On sup-norm bounds part II: GL(2) Eisenstein series

Abstract

In this paper we consider the sup-norm problem in the context of analytic Eisenstein series for GL(2) over number fields. We prove a hybrid bound which is sharper than the corresponding bound for Maaß forms. Our results generalise those of Huang and Xu where the case of Eisenstein series of square-free levels over the base field Q had been considered. theory of Eisenstein series is used to give lower bounds for ζ(1 + it). This last method has vast generalizations. See for example [8]. Our case is reverse. We exploit that the theory of Hecke L-functions over number fields is well developed and use this to produce an improved amplifier. This leads to the remarkable fact that (as in [23]) we get sup-norm bounds which are in some aspects better than the state of the art results for cusp forms. It was already observed in [13] that one can improve the sup-norm bound in the spectral aspect if one has a better understanding of the Hecke eigenvalues. This is exactly the ingredient we gain from the classically well developed GL 1 theory. More precisely, we use highly non-trivial zero free regions for Hecke L-functions to derive an asymptotic expansion for generalized divisor sums. This will serve as a lower bound for the amplifier.Note that recently in [16] it was shown that the quality of the sup-norm bound that we achieve for Eisenstein series holds for SL 2 (Z)-Maaß forms on average.Before we can give a precise statement of our theorems we have to introduce some notation. We assume that the reader is familiar with the results and the notation in [1] and only focus on the aspects where Eisenstein series differ from cusp forms.

Sup-Norm and Nodal Domains of Dihedral Maass Forms

“…Now we need to analyze the residue of the pole from −ζ ′ (s)/ζ(s). The residue at s = 1 contributes N w (1). Hence, by (4.8), we prove the lemma.…”

“…if y ≫ 1, where c 0 (y, s) is the constant term in the Fourier expansion of E(z, s); Assing [1] extended to the number fields case. To prove our results, we will follow the approach in [17].…”

“…where w is a fixed, compactly-support positive function on the positive reals, with supp(w) ⊂[1,2], 0 ≤ w(r) ≤ 1, and…”

Eisenstein series and an asymptotic for the K-Bessel function

“…to tesselate H, the quotient group Γ ∞ \Γ tessellates the unit strip so as to agree with the tessellation of H given by Γ , and we have a canonical 2 fundamental domain F that extends to infinity for the Γ ∞ \Γ action-to determine F, let the real part of the points of F range between 0 and 1, inclusive of 0. Often we will consider the topological closure F. There are, in general, a finite number of inequivalent cusps {κ j } q j=1 ⊂ R ∪ {∞}, and the stabilizer in Γ of a cusp κ j is a parabolic subgroup Γ j (see, for example, [10,Chap.…”

“…In particular, the sup-norm problem for certain eigenfunctions has had much interest (see [4,13,20] for example). Specifically, for Eisenstein series, there are recent results in [2,11,23] of which the most relevant for us is the result by Huang and Xu (generalizing the earlier result of Young) for the modular group Γ = PSL 2 (Z) [11, Theorem 1.1]:…”