Large sieves for ${\mathrm{GL}}_n$ and applications

Abstract

Let ${\mathfrak{F}}_n$ be the set of unitary cuspidal automorphic representations of ${\mathrm{GL}}_n$ over a number field $F$, and let $S\subseteq {\mathfrak{F}}_n$ be an arbitrary finite subset. Given $\ddot{I}{€}_0\in {\mathfrak{F}}_{n_0}$, we establish large sieve inequalities for the families $\{L(s,\ddot{I}€):\ddot{I}€\in S\}$ and $\{L(s,\ddot{I}€\times \ddot{I}{€}_0):\ddot{I}€\in S\}$ that, unlike previous results, are independent of progress towards the generalized Ramanujan conjecture, and simultaneously handle the Dirichlet coefficients of $L$, $L^{-1}$, and $\log L$. We also give the first such result that improves upon the trivial bound for short sums. We present several applications, including: (1) the strongest bound for ${\sum }_{\ddot{I}€\in S}|L(\frac{1}{2},\ddot{I}€){|}^2$ that holds for arbitrary $S$, (2) significant improvements to zero density estimates for families of automorphic and Rankin--Selberg $L$-functions, counting violations to the generalized Riemann hypothesis near $\mathrm{Re}(s)=1$, (3) the removal of all unproven hypotheses in the conditional log-free zero density estimate for families of Rankin--Selberg $L$-functions proved by Brumley, Thorner, and Zaman, and (4) an improvement of the density theorem for non-archimedean Langlands parameters due to Lichtman and Pascadi, counting violations to the generalized Ramanujan conjecture.

33 pages. Title and abstract changed; Theorem 1.6 added