Zeros of Rankin–Selberg $L$-functions at the edge of the critical strip (with an appendix by Colin J. Bushnell and Guy Henniart)

Abstract

Let $\pi$ (respectively ${\pi }_0$) be a unitary cuspidal automorphic representation of ${\mathrm{GL}}_m$ (respectively ${\mathrm{GL}}_{m_0}$) over $\mathbb{Q}$. 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.