Sup norm on $\text{PGL}_n$ in depth aspect

TL;DR

This paper establishes a sub-local upper bound for the supremum norm of automorphic forms on PGL_n with increasing conductor, focusing on minimal vectors at ramified places, advancing understanding in automorphic representation theory.

Contribution

It provides the first sub-local upper bound for automorphic forms on PGL_n in the depth aspect, specifically for forms with minimal vectors at ramified places.

Findings

Derived a sub-local upper bound for the sup norm in the depth aspect

Focused on automorphic forms with minimal vectors at ramified places

Results contribute to the understanding of automorphic forms' growth in high conductor regimes

Abstract

In this short paper we give the sub-local upper bound for the sup norm of an automorphic form on $\text{PGL}_n$, whose associated automorphic representation has finite conductor $C(\pi)=p^c$ with $c\rightarrow \infty$, and its local component at the place of ramification is a minimal vector belonging to an irreducible representation with generic induction datum.