017


  • Amplification
  • Automorphic form
  • Maass form
  • Newform
  • Sup-norm


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