Exceptional zeros of Rankin-Selberg $L$-functions and joint Sato-Tate distributions

Abstract: Let $\chi$ be an idele class character over a number field $F$, and let $\pi,\pi'$ be non-dihedral twist-inequivalent cuspidal automorphic representations of $\mathrm{GL}_2(\mathbb{A}_F)$. We prove the following results. 1. If $m,n\geq 0$ are integers, $m+n\geq 1$, $F$ is totally real, $\chi$ corresponds with a ray class character, and $\pi,\pi'$ correspond with primitive non-CM holomorphic Hilbert cusp forms, then the Rankin-Selberg $L$-function $L(s,\mathrm{Sym}^m(\pi)\times(\mathrm{Sym}^n(\pi')\otimes\chi))$ has a standard zero-free region with no exceptional Landau-Siegel zero. When $m,n\geq 1$ and $m+n\geq 4$, this is new even for $F=\mathbb{Q}$. As an application, we establish the strongest known unconditional effective rates of convergence in the Sato-Tate distribution for $\pi$ and the joint Sato-Tate distribution for $\pi$ and $\pi'$. 2. The Rankin-Selberg $L$-function $L(s,\mathrm{Sym}^2(\pi)\times(\mathrm{Sym}^2 (\pi')\otimes\chi))$ has a standard zero-free region with no exceptional Landau-Siegel zero. Until now, this was only known when $\pi=\pi'$, $\pi$ is self-dual, and $\chi$ is trivial.