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

Abstract

We obtain density theorems for exceptional cuspidal automorphic representations of ${\mathrm{GL}}_n$ over $\mathbb{Q}$ that fail the generalized Ramanujan conjecture at some place. We depart from approaches based on trace formulae, and instead rely on properties of families of Rankin–Selberg $L$-functions. This improves previous density results near the threshold of the pointwise bounds.