Bounding sup-norms of cusp forms of large level

摘要

Let f be an L (2)-normalized weight zero Hecke-Maa cusp form of square-free level N, character chi and Laplacian eigenvalue lambda a parts per thousand yen1/4. It is shown that aEuro-faEuro-(a)a parts per thousand(a) (lambda) N (-1/37), from which the hybrid bound aEuro-faEuro-(a)a parts per thousand(a)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.