An unconditional $\mathrm{GL}(n)$ large sieve

Abstract

Let ${\mathfrak{F}}_n$ be the set of all cuspidal automorphic representations $\pi$ of ${\mathrm{GL}}_n$ over a number field with unitary central character. We prove two unconditional large sieve inequalities for the Hecke eigenvalues of $\pi \in {\mathfrak{F}}_n$, one on the integers and one on the primes. The second leads to the first unconditional zero density estimate for the family of $L$-functions $L(s,\pi )$ associated to $\pi \in {\mathfrak{F}}_n$, which we make log-free. As an application of the zero density estimate, we prove a hybrid subconvexity bound for $L(\frac{1}{2},\pi )$ for a density one subset of $\pi \in {\mathfrak{F}}_n$.

17 pages. Incorporates referee comments