Abstract
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.
Citation
Abhishek Saha. "Hybrid sup-norm bounds for Maass newforms of powerful level." Algebra Number Theory 11 (5) 1009 - 1045, 2017. https://doi.org/10.2140/ant.2017.11.1009
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