#### Vol. 11, No. 5, 2017

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Hybrid sup-norm bounds for Maass newforms of powerful level

### Abhishek Saha

Vol. 11 (2017), No. 5, 1009–1045
##### Abstract

Let $f$ be an ${L}^{2}$-normalized Hecke–Maass cuspidal newform of level $N$, character $\chi$ and Laplace eigenvalue $\lambda$. Let ${N}_{1}$ denote the smallest integer such that $N|{N}_{1}^{2}$ and ${N}_{0}$ denote the largest integer such that ${N}_{0}^{2}|N$. Let $M$ denote the conductor of $\chi$ and define ${M}_{1}=M∕gcd\left(M,{N}_{1}\right)$. We prove the bound $\parallel f{\parallel }_{\infty }{\ll }_{\epsilon }{N}_{0}^{1∕6+\epsilon }{N}_{1}^{1∕3+\epsilon }{M}_{1}^{1∕2}{\lambda }^{5∕24+\epsilon }$, which generalizes and strengthens previously known upper bounds for $\parallel f{\parallel }_{\infty }$.

This is the first time a hybrid bound (i.e., involving both $N$ and $\lambda$) has been established for $\parallel f{\parallel }_{\infty }$ 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 $\parallel f{\parallel }_{\infty }{\ll }_{\lambda ,\epsilon }{N}^{5∕12+\epsilon }$ 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 $\parallel f{\parallel }_{\infty }{\ll }_{\epsilon }{N}^{1∕3+\epsilon }{\lambda }^{5∕24+\epsilon }$, 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 ${GL}_{2}\left(F\right)$ 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