Large sieves for GLₙ and applications

Let Fₙ be the set of unitary cuspidal automorphic representations of GLₙ over a number field F, and let Sₙ be an arbitrary finite subset. Given π₀n₀, we establish large sieve inequalities for the families (s,π) π∈ S\ and (s,π×π₀) π∈ 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 ∑π∈ S|L(1/2,π)|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 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.