The GL(n) large sieve

Jesse Thorner, Asif Zaman

Published 2019-06-18Version 1

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 an unconditional large sieve inequality for the Hecke eigenvalues of $\pi\in\mathfrak{F}_n$. This 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, we prove a subconvexity bound on $L(1/2,\pi)$ for almost all $\pi\in\mathfrak{F}_n$.