On the sup-norm of Maass cusp forms of large level

Abstract

We establish upper bounds for the sup-norm of Hecke-Maass eigenforms on arithmetic surfaces. In a first part, the case of open modular surfaces is studied. Let f be an Hecke-Maass cuspidal newform of square-free level N and bounded Laplace eigenvalue. Recently, V. Blomer and R. Holowinsky [Invent. Math., 179 (3)] provide a non-trivial bound when f is non-exceptional. Our approach is different in that we rely on the geometric side of the trace formula. The improved bound f ∞ N −1/23 f 2 is established. In a second part, we show that a corresponding result holds true for compact arithmetic surfaces and with a better exponent 1/12. The proof requires an estimate for the number of lattice points in a certain annulus domain.A key input is that a congruence subgroup (multiplicative group) is included in an order (ring). This structure enables us to introduce a diophantine argument.

Sup-norms of Eigenfunctions on Arithmetic Ellipsoids

“…We consider an amplified second moment which we transform by a pre-trace formula into a sum over a sort of automorphic kernel, see (5.3). This starting point is similar in most investigations of sup-norms of eigenfunction on arithmetically defined manifolds, see, for example, [2,9,11,16,21]. In all cases one encounters eventually an interesting diophantine problem the solution of which is at the heart of the problem.…”

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

“…Furthermore, for z := σ −1 a z, (8) u(γ z , z ) = u(γz, z) N . We shall now count general matrices, with special attention to square determinants.…”

“…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).…”

“…This paper, a sequel to [8], deals with the problem of bounding the L ∞ norm (or sup-norm) in terms of the L 2 norm on an arithmetic hyperbolic surface. It is natural to restrict this problem for Hecke-Maass cuspidal newforms, that is, square-integrable joint eigenfunctions of the Laplacian and Hecke operators.…”

“…The restriction on N seems difficult to remove: it is needed for a certain application of AtkinLehner theory. In [8] the second named author revisited the proof by making a systematic use of geometric arguments, and derived a stronger exponent: f ∞ λ, N −1/22+ . In the present paper we exploit Atkin-Lehner theory in a more efficient way, which results in a rather elegant proof and a significantly better estimate for the sup-norm.…”