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

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.

53 pages. Farrell Brumley added as coauthor. Added two appendices, one of which is authored by Colin Bushnell and Guy Henniart. Theorem 1.3 added, leads to cleaner versions of Theorems 1.2 and 1.4

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