Hybrid sup-norm bounds for Maass newforms of powerful level

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.