The sup-norm problem for GL(2) over number fields

TL;DR

This paper establishes new bounds for spherical Hecke-Maass newforms over number fields, improving the understanding of their sup-norms with power savings and Weyl-type exponents in both eigenvalue and level aspects.

Contribution

It provides the first power-saving bounds for the sup-norm problem over number fields with square-free level, matching or surpassing known results over the rationals.

Findings

Achieved power savings over local geometric bounds.

Reproduced or improved all known special cases.

Established bounds with Weyl-type exponents in level aspect.

Abstract

We solve the sup-norm problem for spherical Hecke-Maass newforms of square-free level for the group GL(2) over a number field, with a power saving over the local geometric bound simultaneously in the eigenvalue and the level aspect. Our bounds feature a Weyl-type exponent in the level aspect, they reproduce or improve upon all known special cases, and over totally real fields they are as strong as the best known hybrid result over the rationals.