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

TL;DR

This paper develops large sieve inequalities for automorphic L-functions over number fields that are independent of the Ramanujan conjecture, enabling new bounds and applications in zero density estimates and Ramanujan conjecture violations.

Contribution

It introduces novel large sieve inequalities for automorphic L-functions that do not rely on the Ramanujan conjecture, improving bounds and applications.

Findings

Established large sieve inequalities independent of Ramanujan conjecture

Achieved the strongest bound for sums of |L(1/2,π)|^2 over arbitrary sets

Improved zero density estimates and removed unproven hypotheses in existing results

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 $\pi_0\in\mathfrak{F}_{n_0}$, we establish large sieve inequalities for the families $\{L(s,\pi)\colon \pi\in S\}$ and $\{L(s,\pi\times\pi_0)\colon \pi\in S\}$ 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 $\sum_{\pi\in S}|L(\frac{1}{2},\pi)|^2$ 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.