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 $M_1=M/\gcd (M,N_1)$. In this paper, we prove the bound $|f{|}_{\infty }$ ${\ll }_{\epsilon }$ $N_0^{1/6+\epsilon }{N}_1^{1/3+\epsilon }{M}_1^{1/2}{\lambda }^{5/24+\epsilon }$, which generalizes and strengthens previously known upper bounds for $|f{|}_{\infty }$. This is the first time a hybrid bound (i.e., involving both $N$ and $\lambda$) has been established for $|f{|}_{\infty }$ 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 $|f{|}_{\infty }{\ll }_{\lambda ,\epsilon }{N}^{5/12+\epsilon }$ 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 $|f{|}_{\infty }{\ll }_{\epsilon }{N}^{1/3+\epsilon }{\lambda }^{5/24+\epsilon }$, 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 $G{L}_2(F)$ where $F$ is a local field.

Postprint version; to appear in Algebra and Number Theory

References in corpus (2)

• Large values of newforms on GL(2) with highly ramified central character
• Local bounds for $L^p$ norms of Maass forms in the level aspect

Cited by in corpus (18)

• Short geodesic loops and $L^p$ norms of eigenfunctions on large genus random surfaces
• On the order of vanishing of newforms at cusps
• Large values of newforms on GL(2) with highly ramified central character
• On the size of p-adic Whittaker functions
• Analytic properties of spherical cusp forms on GL(n)
• On sup-norm bounds part II: $GL(2)$ Eisenstein series
• Sup-norm and nodal domains of dihedral Maass forms
• Beyond the spherical sup-norm problem
• Theta functions, fourth moments of eigenforms, and the sup-norm problem I
• On the global sup-norm of GL(3) cusp forms
• Sup norms of newforms on $G{L}_2$ with highly ramified central character
• Sup-norms of eigenfunctions in the level aspect for compact arithmetic surfaces, II: newforms and subconvexity
• Hybrid bounds for the sup-norm of automorphic forms in higher rank
• Some analytic aspects of automorphic forms on GL(2) of minimal type
• Explicit Burgess-like subconvex bounds for ${\mathrm{GL}}_2\times {\mathrm{GL}}_1$
• A geometric approach to the sup-norm problem for automorphic forms: the case of newforms on $G{L}_2({\mathbb{F}}_q(T))$ with squarefree level
• $L^p$-Norm Bounds for Automorphic Forms via Spectral Reciprocity
• The sup-norm problem beyond the newform