Effective sup-norm bounds on average for cusp forms of even weight

Abstract

Let Γ ⊂ PSL2(R) be a Fuchsian subgroup of the first kind acting on the upper half-plane H. Consider the d 2k -dimensional space of cusp forms S Γ 2k of weight 2k for Γ, and let {f1, . . . , f d 2k } be an orthonormal basis of S Γ 2k with respect to the Petersson inner product. In this paper we will give effective upper and lower bounds for the supremum of the quantity S Γ 2k (z) := d 2k j=1 |fj (z)| 2 Im(z) 2k as z ranges through H.

Bounds for the Bergman kernel and the sup-norm of holomorphic Siegel cusp forms

“…Among some other results available in the weight aspect in the holomorphic setting, we mention [JK11, FJK16, FJK19, Blo15, BP16, CL11, DS15, DS20]. In fact the results on Bergman kernel estimates of Kramer [JK11,FJK19], especially [FJK16] et. al.…”

“…Remarks and discussions about proofs. When n = 1, Conjecture 1.2 is known from the works [FJK19,JK04] by the heat-kernel method (also see [AB18] for the case of Hilbert modular forms), but it appears with O ǫ (k 3/2−ǫ ) as the lower bound. It also follows from (1.2) by summing over a basis, with k ±ǫ defects in the upper and lower bound respectively.…”

Construction of Poincaré-type series by generating kernels

“…163]. The resolvent kernel asymptotic is used more recently in [FJK19] with k ∈ 1 2 N to establish effective sup-norm bounds on average for weight 2k cusp forms for Γ.…”

The heat kernel of a weighted Maass-Laplacian with real weights

“…More recently, the fact that for any integral or half-integral k, the kernel of the operator D k − k(1 − k) is isomorphic to the space of weight 2k cusp forms for Γ (when both spaces are viewed as C−vector spaces) was used in [6] and [7] to deduce effective sup-norm bounds on average for such cusp forms. Here D k denotes the weighted Maass-Laplacian, defined by (1.1)…”

“…The asymptotic behavior of the heat kernel of the Maass-Laplacian D k on the upper half-plane, evaluated for k ∈ 1 2 Z in a closed form by [16] played a crucial role in the results of [6] and [7].…”