Bounding sup-norms of cusp forms of large level

Abstract

Abstract. Let f be an L 2 -normalized weight zero Hecke-Maaß cusp form of square-free level N , character χ and Laplacian eigenvalue λ 1/4. It is shown that f ∞ ≪ λ N −1/37 , from which the hybrid bound f ∞ ≪ λ 1/4 (N λ) −δ (for some δ > 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.

Hybrid bounds for automorphic forms on ellipsoids over number fields

“…For O varying (of square-free level) a bound simultaneously nontrivial in vol(X (2) (O)) and in |λ| was obtained by the first named author and R. Holowinsky [BH10]. This result was extended by Templier [Tem10] to the case of a totally real number field and for B indefinite at one archimedean place.…”

Sup-norms of Eigenfunctions on Arithmetic Ellipsoids

“…Our result is uniform in both of these directions of variation. In [2], only the O-direction was considered (the corresponding quaternion algebra in that case is the algebra B = M 2 (Q) of 2 × 2 matrices), although this has now been substantially extended and improved by…”

On the Sup-Norm of Maass Cusp Forms of Large Level: II

“…The earlier works [2,4,8] made great use, when estimating |f (z)|, of a fine analysis of the diophantine properties of the point z ∈ H. The present paper bypasses that analysis by focusing on a point z ∈ H with highest imaginary part such that |f (z)| = f ∞ . The key observation is that such a point z ∈ H always has good diophantine properties (Lemma 1) which allows a more efficient treatment of the counting problem that lies at the heart of the argument (Lemma 3).…”

“…In this paper we focus mainly on the N -aspect and, thanks to the new ideas explained below, we are able to provide a short but self-contained treatment. For several reasons, see [1,2,5], the "trivial" bound is f ∞ λ, N , while the most optimistic bound would be f ∞ λ, N −1/2+ . Here and later, the dependence on λ is understood to be continuous.…”