Zeros of Rankin-Selberg $L$-functions at the edge of the critical strip

TL;DR

This paper establishes zero density estimates for Rankin-Selberg L-functions, which are crucial for understanding automorphic forms, zeros distribution, and number theoretic applications, with many results unconditional and others assuming conjectures.

Contribution

It provides new unconditional log-free zero density estimates for Rankin-Selberg L-functions, advancing understanding of their zeros and related number theory problems.

Findings

Unconditional zero density estimates for many cases.

Conditional estimates assuming generalized Ramanujan conjecture.

Applications to mass equidistribution, Landau-Siegel zeros, Chebotarev theorem, and class groups.

Abstract

Let $\pi$ and $\pi_0$ be unitary cuspidal automorphic representations. We prove log-free zero density estimates for Rankin-Selberg $L$-functions of the form $L(s,\pi\times\pi_0)$, where $\pi$ varies in a given family and $\pi_0$ is fixed. These estimates are unconditional in many cases of interest; they hold in full generality assuming an average form of the generalized Ramanujan conjecture. We consider applications of these estimates related to mass equidistribution for Hecke-Maass forms, the rarity of Landau-Siegel zeros of Rankin-Selberg $L$-functions, the Chebotarev density theorem, and $\ell$-torsion in class groups of number fields.