The sup-norm problem beyond the newform

TL;DR

This paper investigates the sup-norm bounds for automorphic forms by analyzing specific small types within twist minimal automorphic representations, achieving improved bounds as the dimension increases.

Contribution

It introduces a new perspective on the sup-norm problem by focusing on small types in automorphic representations and provides bounds that improve with the dimension of these types.

Findings

Achieves non-trivial bounds for automorphic forms using small types.

Provides bounds that improve as the dimension of the type increases.

Offers a new approach to the classical sup-norm problem.

Abstract

In this note we take up the classical sup-norm problem for automorphic forms and view it from a new angle. Given a twist minimal automorphic representation $\pi$ we consider a special small $GL_2(\mathbb{Z}_p)$-type $V$ in $\pi$ and proof global sup-norm bounds for an average over an orthonormal basis of $V$. We achieve a non-trivial saving when the dimension of $V$ grows.