Approximate Converse Theorem

TL;DR

This paper introduces an approximate converse theorem that quantifies the proximity of local representations to genuine cuspidal representations using a new measure and constructs explicit automorphic functions.

Contribution

It provides a novel measure for how close local representations are to cuspidal representations and constructs explicit automorphic functions using a new annihilating operator.

Findings

Developed a formula for the measure of proximity to cuspidal representations

Introduced a quasi-Maass form on the generalized upper half plane

Constructed an explicit cuspidal automorphic function

Abstract

We present an approximate converse theorem which measures how close a given set of irreducible admissible unramified unitary generic local representations of GL(n) is to a genuine cuspidal representation. To get a formula for the measure, we introduce a quasi-Maass form on the generalized upper half plane for a given set of local representations. We also construct an annihilating operator which enables us to write down an explicit cuspidal automorphic function.