Hybrid sup-norm bounds for Maass newforms of powerful level

Abstract

Let $f$ be an $L^2$-normalized Hecke--Maass cuspidal newform of level $N$,\ncharacter $\\chi$ and Laplace eigenvalue $\\lambda$. Let $N_1$ denote the\nsmallest integer such that $N|N_1^2$ and $N_0$ denote the largest integer such\nthat $N_0^2 |N$. Let $M$ denote the conductor of $\\chi$ and define $M_1=\nM/\\gcd(M,N_1)$. In this paper, we prove the bound $|f|_\\infty$ $\\ll_{\\epsilon}$\n$N_0^{1/6 + \\epsilon} N_1^{1/3+\\epsilon} M_1^{1/2} \\lambda^{5/24+\\epsilon}$,\nwhich generalizes and strengthens previously known upper bounds for\n$|f|_\\infty$.\n This is the first time a hybrid bound (i.e., involving both $N$ and\n$\\lambda$) has been established for $|f|_\\infty$ in the case of non-squarefree\n$N$. The only previously known bound in the non-squarefree case was in the\nN-aspect; it had been shown by the author that $|f|_\\infty \\ll_{\\lambda,\n\\epsilon} N^{5/12+\\epsilon}$ provided $M=1$. The present result significantly\nimproves the exponent of $N$ in the above case. If $N$ is a squarefree integer,\nour bound reduces to $|f|_\\infty \\ll_\\epsilon N^{1/3 + \\epsilon}\\lambda^{5/24 +\n\\epsilon}$, which was previously proved by Templier.\n The key new feature of the present work is a systematic use of p-adic\nrepresentation theoretic techniques and in particular a detailed study of\nWhittaker newforms and matrix coefficients for $GL_2(F)$ where $F$ is a local\nfield.\n