Bounding sup-norms of cusp forms of large level

Abstract

Let f be an $L^2$-normalized weight zero Hecke-Maass cusp form of square-free level N, character $\chi$ and Laplacian eigenvalue $\lambda \ge 1/4$. It is shown that $\Vert f{\Vert }_{\infty }{\ll }_{\lambda }{N}^{-1/37}$, from which the hybrid bound $\Vert f{\Vert }_{\infty }\ll {\lambda }^{1/4}(N\lambda {)}^{-\delta }$ (for some $\delta >0$) is derived. The first bound holds also for $f=y^{k/2}F$ where F is a holomorphic cusp form of weight k with the implied constant now depending on k.

version 3: substantially revised version

References in corpus (1)

• Bounds on supremum norms for Hecke eigenfunctions of quantized cat maps

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