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

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.

33 pages. v2: Incorporates a new result (Theorem 1.5), numbering changes, and referees' comments (to appear in JEMS)