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

TL;DR

This paper establishes sharper sup-norm bounds for analytic Eisenstein series on GL(2) over number fields, extending previous results and improving understanding of their growth behavior.

Contribution

It provides a new hybrid bound for Eisenstein series on GL(2) over number fields, generalizing prior work to broader settings.

Findings

Proved a sharper hybrid sup-norm bound for Eisenstein series

Extended previous results from rational to general number fields

Generalized bounds to Eisenstein series of arbitrary levels

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 Maa{\ss} forms. Our results generalise those of Huang and Xu where the case of Eisenstein series of square-free levels over the base field $\mathbb{Q}$ had been considered.