Density theorems for $\text{GL}_n$ via Rankin-Selberg $L$-functions

TL;DR

This paper develops density theorems for automorphic representations of GL_n over Q that do not satisfy the Ramanujan conjecture, using L-function techniques instead of trace formulas, improving existing bounds.

Contribution

It introduces a new approach based on L-function techniques to obtain density theorems, surpassing previous Kuznetsov trace formula methods.

Findings

Improved density bounds for non-Ramanujan automorphic representations.

Enhanced understanding of the distribution of automorphic representations.

Refined estimates near the Ramanujan conjecture threshold.

Abstract

We obtain density theorems for cuspidal automorphic representations of $\text{GL}_n$ over $\mathbb{Q}$ which fail the generalized Ramanujan conjecture at some place. We depart from previous approaches based on Kuznetsov-type trace formulae, and instead rely on $L$-function techniques. This improves recent results of Blomer near the threshold of the pointwise bounds.